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one of the present authors ( n.i . ) together with his collaborators has published series of papers on the calculations of the electrical and thermal conductivities of dense matter ( flowers & itoh 1976 , 1979 , 1981 ; itoh et al . 1983 ; mitake , ichimaru , & itoh 1984 ; itoh et al . 1984 ; itoh & kohyama 1993 ; itoh , hayashi , & kohyama 1993 ) . among these works , the calculation corresponding to the liquid metal case ( itoh et al . 1983 ; hereafter referred to as paper i ) appears to have been most widely used in various fields of stellar evolution studies . therefore , it is important to keep scrutinizing the accuracy of paper i , as this paper is in frequent use among the stellar evolution researchers . in paper i the coulomb scatterings of the electrons off the atomic nuclei have been calculated in the framework of the first born approximation . subsequently yakovlev ( 1987 ) has made an improvement on paper i by taking into account the contributions beyond the first born approximation . here we note that yakovlev ( 1987 ) also used the analytic approach by taking into account the second born term for the coulomb scattering cross section . however , his second born corrections did not include the screening effects . we shall consistently take into account the screening effects in our second born corrections . later works by his group ( potekhin , chabrier , & yakovlev 1997 ; potekhin et al . 1999 ) improved on yakovlev s ( 1987 ) original method by treating the first born term and the non - born term on the same footing , thereby taking into account the screening effects self - consistently . in these works they have made use of the fully numerical values of the cross section for the coulomb scattering of the electron by the atomic nucleus calculated by doggett & spencer ( 1956 ) . here we remark that the numerical calculation of the coulomb scattering cross section by doggett & spencer ( 1956 ) has been carried out for a limited number of @xmath1-values for atomic nucleus @xmath1=6 , 13 , 29 , 50 , 82 , and 92 , and for a limited number of electron energies , 0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , and 10mev . in this paper , we shall take a complementary semi - analytic approach by using the analytic expression for the second born cross section for the coulomb scattering of the electron off the atomic nucleus ( mckinley & feshbach 1948 ; feshbach 1952 ) . for nuclei @xmath0 , the inclusion up to the second born approximation is sufficiently accurate ( eby & morgan 1972 ) . in the following sections , however , we will confirm that the interpolations with respect to @xmath1 and the electron energies done by the previous authors are remarkably accurate . the basic formulae for the calculation of the electrical and thermal conductivities are presented in 2 by generalizing the formulation of paper i. the numerical results and the assessment of the contributions beyond the first born approximation are presented in 3 . the analytic formulae that fit the results of the numerical calculations are given in 4 . the case of the mixtures of nuclear species is dealt with in 5 . the last section is devoted to concluding remarks . in the appendix we evaluate the accuracy of the second born approximation by comparing with the exact results obtained by dogget & spencer ( 1956 ) . we shall closely follow the method described in paper i and generalize it in such a way that it include the second born term for the coulomb scattering of the electron off the atomic nucleus ( mckinley & feshbach 1948 ; feshbach 1952 ) . the reader is referred to paper i for the earlier references in this field of research . we shall consider the case that the atoms are completely pressure - ionized . we further restrict ourselves to the density - temperature region in which electrons are strongly degenerate . this condition is expressed as @xmath4^{1/2 } \ , - \ , 1 \right ] \ , [ \rm k ] \ , , \end{aligned}\ ] ] where @xmath5 is the fermi temperature , @xmath1 the atomic number of the nucleus , @xmath6 the mass number of the nucleus , and @xmath7 the mass density in units of 10@xmath8 g @xmath9 . the reader is referred to the paper by cassisi et al . ( 2007 ) for the case of the partial electron degeneracy . for the ionic system we consider the case that it is in the liquid state . the latest criterion corresponding to this condition is given by ( potekhin & chabrier 2000 ) @xmath10 where @xmath11^{1/3}$ ] is the ion - sphere radius , and @xmath12 the temperature in units of 10@xmath13 k. for the calculation of the electrical and thermal conductivities we use the ziman formula ( 1961 ) as is extended to the relativistically degenerate electrons ( flowers & itoh 1976 ) . on deriving the formula we retain the dielectric screening function due to the degenerate electrons . as to the explicit expressions for the dielectric function , we use the relativistic formula worked out by jancovici ( 1962 ) : @xmath14 where @xmath15 is the thomas - fermi wavenumber for the nonrelativistic electrons , @xmath16 is the momentum transfer measured in units of 2@xmath17 , @xmath18 is the dimensionless relativistic parameter @xmath19 and @xmath20 is the usual electron density parameter given by @xmath21 working on the transport theory for the relativistic electrons given by flowers & itoh ( 1976 ) and taking into account the finite - nuclear - size corrections ( itoh & kohyama 1983 ) and the second born term ( mckinley & feshbach 1948 ; feshbach 1952 ) , we obtain the expression for the electrical conductivity @xmath22 and the thermal conductivity @xmath23 : @xmath24 \ , , \\ \kappa & = & 2.363 \times 10^{17 } \frac { \rho_{6}t_{8}}{a } \frac{1 } { ( 1+b^{2})<s > } \ , \left [ { \rm ergs \ , \ , cm^{-1 } \ , s^{-1 } \ , k^{-1 } } \right ] \ , , \end{aligned}\ ] ] @xmath25^{2 } } \nonumber \\ & - & \frac{b^{2}}{1 + b^{2 } } \int_{0}^{1 } d \left(\frac{k}{2k_{f } } \right ) \left(\frac{k}{2k_{f } } \right)^{5 } \frac{s(k/2k_{f } ) \left|f(k/2k_{f } ) \right|^{2 } } { \left[(k/2k_{f})^{2 } \epsilon(k/2k_{f},0 ) \right]^{2 } } \nonumber \\ & \equiv & < s_{-1 } > \ , - \ , \frac{b^{2}}{1 + b^{2 } } < s_{+1 } > \ , , \\ < s>^{2b } & = & \pi z \alpha \frac{b}{(1 + b^{2})^{1/2 } } \nonumber \\ & \times & \left\ { \int_{0}^{1 } d \left(\frac{k}{2k_{f } } \right ) \left(\frac{k}{2k_{f } } \right)^{4 } \frac{s(k/2k_{f } ) \left|f(k/2k_{f } ) \right|^{2 } } { \left[(k/2k_{f})^{2 } \epsilon(k/2k_{f},0 ) \right]^{2 } } \right . \nonumber \\ & & - \left . \int_{0}^{1 } d \left(\frac{k}{2k_{f } } \right ) \left(\frac{k}{2k_{f } } \right)^{5 } \frac{s(k/2k_{f } ) \left|f(k/2k_{f } ) \right|^{2 } } { \left[(k/2k_{f})^{2 } \epsilon(k/2k_{f},0 ) \right]^{2 } } \right\ } \ , \nonumber \\ & \equiv & \pi z \alpha \frac{b}{(1 + b^{2})^{1/2 } } \left [ < s_{0 } > - < s_{+1 } > \right ] \ , , \end{aligned}\ ] ] where @xmath26 corresponds to the first born term and @xmath27 corresponds to the second born term . in the above , @xmath28 is the momentum transferred from the ionic system to an electron , @xmath29 the ionic liquid structure factor , and @xmath30 the static dielectric screening function due to degenerate electrons . for the ionic liquid structure factor we use the results of young , corey , & dewitt ( 1991 ) calculated for the classical one - component plasma ( ocp ) . in the above formulae we have also taken into account the finite - nuclear - size corrections through the use of the atomic form factor ( itoh & kohyama 1983 ) @xmath31 @xmath17 and @xmath32 being the electron fermi wave number and the charge radius of the nucleus , respectively . the electron fermi wave number is expressed as @xmath33 the charge radius of the nucleus is represented by @xmath34 the present method differs from that of baiko et al . these authors subtracted the contribution corresponding to the elastic scattering in the crystalline lattice phase from the total static structure factor in the liquid . the main motivation for the modification of the structure factor near the melting point by baiko et al . ( 1998 ) is the partial ordering of the coulomb liquid revealed by microscopic numerical simulations . this procedure was followed by potekhin et al . ( 1999 ) , gnedin et al . ( 2001 ) , and cassisi et al . ( 2007 ) . in the field of condensed matter physics , however , the correctness of the original ziman ( 1961 ) method with the use of the full liquid structure factor has long been established ( ashcroft & lekner 1966 ; rosenfeld & stott 1990 ) . part of the motivation for the introduction of baiko et al.s ( 1998 ) suggestion appears to be the finding by itoh , hayashi , & kohyama ( 1993 ) that the conductivity of astrophysical dense matter increases typically by 24 times upon crystallization . regarding this finding , we should note that simple metals in the laboratory show similar phenomena . the electrical conductivity of the simple metals in the laboratory shows significant ( 2 - 4 times ) jumps upon crystallization ( iida & guthrie 1993 ) . for these reasons we shall follow the method of paper i in which we use the full liquid structure factor , which is in accord with the method used in condensed matter physics ( ashcroft & lekner 1966 ; rosenfeld & stott 1990 ) . of course the analogy with simple metals should be examined with future full @xmath35 @xmath36 calculations . we have carried out integrations in equations ( 2.9 ) and ( 2.10 ) numerically for the cases of @xmath37h , @xmath38he , @xmath39c , @xmath40n , @xmath41o , @xmath42ne , @xmath43 mg , @xmath44si , @xmath45s , @xmath46ca , @xmath47fe by using the structure factor of the classical one - component plasma calculated by young , corey , & dewitt ( 1991 ) and jancovici s ( 1962 ) relativistic dielectric function for degenerate electrons . for the neutron star matter , the reader is referred to the paper by gnedin , yakovlev , & potekhin ( 2001 ) . we have made calculations for the parameter ranges , 0.1 @xmath48 , @xmath49 , which cover most of the density - temperature region of the dense matter in the liquid metal phase of astrophysical importance . note that for some elements such as the @xmath47fe matter these parameter ranges include the density - temperature region in which either the condition for the strong electron degeneracy or the condition for the complete pressure ionization does not hold . all of the considered elements are certainly unstable against nuclear reactions or electron captures at extremely high densities ( @xmath50g@xmath9 ) . we have chosen these wide parameter ranges in order to construct fitting formulae that have a wide applicable range . the reader should use our fitting formulae in the density - temperature region in which the conditions in the above are valid . corresponding to the parameter range @xmath51 , we have used the debye - h@xmath52ckel form for the structure factor @xmath53^{-1 } \ , .\end{aligned}\ ] ] here we remark that young , corey , & dewitt s ( 1991 ) calculation has been done for @xmath54 . we have made a smooth extrapolation to the debye - h@xmath52ckel regime @xmath55 . in figure 1 we show the results of the calculation for the case of @xmath39c . we find that the second born corrections amount to about 2% at @xmath56 and @xmath2=10@xmath8 g @xmath9 and about 5% at @xmath56 and @xmath57 g @xmath9 . in figure 2 we show the results of the calculation for the case of @xmath47fe . we find that the second born corrections amount to about 8% at @xmath56 and @xmath58 g @xmath9 and about 17% at @xmath56 and @xmath57 g @xmath9 . these values are in good quantitative agreement with those of potekhin , chabrier , & yakovlev ( 1997 ) . for the case of @xmath47fe at @xmath56 and @xmath57 g @xmath9 , the present second born corrections are significantly smaller than those of these authors who obtain about 22% non - born corrections for this case . significant part of this discrepancy appears to be due to the terms higher than the second born term . in table 1 we compare the present numerical results with the numerical results by potekhin et al . ( 1997 ) for the cases of @xmath59 g @xmath9 ; @xmath60=1 , 10 , 100 . we find generally good agreement between the present numerical results and the numerical results by potekhin et al . the present numerical results appear to underestimate the non - born effects for large values of @xmath1 ( @xmath61 ) . we have carried out the numerical integrations of equations ( 2.9 ) and ( 2.10 ) for @xmath37h , @xmath38he , @xmath39c , @xmath40n , @xmath41o , @xmath42ne , @xmath43 mg , @xmath44si , @xmath45s , @xmath46ca , @xmath47fe . for the convenience of application we have fitted the numerical results of the calculation by analytic formulae . we introduce the following variable @xmath62 the fitting has been carried out for the ranges @xmath63 g @xmath9 , @xmath64 . the fitting formulae are taken as follows : @xmath65 the coefficients are given in tables 25 . the accuracy of the fitting is better than 3% for most of the cases treated in this section . so far we have dealt with the case in which the matter consists of one species of atomic nucleus . in the actual application of the present calculation to the astrophysical studies , we often encounter the case in which the matter consists of more than one species of atomic nucleus . in this section we shall extend our calculation to the case of mixtures of nuclear species . the case of mixtures has been discussed by potekhin et al . ( 1999 ) and also by brown , bildsten , & chang ( 2000 ) and by cassisi et al . their formalism is based on the linear mixing rule . here we shall give expressions according to our notations . let us consider the case in which the mass fraction of the nuclear species @xmath66 is @xmath67 . the electrical resistivity @xmath68 due to the scattering by the nuclear species @xmath66 is given by @xmath69 \ , , \\ < s>_{j } & = & < s>_{j}^{1b } + < s>_{j}^{2b } \ , , \\ < s>_{j}^{1b } & = & < s_{-1}>_{j } \ , - \ , \frac{b^{2}}{1+b^{2 } } < s_{+1}>_{j } \ , , \\ < s>_{j}^{2b } & = & \pi z_{j } \alpha \frac{b}{(1+b^{2})^{1/2 } } \left [ < s_{0}>_{j } - < s_{+1}>_{j } \right ] \ , .\end{aligned}\ ] ] here for the mixture case the parameter @xmath20 in equation ( 2.5 ) is generalized as @xmath70 the total electrical resistivity @xmath71 is given by @xmath72 therefore , the electrical conductivity @xmath22 is given by @xmath73 \ , .\end{aligned}\ ] ] in the same manner , the thermal conductivity @xmath23 is given by @xmath74 \ , .\end{aligned}\ ] ] in the above , the scattering integral @xmath75 corresponding to the nuclear species @xmath66 should be calculated by using the coulomb coupling parameter ( itoh et al . 1979 ; potekhin et al . 1999 ; brown , bildsten , & chang 2002 ; itoh et al . 2004 ) @xmath76 where @xmath77 is the electron - sphere radius , and @xmath78 and @xmath79 are the number densities of the electrons and the @xmath80-th nuclear species @xmath81 , respectively . we have calculated the second born corrections to the electrical and thermal conductivities of the dense matter in the liquid metal phase for various elemental compositions of astrophysical importance by extending the calculations reported in paper i. we have used the semi - analytical approach which is in contrast to that of the previous authors ( yakovlev 1987 ; potekhin , chabrier , & yakovlev 1997 ; potekhin et al . 1999 ) , who made use of the fully numerical values of the cross section for the scattering of the electron by the atomic nucleus calculated by doggett & spencer ( 1956 ) . it should be noted that the numerical calculation of the coulomb scattering cross section by doggett & spencer ( 1956 ) has been carried out for a limited number of @xmath1-values for the atomic nucleus @xmath1=6 , 13 , 29 , 50 , 82 , and 92 , and for a limited number of electron energies 0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , and 10mev , and also for a limited number ( 13 ) of the scattering angles that are related to @xmath82 in equations ( 2.9 ) and ( 2.10 ) . the sparseness of data for light and medium nuclei ( only for @xmath1=6 , 13 , 29 ) is potentially vulnerable in order to obtain results with reliable @xmath1-dependence . however , our study has confirmed that the previous results have sufficiently accurate @xmath1-dependence and @xmath2-dependence , since they are recovered , within about 1% , if our second - born results are multiplied by the ratio of the full non - born @xmath83 to the second - born @xmath84 . the definitions of @xmath83 and @xmath84 are given in the appendix . we have found that our results are in general agreement with those of potekhin , chabrier , & yakovlev ( 1997 ) . our second born corrections are significantly smaller than the non - born corrections of these authors for the case of @xmath47fe at @xmath56 and @xmath57 g @xmath9 . significant part of this discrepancy appears to be due to the terms higher than the second born term . in the present calculation , in contrast to baiko et al . ( 1998 ) , we have used the full liquid structure factor , for the reasons explained in 2 . we have summarized our numerical results by accurate analytic fitting formulae . we have also presented the prescriptions to deal with the cases of mixtures of nuclear species . therefore , the present results should be readily applied to various studies in the field of stellar evolution . we wish to thank our referee for many useful comments that have greatly helped us in revising the manuscript . we also wish to thank d. g. yakovlev and a. y. potekhin for their very informative communication and providing us with the numerical data of their results in table 1 . one of the authors ( n.i . ) wishes to thank n. w. ashcroft , k. hoshino , and h. maebashi for their expert advice regarding the calculations of the conductivities of simple metals in the laboratory . he especially appreciates n. w. ashcroft s lucid reasoning regarding the correctness of ziman s original method with the use of the full liquid structure factor . he also wishes to thank h. e. dewitt and s. hansen for their valuable communication regarding the ocp structure factor . this work is financially supported in part by the grant - in - aid for scientific research of japanese ministry of education , culture , sports , science , and technology under the contract 16540220 . in this appendix we evaluate the accuracy of the second born approximation by comparing with the exact results obtained by doggett & spencer ( 1956 ) . the second born approximation gives a correction factor to the rutherford cross section ( mckinley & feshbach 1948 ; feshbach 1952 ) : @xmath85 where @xmath86^{1/2}}{1 + ( e_{kin}/0.5110{\rm mev } ) } \ , , \end{aligned}\ ] ] @xmath87 being the kinetic energy of the electron , and @xmath88 is the angle of scattering . the @xmath89 factor corresponding to the results by doggett & spencer ( 1956 ) is defined by @xmath90^{2 } } \ , r^{ds}(e_{kin } , k/2k_{f } ) \ , , \end{aligned}\ ] ] where @xmath91 is related to @xmath88 by @xmath92 in order to make the comparison self - consistent , in this appendix we define @xmath93^{2 } } \ , r^{1b+2b } \nonumber \\ & = & < s>^{1b } + < s>^{2b } \ , , \end{aligned}\ ] ] which of course coincides with our previous equations ( 2.8 ) , ( 2.9 ) , ( 2.10 ) . here we have used the relationship @xmath94^{1/2 } \ , - \ , 1 \right\ } \ , .\end{aligned}\ ] ] in table 6 we compare the results corresponding to the second born approximation with those corresponding to doggett & spencer ( 1956 ) for the cases of @xmath60=10 ; @xmath1=6 , 13 , 29 ; and @xmath87=0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , 10mev . we find the accuracy of the second born correction is better than 0.4% for @xmath1=6 , better than 1.4% for @xmath1=13 , and better than 6.0% for @xmath1=29 . ashcroft , n. w. , & lekner , j. 1966 , phys . , 145 , 83 baiko , d. a. , kaminker , a. d. , potekhin , a. y. , & yakovlev , d. g. 1998 , phys . letters , 81 , 5556 brown , e. f. , bildsten , l. , & chang , p. 2002 , apj , 574 , 920 cassisi , s. , potekhin , a. y. , pietrinferni , a. , catelan , m. , & salaris , m. 2007 , apj , 661 , 1094 doggett , j. a. , & spencer , l. v. 1956 , phys . , 103 , 1597 eby , p. b. , & morgan , s. h. , jr . 1972 , phys . a , 5 , 2536 feshbach , h. 1952 , phys . , 88 , 295 flowers , e. , & itoh , n. 1976 , apj , 206 , 218 flowers , e. , & itoh , n. 1979 , apj , 230 , 847 flowers , e. , & itoh , n. 1981 , apj , 250 , 750 gnedin , o. y. , yakovlev , d. g. , & potekhin , a. y. 2001 , mnras , 324 , 725 iida , t. , & guthrie , r. i. l. 1993 , the physical properties of liquid metals ( oxford univ . press ) itoh , n. , asahara , r. , tomizawa , n. , wanajo , s. , & nozawa , s. 2004 , apj , 611 , 1041 itoh , n. , hayashi , h. , & kohyama , y. 1993 , apj , 418 , 405 ; erratum 436 , 418 ( 1994 ) itoh , n. , & kohyama , y. 1983 , apj , 275 , 858 itoh , n. , & kohyama , y. 1993 , apj , 404 , 268 itoh , n. , kohyama , y. , matsumoto , n. , & seki , m. 1984 , apj , 285 , 758 ; erratum 404 , 418 ( 1993 ) itoh , n. , mitake , s. , iyetomi , h. , & ichimaru , s. 1983 , apj , 273 , 774 itoh , n. , totsuji , h. , ichimaru , s. , & dewitt , h. e. 1979 , apj , 234 , 1079 ; erratum 239 , 415 ( 1980 ) jancovici , b. 1962 , nuovo cimento , 25 , 428 mckinley , w. a. , jr . , & feshbach , h. 1948 , phys . , 74 , 1759 mitake , s. , ichimaru , s. , & itoh , n. , 1984 , apj , 277 , 375 potekhin , a. y. , baiko , d. a. , haensel , p. , & yakovlev , d. g. 1999 , a & a , 346 , 345 potekhin , a. y. , chabrier , g. 2000 , phys . e. , 62 , 8554 potekhin , a. y. , chabrier , g. , & yakovlev , d. g. 1997 , a & a , 323 , 415 rosenfeld , a. m. , & stott , m. j. 1990 , phys . rev . b , 42 , 3406 yakovlev , d. g. 1987 , sov . astron . , 31 , 347 young , d. a. , corey , e. m. , & dewitt , h. e. 1991 , phys . rev . a , 44 , 6508 ziman , j. 1961 , phil . mag . , 6 , 1013 cccc 1 & 6 & 1.0841 & 1.087 + & 7 & 1.1297 & 1.133 + & 8 & 1.1701 & 1.175 + & 10 & 1.2400 & 1.248 + & 12 & 1.2996 & 1.311 + & 14 & 1.3521 & 1.368 + & 16 & 1.3995 & 1.420 + & 20 & 1.4834 & 1.516 + & 26 & 1.5925 & 1.649 + 10 & 6 & 0.6490 & 0.651 + & 7 & 0.6975 & 0.701 + & 8 & 0.7407 & 0.745 + & 10 & 0.8159 & 0.823 + & 12 & 0.8806 & 0.891 + & 14 & 0.9379 & 0.953 + & 16 & 0.9898 & 1.009 + & 20 & 1.0819 & 1.113 + & 26 & 1.2017 & 1.256 + 100 & 6 & 0.5236 & 0.526 + & 7 & 0.5717 & 0.575 + & 8 & 0.6152 & 0.620 + & 10 & 0.6929 & 0.700 + & 12 & 0.7609 & 0.771 + & 14 & 0.8209 & 0.836 + & 16 & 0.8750 & 0.880 + & 20 & 0.9700 & 1.001 + & 26 & 1.0931 & 1.148 + crrrrrrrrrrr @xmath95 & 0.6496 & 0.7407 & 0.8981 & 0.9232 & 0.9457 & 0.9848 & 1.0181 & 1.0471 & 1.0729 & 1.1171 & 1.1690 + @xmath96 & 0.0471 & @xmath970.0007 & @xmath970.0666 & @xmath970.0781 & @xmath970.0884 & @xmath970.1065 & @xmath970.1221 & @xmath970.1357 & @xmath970.1477 & @xmath970.1684 & @xmath970.1970 + @xmath98 & @xmath970.0056 & @xmath970.0165 & @xmath970.0071 & @xmath970.0045 & @xmath970.0019 & 0.0031 & 0.0076 & 0.0117 & 0.0155 & 0.0222 & 0.0297 + @xmath99 & @xmath970.0284 & @xmath970.0376 & @xmath970.0558 & @xmath970.0588 & @xmath970.0615 & @xmath970.0663 & @xmath970.0703 & @xmath970.0737 & @xmath970.0767 & @xmath970.0818 & @xmath970.0869 + @xmath100 & 0.0054 & 0.0114 & 0.0247 & 0.0270 & 0.0291 & 0.0326 & 0.0356 & 0.0382 & 0.0404 & 0.0440 & 0.0481 + @xmath18 & 0.0921 & 0.1037 & 0.1068 & 0.1064 & 0.1059 & 0.1046 & 0.1032 & 0.1018 & 0.1004 & 0.0977 & 0.0946 + @xmath101 & 0.4531 & 0.3959 & 0.4040 & 0.4047 & 0.4053 & 0.4063 & 0.4069 & 0.4074 & 0.4078 & 0.4084 & 0.4017 + @xmath102 & 0.0268 & 0.2196 & 0.4347 & 0.4753 & 0.5166 & 0.5930 & 0.6520 & 0.6976 & 0.7358 & 0.8004 & 0.8856 + @xmath103 & 0.0012 & 0.0006 & 0.0084 & 0.0052 & 0.0006 & @xmath970.0094 & @xmath970.0176 & @xmath970.0243 & @xmath970.0304 & @xmath970.0423 & @xmath970.0612 + @xmath104 & 0.0051 & 0.0440 & 0.0741 & 0.0796 & 0.0854 & 0.0962 & 0.1035 & 0.1082 & 0.1116 & 0.1169 & 0.1224 + @xmath105 & @xmath970.0007 & @xmath970.0081 & @xmath970.0189 & @xmath970.0224 & @xmath970.0263 & @xmath970.0338 & @xmath970.0399 & @xmath970.0449 & @xmath970.0494 & @xmath970.0571 & @xmath970.0679 + @xmath106 & 0.0018 & 0.0155 & 0.0228 & 0.0241 & 0.0257 & 0.0287 & 0.0305 & 0.0316 & 0.0323 & 0.0333 & 0.0344 + @xmath107 & 0.0011 & 0.0056 & 0.0174 & 0.0184 & 0.0189 & 0.0194 & 0.0200 & 0.0208 & 0.0214 & 0.0223 & 0.0230 + @xmath108 & 0.0621 & 0.3641 & 0.3604 & 0.3670 & 0.3787 & 0.4011 & 0.4097 & 0.4097 & 0.4074 & 0.4036 & 0.3986 + crrrrrrrrrrr @xmath109 & 0.2781 & 0.3281 & 0.4042 & 0.4170 & 0.4286 & 0.4489 & 0.4662 & 0.4813 & 0.4946 & 0.5173 & 0.5452 + @xmath110 & 0.0357 & 0.0222 & @xmath970.0077 & @xmath970.0131 & @xmath970.0180 & @xmath970.0266 & @xmath970.0339 & @xmath970.0404 & @xmath970.0460 & @xmath970.0556 & @xmath970.0672 + @xmath111 & 0.0224 & 0.0249 & 0.0396 & 0.0423 & 0.0448 & 0.0491 & 0.0528 & 0.0560 & 0.0588 & 0.0635 & 0.0684 + @xmath112 & @xmath970.0072 & @xmath970.0134 & @xmath970.0245 & @xmath970.0264 & @xmath970.0280 & @xmath970.0309 & @xmath970.0333 & @xmath970.0353 & @xmath970.0370 & @xmath970.0397 & @xmath970.0426 + @xmath113 & 0.0059 & 0.0082 & 0.0152 & 0.0163 & 0.0174 & 0.0191 & 0.0205 & 0.0217 & 0.0227 & 0.0241 & 0.0254 + @xmath114 & 0.0303 & 0.0323 & 0.0269 & 0.0258 & 0.0247 & 0.0227 & 0.0209 & 0.0193 & 0.0179 & 0.0154 & 0.0127 + @xmath80 & 0.3087 & 0.2749 & 0.2790 & 0.2794 & 0.2797 & 0.2802 & 0.2805 & 0.2808 & 0.2810 & 0.2813 & 0.2773 + @xmath115 & 0.0225 & 0.1881 & 0.2913 & 0.3121 & 0.3353 & 0.3787 & 0.4085 & 0.4283 & 0.4436 & 0.4687 & 0.5053 + @xmath116 & 0.0007 & @xmath970.0008 & 0.0011 & @xmath970.0026 & @xmath970.0072 & @xmath970.0166 & @xmath970.0233 & @xmath970.0281 & @xmath970.0321 & @xmath970.0395 & @xmath970.0505 + @xmath117 & 0.0045 & 0.0385 & 0.0519 & 0.0551 & 0.0590 & 0.0664 & 0.0711 & 0.0736 & 0.0754 & 0.0781 & 0.0819 + @xmath118 & @xmath970.0006 & @xmath970.0071 & @xmath970.0133 & @xmath970.0156 & @xmath970.0182 & @xmath970.0229 & @xmath970.0263 & @xmath970.0288 & @xmath970.0307 & @xmath970.0339 & @xmath970.0383 + @xmath119 & 0.0016 & 0.0136 & 0.0164 & 0.0173 & 0.0185 & 0.0208 & 0.0221 & 0.0228 & 0.0231 & 0.0236 & 0.0243 + @xmath91 & 0.0007 & 0.0041 & 0.0095 & 0.0093 & 0.0087 & 0.0074 & 0.0066 & 0.0061 & 0.0056 & 0.0047 & 0.0033 + @xmath120 & 0.0557 & 0.3200 & 0.2571 & 0.2604 & 0.2694 & 0.2873 & 0.2923 & 0.2897 & 0.2856 & 0.2799 & 0.2763 + crrrrrrrrrrr @xmath121 & 0.1543 & 0.1881 & 0.2380 & 0.2466 & 0.2544 & 0.2679 & 0.2794 & 0.2893 & 0.2980 & 0.3126 & 0.3306 + @xmath122 & 0.0202 & 0.0137 & @xmath970.0068 & @xmath970.0105 & @xmath970.0138 & @xmath970.0196 & @xmath970.0244 & @xmath970.0287 & @xmath970.0323 & @xmath970.0384 & @xmath970.0451 + @xmath123 & 0.0205 & 0.0248 & 0.0367 & 0.0388 & 0.0406 & 0.0438 & 0.0465 & 0.0487 & 0.0506 & 0.0537 & 0.0569 + @xmath124 & @xmath970.0024 & @xmath970.0063 & @xmath970.0140 & @xmath970.0152 & @xmath970.0164 & @xmath970.0183 & @xmath970.0198 & @xmath970.0211 & @xmath970.0222 & @xmath970.0238 & @xmath970.0254 + @xmath125 & 0.0057 & 0.0074 & 0.0123 & 0.0131 & 0.0137 & 0.0149 & 0.0158 & 0.0165 & 0.0170 & 0.0178 & 0.0183 + @xmath126 & 0.0133 & 0.0135 & 0.0084 & 0.0074 & 0.0065 & 0.0049 & 0.0036 & 0.0024 & 0.0013 & @xmath970.0005 & @xmath970.0024 + @xmath127 & 0.2293 & 0.2069 & 0.2095 & 0.2097 & 0.2099 & 0.2102 & 0.2104 & 0.2105 & 0.2106 & 0.2108 & 0.2082 + @xmath128 & 0.0196 & 0.1636 & 0.2027 & 0.2148 & 0.2302 & 0.2597 & 0.2776 & 0.2872 & 0.2938 & 0.3049 & 0.3248 + @xmath129 & 0.0004 & @xmath970.0017 & @xmath970.0028 & @xmath970.0064 & @xmath970.0108 & @xmath970.0191 & @xmath970.0245 & @xmath970.0279 & @xmath970.0305 & @xmath970.0352 & @xmath970.0422 + @xmath130 & 0.0041 & 0.0339 & 0.0375 & 0.0397 & 0.0428 & 0.0487 & 0.0521 & 0.0536 & 0.0545 & 0.0561 & 0.0589 + @xmath131 & @xmath970.0006 & @xmath970.0064 & @xmath970.0099 & @xmath970.0115 & @xmath970.0134 & @xmath970.0168 & @xmath970.019 & @xmath970.0202 & @xmath970.0211 & @xmath970.0226 & @xmath970.0246 + @xmath132 & 0.0015 & 0.0121 & 0.0121 & 0.0129 & 0.0139 & 0.0159 & 0.0170 & 0.0174 & 0.0175 & 0.0177 & 0.0181 + @xmath133 & 0.0005 & 0.0032 & 0.0051 & 0.0044 & 0.0035 & 0.0017 & 0.0005 & @xmath970.0002 & @xmath970.0008 & @xmath970.0019 & @xmath970.0035 + @xmath134 & 0.0507 & 0.2834 & 0.1902 & 0.1933 & 0.2021 & 0.2188 & 0.2226 & 0.2192 & 0.2147 & 0.2089 & 0.2068 + crrrrrrrrrrr @xmath135 & 0.4288 & 0.1778 & 0.2634 & 0.2661 & 0.2619 & 0.2489 & 0.2461 & 0.2513 & 0.2583 & 0.2700 & 0.2760 + @xmath136 & @xmath970.5654 & @xmath970.5446 & @xmath970.5073 & @xmath970.5144 & @xmath970.5251 & @xmath970.5454 & @xmath970.5525 & @xmath970.5517 & @xmath970.5495 & @xmath970.5480 & @xmath970.5561 + @xmath137 & 0.0769 & 0.3337 & 0.2504 & 0.2474 & 0.2512 & 0.2638 & 0.2674 & 0.2632 & 0.2567 & 0.2453 & 0.2389 + @xmath138 & 0.0662 & 0.0467 & 0.0083 & 0.0155 & 0.0264 & 0.0472 & 0.0549 & 0.0546 & 0.0527 & 0.0512 & 0.0594 + @xmath139 & 0.5283 & 0.0090 & 0.3358 & 0.3352 & 0.3056 & 0.2147 & 0.1748 & 0.1905 & 0.2221 & 0.2725 & 0.2749 + @xmath140 & @xmath970.5546 & @xmath970.7314 & @xmath970.4548 & @xmath970.4723 & @xmath970.5089 & @xmath970.5916 & @xmath970.6101 & @xmath970.5802 & @xmath970.5437 & @xmath970.5001 & @xmath970.5121 + @xmath141 & @xmath970.0256 & 0.5152 & 0.1868 & 0.1849 & 0.2120 & 0.3007 & 0.3446 & 0.3334 & 0.3045 & 0.2539 & 0.2464 + @xmath142 & 0.0558 & 0.2428 & @xmath970.0421 & @xmath970.0245 & 0.0124 & 0.0973 & 0.1193 & 0.0915 & 0.0555 & 0.0106 & 0.0208 + @xmath143 & 0.6074 & @xmath974.1967 & 0.4740 & 0.4663 & 0.3642 & @xmath970.2350 & @xmath970.9551 & @xmath970.6418 & @xmath970.1656 & 0.2635 & 0.2676 + @xmath144 & @xmath970.5134 & @xmath972.8194 & @xmath970.1732 & @xmath970.2319 & @xmath970.3358 & @xmath970.7496 & @xmath971.0234 & @xmath970.6404 & @xmath970.3168 & @xmath970.1219 & @xmath970.1927 + @xmath145 & @xmath970.1064 & 4.9319 & 0.0621 & 0.0612 & 0.1539 & 0.7369 & 1.4929 & 1.2219 & 0.7513 & 0.2992 & 0.2587 + @xmath146 & 0.0142 & 2.4604 & @xmath970.3263 & @xmath970.2676 & @xmath970.1633 & 0.2629 & 0.5802 & 0.2094 & @xmath970.1265 & @xmath970.3460 & @xmath970.2894 + ccccc 6 & 0.05 & 0.9324 & 0.9299 & 0.9973 + & 0.1 & 0.8921 & 0.8897 & 0.9973 + & 0.2 & 0.8263 & 0.8239 & 0.9971 + & 0.4 & 0.7489 & 0.7465 & 0.9969 + & 0.7 & 0.6950 & 0.6929 & 0.9970 + & 1 & 0.6698 & 0.6677 & 0.9969 + & 2 & 0.6410 & 0.6386 & 0.9963 + & 4 & 0.6288 & 0.6268 & 0.9968 + & 10 & 0.6224 & 0.6202 & 0.9965 + 13 & 0.05 & 1.1805 & 1.1668 & 0.9883 + & 0.1 & 1.1506 & 1.1365 & 0.9878 + & 0.2 & 1.0914 & 1.0778 & 0.9876 + & 0.4 & 1.0182 & 1.0048 & 0.9868 + & 0.7 & 0.9658 & 0.9529 & 0.9867 + & 1 & 0.9412 & 0.9283 & 0.9863 + & 2 & 0.9125 & 0.8998 & 0.9860 + & 4 & 0.9008 & 0.8879 & 0.9857 + & 10 & 0.8915 & 0.8788 & 0.9858 + 29 & 0.05 & 1.5000 & 1.4295 & 0.9530 + & 0.1 & 1.4999 & 1.4239 & 0.9493 + & 0.2 & 1.4664 & 1.3875 & 0.9462 + & 0.4 & 1.4113 & 1.3321 & 0.9439 + & 0.7 & 1.3685 & 1.2900 & 0.9426 + & 1 & 1.3485 & 1.2695 & 0.9414 + & 2 & 1.3240 & 1.2453 & 0.9405 + & 4 & 1.3125 & 1.2343 & 0.9405 + & 10 & 1.2983 & 1.2204 & 0.9400 +

the second born corrections to the electrical and thermal conductivities are calculated for the dense matter in the liquid metal phase for various elemental compositions of astrophysical importance . inclusion up to the second born corrections is sufficiently accurate for the coulomb scattering of the electrons by the atomic nuclei with @xmath0 . our approach is semi - analytical , and is in contrast to that of the previous authors who have used fully numerical values of the cross section for the coulomb scattering of the electron by the atomic nucleus . the merit of the present semi - analytical approach is that this approach affords us to obtain the results with reliable @xmath1-dependence and @xmath2-dependence . the previous fully numerical approach has made use of the numerical values of the cross section for the scattering of the electron off the atomic nucleus for a limited number of @xmath1-values , @xmath1=6 , 13 , 29 , 50 , 82 , and 92 , and for a limited number of electron energies , 0.05mev , 0.1mev , 0.2mev , 0.4mev , 0.7mev , 1mev , 2mev , 4mev , and 10mev . our study , however , has confirmed that the previous results are sufficiently accurate . they are recovered , if the terms higher than the second born terms are taken into account . we make a detailed comparison of the present results with those of the previous authors . the numerical results are parameterized in a form of analytic formulae that would facilitate practical uses of the results . we also extend our calculations to the case of mixtures of nuclear species . the corresponding subroutine can be retrieved from http://www.ph.sophia.ac.jp/@xmath3itoh-ken/subroutine/subroutine.htm .

at present , graphite and its electronic properties attract considerable attention due to the discovery of novel carbon - based materials such as fullerenes and nanotubes constructed from wrapped graphite sheets . @xcite besides , thin films of graphite give promise of device applications . @xcite the attention to graphite is also caused by specific features of its electron energy spectrum which result in interesting physical effects . @xcite the electronic spectrum of graphite is described by the slonzewski - weiss - mcclure ( swm ) model , @xcite and values of the main parameters of this model were found sufficiently accurately from an analysis of various experimental data ; see , e.g. , the review of brandt _ et al_. @xcite and references therein . the fermi surface of graphite consists of elongated pockets enclosing the edge hkh of its brillouin zone ( see figures below ) . these pockets are formed by the two majority groups of electrons ( e ) and holes ( h ) which are located near the points k and h of the brillouin zone , respectively . there is also at least one minority ( m ) low - concentration group of charge carriers in graphite , and this group seems to be located near the point h. however , it is necessary to emphasize that in spite of the considerable attention attracted to graphite an unresolved problem concerned with its spectrum still exists . it is well known @xcite that in the edge hkh of the brillouin zone of graphite , two electron energy bands are degenerate , and in a small vicinity of the edge these bands split linearly in a deviation of the wave vector @xmath0 from the edge . in other words , the edge is the band - contact line . but , it was shown in our paper @xcite that if in the @xmath0-space a closed semiclassical orbit of a charge carrier surrounds a contact line of its band with some other band ( and lifting of the degeneracy is linear in @xmath0 ) , the wave function of this carrier after its turn over the orbit acquires the addition phase @xmath1 as compared to the case without the band - contact line . this @xmath2 is the so - called berry phase , @xcite and it modifies the constant @xmath3 in the well - known semiclassical quantization rule @xcite for the energy @xmath4 of a charge carrier in the magnetic fields @xmath5 : @xmath6 where @xmath7 is the cross - sectional area of the closed orbit in the @xmath0 space ; @xmath8 is the component of @xmath0 perpendicular to the plane of this orbit ; @xmath9 is a large integer ( @xmath10 ) ; @xmath11 is the absolute value of the electron charge , and the constant @xmath3 is now given by the formula:@xcite @xmath12 when the magnetic field is applied along the hkh axis , orbits of electrons and holes in the brillouin zone of graphite surround this axis . thus , one might expect to find @xmath13 for these orbits instead of the usual value @xmath14 ( the values @xmath13 and @xmath15 are equivalent ) . a value of @xmath3 can be measured using various oscillation effects and in particular , with the de haas - van alphen effect . @xcite for example , the first harmonic of the de haas - van alphen oscillations of the magnetic susceptibility has the form , @xcite @xmath16 where @xmath17 , @xmath18 is some extremal cross section of the fermi surface of a metal in @xmath8 , a positive @xmath19 is the amplitude of this first harmonic , and @xmath20 is its phase which is given by @xmath21 with @xmath22 for a minimum and maximum cross - section @xmath18 , respectively , and @xmath23 in the case of a two - dimensional fermi surface . @xcite it follows from eq . ( [ 4 ] ) that one has to obtain @xmath24 for the maximum cross - sections of the electron and hole majorities in graphite . however , the phases @xmath25 , @xmath26 measured long ago @xcite agree with the usual value @xmath27 ; see table i. recently a new method of determining the phase @xmath20 of the de haas - van alphen oscillations was elaborated , @xcite and the authors of that paper found @xmath13 for the cross section of the hole majority in graphite . however , in this determination they assumed the fermi surface of the holes to be two - dimensional ( @xmath28 ) ; see table i. besides this , they found @xmath27 for the maximum cross section of the electron majority , assuming the three - dimensional fermi surface for this majority ( @xmath29 ) . although the obtained value @xmath13 for the holes agrees with the above prediction , the results of ref . give rise to the following new problems : first , since the band - contact line in graphite penetrates both the electron and hole extremal cross sections , these cross sections must have the same @xmath3 . second , using the values of the parameters of swm model , @xcite one might expect that in graphite the electrons and holes of the extremal cross sections are both three - dimensional . in this paper we show that in graphite , apart from the band - contact line coinciding with the edge hkh , _ three additional _ band - contact lines exist near this edge . the existence of these lines leads to the usual value @xmath27 for the maximum cross sections of the electron and hole majority groups in graphite . in other words , we resolve the above - mention contradiction between the theoretical value of @xmath3 and the data of refs . . we also discuss the data of ref . . [ cols="<,^,>,>,^,^,>,>,^,^ , < " , ] when the magnetic field @xmath5 is directed along the @xmath30 axis , the maximum electron cross section in @xmath31 is located at @xmath32 , while the maximum cross section of the hole majority is between the points k and @xmath33 , viz . , at @xmath34 where @xmath35 is the fermi energy in graphite , see fig . 2 . thus , both these cross sections are penetrated by the four band - contact lines . however , an _ even _ number of the band - contact lines do not change @xcite the usual value @xmath14 . thus , we find @xmath14 for the maximum cross sections of the majority groups , which agrees with the experimental results of refs . . we now discuss briefly the value of @xmath3 for the minority group . for the parameters presented in table ii , the hole minority is located near the point h and it results from the band @xmath36 . at this point the minority and the hole majority produced by the band @xmath37 have equal cross sections when the magnetic field is along the hkh axis . since no contact lines of the bands @xmath36 and @xmath37 penetrate this common cross section , one might expect to find the usual value @xmath14 in this case . however , the semiclassical approximation which is used in deriving eqs . ( [ 1 ] ) and ( [ 2 ] ) fails for the hole orbits corresponding to this cross section since for this approximation to be valid , the orbits must be sufficiently far away from each other . the analysis carried out beyond the scope of the semiclassical approximation @xcite led to @xmath13 and @xmath23 for the `` degenerate '' orbit . in experiments this orbit is ascribed to the hole minority , and the phase @xmath38 measured in ref . agrees with these @xmath3 and @xmath39 , see table i. in ref . a new method was developed to determine the phase @xmath20 of the de haas -van alphen oscillations of the magnetic susceptibility . the appropriate results for @xmath20 and @xmath3 in graphite are presented in table i. however , authors of ref . implied in their analysis of @xmath3 that the sign of @xmath19 in formula ( [ 3 ] ) is positive in the case of electrons and negative for holes . this is not correct ; the sign is always positive . a re - examination of the derivation of the lifshits - kosevich formula @xcite proves this statement . @xcite with this in mind we have corrected @xmath39 and @xmath3 of ref . , and the obtained results are also presented in table i. for the hole minority and for the electron majority @xcite the corrected results coincide with those of williamson _ et al_.@xcite ( but @xmath40 can be caused by the above - mentioned degeneracy of the hole orbits rather than by the two - dimensional spectrum of the hole minority ) . for the hole majority the phases @xmath26 measured in refs . and the phase @xmath41 obtained by lukyanchuk and kopelevich @xcite means that either the spectrum of these carriers is two dimensional , or if @xmath42 , one obtains @xmath43 . however , in the semiclassical approximation , @xmath3 can be equal to @xmath44 or to @xmath45 only . @xcite intermediate values can occur in situations close to the magnetic breakdown . @xcite in principle , such the situation is possible for the swm model , but it does not occur for the parameters presented in table ii . the parameters of table ii correspond to three dimensional spectrum of graphite and lead to a consistent description of the experimental data @xcite obtained many yeas ago . however , lukyanchuk and kopelevich @xcite used the highly oriented pyrolytic graphite ( hopg ) with very high ratio of the out - of - plane to basal - plane resistivities ( @xmath46 ) , and in this sample , quantum - hall - effect features were observed which indicate a quasi two dimensional nature of this hopg . @xcite it was also argued @xcite that in similar samples of hopg an incoherent transport occurs in the direction perpendicular to the graphite layers , and the three dimensional spectrum of carriers seems to fail . if this conclusion is valid only for the hole majority , it could explain the above - mentioned disagreement . this also means that the parameters of swm model should be reconsidered to describe the spectrum of such hopg . to conclude , the phases of the de haas - van alphen oscillations in graphite were measured in refs . . the data of refs . can be completely explained in the framework of the known band structure of graphite @xcite if one takes into account that four band - contact lines exist near the hkh edge of its brillouin zone . the data of lukyanchuk and kopelevich @xcite obtained for hopg disagree with the experimental results of refs . for one of the two large cross sections and probably imply that a reconsideration of the energy - band parameters for such hopg is required .

we discuss the known experimental data on the phase of the de haas -van alphen oscillations in graphite . these data can be understood if one takes into account that four band - contact lines exist near the hkh edge of the brillouin zone of graphite .

the non - mesonic weak decay ( nmwd ) process of a @xmath11 hypernucleus , @xmath12 , gives a unique opportunity to study the weak interaction between baryons since this strangeness non - conserving process is purely attributed to the weak interaction . in the nmwd , there are two decay channels , @xmath13 @xmath14 ( @xmath9 ) and @xmath15 @xmath16 ( @xmath7 ) . the ratio of those decay widths , @xmath7/@xmath9 , is an important observable used to study the isospin structure of the nmwd mechanism . for the past 40 years , there has been a longstanding puzzle that the experimental @xmath7/@xmath9 ratio disagrees with that of theoretical calculations based on the most natural and simplest model , the one - pion exchange model ( ope ) . in this model , the @xmath12 reaction is expressed as a pion absorption process after the @xmath17 decay inside the nucleus . since the ope process is tensor - dominant and the tensor transition of the initial @xmath18 pair in the @xmath19-state requires the final @xmath20 pair to have isospin zero , the @xmath21 ratio in the ope process becomes close to 0 . however , previous experimental results have indicated a large @xmath21 ratio ( @xmath221 ) @xcite . this large discrepancy between the ope - model predictions and the experimental results has stimulated many theoretical studies : the heavy meson exchange model , the direct quark model and the two - nucleon ( 2@xmath23 ) induced model ( @xmath24 ) . after k. sasaki @xmath25 pointed out an error in the sign of the kaon exchange amplitudes in 2000 @xcite , those theoretical values of the @xmath21 ratio have increased to the level of 0.4@xmath220.7 @xcite . on the other hand , the experimental data still have large errors ( @xmath21 = 0.93 @xmath26 0.55 for @xmath0he @xcite ) , and it is hard to draw a definite conclusion on the @xmath21 ratio . when we compare the measured @xmath21 ratio with that obtained in theoretical calculations , the most serious technical problem was a treatment of the re - scattering effect in the residual nucleus , the so - called final state interaction ( fsi ) . moreover , the possible existence of a multi - nucleon induced process has been discussed theoretically ( such as 2@xmath23-induced process ) , though there has been no experimental evidence . several nucleon energy spectra from hypernuclear decay have been reported so far @xcite , in which it is however difficult to extract the @xmath21 ratio without theoretical assumptions on the effects of fsi and possible multi - nucleon induced processes . since the 1@xmath23-induced decay `` @xmath27 '' is two - body process , the outgoing nucleon - nucleon pair suffering no fsi effect must have about 180 degree opening angle and clear energy correlation . in the present experiment , we performed a coincident measurement of the two nucleons , @xmath4 and @xmath5-pairs , in the decay for the first time . the 1@xmath23-induced processes could be clearly observed by measuring yields of the back - to - back @xmath4- and @xmath5-pairs and confirming that the energy sums roughly correspond to their @xmath28-values ( @xmath22150 mev ) . the measured yields of the coincident back - to - back @xmath4- and @xmath5-pairs , @xmath29 , are represented as @xmath30 , where @xmath31 are the number of back - to - back @xmath4(@xmath5)-pair events from the decay ; @xmath32 , @xmath33 and @xmath34 stand for decay - counter acceptances and detection efficiencies and reduction factors ( due to the fsi or / and other non back - to - back processes ) for the @xmath4(@xmath5)-pair , respectively . it is noteworthy that the reduction factors are approximately canceled out with assumption of the charge symmetry , @xmath35 , when we take the ratio of the @xmath4- and @xmath5-pair yields , @xmath6 . in order to minimize the fsi effect , we selected a light @xmath19-shell hypernucleus , @xmath0he . in @xmath19-shell hypernucleus , initial relative @xmath18 states must be @xmath36 states , whereas in a @xmath37-shell hypernucleus they may be @xmath38 states . to investigate the @xmath37-wave effect , we also performed the same experiment for a typical light @xmath37-shell hypernucleus , @xmath1c . in this letter , we show the opening angle and the energy sum distributions of @xmath4- and @xmath5-pairs from the nmwd of @xmath0he and the @xmath39 ratio for both hypernuclei . the present experiments ( kek - ps e462/e508 ) were carried out at the 12-gev proton synchrotron ( ps ) in the high energy accelerator research organization ( kek ) . hypernuclei , @xmath0he and @xmath1c , were produced via the ( @xmath2,@xmath3 ) reaction at 1.05 gev/@xmath40 on @xmath41li and @xmath42c targets , respectively . since the ground state of @xmath43li is above the threshold of @xmath0he @xmath44 , it promptly decays into @xmath0he emitting a low - energy proton . the @xmath45li ( @xmath2,@xmath3 ) @xmath43li reaction was therefore employed to produce @xmath0he . the hypernuclear mass spectra were calculated by reconstructing the momenta of incoming @xmath2 and outgoing @xmath3 using a beam - line spectrometer composed of the qqdqq system and the superconducting kaon spectrometer ( sks ) @xcite , respectively . particles emitted from the decays of @xmath11 hypernuclei were detected by the decay - particle detection system installed symmetrically in the direction to the target in order to maximize acceptance of the back - to - back event for @xmath4- and @xmath5-pairs from the nmwd process , as shown in ref.@xcite ( fig . 1 ) . it was composed of plastic scintillation counters and multi - wire drift chambers . the decay particles were identified by the time - of - flight and the range . r0.65 the ground state yields of @xmath0he and @xmath1c are , respectively , about 4.6 @xmath46 10@xmath47 and 6.2 @xmath46 10@xmath47 events , which were one order - of - magnitude higher than those of previous experiments . the inclusive excitation - energy spectra of @xmath43li and @xmath1c are shown in ref.@xcite ( fig . 2 ) . upper figures of fig . [ coinfig ] , ( a ) and ( b ) , show opening angle distributions of @xmath4- and @xmath5-pairs at the energy threshold level of 30 mev for both of proton and neutron . they seem to have clear back - to - back correlations , though these are not corrected the angular dependent acceptance . the shaded histogram shows estimated nucleon contaminations due to the pion absorption process in which @xmath48 s from the mesonic decay of @xmath11 hypernucleus are absorbed by the materials around the target . the background was estimated by assuming that the shape of the angular distribution from this @xmath48 absorption process is the same as that from the @xmath48 decay of @xmath11 ( @xmath11 @xmath49 ) formed via the quasi - free formation process ( see ref.@xcite for the detail ) . the angular distributions of middle of fig . [ coinfig ] , ( c ) and ( d ) , are corrected for acceptances and efficiencies for @xmath4- and @xmath5-pairs , and normalized per nmwd . the estimated contamination due to the pion absorption stated above are subtracted . they still have back - to - back correlation , which indicates that the fsi effect is not so severe and 1@xmath23-induced nmwd ( two body process ) is the major one . lower figures of fig . [ coinfig ] , ( e ) and ( f ) , show energy sum distributions of the @xmath4- and @xmath5-pairs by gating back - to - back events as shown in the upper figures ( @xmath50 ) . we confirmed that those energy sum distributions have broad peak around these @xmath28-values as expected . the shaded histogram shows estimated contaminations due to pion absorption as described above , which distributes to lower energy region . also for @xmath1c , similar distributions of the angle and energy sum of the @xmath4- and @xmath5-pairs were obtained in a same way . we successfully observed @xmath4- and @xmath5-pairs from the nmwd of @xmath0he and @xmath1c . the ratio of the back - to - back @xmath4- and @xmath5-pair yields , @xmath51 , for @xmath0he and @xmath1c were obtained as @xmath52 where the quoted systematic errors mainly come from the neutron detection efficiency ( @xmath22 6 % ) . they can be approximately regarded as the @xmath21 with assumption of the charge symmetry . it is now revealed that the @xmath21 ratio is significantly less than unity , thus excluding the earlier claim that the ratio is close to unity @xcite . on the contrary , recent theoretical calculations seem to be supportive to our results being on the increase of the ratio toward 0.5 . the present results have finally given the answer to the longstanding @xmath21 ratio puzzle , and have made a significant contribution to the study of the nmwd .

we have measured both yields of neutron - proton and neutron - neutron pairs emitted from the non - mesonic weak decay process of @xmath0he and @xmath1c hypernuclei produced via the ( @xmath2,@xmath3 ) reaction for the first time . we observed clean back - to - back correlation of the @xmath4- and @xmath5-pairs in the coincidence spectra for both hypernuclei . the ratio of those back - to - back pair yields , @xmath6 , must be close to the ratio of neutron- and proton - induced decay widths of the decay , @xmath7(@xmath8)/@xmath9(@xmath10 ) . the obtained ratios for each hypernuclei support recent calculations based on short - range interactions .

* the sequence . * in blazars , radio loud active galactic nuclei ( agn ) with their relativistic jet axis pointing to our line of sight , the synchrotron peak frequency ( @xmath3 ) covers a wide range ( @xmath4 hz ) , with bllacs ( bll , lineless blazars ) spanning the entire range and fsrqs ( flat spectrum radio quasars , sources with strong broad emission lines ) having lower @xmath3 ( @xmath5 hz ) . following @xcite , we adopt the generic terms for low , intermediate , and high _ synchrotron - peaking _ ( lsp , isp , hsp ) blazars independently of the spectroscopic type . @xcite found that as the source synchrotron power @xmath6 increases , @xmath3 decreases , with predominantly fsrq sources at the low @xmath3 , high @xmath6 end through lsp , isp , and finally hsp bllacs at the low @xmath6 end . they also used the sparse _ egret _ data to argue that the same reduction of the peak frequency happens in the high energy - presumably inverse compton ( ic ) component - component and that the compton dominance ( the ratio of ic to synchrotron power ) increases with source power . @xcite suggested that more efficient cooling of particles in the jets of high luminosity blazars is responsible for the lower peak frequencies . * from sequence to envelope . * @xcite and @xcite identified relatively powerful sources with a radio to x - ray spectral index @xmath7 typical of weak sources with @xmath3 in the x - rays . such sources , if confirmed , challenge the sequence . upon close study , however , their x - ray emission was found not to be of synchrotron origin @xcite and as of now sources with high @xmath6 - high @xmath3 have not been found @xcite . sources below the blazar sequence are expected from jets less aligned to the line of sight . indeed , @xcite found that new sources they identified modify the blazar sequence to an _ envelope_. * challenges . * @xcite found several low @xmath6 - low @xmath3 sources that , because they have a high core dominance ( @xmath8 , ratio of core and therefore beamed to extended and therefore isotropic radio emission ) , are not intrinsically bright sources at a larger jet angle . these sources challenge the sequence because ( @xmath0 ) both intrinsically weak and intrinsically powerful jets can have similar @xmath3 and ( @xmath1 ) intrinsically weak jets can produce a wide range of @xmath3 from ( @xmath9 - @xmath10 hz ) . another challenge came from @xcite who showed that , contrary to what is anticipated by the sequence , high and low synchrotron peak frequency ( hsp and lsp ) bl lacertae objects ( blls , blazars with emission line ew @xmath11 ) have similar @xmath12 . these findings challenge the sequence , even after being extended to include the sources in the envelope as de - beamed analogs of the blazar sequence sources . @xcite argued that at a critical value of the accretion rate @xmath13 , the accretion switches from a standard radiatively efficient thin disk with accretion - related emission power @xmath14 for @xmath15 , to a radiatively inefficient mode where @xmath16 . this critical point may be connected to the transition between fanaroff riley ( fr ; * ? ? ? * ) ii to fr i radio galaxies ( rg ) : the level of the low frequency extended radio emission ( coming mostly from the radio lobes and considered to be isotropic ) that separates fr i and fr ii rg , has been shown to be a function of the host galaxy optical magnitude @xcite : the division between fr i and fr ii is at higher radio luminosities for brighter galaxies . @xcite argued that , because the optical magnitude of a galaxy is related to the central black home mass @xcite and the extended radio luminosity is related to the jet kinetic luminosity ( following the scaling of * ? ? ? * ) , this division can be casted as a division in terms of the fraction of the eddington luminosity carried by the jet : jets with kinetic luminosity @xmath17 give rise to fr i rg , while jets with @xmath18 are predominantly fr ii sources . interestingly , and in agreement with the unification scheme , @xcite and @xcite find that the same dichotomy applies to separating blls and fsrq , the aligned versions of fr i and fr ii respectively . finally , it is very intriguing that @xcite argue that there is a paucity of sources around @xmath19 . fr i , low line excitation fr ii and some high line excitation fr ii were found to occupy the low @xmath20 regime , while the high @xmath20 regime was occupied by high line excitation fr ii , broad line radio galaxies and powerful radio quasars . track ( a ) shows the path of a synchrotron peak for a single speed jet in an environment of radiatively efficient accretion and ( b ) for a decelerating jet of the type hypothesized to exist in fri sources as the jet orientation changes.__,width=268 ] recently , we ( * ? ? ? * heretofore m11 ) compiled the largest sample of radio loud agn for which sufficient data existed to determine variability - averaged @xmath3 @xmath6 , as well as the extended low frequency radio emission @xmath12 . this is an important quantity in our study , because it has been shown to be a good proxy for the jet kinetic power @xmath21 , as measured by the energy required to inflate the x - ray cavities seen to coincide with the radio lobes of a number of sources ( e.g. * ? ? ? * ; * ? ? ? * ) . the picture that emerges ( figure [ m11 ] ) exhibits some important differences with the blazar sequence . in particular , isp blls have @xmath21 comparable to that of hsp and lsp blls . also , although all the fr i galaxies were found to have similar @xmath21 with blls , no fr i galaxies were found with @xmath22 hz . because there is no obvious selection acting against the detection of fr i galaxies with core sed peaking at higher energies , we are lead to conclude that the un - aligned versions of hsp blazars have @xmath3 smaller by a factor of least @xmath23 compared to their aligned equivalent , something that agrees with the existence of velocity profiles in the emitting plasma , as supported by other investigations ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? in m11 we suggested that _ extragalactic jets can be described in terms of two families_. the first is that of weak jets characterized by velocity profiles and weak or absent broad emission lines . hsps ( @xmath24 hz ) , isps ( @xmath25 hz ) , and fr i rg belong to this family . on the basis of having similar @xmath21 with hsps and fr i rg , the isp sources were argued to be somewhat un - aligned hsps . the second family is that of more powerful jets having a single lorentz factor emitting plasma and , in most cases , stronger broad emission lines . interestingly , the two families divide at @xmath26 erg s@xmath27 , which for @xmath28 , corresponds to @xmath29 , similar to the @xmath30 of @xcite . aligned sources are found along the broken power sequence depicted by the solid lines a and b with a ( b ) corresponding to jets in radiatively efficient ( inefficient ) accretion environments with @xmath31 ( @xmath32 ) . the broken lines a and b depict the tracks followed by two sources as they depart from the power sequence and their orientation angle @xmath33 increases . while in the first case a single velocity flow is assumed , in the second case emission from a decelerating flow is considered @xcite . . see text for the description of boxes a , b , c and zones @xmath34.__,width=268 ] we now discuss some predictions of the new unification scheme and their confirmation from the current data . * @xmath35 increases along the two branches of the broken power sequence . * we examine now if along the two branches of the power sequence , depicted schematically by the red and blue arrows in the figure [ prelim ] , @xmath21 increases . to do that , we select those sources that are close to the sequence of powerful aligned objects and split them in three groups a , b , c , as seen in figure [ prelim ] . in figure [ powerful ] we plot the @xmath21 distribution of sources in these three groups . as expected , the average @xmath21 increases from group c to a. running the same test for jets with inefficient accretion requires to use sources that are not aligned , because of the small number of sources . for this reason , we select all low power sources with @xmath36 to insure that we do not have any mixing with sources of the other branch and we separate them in the three groups @xmath37 ( figure [ prelim ] ) separated by the de - beaming tracks of a decelerating jet depicted also in figure [ m11 ] . as can be seen in figure [ weak ] , the average @xmath21 increases from group @xmath38 to @xmath39 , according to our expectations . * as @xmath35 increases , the fraction of the blls decreases along the powerful sequence . * as @xmath21 increases along the powerful sequence , @xmath6 increases , but @xmath3 decreases . at the same time , if we assume that @xmath21 scales with accretion power , we expect that the blr luminosity increases . if @xmath3 did not change , we would expect that the ratio of the blr to optical synchrotron emission would not change . but @xmath3 does decrease as @xmath21 increases , shifting the synchrotron component to lower frequencies and revealing more of the blr . thus we expect that the fraction of sources that is classified as blls will become smaller as @xmath21 increases along the powerful sequence . this is clearly seen in figure [ powerful ] , with the fraction of blls clearly decreases as @xmath21 decreases . * for powerful sources , the fraction of blls increases for less aligned sources . * in our scheme , we expect that for powerful sources of a given @xmath21 , as they become more un - aligned , the beamed synchrotron emission will decrease , while the blr luminosity will be much less affected , resulting to a decreasing fraction of blls for more un - aligned sources . to address this , we selected sources with @xmath40 erg s@xmath27 ( orange sources in figure [ prelim ] ) and we plotted the fraction of blls as a function of radio core dominance @xmath41 which is an orientation indicator . as can be seen in figure [ bll_fraction ] , as the core dominance decreases , the fraction of blls quickly decreases , in agreement with our expectations . * a given accretion power @xmath42 corresponds to a narrow @xmath35 range . * we collected black hole masses from the literature for most of the sources of m11 and used them to calculate the ratio of @xmath43 . we plot our results in figure [ mcrit ] : in blue sources with @xmath44 hz , almost exclusively blls , therefore radiatively inefficient accretors ; in red sources with @xmath45 hz and @xmath46 erg s@xmath27 , almost all fsrqs , therefore radiatively efficient accretors . the separation of red and blue sources at @xmath47 suggests that there is a transition at @xmath48 with radiatively efficient accretion at @xmath49 and that sources with a given accretion power do not produce jets with @xmath21 significantly smaller or larger than their accretion power .

we recently argued @xcite that the collective properties of radio loud active galactic nuclei point to the existence of two families of sources , one of powerful sources with single velocity jets and one of weaker jets with significant velocity gradients in the radiating plasma . these families also correspond to different accretion modes and therefore different thermal and emission line intrinsic properties : powerful sources have radiatively efficient accretion disks , while in weak sources accretion must be radiatively inefficient . here , after we briefly review of our recent work , we present the following findings that support our unification scheme : ( @xmath0 ) along the broken sequence of aligned objects , the jet kinetic power increases . ( @xmath1 ) in the powerful branch of the sequence of aligned objects the fraction of blls decreases with increasing jet power . ( @xmath2 ) for powerful sources , the fraction of blls increases for more un - aligned objects , as measured by the core to extended radio emission . our results are also compatible with the possibility that a given accretion power produces jets of comparable kinetic power .

the tremendous progress in the last decade has made it possible to pin down , with impressive accuracy , many of the fundamental parameters in the neutrino sector . a complete picture , however , is still not available . chief among the missing information is the determination of the @xmath1 element of the neutrino mixing matrix @xmath2 , which , in turn , is crucial in ascertaining the cp violation effects in the leptonic sector . given that direct cp violations in the quark sector @xcite have been well - established and accurately measured , it is imperative , from both the theoretical and experimental points of view , to assess the corresponding situation in the leptonic sector . another unsolved puzzle concerns the neutrino mass spectrum , in that there are the possibilities of either the normal " or inverted " orderings . it is certainly important to settle this question . while the fundamental parameters refer to those in vacuum , it has been well - established ( see , _ e.g. _ , ref.@xcite ) that they are modified when neutrinos propagate through matter , by giving the neutrino an induced mass , which is proportional to its energy and to the medium density . indeed , in the analyses of the solar neutrinos , certain features of the data , such as the modification of the energy spectra from the original , can only be understood by the inclusion of matter effects . with the advent of long baseline experiments ( lbl , for an incomplete list , see , _ e.g. _ , ref.@xcite ) , the induced mass can actually be tuned " by changing the neutrino energy ( @xmath3 ) . this provides a powerful tool which can be used to extract fundamental neutrino parameters from measurements . in this work , we will use a rephasing invariant parametrization which enables us to obtain simple formulas for the transition probabilities of neutrinos propagating through matter of constant density . it was shown earlier that these parameters obey evolution equations as a function of the induced mass . in addition , these equations preserve the approximate @xmath0 symmetry @xcite which characterizes the neutrino mixing in vacuum . incorporation of the @xmath0 symmetry for all induced mass values results in a set of very simple transition probabilities @xmath4 . in general , these formulas offer quick estimates of the various oscillation probabilities , using the known solutions obtained earlier . as an example , we will analyze @xmath5 in detail , emphasizing its dependence on the neutrino parameters . neutrino oscillations , being lepton - number conserving , are described in terms of a mixing matrix whose possible majorana phases are not observable . thus it behaves just like the ckm matrix under rephasing transformations , which leave physical observables invariant @xcite . to date , however , such observables are often given in terms of parameters which are not individually invariant . so it seems that the use of manifestly invariant parameters may be more physically relevant . two such sets are known to be @xmath6 @xcite and @xmath7 @xcite . recently , by imposing the condition @xmath8 ( without loss of generality ) , another set was found , given by @xcite @xmath9 where the common imaginary part can be identified with the jarlskog invariant @xmath10 @xcite . their real parts are labeled as @xmath11 the variables are bounded by @xmath12 with @xmath13 for any ( @xmath14 ) , and satisfy two constraints : @xmath15 @xmath16 eq . ( [ con2 ] ) , together with the relation @xmath17 follow @xcite from ( the imaginary and real parts of ) the identity @xmath18 . thus , flavor mixing is specified by the set @xmath19 plus a sign , according to @xmath20 . this sign arises since the transformation @xmath21 , corresponding to a cp conjugation , leaves the real part @xmath19 of @xmath22 invariant , but changes the sign of its imaginary part @xmath23 . note that , using @xmath24 , a complete parametrization also requires four @xmath24 elements plus a sign . the parameters @xmath19 are related to the rephasing invariant elements @xmath24 by @xmath25 = \left(\begin{array}{ccc } x_{1}-y_{1 } & x_{2}-y_{2 } & x_{3}-y_{3 } \\ x_{3}-y_{2 } & x_{1}-y_{3 } & x_{2}-y_{1 } \\ x_{2}-y_{3 } & x_{3}-y_{1 } & x_{1}-y_{2 } \\ \end{array}\right).\ ] ] one can readily obtain the parameters @xmath19 from @xmath26 by computing its cofactors , which form the matrix @xmath27 with @xmath28 , and is given by @xmath29 the relations between @xmath19 and @xmath30 are given by ( using @xmath31 ) : @xmath32 the second term in either expression is one of the @xmath33 s ( @xmath34 s ) defined in eq . ( [ eq : g ] ) . also , by using the constraint in eq . ( [ cons ] ) , @xmath35 can be expressed in terms of quadratics in @xmath19 , a result which will be used later in tables i and ii . for neutrinos in matter ( of constant density ) , it was shown @xcite that , as a function of the induced mass @xmath36 , the neutrino parameters satisfy a set of evolution equations which are greatly simplified by using the @xmath19 variables . it was found that @xmath37 where @xmath38 are the eigenvalues of the hamiltonian . also , the evolution equations for all @xmath39 can be obtained and are collected in table i of ref . @xcite . of particular interest for our purposes are the equations : @xmath40 and @xmath41 note that the quantities @xmath42 and @xmath43 form a closed system under the evolution equations , independent of other possible combinations of these variables . there remain two more independent evolution equations , which may be chosen as those for ( @xmath44 ) . we define @xmath45 @xmath46 then @xmath47.\end{aligned}\ ] ] it follows that @xmath48 if @xmath49 . this condition is equivalent to @xmath50 , @xmath51 , @xmath0 exchange symmetry . thus , the evolution equations preserve the @xmath0 symmetry , which was established ( approximately ) for neutrino mixing in vacuum . another useful property of the evolution equations is to establish matter invariants . for instance @xcite , @xmath52=0,\ ] ] where @xmath53 is defined in eq . ( [ eq : xi ] ) and @xmath54 ( also , @xmath55 , as mentioned before @xcite ) . in addition , there is a simple relation @xmath56=1.\ ] ] eqs . ( [ xd ] ) and ( [ sxd ] ) are three - flavor generalizations of the two - flavor results @xcite : @xmath57 @xmath58=-1,\ ] ] where @xmath59 , @xmath60 , @xmath61 , in the usual notation . the vacuum neutrino masses are known to be hierarchical , @xmath62 , @xmath63 , @xmath64 . there are two possibilities , the normal hierarchy ( @xmath65 ) , or the inverted hierarchy ( @xmath66 ) . in matter of constant density , @xmath67 , which are @xmath68-dependent . for the case of normal hierarchy , there are two @xmath68-values where the levels cross " , at the lower resonance , @xmath69 , @xmath70_{a_{l}}=0 $ ] , and at the higher resonance , @xmath71 , @xmath72_{a_{h}}=0 $ ] . from eqs . ( [ eq : ln ] ) , one finds that rapid variations occur only for @xmath68 to be near @xmath73 or @xmath74 . let us denote by @xmath75 the values of @xmath68 in vacuum @xmath76 , at the lower resonance @xmath77 , in the intermediate range @xmath78 , at the higher resonance @xmath79 , and in dense medium @xmath80 . then , the solutions for @xmath81 are well - approximated @xcite by two - flavor resonance solutions . for @xmath82 , @xmath83^{1/2 } , \nonumber \\ x_{1 } & = & \frac{1}{2}[p_{l}-(p^{2}_{l}a - q_{l}\delta_{0})/\delta_{21 } ] , \nonumber \\ x_{2 } & = & \frac{1}{2}[p_{l}+(p^{2}_{l}a - q_{l}\delta_{0})/\delta_{21 } ] , \nonumber \\ x_{3 } & \cong & ( x_{3})_{0},\end{aligned}\ ] ] where @xmath84 in matter , @xmath85 , @xmath86 , @xmath87 . note that @xmath88 , @xmath89 , and @xmath90 . for @xmath91 , @xmath92^{1/2 } , \nonumber \\ x_{1 } & \cong & ( x_{1})_{i } , \nonumber \\ x_{2 } & = & \frac{1}{2}[p_{h}-(p^{2}_{h } \bar{a}-q_{h } \delta_{i})/\delta_{32 } ] , \nonumber \\ x_{3 } & = & \frac{1}{2}[p_{h}+(p^{2}_{h } \bar{a}-q_{h } \delta_{i})/\delta_{32}].\end{aligned}\ ] ] here , @xmath93 , and @xmath94 , @xmath95 , @xmath96 are taken at @xmath97 . note that @xmath98 , @xmath99 , @xmath100 . also , @xmath101 is an invariant as @xmath68 varies . thus , the product @xmath102 has a resonance behavior near @xmath103 . note also that the minimum of @xmath104 is at @xmath105 . to obtain @xmath106 for @xmath91 and @xmath104 for @xmath82 , one first notes from eq . ( [ eq : di ] ) that @xmath107 for high @xmath68 . thus , a direct integration leads to @xmath108\ ] ] for @xmath91 , where @xmath109 . similarly , a direct integration of @xmath110 for low @xmath68 gives @xmath111.\ ] ] the solutions for @xmath112 in both regions of @xmath68 are obtained from @xmath113 . note that the solutions for @xmath82 and for @xmath91 should agree for @xmath114 . this condition leads to @xmath115 and @xmath116 . for inverted hierarchy , the behaviors of @xmath53 near @xmath73 are given by the same eq . ( [ low ] ) . however , for @xmath117 , there is no longer a resonance . instead , all @xmath53 change slowly , so that @xmath118 , @xmath119 , @xmath120 , for @xmath117 . the solutions for @xmath121 are obtained by @xmath122 . thus , there is a resonance behavior near @xmath74 , for the inverted hierarchy scenario . otherwise all the changes are small . the accuracy of the approximate formulas in eqs . ( [ low]-[high ] ) can be assessed by numerical integrations of the exact equations , eqs . ( [ eq : di ] ) and ( [ eq : ln ] ) . to do that we write @xmath123 where @xmath124 in vacuum , and @xmath26 reduces to the tribimaximal @xcite matrix when @xmath125 . it should be emphasized that the parameters @xmath126 carry quite distinct behaviors as @xmath68 varies , as shown in the following . ( [ eq : w ] ) and ( [ w0 ] ) give rise to @xmath127 and from @xmath128 , we have @xmath129 with the constancy of @xmath130 , one concludes that @xmath131 as @xmath68 varies . in addition , since @xmath132 , we have @xmath133,\ ] ] and @xmath134 furthermore , one obtains from @xmath135 that @xmath136,\ ] ] and @xmath137 thus , @xmath138 and @xmath139 can change considerably as functions of @xmath68 , but @xmath140 throughout . for numerical integrations , eqs . ( [ eq : w ] ) and ( [ w0 ] ) suggest the following initial values in vacuum : @xmath141 where @xmath142 is chosen and the terms in @xmath143 are ignored . we shall choose the initial values @xmath144 and @xmath145 , which correspond to the experimental bounds @xmath146 @xcite and an assumed cp violation phase @xmath147 , respectively . the numerical solutions for the @xmath19 parameters , the squared elements of the mixing matrix , and @xmath10 in matter follow directly and are shown in figs . 2 - 5 in ref . our choice of @xmath148 signifies a small @xmath0 symmetry breaking , the solutions verify that @xmath149 remain negligible for all a values . in addition , we show in fig . 1 both the numerical and the approximate solutions for @xmath150 in matter . note that the hierarchical relation among the @xmath150 s varies in matter and plays an important role in the oscillatory factor @xmath151 of the probability functions . it is seen that @xmath152 ( normal hierarchy ) and @xmath153 ( inverted hierarchy ) for @xmath154 . while in @xmath155 , the @xmath150 s are less hierarchical : @xmath156 ( normal ) and @xmath157 ( inverted ) . [ cols="^,^,^,^,^",options="header " , ] our results may be compared to formulas in terms of the standard parametrization " @xcite , given , @xmath158 , in kimura @xmath159 @xcite . the relations between @xmath19 and the standard parametrization " are given by @xmath160 where @xmath161 , @xmath162 , and @xmath163 is the dirac cp phase . it can be shown that the functions @xmath164 here in terms of @xmath19 are simply @xmath165 in eqs.(15 - 23 ) of ref . @xcite , and the resultant probability functions are identical . eq ( [ eq : kk ] ) also offers some insight on the @xmath68-independence of the approximate @xmath0 symmetry . it is seen that the conditions @xmath166 are fulfilled if 1 ) @xmath167 , and 2 ) @xmath168 . the behaviors of @xmath169 were given in fig . . @xcite . while @xmath170 is almost independent of @xmath68 , @xmath171 for low @xmath68 , and @xmath172 for high @xmath68 . they combine to validate conditions 1 ) and 2 ) , for all @xmath68 values . the other possibility is that @xmath173 . here , @xmath163 itself is largely @xmath68-independent because of the matter invariant @xmath174 @xcite . exact @xmath0 symmetry was studied earlier by harrison and scott @xcite . their formulation uses the mixing matrix @xmath2 ( with specific choice of phases ) , while our results are in terms of rephasing invariant ( and observable ) variables , making it possible to calculate transition probabilities directly . in addition , by comparing with the exact formulas in table i , one can quickly compute corrections to the presumed exact symmetry . the unique features of the @xmath19 parametrization can be used to facilitate , @xmath158 , the analyses of the lbl experiments . as an example , let us consider the probability @xmath5 explicitly . according to table i , with the approximation @xmath166 , @xmath175 \nonumber \\ & -&8j\sin\phi_{21}\sin\phi_{31}\sin\phi_{32},\end{aligned}\ ] ] with @xmath176 . using the solutions in eqs . ( [ low],[high ] ) , it is straightforward to infer the behaviors of @xmath5 . in the following , let us focus on the region of high @xmath68 values @xmath177 . here , @xmath178 so that ( excluding the case @xmath179 ) @xmath180 it is useful to examine the qualitative properties of @xmath181 and @xmath182 separately . if the mass hierarchy is normal , the solutions in eq . ( [ high ] ) suggest a higher resonance for @xmath181 at @xmath183 , where @xmath184 . with @xmath185[(e/\mbox{gev})]$ ] , @xmath186 @xmath187 , and @xmath188 @xmath189 , the location of resonance @xmath74 corresponds to an energy @xmath190 gev , which is independent of the baseline length . ( [ pemu ] ) shows that , in the high @xmath68 region , @xmath5 @xmath191 is two - flavor like . however , it does not mean that the three - flavor problem is reduced to a single two - flavor problem . this is because the probability @xmath192 , according to table i , would have contributions from all the @xmath193 s . as an illustration , we show @xmath194 , @xmath195 , and @xmath196 as functions of @xmath3 in fig . 2 , with @xmath197 km . it is seen that a resonance for @xmath194 occurs near @xmath198 gev as expected . however , the smallness of @xmath182 near @xmath198 gev suppresses the probability even if @xmath194 is at a resonance . on the other hand , the probability at the first peak of @xmath182 ( near @xmath199 gev ) also gets suppressed by the smallness of @xmath194 . as a result , a significant flavor transition only occurs when @xmath200 is adjusted so that the peak of @xmath195 is located near the resonance of @xmath194 . the first maximum of @xmath182 occurs if @xmath201 is properly chosen : @xmath202=\frac{\pi}{2}.\ ] ] for the first maximum to coincide with the resonance of @xmath181 , the value of @xmath104 is taken at @xmath74 : @xmath203 . it leads to @xmath204 using the current upper bound @xmath205 . one concludes that if the mass hierarchy is normal , an extra long baseline ( @xmath206 km ) can lead to a greatly enhanced probability for the neutrino beam near @xmath207 gev , at which energy both @xmath194 and @xmath182 reach the maximal values . the probability will be suppressed when @xmath200 starts to vary and @xmath182 moves away from the maximum . note that for the maxima of @xmath181 and @xmath182 to coincide near @xmath207 gev , the baseline @xmath200 and the undetermined @xmath208 are related by @xmath209 . on the other hand , since @xmath194 does not go through the higher resonance under the inverted hierarchy , the probability is in general suppressed even if @xmath182 reaches its maximum . one further concludes that under the inverted hierarchy , the transition probability remains small and is insensitive to variation of the baseline length @xmath200 . thus , if the mass hierarchy is normal , one would expect to observe sizable probability difference at high energy for experiments involving two baselines with sizable difference in length . on the other hand , the probability would be small and nearly independent of the baseline at high energy if the mass hierarchy is inverted . we show in fig . 3 the probability function under both hierarchies for two arbitrarily chosen baselines . note that the peak locations and the peak values vary as @xmath200 . it is seen that for the normal hierarchy , @xmath210 km ) @xmath211 km ) near the first peak is expected , while @xmath210 km ) @xmath212 km ) @xmath213 if the mass hierarchy is inverted . this result may provide useful hints to the determination of the mass hierarchy . note that the probabilities can be deduced if the details of the experiments are considered . if the neutrino energy can be reconstructed accurately from the secondary particles involved in an experiment , the observed spectrum will tell how the magnitude of the transition probability plays a role . on the other hand , if reliable measurement of the energy spectrum is not available , a collection of the event rates should also be useful in comparing the probabilities . another possible application is to look for both @xmath214 and @xmath215 for a single , but very long baseline . since the @xmath121 s only go through the higher resonance under the inverted hierarchy , one would expect to observe in the vicinity of the peak either @xmath216 if the hierarchy is normal , or @xmath217 if the hierarchy is inverted . we show an example in fig . 4 . note that although the peak value of the probability varies with the baseline length , the relative and qualitative features of the above observation remain valid for a chosen baseline . neutrino transition probabilities are usually given in terms of the simple expression @xmath218 , although the individual @xmath219 s are not directly observable . when one rewrites them using physical observables , such as those in the standard parametrization " , the resulting formulas are often very complicated . it is thus not easy to obtain general properties of these probabilities in experimental situations . in this paper we express the probabilities as functions of rephasing invariant parameters . in addition , we incorporate the @xmath0 symmetry , valid ( approximately ) for any value of the induced neutrino mass ( @xmath68 ) . the resulting formulas are very simple , and are listed in tables i and ii . they offer a quick quantitative assessment for any physical process at arbitrary @xmath68 values . as an illustration , we analyzed the probability @xmath214 , with emphasis on its dependence on @xmath3 , @xmath200 , and @xmath220 . by changing the value of @xmath3 and @xmath200 in various lbl experiments , one can hope not only to test the theory used to establish @xmath4 , but also to help in the efforts to determine the unknown parameter @xmath208 .

the vacuum neutrino mixing is known to exhibit an approximate @xmath0 symmetry , which was shown to be preserved for neutrino propagating in matter . this symmetry reduces the neutrino transition probabilities to very simple forms when expressed in a rephasing invariant parametrization introduced earlier . applications to long baseline experiments are discussed .

a spin glass ( sg ) is a complex system characterized by both quenched randomness and frustration , which lead to the irreversible freezing of spins to states without the long - range spatial order below the glass transition temperature ( @xmath0 ) @xcite . theoretical approaches for understanding sg transitions are generally concerned with the study of mean - field level calculations performed using infinite - range interaction models , of which the sherrington - kirkpatrick ( sk ) model @xcite is a prototype . some infinite - range interaction sg models have recently sparked interest in relation to the so - called inverse transitions . since tammann s hypothesis @xcite a century ago , there has been substantial interest in a different class of phase transitions known as inverse transitions ( melting or freezing ) . in these phase transitions , an ordered phase is more entropic than a disordered one , whereby the ordered phase may appear at a higher temperature than the disordered one . such inverse transitions have already been observed experimentally in physical systems such as of liquid crystals @xcite , polymers @xcite , high-@xmath1 superconductors @xcite , magnetic thin films @xcite , and organic monolayers @xcite . meanwhile , from a theoretical point of view , there have been various attempts to identify a suitable model for inverse transitions . spin - glass models have been suggested to be candidates for inverse freezing , wherein the sg phase becomes one with higher entropy . the ghatak - sherrington ( gs ) model @xcite is a spin-1 spin - glass model with a crystal field and it is especially well known as a prototypical sg model for inverse freezing @xcite . in ordinary sg systems , in general , the second - order phase transition from paramagnetic ( pm ) to sg occurs as temperature is decreased . however , according to crisanti and leuzzi @xcite , there seems to be a second reentrance as well as inverse freezing in the gs model . ( see fig . 2 in refs . this implies that phase transitions are likely when the phase is varied successively in the order pm @xmath2 sg @xmath2 pm @xmath2 sg as the temperature is reduced . in other words , there seems to exist two different sgs , i.e. , a sg in the higher - temperature region [ higher - temperature spin glass ( htsg ) ] and a sg in the lower - temperature region [ lower - temperature spin glass ( ltsg ) ] . the aim of this paper is to investigate the theoretical validity for the existence of such separated sgs using a simple gs - like model . for this purpose , we study a quantum version of the gs model by adding a transverse tunneling field , similar to the manner in which the quantum version of the sk model has been studied by considering quantum tunneling with a transverse field @xcite . we expect the quantum gs model to clarify the changes in the existence and features of the two sgs with respect to the transverse field . herein we use one - step replica symmetry breaking ( 1rsb ) for theoretical investigations instead of the replica symmetry ( rs ) @xcite and the full replica symmetry breaking ( frsb ) @xcite . we select the 1rsb because it provides more physically meaningful results than rs does and numerical values of order parameters more easily than frsb does . although 1rsb is approximated with respect to the exact frsb ansatz , it is a good approximation around transition lines because at criticality the thermodynamics is not very sensitive to the ansatz chosen , as shown in refs . the hamiltonian of the quantum gs model is @xmath3 where ( @xmath4 ) means all the distinct pairs of spins with the total number @xmath5 , @xmath6 are quenched random exchange interaction variables , @xmath7 is the crystal field , and @xmath8 is the transverse tunneling field . the spin-1 quantum spin operators @xmath9 and @xmath10 are defined by @xmath11 respectively . the distribution of @xmath6 is taken to be gaussian with a mean zero and a variance of @xmath12 . when @xmath13 and @xmath14 are the degeneracy of the filled or interacting states of @xmath9 and of the empty or noninteracting states of @xmath9 , respectively , we can define the relative degeneracy of the filled states as @xmath15 @xcite . by the imaginary - time formalism @xcite , the partition function of the system can be written as @xmath16 \mathcal{t } \exp \big[\int_{0}^{\beta } d\tau \nonumber\\ & & \big\ { \sum_{ij}^{n } j_{ij}s_{iz}(\tau)s_{jz}(\tau ) - d \sum_{i}^{n } ( s_{iz}(\tau))^{2 } \big\}\big]\end{aligned}\ ] ] where @xmath17 is the imaginary time , @xmath18 is the time - ordering operator , @xmath19 are the operators under the interaction representation introduced in the quantum physics , [ i.e. , @xmath20 where @xmath21 and @xmath22 ( where @xmath23 for simplicity ) . for this model , the free energy is calculated as @xmath24_{j } = \int \prod_{i , j}^{n } dj_{ij } p(j_{ij } ) \ln z ( \{j_{ij}\})$ ] , where @xmath25_{j}$ ] indicates an average over the quenched disorder of @xmath6 . for the quenched random system the free energy can be evaluated using the replica method @xmath26 $ ] . by averaging @xmath27 over @xmath28 , rearranging terms , and taking the method of steepest descent in the thermodynamic limit ( @xmath29 ) , the intensive free energy @xmath30 can be written as @xmath31 - \ln \textrm{tr } \exp ( \tilde{\mathcal{h } } ) \bigg\}\end{aligned}\ ] ] with the effective hamiltonian @xmath32 ~\mathcal{t } \exp \bigg\ { \int_{0}^{\beta } d\tau \int_{0}^{\beta } d\tau ' \big [ \frac{1}{2 } \sum_{(\alpha \beta)}^{n } q^{\alpha \beta}(\tau , \tau ' ) s_{z}^{\alpha}(\tau)s_{z}^{\beta}(\tau ' ) \nonumber\\ + \frac{1}{2 } \sum_{\alpha}^{n } r^{\alpha \alpha}(\tau , \tau ' ) s_{z}^{\alpha}(\tau)s_{z}^{\alpha}(\tau ' ) \big ] - d \int_{0}^{\beta } d\tau \sum_{\alpha}^{n } ( s_{z}^{\alpha}(\tau))^{2 } \bigg\}\end{aligned}\ ] ] where @xmath33 denotes a summation over replica indices @xmath34 and @xmath35 running from 1 to @xmath36 , and the trace @xmath37 is over @xmath36 replicas at a single spin site . here two order parameters are introduced : the spin - glass order parameter @xmath38 and the spin self - interaction @xmath39 , where @xmath40/\textrm{tr}~ e^{\tilde{\mathcal{h}}}$ ] . we take the static approximation @xcite by @xmath41 and @xmath42 . then the free energy @xmath43 is given by @xmath44~~\end{aligned}\ ] ] with the effective hamiltonian @xmath45 next , we use parisi s 1rsb scheme as in the case of the sk model @xcite : for the @xmath46 matrix @xmath47 in the replica spin space , the @xmath36 replicas of @xmath47 are divided into @xmath48 groups of @xmath49 replicas , assuming that @xmath36 must be a multiple of @xmath49 , so that @xmath47 consists of @xmath48 diagonal matrices of @xmath50 elements each ( in which all the diagonal elements are zero and off - diagonal elements are @xmath51 ) and @xmath52 matrices of @xmath50 elements ( in which all the elements are @xmath53 ) . then the free energy obtained by the 1rsb ansatz is given as follows : @xmath54^{m } \bigg]\end{aligned}\ ] ] we can complete phase diagrams of the present model from these equations . first , let us consider the @xmath61 case in order to check whether the result of crisanti and leuzzi @xcite is correct . the graphs in fig . 1 show the @xmath62 phase diagrams obtained for specific @xmath8 values . as shown in fig . 1(a ) , the @xmath62 phase diagram of the @xmath63 case ( gs model ) at @xmath61 is nearly the same as that of the model used by crisanti and leuzzi @xcite . the locations of the first - order phase boundary and tricritical point ( tcp ) , i.e. , the cross - point between first- and second - order phase boundaries , were determined by the same criteria proposed in ref . the tcp of fig . 1(a ) is located at ( 0.962 , 0.333 ) , as analytically obtained in ref . @xcite . in the region @xmath64 , the second - order phase transition from pm to sg occurs as the temperature is decreased , which is generally observed in ordinary sg systems . however , in the region @xmath65 , successive phase transitions occur for which the phase is varied in the order pm @xmath66 htsg @xmath67 pm @xmath67 ltsg , as the temperature is reduced . this result shows clearly the second reentrance that crisanti and leuzzi referred to previously @xcite . in the region @xmath68 , inverse freezing is shown through the phase transitions in the order pm @xmath66 sg @xmath67 pm , as the temperature is decreased . therefore , we have verified that inverse freezing , which many investigators of the gs model have focused upon , occurs only in a narrow region . figure . 1(b ) shows the @xmath62 phase diagrams for several values of @xmath8 , including the result of the @xmath63 case ( gs model ) . as @xmath8 is gradually increased , the glass transition temperatures decrease . in the range @xmath69 , only the second - order phase transition from pm to sg occurs as the temperature is reduced , and the glass transition temperatures decrease as @xmath8 is increased . however , when @xmath7 is larger than 0.7 , the first - order phase transitions occur and the position of each tcp depends on each @xmath8 value . the shapes of the phase boundaries in this range are rather complex , as can be checked in fig . 2(b ) . phase diagram for the @xmath63 case ( gs model ) and ( b ) @xmath62 phase diagrams for several values of @xmath8 . the solid - line ( dotted - line ) part of each phase boundary indicates the second - order ( first - order ) phase transition and each circle between the two kinds of lines denotes a tcp.,title="fig:",scaledwidth=50.0% ] phase diagram for the @xmath63 case ( gs model ) and ( b ) @xmath62 phase diagrams for several values of @xmath8 . the solid - line ( dotted - line ) part of each phase boundary indicates the second - order ( first - order ) phase transition and each circle between the two kinds of lines denotes a tcp.,title="fig:",scaledwidth=50.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] phase diagrams for several values of @xmath7 between ( a ) 0.0 and 0.697 and ( b ) 0.697 and 0.879 . as @xmath7 increases gradually , the phase boundary is kinked in the direction of the dashed arrow of the figure . ( c ) the case of @xmath70 . when @xmath7 is larger than 0.879 , the phase boundary becomes split . ( d ) three cases with @xmath7 larger than 0.88 . , title="fig:",scaledwidth=42.0% ] the graphs of fig . 2 show the @xmath71 phase diagrams obtained for specific @xmath7 values . figure 2(a ) represents the temperature - dependent variations in the phase boundaries , which are obtained for @xmath7 between 0.0 and 0.697 . in the ising spin - glass model with a transverse field , the glass transition temperature at @xmath63 is 1.0 @xcite , whereas in our model the transition temperature at @xmath63 is 0.86 . the difference between the two values can be attributed to the fact that our model includes the eigenvalues of @xmath72 as well as @xmath73 and @xmath74 . when @xmath75 is increased , the glass transition temperature gradually increases to 1.0 . as expected , the phase boundary is shifted to a lower temperature with the increase in @xmath7 . according to our detailed numerical calculation , the first - order phase transition first arises at @xmath76 , where the tcp is located at @xmath77 . when @xmath7 is larger than 0.697 , the shift becomes more complex , as shown in fig . as @xmath7 is larger than 0.697 , one tcp is separated into two new tcps and a first - order phase transition lies between these two tcps @xcite . as @xmath7 is gradually increased , the phase boundary is kinked in the direction of the dashed arrow of fig . 2(b ) and the region of the first - order phase transition simultaneously broadens . as @xmath7 increases further , one of the tcps collapses with the @xmath8 axis . when @xmath7 becomes 0.879 , the phase boundary starts to split . the second reentrance of the gs model [ fig . 1(a ) ] is a zero-@xmath8 case reflecting this splitting of the phase boundary . the two sg phases generated by the splitting are the htsg and the ltsg . the htsg is inside the extremely narrow region of @xmath8 and surrounded by the @xmath78 axis , the second - order phase boundary , one tcp , and the first - order phase boundary . however , the ltsg is spread along the @xmath8 axis and is surrounded only by the axis and the first - order phase boundary . in the case of @xmath70 of fig . 2(c ) , two types of sgs ( htsg and ltsg ) exist between @xmath79 . however , for values of @xmath8 greater than 0.045 , only one type of sg ( ltsg ) exists under the pm phase . the case of @xmath80 in fig . 2(d ) is characterized by the clear occurrence of inverse freezing in the extremely narrow region of @xmath81 . however , for @xmath82 , there is no other phase except the pm phase at any temperature . for the @xmath83 region , the ltsg exists under the pm phase . when @xmath7 reaches the value of 0.962 , the htsg converges to one point @xmath84 , which is the tcp of the gs model . therefore , as @xmath7 increases , one tcp corresponding to the @xmath7 value greater than 0.697 gradually shifts to the tcp of the gs model , and the area of the htsg reduced throughout this process , until the htsg converges to the tcp of the gs model . during the same process , the area of the ltsg also decreases gradually . when @xmath7 reaches the value of 1.024 , the ltsg converges to a point @xmath85 . when @xmath7 is larger than 1.024 , no sg phase exists for any temperature or @xmath8 field . the appearance and disappearance of the htsg and ltsg thus depend on the value of @xmath7 . note that the two sg phases ( htsg and ltsg ) originate from the @xmath7 field , irrespective of the @xmath8 field . as shown in fig . 1(a ) , in the region @xmath65 , the two sg phases occur even when @xmath86 . the role of the @xmath8 field is to lower the glass transition temperature through quantum tunneling in proportion to the @xmath8 value , as already checked in refs . @xcite . in particular , in our model , the @xmath8 field plays a role in the sudden lowering of the second - order transition temperature of the sg ( at @xmath87 ) or htsg ( at @xmath88 ) . thus even a small value of the @xmath8 field ( about 0.05 ) makes the htsg disappear in the region @xmath88 . and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] and @xmath51 for ( a ) @xmath89 , ( b ) @xmath90 , ( c ) @xmath91 , and ( d ) @xmath92 . here @xmath7 is fixed at 0.88 . , title="fig:",scaledwidth=42.0% ] our previous results can be directly checked by numerical analysis of the free energy @xmath59 , @xmath53 , and @xmath51 . all values of @xmath53 and @xmath51 shown in fig . 3 are obtained for @xmath70 , which is given for comparison with fig . 2(c ) . for @xmath89 , as shown in fig . 3(a ) , phase transitions occur in the order pm @xmath66 htsg @xmath67 pm @xmath67 ltsg as the temperature is reduced . here the first - order phase transitions can be easily confirmed as sudden changes in the free energy @xmath59 or discontinuities of the entropy @xmath93 , which is the temperature - derivative of the free energy @xmath59 . the pm phase gap between htsg and ltsg , i.e. , the difference between the first - order transition temperature of the htsg - to - pm transition and that of the pm - to - ltsg transition , is an extremely small value of 0.04 . in fig . 3(b ) , when @xmath8 is increased to 0.04 , the pm phase gap between the htsg and the ltsg widens to 0.1 , and the phase transitions occur in the order pm @xmath66 htsg @xmath67 pm @xmath67 ltsg as the temperature is reduced . when @xmath8 is increased to 0.1 , as shown in fig . 3(c ) , the htsg disappears and a first - order phase transition occurs from pm to ltsg as the temperature is decreased . this feature is maintained even when @xmath8 is increased to 0.55 , which is shown in fig . 3(d ) . for @xmath94 and @xmath70.,scaledwidth=50.0% ] for @xmath63 , @xmath70 , and several @xmath75 values.,scaledwidth=50.0% ] in the inverse freezing among pm - sg - pm phases in the gs model , there has been a discovery that the higher - temperature pm phase is characterized by a low density of empty states , whereas the lower - temperature pm phase has a higher density of empty states @xcite . here the density of empty states @xmath95 plays a crucial role in distinguishing the two pm phases . similarly , in order to clarify a difference between two sg phases , we draw a graph of the spin self - interaction @xmath58 [ eq.(10 ) ] , which signifies the density of filled states . as shown in fig . 4 , @xmath58 shows the difference between two sg phases clearly : the htsg has lower @xmath58 values than the ltsg does . thus , we can infer that the ltsg is characterized by a higher density of filled states . we finally examine whether the second reentrance or the splitting between the htsg and the ltsg occur at @xmath96 . as shown in fig . 5 , at @xmath97 , the pm phase gap between the htsg and the ltsg is wider than that of the @xmath61 case . at @xmath98 , there is an extremely narrow gap near @xmath99 . when @xmath75 is larger than 1.015 , there exists only one sg phase , instead of the two separated sg phases . since schupper and shnerb @xcite focused on the inverse freezing of the gs model , they selected large values of @xmath75 ( e.g. , 6.0 ) . in order to observe sg splitting , however , it is better to select @xmath75 values smaller than 1.0 because when the degeneracy of the empty states of @xmath9 ( @xmath14 ) is larger than one of the filled states of @xmath9 ( @xmath13 ) , the pm phase gap generating the sg splitting becomes wider . in the present work , we proposed an expanded spin - glass model , the quantum gs model , in order to obtain more meaningful evidence for the second reentrance observed in the gs model . by obtaining the 1rsb solutions of the quantum gs model , we could check the detailed pm - sg phase boundaries depending on the crystal field @xmath7 and the transverse field @xmath8 . we first confirmed that a second reentrance occurs in the gs model ( @xmath89 case ) , as reported by crisanti and leuzzi @xcite . we can thus describe the gs model as a prototypical model that can be used to verify the second entrance as well as inverse freezing . furthermore , there exist first - order phase transitions and tcps for @xmath100 and large values of @xmath7 . this is clearly observable from the @xmath71 phase diagrams for @xmath101 , which are shown in fig . 2(b ) . in particular , when @xmath7 is larger than 0.879 , one sg phase is split into two sg phases ( htsg and ltsg ) . we can distinguish the two sg phases by the spin self - interaction @xmath58 . the htsg and ltsg show certain differences in shape and phase boundaries . such sg splitting becomes more distinctive when @xmath75 is less than 1 . we verified that the empty states of @xmath9 are thus crucial for the occurrence of sg splitting . it is well known that the sk model with a transverse field @xcite has been successfully applied to the quantum spin glass @xmath102 @xcite , a site - diluted and isostructural derivative of the dipolar - coupled ising ferromagnet @xmath103 ( @xmath1=1.53k ) . in the absence of a magnetic field , @xmath102 is a conventional spin glass with the glass transition temperature @xmath104 . when an externally tunable magnetic field is induced transverse to the magnetic easy axis , quantum tunneling occurs . provided we can identify a suitable candidate spin - glass material with @xmath105 and crystal field and provided quantum tunneling by an externally tunable transverse magnetic field occurs in the material , we may be able to observe and verify sg splitting through experimental results . in contrast , it would be of interest to extend our theory beyond the static approximation used in this work in order to obtain analytic solutions for free energy and order parameters . it would also be interesting theoretically to search for other sg models for sg splitting . we believe that these topics will extend our viewpoint on sg systems . two tcps and a first - order phase transition lying between the two tcps were also found in the sk model under a bimodal random field . see e. nogueira , jr . , f. d. nobre , f. a. da costa , and s. coutinho , phys . e * 57 * , 5079 ( 1998 ) .

we propose an expanded spin - glass model , called the quantum ghatak - sherrington model , which considers spin-1 quantum spin operators in a crystal field and in a transverse field . the analytic solutions and phase diagrams of this model are obtained by using the one - step replica symmetry - breaking ansatz under the static approximation . our results represent the splitting within one spin - glass ( sg ) phase depending on the values of crystal and transverse fields . the two separated sg phases , characterized by a density of filled states , show certain differences in their shapes and phase boundaries . such sg splitting becomes more distinctive when the degeneracy of the empty states of spins is larger than one of their filled states .

a result of this work is that there is a simple and illuminating formula for the rate . there is a quantity @xmath2 , given by the flux of the surrounding particles or excitations , and the @xmath3 matrix for the interaction of our system ( e.g. the chiral molecule ) with these surroundings : @xmath4 the imaginary part gives the rate or loss of phase coherence per unit time @xmath5 : @xmath6 ( the real part also has a significance , a level shift induced by the surroundings . this turns out to be a neat way to find the index of refraction formula for a particle in a medium @xcite . ) the labels ( l , r ) on the @xmath3 refer to which state of the molecule ( or other system ) is doing the interacting with the surroundings . here with ( l , r ) we have taken the case of the simplest non - trivial system , the two - level system . these equations may be derived@xcite by thinking of the s - matrix as the operator which transforms the initial state of an incoming object into the final state . if the different states ( l , r ) of our system scatter the object differently , a `` lack of overlap '' or `` unitarity deficit '' as given by eq [ [ lam ] ] arises . these intuitive arguments can also supported by more formal manipulations @xcite . an important point that we see here , in eq [ [ lam ] ] , is that the environment `` chooses a direction in hilbert space''@xcite . that is , there is some direction ( here l , r ) in the internal space of the system under study ( the molecule ) that is left unchanged is not `` flipped'' by the interaction with the surroundings . such states however get a phase factor by the interaction , and this is the . if the interaction did not distinguish some direction , if we had @xmath7 then the formula tells us there would be no . this is intuitively correct in accord with one s ideas about `` measurement '' . if the probe does not distinguish any state there are no `` wavefunction collapses '' and no takes place . ( this is not meant to imply sanctioning of `` wavefunction collapses '' in any way . ) another simple limit for the formula occurs when only one state interacts , say no interaction for l , or @xmath8 . then one finds that the rate is 1/2 the scattering rate for the interacting component @xcite . thus eqs [ [ lam],[d ] ] have two interesting limits : @xmath9 and @xmath10 the latter followed from an application of the optical theorem . with appropriate evaluation of the s - matrices , eqs [ [ lam],[d ] ] can be applied to many types of problems , like quantum dots @xcite or neutrinos @xcite , or even gravity @xcite . eq [ [ no ] ] is quite interesting in that it says the system can interact but nevertheless retain its internal coherence . a lesson here is that one should nt think that every interaction or disturbance `` decoheres '' or `` reduces '' the system . the system can interact quite a bit as long as the interactions do nt distinguish the different internal states . the fact that the interaction responsible for the must `` choose a direction in hilbert space '' has some interesting implications . one of these has to do with the of a free particle in some background environment . eq [ [ lam ] ] was for a two - state system , and the extension to a larger number of states , as long as it is a finite number , can be easily envisioned as following the logic@xcite used in finding eq [ [ lam ] ] . however if we go to the continuum , that is if we have a infinite number of states , the problem becomes more subtle . the most common example of this is the free particle which , say in the limit of an infinitely large `` box '' , is described as system of continuous , dense , levels . a number of authors , in talking about this system , have automatically assumed , as indeed first seems plausible , that at long times the particle under the influence of some continually interacting environment becomes totally `` decohered '' ; in the sense that the density matrix of the particle @xmath11 approaches the situation of no off - diagonal elements , that @xmath12 approaches a @xmath13 function . although this may seem plausible , that under the repeated bombardment by the surroundings the particle becomes more and more `` decohered '' , it is in fact wrong @xmath14 consider the simplest case , that of a thermal environment . on general grounds we expect the particle in a thermal environment to be described by the boltzmann factor , to be given by a density matrix operator @xmath15 , where t is the temperature and h the hamiltonian , say @xmath16 for a non - relativistic particle . now evaluate this operator in the position representation : @xmath17 this is the stationary , long time value of @xmath12 . it applies for nearly any state we care to initially throw into the medium . evidently it shows no signs of changing and certainly no sign of turning into a @xmath18 function . of course at high temperature our expression will resemble a delta function . the practical importance of this will depend on the other length scales in the problem at hand . the point we wish to make , however , is of a conceptual nature , namely that repeated interactions with the environment do nt necessarily lead to more `` decoherence '' . indeed eq [ [ boltz ] ] says if we were initially to put @xmath19 or some other `` highly incoherent '' density matrix into the medium , the density matrix of the particle would become _ more coherent _ with time until it reached the value eq [ [ boltz ] ] . apparently the medium can `` give coherence '' to a state that never had any to start with . `` creating coherence '' by an outside influence is not as mysterious as it may sound , there are familiar cases where we know this already . for example , using a high resolution detector can `` create a long wavepacket '' @xcite or in particle physics neutral @xmath0 oscillations and the like may be enhanced or `` created '' by using some subset of our total event sample , such as a `` flavor tag '' . where did the seemingly plausible argument or feeling about the indefinitely increasing go wrong ? it s the question of the `` direction chosen in hilbert space '' . the feeling is right , but we must know where to apply it . as we can see from the boltzmann factor , thermodynamics likes to work in momentum ( actually energy ) space . the intuition would have been right there , in momentum space but this then means something non - trivial in position space . the lesson here is that the notion of `` by the environment '' must be understood to include a statement about the `` direction chosen in hilbert space '' by that environment @xcite . the interest in these issues has had a revival with the advances made possible by the technologies of mesoscopic systems . in one such system , the `` quantum dot observed by the qpc '' , one has a complete model of the measurement process , including the `` observer '' , `` who '' in this case is a quantum point contact ( qpc ) @xcite . in a slight generalization of the original experiment @xcite one can see how not only the density matrix of the object being observed is `` reduced '' by the observing process , but also see how the readout current the `` observer '' responds . in particular one may see how effects looking very much like the `` collapse of the wavefunction '' , that is sequences of repeated or `` telegraphic '' signals indicating one or another of the two states of the quantum dot , arise . all this without putting in any `` collapses '' by hand @xcite . we should stress that what we are not only talking about a reduction of fringe contrast due to `` observing '' or disturbing an interference experiment , as in @xcite ; and also in interesting experiments in quantum optics where an environment is simulated @xcite or different branches of the interferometer @xcite interact differently and adjustably with the radiation in a cavity ( like our two s - matrices ) . by the `` collapses '' however , we are referring not so much to the interferometer itself as to the signal from some `` observing '' system , like the current in the qpc . with repeated probing of the _ same _ object ( say electron or atom ) , in the limit of strong `` observation '' this signal repeats itself -this is the `` collapse '' . for not too strong observation there is an intermediate character of the signal , and so on . all this may be understood by considering the amplitude for the interference arrangement and the readout procedure to give a certain result @xcite . the properties of the readout signal naturally stand in some relation to the loss of coherence or `` fringe contrast '' of the interference effect under study . following this line of thought we come to the idea that there should be some relation between the fluctuations of a readout signal and decoherence . indeed the decoherence rate , the imaginary part of eq [ [ lam ] ] is a dissipative parameter in some sense ; it characterizes the rate of loss of coherence . now there is the famous `` dissipation- fluctuation theorem '' , which says that dissipative parameters are related to fluctuations in the system . is there some such relationship here ? indeed , one is able to derive a relation between the fluctuations of the readout current and the value of @xmath5@xcite . the interesting and perhaps practical lesson here is that the parameter can be observed in two ways . one is the direct way , just observe the damping out of the coherent oscillations of the system in question . experimentally , this involves starting the system in a definite , selected state . however , as just explained , there is a second way ; namely observe the fluctuations of the readout . this can be done even if the system is in the totally `` decohered '' @xmath20 state . another mesoscopic system , the squid and in particular the rf squid , has been long discussed@xcite as a candidate for showing that even macroscopic objects are subject to the rules of quantum mechanics . the rf squid , a josephson device where a supercurrent goes around a ring , can have two distinct states , right- or left- circulation of the current . these two conditions apparently differ greatly , since a macroscopic number of electrons change direction . it would be a powerful argument for the universality of the quantum rules if one could demonstrate the meaningfulness of quantum linear combinations of these two states . such linear combinations can in principle be produced since there is some amplitude for a tunneling between the two configurations . in fact this was recently manifested through the observation of the `` repulsion of levels '' to be anticipated if the configurations of opposite current do behave as quantum states@xcite . another approach , where we would directly `` see '' the meaningfulness of the relative quantum phase of the two configurations , is the method of `` adiabatic inversion '' @xcite . this method also offers the possibility of a direct measurement of the time . in adiabatic inversion the `` spin '' representing a two - level system @xcite , @xcite is made to `` follow '' a slowly moving `` magnetic field '' ( meant symbolically , as an analogy to spin precession physics ) , which is swept from `` up '' to `` down '' . in this way the system can be made to invert its direction in `` spin space '' , that is to reverse states and go from one direction of circulation of the current to the other . this inversion is an intrinsically quantum phenomenon . if it occurs it shows that the phases between the two configurations were physically meaningful and that they behave quantum mechanically . this may be dramatically manifested if we let destroy the phase relation between the two configurations . now the configurations act classically and the inversion is blocked . we thus predict that when the rate is low the inversion takes place , and when it is high it does not . figs 1 and 2 show the idea of this procedure . .3 cm since in such an experiment we have the sweep speed at our disposal , we have a way of determining the time . it is simply the slowest sweep time for which the inversion is successful . we must only be sure that for the sweep speeds in question the the conditions remain adiabatic . setting up the adiabatic condition and taking some estimates for the time , it appears that the various requirements can be met @xcite , @xcite when operating at low temperature . hence it may be realistically possible to move between the classical and quantum mechanical worlds to turn quantum mechanics `` on and off '' in one experiment . this would be a beautiful experiment , the main open question being if the estimates of the rate are in fact realistic , since we are entering a realm which has not been explored before . a two - state system behaving quantum mechanically can serve as the physical embodiment of a quantum mechanical bit , the `` qbit '' . furthermore , the adiabatic inversion procedure just described amounts to a quantum realization of one of the basic elements of computer logic : the not . if one configuration is identified as 1 and the other as 0 , then the inversion turns a linear combination of 1 and 0 into a linear combination of 0 and 1 with reversed weights . we can try to push this idea of `` adiabatic logic '' a step further . not was a one bit operation . the next most complicated logic operation is a two bit operation , which we may take to be `` controlled not '' or cnot . in cnot the two bits are called the control bit and the target bit , and the operation consists of performing or not performing a not on the target bit , according to the state of the control bit . to realize cnot , an idea which suggests itself @xcite as a generalization of adiabatic inversion is the following . we have a two bit operation and so two squids . these are devices with magnetic fields . now if one squid , the target bit , is undergoing a not operation , it can be influenced by the control bit , a second nearby squid , through its linking flux . we could imagine that this linking flux can be arranged so that it helps or hinders the not operation according to the state of the second squid . this would amount to a realization of `` controlled not '' , again by means of an adiabatic sweep . to analyze this proposal we must set up the two - variable schroedinger equation describing the two devices and their interaction . the result is a hamiltonian with the usual kinetic energy terms and a potential energy term in the two variables , which in this case are the fluxes in the squids , @xmath21 : @xmath22+\beta_1f(\phi_1)+\beta_2f(\phi_2 ) \bigr\ } \ ; .\ ] ] the @xmath23 are external biases which in general will be time varying . the @xmath24 are dimensionless inductances and @xmath25 represents the coupling between the two devices . the @xmath26 are symmetric functions starting at one and decreasing with increasing @xmath27 so as to produce a double well potential when combined with the quadratic term ; in the squid @xmath28 . 3 shows this `` potential landscape '' for some typical values of the parameters . given the hamiltonian , we must search for values of the control parameters @xmath23 , the `` external fields '' , which can be adiabatically varied in such a way as to produce cnot . preliminary analysis indicates favorable regimes of the rather complex parameter space where this can in fact be done @xcite . finally we would like to recall that there are still some fundamental and beautiful experiments waiting to be done in these areas . \a ) one is the demonstration of the large effects of parity violation for appropriately chosen and contained handed molecules @xcite . because of what we now call this seemed very remote at the time . but now with the existence of single atom / molecule traps and related techniques , perhaps it s not so hopeless . \b ) another , concerned with fundamentals of quantum mechanics , could be called the `` adjustable collapse of the wavefunction '' where the `` strength of observing '' can be varied , leading to effects like washing out of interferences , as already seen in @xcite and a number of further predictions where we vary the qualities of the `` observer '' @xcite , or slowing down of relaxation according to the rate of probing of the object @xcite . many of the questions we have briefly touched upon had their origins in an unease with certain consequences of quantum mechanics , often as `` paradoxes '' and `` puzzles '' . it is amusing to see how , as we get used to them , the `` paradoxes '' fade and yield to a more concrete understanding , sometimes even with consequences for practical physics or engineering . if we avoid overselling and some tendency to an inflation of vocabulary , we can anticipate a bright and interesting future for `` applied fundamentals of quantum mechanics '' . on the time dependence of optical activity , r.a . harris and , j. chem . phys . * 74 * ( 4 ) , 2145 ( 1981).[decoherence by environment , formula for decoherence rate , application to chiral molecules . ] the notion that handed molecules could `` decohere '' or be stabilized by the environment somehow was raised by h.d.zeh , found . phys . 1,69 ( 1970 ) , m. simonius , phys rev . 40 , 980 ( 1978 ) . two level systems in media and ` turing s paradox ' , r.a . harris and , phys . b 116(1982)464.[decoherence by environment , formula for decoherence rate , quantitative explanation of `` zeno '' , prediction of anti - intuitive relaxation , application to neutrinos . ] on the treatment of neutrino oscillations in a thermal environment , , phys d 36(1987)2273 .[method for decoherence in neutrino oscillations . spin precession picture for neutrinos . ] see chapter 9 of g.g . raffelt , _ stars as laboratories for fundamental physics _ ( univ . chicago press , 1996 ) measurement process in a variable - barrier system , , phys . * b459 * pages 193 - 200 , ( 1999 ) . [ formalism for quantum dot - qpc system . prediction of novel phase effect . prediction of `` collapse - like '' behavior of readout . ] decoherence - fluctuation relation and measurement noise , , physics reports * 320 * 51 - 58 ( 1999 ) , quant - ph/9903075 . [ suggestion of decoherence - fluctuation relation connecting decoherence rate and fluctuations of readout signal . ] see a. j. leggett , les houches , session xlvi ( 1986 ) _ le hasard et la matiere _ ; north -holland ( 1987 ) , references cited therein , and introductory talk , conference on macroscopic quantum coherence and computing , naples , june 2000 , proceedings published by academic - plenum . ) j. friedman , v. patel , w.chen , s.k . tolpygo and j.e . lukens , nature * 406 * , 43 ( 2000 ) . van der wal , a.c . ter haar , f. k. wilhelm , r. n. schouten , c.j.p.m . harmans , t.p . orlando , seth lloyd , and j.e . mooij , science * 290 * 773 ( 2000 ) and in mqc2 : conference on macroscopic quantum coherence and computing , naples , june 2000 , proceedings published by academic - plenum . study of macroscopic coherence and decoherence in the squid by adiabatic inversion , paolo silvestrini and , physics letters * a280 * 17 - 22 ( 2001).[linear combinations of macroscopic states . measuring decoherence time . relation to not operation . ] cond - mat/0004472 adiabatic inversion in the squid , macroscopic coherence and decoherence , paolo silvestrini and , _ macroscopic quantum coherence and quantum computing _ , pg.271 , eds . d. averin , b. ruggiero and p. silvestrini , kluwer academic / plenum , new york ( 2001 ) . [ linear combinations of macroscopic states . measuring decoherence time.]cond - mat/0010129 . averin , solid state communications * 105 * , 659 ( 1998 ) , has discussed related ideas using adiabatic operations on the charge states of small josephson junctions . adiabatic methods for a quantum cnot gate , valentina corato , paolo silvestrini , , and jacek wosiek , cond - mat/0205514 , and contribution to _ macroscopic quantum coherence and quantum computing 2002_. [ principles of quantum gates using adiabatic inversion . design parameters for cnot . ]

we indicate some of the lessons learned from our work on coherence and decoherence in various fields and mention some recent work with solid state devices as elements of the `` quantum computer '' , including the realization of simple logic gates controlled by adiabatic processes . we correct a commonly held misconception concerning decoherence for a free particle . = -0.35 cm = 0.3 cm presented at the xxii solvay conference , the physics of information delphi , nov 2001 2.0pc the subject of `` quantum information '' and in particular its realization in terms of real devices revolves in large measure around the problems of coherence and . thus it may be of interest here to review the origins of the subject and see what has been learned in applications to various areas . we first got involved in these issues through the attempt to see the effects of parity violation ( `` weak neutral currents '' ) in handed molecules @xcite . the method we found an analogy to the famous neutral @xmath0 meson behavior with chiral molecules seemed too good to be true : we had a way of turning @xmath1 ev into a big effect ! there must be some difficulty , we felt . indeed there was ; it turned out to be what we called `` quantum damping '' and what now - a - days is called `` decoherence '' . the lessons from this work were several and interesting . first , concerning parity violation , we realized that this could solve hund s `` paradox of the optical isomers '' as to why we observe handed molecules when the true ground state should be parity even- or- odd linear combinations . we realized that for molecules where tunneling between chiral isomers is small , parity violation dominates and the stationary state of the molecule becomes a handed or chiral state , and not a 50 - 50 linear combination of chiral states . this holds for a perfectly isolated molecule , and in itself has nothing to do with . however , and this is very related , even a very small interaction with the surroundings suffices to destroy the coherence necessary for the aforesaid linear combination , in effect the environment can stabilize the chiral states . this now goes under the catch - word `` by the environment '' . the limit of strong damping or stabilization is often called the zeno or `` watched pot '' effect , an idea which as far as i can tell , goes back to turing . we were able to show how this just arises as the strong damping limit of some simple `` bloch - like '' equations @xcite .

mixtures of water and organic solutes are of fundamental importance for understanding biological and chemical processes as well as transport properties of fluids . even though the simplicity , of these solutions some of them show a complex behavior of their thermodynamic and structural properties @xcite . for example , close to the ambient conditions , around @xmath0 , @xmath1 , the excess volume in binary mixtures of water and alcohols @xcite and of water and alkanolamines @xcite is negative and it exhibits a minimum as the fraction of the solute is increased . in the case of water - ionic liquids , however , the excess volume depend on the hydrophobicity of the solute . simulation results suggest that for hydrophilic solutes as the 1,3-dimethylimidazolium chloride the excess volume has a minimum as in the case of the alcohols , whereas for the case of more hydrophobic liquids as the 1,3-dimethylimidazolium hexafluorophosphate the excess volume is positive @xcite . the excess enthalpy of the aqueous organic mixtures also show a distinct behavior . while the mixtures of water with small alcohol molecules as methanol @xcite and ethanol @xcite exhibit a negative excess enthalpy , the mixtures of water with large alcohol molecules as propanol and butanol isomers show a positive excess enthalpy @xcite . similarly to the small alcohol - water mixtures the excess enthalpy for the water - alkanolamine solutions also show a minimum @xcite . in the case of ionic liquids the excess enthalpy also show two types of behavior . for the less hydrophobic ionic liquids in which the excess volume is negative , the excess enthalpy is negative and shows a minimum at the same solute fraction of the minimum of the excess volume . for the hydrophobic ionic liquids the excess enthalpy is positive and shows a maximum for the same fraction of the solute of the maximum of the excess volume @xcite . the excess isobaric specific heat for the methanol at ambient conditions increases with the fraction of the solute and exhibits a maximum value around the solute concentration @xmath2 @xcite . the excess free energy presents a harmonic dependence on the methanol fraction @xcite and the excess entropy of mixing , differently from the ideal mixtures @xcite , assumes negative values and decrease its value as the increasing methanol concentrations @xcite . in the case of the ionic liquids the constant pressure heat capacity also shows an oscillatory behavior but the peak occurs at higher concentrations of the solute @xmath3 @xcite . the description of this complex behavior of the organic solutes in water in can be made , in principle , in the framework of the frank and evans @xcite iceberg theory . these authors proposed that water is able to form microscopic icebergs around solute molecules depending on their size and the water - solute interactions . recent experiments @xcite using neutron diffraction support frank and evans @xcite scenario for the methanol . the diffraction of a concentrated alcohol - water mixture ( @xmath4 ) suggests that at these conditions most of the water molecules ( @xmath5 ) are organized in water clusters bridging methanol hydroxyl groups through hydrogen bonds . in the same direction an experimental result from x - ray emission spectroscopy for an equimolar mixture of methanol and water carried out by guo _ et.al . _ @xcite suggests that in the mixture the hydrogen bonding network of the pure components would persist to a large extent , with some water molecules acting as bridges between methanol chains . consistent with these results , recent experimental work for the methanol @xcite suggests that the negative excess the entropy of mixing arises due to a relatively small degree of the interconnection between the hydrogen bonding networks of the different components rather than from a water restructuring @xcite . motivated by these experimental results and by the huge number of applications , water - methanol mixtures have been intensively studied by computer simulations . in these simulations , water molecules are represented by one of well known classical models spc@xmath6e @xcite , st4 @xcite , tip5p @xcite and methanol molecules are frequently modeled by opls force field @xcite . using molecular dynamics simulation , bako _ et.al . _ @xcite found that on increasing the methanol fraction in the mixture , water essentially maintains its tetrahedral structure , whereas the number of hydrogen - bonds is substantially reduced . allison _ et.al . _ @xcite showed that not only the number hydrogen - bonds decreases , but the water molecules become eventually distributed in rings and clusters in accordance with the experimental results @xcite . analyzing the spatial distribution function of the water , laaksonen _ et.al . _ @xcite observed that the system is highly structured around the hydroxyl groups and that the methanol molecules are solvated by water molecules , in accordance with well known iceberg theory @xcite . in addition to the atomistic approaches , water - methanol mixture has been modeled by continuous potentials in which the water is represented by a spherical symmetric two length scale potential while the methanol is represented by a dimer in which the methyl group is characterized by a hard sphere and the hydroxyl is a water - like group @xcite . numerical simulations for this system displays good qualitative agreement with the response functions for different temperatures @xcite but fails to produce the heat capacity behavior and does not provide the structural network observed in experiments and predicted by the iceberg theory . due to the variety and complexity of the ionic liquids , very few theoretical studies have been made for analyzing the ionic liquids aqueous solutions . for example , there is no clear picture explaining why the excess volume of some ionic liquids is negative while for others is positive . in addition , it is not clear why for large alcohols the excess enthalpy is positive while for the methanol is negative . the explanation for these different behaviors both in the alcohols and in the ionic liquids might rely in the disruption of the iceberg theory as the solute is large of hydrophobic . in order to test this idea , here we explore how the excess properties of the water - solute mixture is affected by the change of the water - solute interaction from attractive to repulsive . in order to allow for the water to form a structure not present in the continuous effective potentials , our model exhibits a tetrahedral structure . in this work the water and the solute are modeled following the associating lattice gas model ( alg ) @xcite scheme . the two molecules are specified by adapting the hydrogen bond and the attractive interactions for each molecule . the excess volume and enthalpy are computed for various types of water - solute interactions . the remaining of the paper goes as follows . in the section [ sec : model ] the models for water , solute and mixture are outlined and the ground state behavior is presented . the technical details about the calculations of ground state are presented in the appendix [ ap : entropy ] . in the section [ sec : methods ] the computational methods are described and the technical aspects can be found in appendix [ sec.isop ] and [ sec.isopmix ] . in the section [ sec : results ] results are presented . section [ sec : conclusions ] ends the paper with the conclusions . we consider three systems : pure water , pure solute and water - solute mixture . in the three cases the system is defined on a body - centered cubic ( bcc ) lattice . sites on the lattice can be either empty or occupied by a water or by a solute molecule . particles representing both water and solute molecules carry four arms that point to four of the nearest neighbor ( nn ) sites on the bcc lattice as illustrated by the figure [ fig : model ] . the interactions between nn molecules are described in the framework of the lattice patchy models @xcite . the particles carry eight patches ( four of them corresponding to the arms in the alg model ) , and each of the patches points to one of the nn sites in the bcc lattice as illustrated in the figure [ fig : model ] . the water molecules have two patches of the type @xmath7 ( acceptors ) , two patches of the type @xmath8(donors ) and four patches of the type @xmath9 ( which do not participate in bonding interactions ) . since the patches of the types @xmath7 and @xmath8 participate in the hydrogen bonding , a water molecule can participate in up to four hydrogen bonds . the structure of the solute is similar to the structure of the water , but it has only one patch of type @xmath7 , the other patch @xmath7 is replaced by a patch of the type @xmath10 that represents the anisotropic group which makes water and the solute different . in the case in which the solute is the methanol @xmath10 is the methyl group while for other alcohols and ionic liquids it does represent larger chains . and @xmath8 represent the acceptors and donors arms respectively . the red sphere represent the solute particle and the arms @xmath7 and @xmath8 represent the acceptor and donors , and the patch @xmath10 represents the anisotropic group , title="fig:",width=264 ] and @xmath8 represent the acceptors and donors arms respectively . the red sphere represent the solute particle and the arms @xmath7 and @xmath8 represent the acceptor and donors , and the patch @xmath10 represents the anisotropic group , title="fig:",width=264 ] and @xmath11 represent particle on its respective positions @xmath12 and @xmath13 . blue sphere represent a water and red , a solute particle . @xmath14 and @xmath15 represent the patches b of water and solute respectively . the patch @xmath9 is not represented here for the clarity of the image.,width=302 ] the distinction between patches implies @xmath16 possible orientations for the water molecules and @xmath17 possible orientations for the solute molecules . the potential energy is defined as a sum of interactions between pairs of particles located at sites which are nn on the bcc lattice . the interaction between particles @xmath12 and @xmath13 , which are nn , only depends on the type of patch of particle @xmath12 that points to particle @xmath13 , and on the type of patch of particle @xmath13 that points to particle @xmath12 . the values of the interaction as a function of the types of the two interacting patches are summarized in the table [ tabla1 ] . the interaction between occupied neighbor sites is repulsive with an increase of energy by @xmath18 with the exception of three cases . for patch - patch interaction of type @xmath19 the energy interaction is taken as : @xmath20 . if the interaction is of type @xmath21 , with the @xmath8 patch belonging to a solute molecule there is also an attractive interaction @xmath22 ( with @xmath23 , whereas if the patch @xmath8 belongs to a water molecule the interaction energy is given by @xmath24 . we have considered @xmath25 , and three cases for the @xmath8-@xmath10 water - solute interaction : attraction with @xmath26 , non - interacting with @xmath27 and repulsion with @xmath28 . the first case represents systems dominated by the water - solute attraction . this is the case of the methanol in which it is assumed that the methyl group shows a small but attractive interaction with the water . this also represents the ionic liquids in which the anions groups are hydrophilic and the cationic chains are not too long @xcite . the second case represents alcohols with larger non - polar alkyl substituents @xcite . the third case represents the ionic liquids in which the combination of the anions and cations lead to an hydrophobic interaction @xcite . due to the simplicity of our model solute , size and hydrophobicity effects are not taken into account independently , but both are considered through the @xmath29 parameter . .interactions between nn particles of the same type ( solute or water ) . the interaction depends on the patches of both particles involved in the interparticle bond . the interaction between patches of type c and b depends on the type of molecule : water ( w ) or solute ( s ) that provides the patch b. we consider @xmath30 ; and @xmath31 . patches of types a , b , and c correspond to the four arms of the standard alg model . [ cols="^,^,>,>,>,>,>,>,>,>,>,>",options="header " , ] [ table - s0 ] in order to estimate the value of @xmath32 in the thermodynamic limit we have considered the scaling relations used by berg _ et al . _ @xcite , @xmath33 the fitting of the simulation results given in table [ table - s0 ] to eq . ( [ eq - s0 m ] ) , with @xmath34 , @xmath35 , and @xmath36 being adjustable parameters leads to : @xmath37 where the label @xmath38 refers to water . considering the quantities @xmath39 $ ] , and fitting the results to @xmath40 we get @xmath41 the values of the exponent @xmath36 agree within statistical uncertainty with the results of berg et al . @xcite . for the residual entropy of the ordinary ice . interestingly , our estimate of @xmath42 for our model defined over a system with cubic symmetry and the estimate of for the ordinary ice of berg et al . @xcite : @xmath43 ; @xmath44 , seem to coincide ( at least within error bars ) in spite of the different structures of the underlying lattices . in principle , we could apply the same simulation techniques used for the water in the determination of the residual entropy of the lattice gas model of the solute . however , the value of @xmath32 for methanol can be deduced directly from the water results . given a ground state , the configuration of the water for a system with @xmath45 molecules ( occupied positions ) one can build up @xmath46 directly related ground states for the methanol model , since the two ( undistinguishable ) @xmath7 patches of each particle in the water model correspond to two distinguishable ( @xmath7 and @xmath10 ) patches in the methanol model . therefore , we get : @xmath47 the excess properties of binary mixtures are usually measured experimentally at fixed conditions of temperature and pressure @xcite . for lattice gas models it is neither straightforward not practical the use of simulation in the npt ensemble . the usual alternative is to carry out simulations in the grand canonical ensemble and compute the pressure by means of thermodynamic integration . since we are interested in analyzing the excess properties at fixed pressure , we have developed a procedure to build up the lines @xmath48 for pure components , i.e. we fix the pressure and compute the chemical potential as a function of temperature at fixed pressure . the objective is to apply this to the ordered phases : ldl and hdl . the pressure at ( very ) low temperature for these phases can be computed from the ground state analysis . in the gce the change of the pressure for transformations at constant @xmath49 and @xmath50 , is given by @xmath51 . the density of the condensed phases at very low temperature hardly changes with @xmath52 , therefore , we can integrate the pressure to get . @xmath53 where the values of @xmath54 , @xmath55 , and @xmath56 can be taken as those corresponding to the phase coexistence at low temperature ( eqs . [ eq.gsw]-[eq.coexm ] ) . once we now how to compute the chemical potential for a given pressure @xmath57 at a ( low ) temperature @xmath58 , we will develop the integration scheme to move on the @xmath59 plane at the fixed pressure @xmath57 . imposing @xmath60 in the differential form for the thermodynamic potential of the gce we get : @xmath61 we typically considered systems with @xmath62 . the excess properties of mixing are usually defined as the differences between the values of the property of the mixture at a given composition , @xmath63 , and the value of the same property for an _ ideal _ mixture of the components at the same conditions of @xmath63 , @xmath49 , and @xmath57 . it is , therefore , desirable to develop simulation strategies to sample in an efficient way different compositions of a given mixture for fixed conditions of temperature and pressure . in order to achieve this aim for our lattice model we have borrowed ideas to form the gibbs - duhem integration procedures , as we did for computing isobars of pure components . the differential form for the grand canonical potential of a binary mixture can be written as : @xmath64 where @xmath65 is the number of molecules of component @xmath12 , and @xmath66 is the chemical potential of component @xmath12 . if we fix @xmath49 , @xmath57 , and @xmath50 , the chemical potential of the two components can not vary independently when modifying the composition . it should be fulfilled : @xmath67 using activities @xmath68 $ ] to carry out the integration of eq . ( [ gdi ] ) we get : @xmath69 let us assume that for some values of @xmath49 , and @xmath57 , we know the values of the activities of the pure components @xmath70 , and @xmath71 . we can integrate numerically ( using simulation results ) the differential equation : @xmath72 for instance , using as starting point @xmath73 and considering @xmath74 as the independent variable and integrating eq . ( [ eq.gdi3 ] ) up to @xmath75 , we should reach @xmath76 . this condition provides a powerful consistency check of the thermodynamic integration schemes at constant pressure . the numerical integration of ( [ eq.gdi3 ] ) can be carried out using the same numerical procedures as in sec . [ sec.isop ] . there is still , a minor technical problem , that appears in the limits @xmath77 ; where @xmath78 , and therefore the ratio @xmath79 can not be directly computed from the simulation . this problem can be solved by applying the widom - insertion test technique@xcite to compute the activity of the minority component ( which actually has mole fraction @xmath80 ) as a function of its density . the result can be written as : @xmath81 } \right\rangle } ; \label{eq.int - mix}\ ] ] where @xmath82 } \right\rangle}$ ] represents the average of the boltzmann exponential over attempts of insertion of a test particle of type @xmath12 with random position and random orientation on a pure component system of the other component and @xmath83 is the number of possible orientations for molecules of type @xmath12 . results were obtained from simulations of systems with @xmath62 . a. p. furlan and m. c. barbosa acknowledge the brazilian agency capes ( coordenao de aperfeicoamento de pessoal de nvel superior ) for the financial support and centro de fsica computacional - cfcif ( if - ufrgs ) for computational support . partial financial support from the direccin general de investigacin cientfica y tcnica ( spain ) under grant no . fis2013 - 47350-c5 - 4-r is acknowledged . in the course of writing this article , no g. almarza unexpectedly passed away . the authors would like to dedicate this work to his memory .

a lattice model for the study of mixtures of associating liquids is proposed . solvent and solute are modeled by adapting the associating lattice gas ( alg ) model . the nature of interaction solute / solvent is controlled by tuning the energy interactions between the patches of alg model . we have studied three set of parameters , resulting on , hydrophilic , inert and hydrophobic interactions . extensive monte carlo simulations were carried out and the behavior of pure components and the excess properties of the mixtures have been studied . the pure components : water ( solvent ) and solute , have quite similar phase diagrams , presenting : gas , low density liquid , and high density liquid phases . in the case of solute , the regions of coexistence are substantially reduced when compared with both the water and the standard alg models . a numerical procedure has been developed in order to attain series of results at constant pressure from simulations of the lattice gas model in the grand canonical ensemble . the excess properties of the mixtures : volume and enthalpy as the function of the solute fraction have been studied for different interaction parameters of the model . our model is able to reproduce qualitatively well the excess volume and enthalpy for different aqueous solutions . for the hydrophilic case , we show that the model is able to reproduce the excess volume and enthalpy of mixtures of small alcohols and amines . the inert case reproduces the behavior of large alcohols such as , propanol , butanol and pentanol . for last case ( hydrophobic ) , the excess properties reproduce the behavior of ionic liquids in aqueous solution .

it is known that lipid bilayers ( abound in living cells membranes ) exhibit a ripple phase of the bilayer - water interface , in a narrow temperature range @xcite . the ripple phase is characterized by permanent wave - like deformations of the interfaces , the origin of which is still a widely debated topic . however , it was observed that a giant dielectric dispersion can occur in the radiofrequency range , signified by a large dielectric increment in that frequency range @xcite . in practice , these membrane systems are under intensive research because they are responsible for delivery and retention of drugs @xcite . we aim to obtain some results on the dielectric dispersion spectrum of corrugated membranes , by using our recently developed green s function formalism of periodic interfaces @xcite . in that formalism , we obtained an analytic expression for the greenian that includes the effect of periodicity . in this work , we extend the green s function formalism to compute the local field distribution for a lipid bilayer membrane of arbitrary shape , separating two media of different dielectric constants . we will calculate the effective dielectric constant of the membrane subject to an ac applied field . the green s function formalism has been published recently @xcite . here we re - iterate the formalism to establish notation . we will apply the formalism to a single interface and then extend to a bilayer membrane . the electrostatic potential satisfies the laplace s equation : @xmath0 = -4\pi \rho({\bf r } ) , \label{electrostatics}\end{aligned}\]]with standard boundary conditions on the interface , where @xmath1 is the free charge density , @xmath2 equals @xmath3 in the host and @xmath4 in the embedding medium . let @xmath5 and @xmath6 be the volume of the embedding and host medium , separated by an interface @xmath7 . denoting @xmath8 if @xmath9 and 0 otherwise , leads to an integral equation @xcite : @xmath10\phi({\bf r } ) = \phi_0({\bf r } ) + { u\over 4\pi } \oint_s ds'\left [ \hat{\bf n } ' \cdot \nabla ' g({\bf r } , { \bf r } ' ) \right ] \phi({\bf r } ' ) , \label{integral}\end{aligned}\]]where @xmath11 , @xmath12 is unit normal to @xmath7 , @xmath13 and @xmath14 is the solution of @xmath15 . accordingly , our approach aims to solve a surface integral equation for the potential at the expense of a two - step solution @xcite : 1 . step 1 : determine @xmath16 for all * r * @xmath17 by solving eq.([integral ] ) , and then 2 . step 2 : obtain @xmath16 for all * r * by using eq.([integral ] ) and the results of step 1 . in step 1 , we encounter a singularity when the integration variable @xmath18 approaches the point of observation * r*. to circumvent the problem , we take an infinitesimal volume around * r * and perform the surface integral analytically , we find @xcite @xmath19 \phi({\bf r } ' ) , \ \ \ { \bf r } \in s , \label{r_in_s}\end{aligned}\]]where `` prime '' denotes a restricted integration which excludes @xmath20 . the ( surface ) integral equation ( [ r_in_s ] ) can be solved for @xmath21 . here we apply the integral equation formalism to a periodic interface . suppose the interface profile depends only on @xmath22 , described by @xmath23 , where @xmath24 is a periodic function of @xmath22 with period @xmath25 : @xmath26 . without loss of generality , we will let @xmath27 in subsequent studies . thus medium 1 occupies the space @xmath28 while medium 2 occupies the space @xmath29 separated by the interface at @xmath23 . the external field is @xmath30 and @xmath31 is the potential . for a periodic system , @xmath16 is a periodic function of the lattice vector @xmath32 . in what follows , we adopt similar treatment as the korringa , kohn and rostoker ( kkr ) method @xcite and rewrite the integral equation as : @xmath33where the integration is performed within a _ unit cell_. the structure green s function ( greenian ) is given by @xcite : @xmath34we were able to evaluate the greenian analytically @xcite : @xmath35 \over \cos 2\pi ( x - x ' ) - \cosh 2\pi ( y - f(x ' ) ) } . \label{greenian}\end{aligned}\]]eq.([greenian ] ) is a truly remarkable result the analytic expression is valid for an arbitrary interface profile . if the point of observation @xmath36 is located at the interface , the greenian has a finite limit as @xmath37 : @xmath38we first solve eq.([unit - cell ] ) for the potential @xmath39 right at the interface : @xmath40then we use eq.([integral ] ) to find the potential at any arbitrary point @xmath41 , using the potential at the interface . @xmath42for @xmath43 and @xmath44 respectively . here we extend the formalism to a bilayer membrane . consider two interface profiles described by @xmath45 , where @xmath46 denote the lower and upper interface profiles respectively . again @xmath47 is a periodic function of @xmath22 . thus medium 1 occupies the space @xmath48 while medium 2 occupies the space @xmath49 and @xmath50 . for the upper ( lower ) profile , @xmath51 ( @xmath52 ) , thus the greenian becomes @xmath53 \over \cos 2\pi(x - x ' ) - \cosh2\pi(f_t(x ) - f_{t'}(x'))}.\end{aligned}\]]the effective dielectric constant @xmath54 of the bilayer membrane satisfies the relation : @xmath55 where @xmath5 is the volume of the embedded medium . as @xmath56 , the volume integration can be converted into a surface integration by the green s theorem . moreover , for the upper ( lower ) profile , @xmath51 ( @xmath52 ) , thus the effective dielectric constant becomes @xmath57.\end{aligned}\ ] ] to solve the integral equation , we express the potential at an arbitrary point into a mode expansion : @xmath58where @xmath59 and @xmath60 are mode functions . the potential on the interfaces suffices : @xmath61where @xmath62 . here we make a few remarks on the choice of the mode functions . the choice of the mode function is somewhat arbitrary in theory . in practice , these functions should be simple and easy to use . common choice ranges from extended mode functions like the fourier series expansions to localized mode functions like the step and triangular functions @xcite . substituting the mode expansion eq.([mode - a ] ) into eq.([unit - cell ] ) , the coefficients @xmath63 satisfy the matrix equation : @xmath64 { \bf a } = -e_0 { \bf v},\end{aligned}\]]where @xmath65it should be remarked that the mode functions need not be orthonormal and the matrix b is non - diagonal in general . as a model bilayer membrane , we adopt the interface profiles : @xmath66 where @xmath67 is the amplitude of corrugation , and the sine function is added to upset the reflection symmetry about @xmath68 @xcite . the width of the bilayer membrane is thus unity . we adopt the step functions for the mode expansions : @xmath69 where @xmath70 is the width of the step function . in what follows , we adopt 100 step functions both for the lower and upper profiles , equally spaced in the unit interval @xmath71 $ ] . the integrals eqs.(16)(18 ) can be readily performed . to study the dielectric behavior , we apply an ac field at a frequency @xmath72 . the embedded medium inside the membrane has a complex dielectric constant @xmath73 where @xmath4 and @xmath74 are the dielectric constant and conductivity of the embedded medium respectively , with @xmath72 being the frequency of the applied field . we adopt the following parameters in the calculation : @xmath75 , while @xmath76 . also let @xmath77 . the maxwell - wagner relaxation time of a planar interface is given by : @xmath78 in fig.[fig1 ] , we plot ( a ) the real and ( b ) imaginary parts of the complex effective dielectric constant @xmath54 normalized to @xmath3 as function of frequency for various amplitude of corrugation @xmath67 ranging from 0.1 to 1.0 . as is evident from fig.[fig1 ] , there is a giant dielectric dispersion as the amplitude of corrugation becomes large ( @xmath79 ) . in summary , we have employed the green s function formalism to study the dielectric behavior of a corrugated membrane . the integral equation is solved and the dielectric dispersion spectrum is obtained for a periodic corrugated membrane . we should remark that the present formalism can readily be generalized to multi - layers systems as well as corrugations in two dimensions @xcite . this work was supported by the research grants council of the hong kong sar government under grant cuhk 4245/01p . k. w. yu acknowledges useful conversation with professor hong sun . g. s. smith , e. b. sirota , g. r. safinya and n. a. clark , phys . lett . * 60 * , 813 ( 1988 ) ; m. p. hentschel and f. rustichelli , phys . lett . * 66 * , 903 ( 1991 ) . a. raudino , f. castelli , g. briganti and c. cametti , j. chem . phys . * 115 * , 8238 ( 2001 ) . k. sugano , h. hamada , m. machida , et al . * 228 * , 181 ( 2001 ) . k. w. yu and jones t. k. wan , _ proceedings of the conference on computational physics ( ccp2000 ) _ , comput . . commun . * 142 * , 368 ( 2001 ) ; see also cond - mat/0102059 . k. w. yu , hong sun and jones t. k. wan , _ proceedings of the 5th international conference on electrical transport and optical properties of inhomogeneous media _ , physica b * 279 * , 78 ( 2000 ) . j. korringa , physica * 13 * , 392 ( 1947 ) ; w. kohn and n. rostoker , phys . rev . * 94 * , 1111 ( 1954 ) . f. c. mackintosh , current opinion in colloid and interface science * 2 * , 382 ( 1997 ) .

we have employed our recently developed green s function formalism to study the dielectric behavior of a model membrane , formed by two periodic interfaces separating two media of different dielectric constants . the maxwell s equations are converted into a surface integral equation ; thus it greatly simplifies the solutions and yields accurate results for membranes of arbitrary shape . the integral equation is solved and dielectric dispersion spectrum is obtained for a model corrugated membrane . we report a giant dielectric dispersion as the amplitude of corrugation becomes large .

consider any system of chemical reactions , in which certain molecule types catalyse reactions and where there is a pool of simple molecule types available from the environment ( a ` food source ' ) . one can then ask whether , within this system , there is a subset of reactions that is both self - sustaining ( each molecule can be constructed starting just from the food source ) and collectively autocatalytic ( every reaction is catalysed by some molecule produced by the system or present in the food set ) @xcite,@xcite . this notion of ` self - sustaining and collectively autocatalytic ' needs to be carefully formalised ( we do so below ) , and is relevant to some basic questions such as how biochemical metabolism began at the origin of life @xcite , @xcite , @xcite . a simple mathematical framework for formalising and studying such self - sustaining autocatalytic networks has been developed so - called ` raf ( reflexively - autocatalytic and f - generated ) theory ' . this theory includes an algorithm to determine whether such networks exists within a larger system , and for classifying these networks ; moreover , the theory allows us to calculate the probability of the formation of such systems within networks based on the ligation and cleavage of polymers , and a random pattern of catalysis . however , this theory relies heavily on the system being closed and finite . in certain settings , it is useful to consider polymers of arbitrary length being formed ( e.g. in generating the membrane for a protocell @xcite ) . in these and other unbounded chemical systems , interesting complications arise for raf theory , particularly where the catalysis of certain reactions is possible only by molecule types that are of greater complexity / length than the reactants or product of the reactions in question . in this paper , we extend earlier raf theory to deal with unbounded chemical reaction systems . as in some of our earlier work , our analysis ignores the dynamical aspects , which are dealt with in other frameworks , such as ` chemical organisation theory ' @xcite ; here we concentrate instead on just the pattern of catalysis and the availability of reactants . in this paper , a _ chemical reaction system _ ( crs ) consists of ( i ) a set @xmath0 of molecule types , ( ii ) a set @xmath1 of reactions , ( iii ) a pattern of catalysis @xmath2 that describes which molecule(s ) catalyses which reactions , and ( iv ) a distinguished subset @xmath3 of @xmath0 called the _ food set_. we will denote a crs as a quadruple @xmath4 , and encode the pattern of catalysis @xmath2 by specifying a subset of @xmath5 so that @xmath6 precisely if molecule type @xmath7 catalyses reaction @xmath8 . see fig . [ fig1 ] for a simple example ( from @xcite ) . and seven reactions . dashed arrows indicate catalysis ; solid arrows show reactants entering a reaction and products leaving . in this crs there are exactly four rafs ( defined below ) , namely @xmath9 , @xmath10 , @xmath11 , and @xmath12 . ] in certain applications , @xmath0 often consist of or at least contain a set of polymers ( sequences ) over some finite alphabet @xmath13 ( i.e. chains @xmath14 , @xmath15 , where @xmath16 ) , as in fig . [ fig1 ] ; such polymer systems are particularly relevant to rna or amino - acid sequence models of early life . reactions involving such polymers typically involve cleavage and ligation ( i.e. cutting and/or joining polymers ) , or adding or deleting a letter to an existing chain . notice that if no bound is put on the maximal length of the polymers , then both @xmath0 and @xmath1 are infinite for such networks , even when @xmath17 . in this paper we do not necessarily assume that @xmath0 consists of polymers , or that the reactions are of any particular type . thus , a reaction can be viewed formally as an ordered pair @xmath18 consisting of a multi - set @xmath19 of elements from @xmath0 ( the reactants of @xmath8 ) and a multi - set @xmath20 of elements of @xmath0 ( the products of @xmath8 ) ; but we will mostly use the equivalent and more conventional notation of writing a reaction in the form : @xmath21 where the @xmath22 s ( reactants of @xmath8 ) and @xmath23 s ( products of @xmath8 ) are elements of @xmath0 , and @xmath24 ( e.g. @xmath25 and @xmath26 are reactions ) . in this paper , we extend our earlier analysis of rafs to the general ( finite or infinite ) case and find that certain subtleties arise that are absent in the finite case . we will mostly assume the following conditions ( a1 ) and ( a2 ) , and sometimes also ( a3 ) . * @xmath3 is finite ; * each reaction @xmath27 has a finite set of reactants , denoted @xmath28 , and a finite set of products , denoted @xmath29 ; * for any given finite set @xmath30 of molecule types , there are only finitely many reactions @xmath8 with @xmath31 . given a subset @xmath32 of @xmath1 , we say that a subset @xmath33 of molecule types is _ closed _ relative to @xmath32 if @xmath34 satisfies the property @xmath35 in other words , a set of molecule types is closed relative to @xmath32 if every molecule that can be produced from @xmath34 using reactions in @xmath32 is already present in @xmath34 . notice that the full set @xmath0 is itself closed . the _ global closure _ of @xmath3 relative to @xmath32 , denoted here as @xmath36 , is the intersection of all closed sets that contain @xmath3 ( since @xmath0 is closed , this intersection is well defined ) . thus @xmath36 is the unique minimal set of molecule types containing @xmath3 that is closed relative to @xmath32 . we can also consider a _ constructive closure _ of @xmath3 relative to @xmath32 , denoted here as @xmath37 , which is union of the set @xmath3 and the set of molecule types @xmath7 that can be obtained from @xmath3 by carrying out any finite sequence of reactions from @xmath32 where , for each reaction @xmath8 in the sequence , each reactant of @xmath8 is either an elements of @xmath3 or a product of a reaction occurring earlier in the sequence , and @xmath7 is a product of the last reaction in the sequence . note that @xmath36 always contains @xmath37 ( and these two sets coincide when the crs is finite ) but , for an infinite crs , @xmath37 can be a strict subset of @xmath36 , even when ( a1 ) holds . to see this , consider the system @xmath38 where @xmath39 , @xmath40 , where @xmath41 is defined as follows : @xmath42 @xmath43 @xmath44 then @xmath45 . in this example , notice that @xmath46 has infinitely many reactants , which violates ( a2 ) . by contrast , when ( a2 ) holds , we have the following result . [ lem1 ] suppose that ( a2 ) holds . then @xmath47 . moreover , under ( a1 ) and ( a2 ) , if @xmath32 is countable , then this ( common ) closure of @xmath3 relative to @xmath32 is countable also . suppose the condition of lemma [ lem1 ] holds but that @xmath37 is not closed ; we will derive a contradiction . lack of closure means there is a molecule @xmath7 in @xmath48 which is the product of some reaction @xmath49 that has all its reactants in @xmath37 . by ( a2 ) , the set of reactants of @xmath8 is finite , so we may list them as @xmath50 , and , by the definition of @xmath37 , for each @xmath51 , either @xmath52 or there is a finite sequence @xmath53 of reactions from @xmath32 that generates @xmath54 starting from reactants entirely in @xmath3 and using just elements of @xmath3 or products of reactions appearing earlier in the sequence @xmath53 . by concatenating these sequences ( in any order ) and appending @xmath8 at the end , we obtain a finite sequence of reactions that generate @xmath7 from @xmath3 , which contradicts the assumption that @xmath37 is not closed . if follows that @xmath37 is closed relative to @xmath32 , and since it is clearly a minimal set containing @xmath3 that is closed relative to @xmath32 , it follows that @xmath55 . that @xmath37 is countable under ( a1 ) and ( a2 ) follows from the fact that any countable union of finite sets is countable . @xmath56 in view of lemma [ lem1 ] , whenever ( a2 ) holds , we will henceforth denote the ( common ) closure of @xmath3 relative to @xmath32 as @xmath57 . * definition [ raf , and related concepts ] * suppose we have a crs @xmath58 , which satisfies condition ( a2 ) . an raf for @xmath59 is a non - empty subset @xmath32 of @xmath1 for which * for each @xmath60 , @xmath61 ; and * for each @xmath60 , at least one molecule type in @xmath57 catalyses @xmath8 . in words , a non - empty set @xmath32 of reactions forms an raf for @xmath59 if , for every reaction @xmath8 in @xmath32 , each reactant of @xmath8 and at least one catalyst of @xmath8 is either present in @xmath3 or able to be constructed from @xmath3 by using just reactions from within the set @xmath32 . an raf @xmath32 for @xmath59 is said to be a _ finite raf _ or an _ infinite raf _ depending on whether or not @xmath62 is finite or infinite . the concept of an raf is a formalisation of a ` collectively autocatalytic set ' , pioneered by stuart kauffman @xcite and @xcite . since the union of any collection of rafs is also an raf , any crs that contains an raf necessarily contains a unique maximal raf . irrraf _ is an ( infinite or finite ) raf that is minimal i.e. it contains no raf as a strict subset . in contrast to the uniqueness of the maximal raf , a finite crs can have exponentially many irrrafs @xcite . the raf concept needs to be distinguished from the stronger notion of a _ constructively autocatalytic and f - generated _ ( caf ) set @xcite which requires that @xmath32 can be ordered @xmath63 so that all the reactants and at least one catalyst of @xmath64 are present in @xmath65 for all @xmath66 ( in the initial case where @xmath67 , we take @xmath68 ) . this condition essentially means that in a caf , a reaction can only proceed if one of its catalysts is already available , whereas an raf could become established by allowing one or more reactions @xmath8 to proceed uncatalysed ( presumably at a much slower rate ) so that later , in some chain of reactions , a catalyst for @xmath8 is generated , allowing the whole system to ` speed up ' . notice that although the crs in fig . [ fig1 ] has four rafs it has no caf . , and @xmath69 are products ) , reactions are hollow squares , and dashed arrows indicate catalysis . ] the raf concept also needs to be distinguished from the weaker notion of a _ pseudo - raf _ @xcite , which replaces condition ( ii ) with the relaxed condition : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( ii)@xmath70 : for all @xmath60 , there exists @xmath71 or @xmath72 for some @xmath60 such that @xmath6 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in other words , a pseudo - raf that fails to be an raf is an autocatalytic system that could continue to persist once it exists , but it can never form from just the food set @xmath3 , since it is not @xmath3-generated . these two alternatives notions to rafs are illustrated ( in the finite setting ) in fig . [ fig1b ] . notice that every caf is an raf and every raf is a pseudo - raf , but these containments are strict , as fig . [ fig1b ] shows . while the notion of a caf may seem reasonable , it is arguably too conservative in comparison to an raf , since a reaction can still proceed if no catalyst is present , albeit it at a much slower rate , allowing the required catalyst to eventually be produced . however relaxing the raf definition further to a pseudo - raf is problematic ( since a reaction can not proceed at all , unless all its reactants are present , and so such a system can not arise spontaneously just from @xmath3 ) . this , along with other desirable properties of rafs ( their formation requires only low levels of catalysis in contrast to cafs @xcite ) , suggests that rafs are a reasonable candidate for capturing the minimal necessary condition for self - sustaining autocatalysis , particularly in models of the origin of metabolism . as in the finite crs setting , the union of all rafs is an raf , so any crs that contains an raf has a unique maximal one . it is easily seen that an infinite crs that contains an raf need not have a maximal finite raf , even under ( a1)(a3 ) , but in this case , the crs would necessarily also contain an infinite raf ( the union of all the finite rafs ) . a natural question is the following : if an infinite crs contains an infinite raf , does it also contain a finite one ? it is easily seen that even under conditions ( a1 ) and ( a2 ) , the answer to this last question is ` no ' . we provide three examples to illustrate different ways in which this can occur . this is in contrast to cafs , for which exactly the opposite holds : if a crs contains an infinite caf , then it necessarily contains a sequence of finite ones . moreover , two of the infinite rafs in the following example contain no irrrafs ( in contrast to the finite case , where every raf contains at least one irrraf ) . * example 1 : * let @xmath73 , @xmath74 and @xmath75 . let @xmath76 . we will specify particular crs s by describing @xmath77 , and the pattern of catalysis as follows . * @xmath78 has a reaction @xmath79 for each @xmath80 and @xmath64 is catalysed by @xmath81 for each @xmath82 . * @xmath83 has a reaction @xmath84 \rightarrow x_i)$ ] for each @xmath80 and @xmath64 is catalysed by @xmath81 for each @xmath82 . * @xmath85 has the same reactions as @xmath83 but @xmath64 is now catalysed by every @xmath86 . [ figx ] illustrates the three crs s . each of @xmath87 satisfy ( a1 ) and ( a2 ) , but only @xmath78 satisfies ( a3 ) . all three crss contain infinite rafs , but no finite raf , and no caf . more precisely : * @xmath78 has @xmath1 as its unique raf ( which is therefore an irrraf ) . * the rafs of @xmath83 consist precisely of all subsets of @xmath88 for some @xmath89 . thus @xmath83 has a countably infinite number of rafs but no irrraf . * the rafs of @xmath85 consist precisely of all infinite subsets of @xmath1 . thus , the set of rafs for @xmath85 in uncountably infinite , and it contains no irrraf . in this section , we assume that both ( a1 ) and ( a2 ) hold . given a crs @xmath58 , consider the following nested decreasing sequence of reactions : @xmath90 defined by @xmath91 and for each @xmath80 : @xmath92 thus , @xmath93 is obtained from @xmath94 by removing any reaction that fails to have either all its reactants or at least one catalyst in the closure of @xmath3 relative to @xmath94 . let @xmath95 . it is easily shown that any raf @xmath32 present in @xmath59 is necessarily a subset of @xmath96 ( since @xmath97 for all @xmath82 by induction on @xmath98 ) . thus if @xmath99 then @xmath59 does not have an raf . in the finite case there is a strong converse if @xmath100 then @xmath59 has an raf , and @xmath96 is the unique maximal raf for @xmath59 ( this is the basis for the ` raf algorithm ' @xcite and @xcite ) . however , in contrast , this result can fail for an infinite crs , as we now show with a simple example , which also satisfies ( a1)(a3 ) . * example 2 : * consider the following infinite crs , @xmath101 , where @xmath74 , and @xmath102 where @xmath103 ( this set can be thought of as all polymers of @xmath104 ) . the reaction set is @xmath105 , where , for all @xmath106 @xmath107 @xmath108 the pattern of catalysis is defined as follows : @xmath109 catalyses @xmath110 and @xmath111 catalyses @xmath112 , and for all @xmath80 @xmath113 catalyses @xmath64 and @xmath54 catalyses @xmath114 . this crs is illustrated in fig [ figy ] . which has no raf even though @xmath115 is non - empty ( equal to @xmath116 ) . this crs satisfies ( a1)(a3 ) and ( a5 ) , but not ( a4 ) . ] notice that @xmath117 satisfies ( a1 ) , ( a2 ) and ( a3 ) . however , if we construct the sequence @xmath94 described above , then as the sole catalyst ( @xmath111 ) of @xmath112 is neither in the food set , nor generated by any other reaction , it follows that @xmath112 will be absent from @xmath118 , and so @xmath119 will also be absent from @xmath120 ( since the only catalyst of @xmath119 is produced by @xmath112 ) . continuing in this way , we obtain @xmath121 , but this set is not an raf , since the sole catalyst @xmath109 of @xmath110 does not lie lie in the closure of @xmath3 relative to @xmath122 it was produced by the @xmath123 reactions and in these have all disappeared in the limit ; moreover it is clear that no subset of @xmath117 is an raf . @xmath56 thus , we require slightly stronger hypotheses than just ( a1)(a3 ) in order to ensure that @xmath59 has an raf when @xmath100 . this , is provided by the following result . [ infp ] let @xmath58 satisfy ( a1 ) and ( a2 ) . the following then hold : * @xmath96 contains every raf for @xmath59 ; in particular , if @xmath99 , then @xmath59 has no raf . * suppose that @xmath59 satisfies both of the following further conditions : * * @xmath124 , for the sequence @xmath94 defined in ( [ r1eqx ] ) . * * each reaction @xmath27 is catalysed by only finitely many molecule types . + then @xmath59 contains an raf if and only if @xmath96 is non - empty ( in which case , @xmath96 is the maximal raf for @xmath59 ) . before proving this result , we pause to make some comments and observations concerning the new conditions ( a4 ) and ( a5 ) . regarding condition ( a4 ) , containment in the opposite direction is automatic ( by virtue of the fact that @xmath125 for any function @xmath104 and sets @xmath126 ) , so ( a4 ) amounts to saying that the two sets described are equal . notice also that @xmath117 in example 2 ( fig . [ figy ] ) satisfies ( a5 ) but it violates ( a4 ) , as it must , since @xmath117 does not have an raf . to see how @xmath117 violates ( a4 ) , notice that @xmath127 , while @xmath128 . condition ( a5 ) is quite strong , but proposition [ infp ] is no longer true if it is removed . to see why , consider the following modification @xmath129 of @xmath117 in which the only product of @xmath130 ( for @xmath80 ) is @xmath54 , and @xmath54 catalyses @xmath110 for all @xmath80 ( in addition to @xmath114 ) , as shown in fig . then @xmath131 so ( a4 ) holds ; however @xmath132 which , as before , is not an raf for @xmath129 since there is no catalyst of @xmath110 in @xmath133 . notice that ( a5 ) fails for @xmath129 since @xmath110 has infinitely many catalysts . nevertheless , it is possible to obtain a result that dispenses with ( a5 ) at the expense of a strengthening ( a4 ) , which we will do shortly in proposition [ infpro ] . which has no raf even though @xmath134 is non - empty ( equal to @xmath116 ) . this crs satisfies ( a1)(a3 ) and ( a4 ) , but not ( a5 ) , nor ( a4)@xmath70 . ] _ proof of proposition [ infp ] : _ suppose @xmath32 is any raf for @xmath59 . induction on @xmath135 shows that @xmath97 for all @xmath98 , so that @xmath136 ; in particular , if @xmath99 , then @xmath59 has no raf . the proof of part ( ii ) of proposition [ infp ] relies on a simple lemma . [ simlem ] suppose that @xmath137 is any nested decreasing sequence of subsets and @xmath20 is a finite set for which @xmath138 for all @xmath82 . then some element of @xmath20 is present in every set @xmath139 . _ proof of lemma : _ suppose , to the contrary , that for every element @xmath140 , there is some set @xmath141 in the sequence that fails to contain @xmath142 ( we will show this is not possible by deriving a contradiction ) . let @xmath143 . since @xmath20 is a finite set , @xmath144 is a finite integer , and since the sequence @xmath137 is a nested decreasing sequence , it follows that @xmath145 , a contradiction . @xmath56 returning to the proof of part ( ii ) , suppose that @xmath100 ; we will show that @xmath96 is an raf for @xmath59 ( and so , by part ( i ) , the unique maximal raf for @xmath59 ) . for @xmath146 for each @xmath98 ( otherwise @xmath8 would not be an element of @xmath93 and thereby fail to lie in @xmath96 ) . thus @xmath147 by ( a4 ) . it remains to show that @xmath8 is catalysed by at least one element of @xmath148 . let @xmath149 . by ( a5 ) , @xmath150 is finite . moreover , for each @xmath135 , @xmath151 ( otherwise @xmath8 would fail to be in @xmath93 and thereby not lie in @xmath96 ) . by lemma [ simlem ] , there is a molecule type @xmath152 that lies in @xmath153 and this latter set is contained in @xmath154 by ( a4 ) . in summary , every reaction in @xmath96 has all its reactants and at least one catalyst present in @xmath148 and so @xmath96 is an raf for @xmath59 , as claimed . @xmath56 suppose we now remove condition ( a5 ) in proposition [ infp ] . in this case , by a slight strengthening of ( a4 ) , we obtain a positive result ( proposition [ infpro ] ) . to describe this , we first require a further definition . recall that @xmath2 is the set of pairs @xmath155 where molecule type @xmath7 catalyses reaction @xmath8 . given a subset @xmath156 of @xmath2 , let @xmath157 = \{r \in { { \mathcal r } } : ( x , r ) \in c ' \mbox { for some } x\in x\}.\ ] ] define a nested decreasing sequence of subsets @xmath158 by @xmath159 and for each @xmath82 , @xmath160}(f)\},\ ] ] and let @xmath161 . [ infpro ] let @xmath59 satisfy ( a1 ) and ( a2 ) , as well as the following property : @xmath162}(f ) \subseteq { \rm cl}_{r[c_\infty]}(f ) , \mbox { for the sequence $ c_i$ defined in ( \ref{cieqx})}.\ ] ] then @xmath59 has an raf if and only if @xmath163 , in which case @xmath164 $ ] is a maximal raf for @xmath59 . suppose that @xmath165 . then for any @xmath166 $ ] there exists @xmath167 such that @xmath168 . it follows that @xmath169 for all @xmath98 . by definition , this means that @xmath170}(f)$ ] for all @xmath98 , and so @xmath171}(f).$ ] now , by ( a4)@xmath70 , this means that @xmath172}(f)$ ] . in summary , every reaction in the non - empty set @xmath164 $ ] has all its reactants and at least one catalyst in the closure of @xmath3 with respect to @xmath164 $ ] and so @xmath164 $ ] forms an raf for @xmath59 . conversely , suppose that @xmath59 contains an raf @xmath32 ; we will show that @xmath165 . for each @xmath49 , select a catalyst @xmath173 for @xmath8 for which @xmath174 . let @xmath175 . we use induction on @xmath98 to show that @xmath176 for all @xmath135 . clearly @xmath177 , so suppose that @xmath176 and select an element @xmath178 . by definition , @xmath179}(f ) \subseteq { \rm cl}_{{{\mathcal r}}[c_i]}(f),\ ] ] which means that @xmath180 , establishing the induction step . it follows that @xmath181 and so @xmath165 as claimed . @xmath56 notice that , just as for condition ( a4 ) , the condition ( a4)@xmath70 is equivalent to requiring that the two sets described be identical . notice also that , although condition ( a4 ) applies to the crs @xmath182 , condition ( a4)@xmath70 fails , since @xmath183 and so @xmath184}(f ) = { { \mathcal f}}= \{f , ff , fff , \ldots\}$ ] , while @xmath185}(f)$ ] for all @xmath82 , and so @xmath186}$ ] is not a subset of @xmath187}(f)$ ] . in summary , a single application of @xmath188 allows us to determine when @xmath59 has an raf , provided the additional condition ( a4)@xmath70 holds . example 2 showed that some additional assumption of this type is required , however one could also consider other approaches for determining the existence rafs that do not assume a further condition like ( a4)@xmath70 , but instead iterate the map @xmath188 . in other words , consider the following ` higher level ' sequence of subsets of @xmath1 : @xmath189 where @xmath190 for each @xmath191 . again , this forms a decreasing nested sequence of subsets of @xmath1 and so we can consider the set : @xmath192 in the example above for @xmath117 where @xmath100 , notice that @xmath193 ( and so @xmath194 ) . it follows from proposition [ infp ] that if @xmath195 for any @xmath196 then @xmath59 has no raf . however , just because @xmath197 , this does not imply that @xmath59 contains an raf as the next example shows . * example 3 : * consider the infinite crs @xmath198 which is obtained by taking a countably infinite number of ( reaction and molecule disjoint ) copies of @xmath117 ( from example 2 ) and letting the molecule type @xmath109 in the @xmath98-th copy of @xmath117 play the role of the molecule @xmath111 in the @xmath199th copy of @xmath117 . in addition , let @xmath46 be the reaction @xmath200 ( where @xmath201 is an additional molecule ) catalysed by the @xmath109-products of all the copies of @xmath117 . now @xmath202 contains all but the first @xmath203 copies of @xmath117 , plus @xmath46 . consequently , @xmath204 but , as before , this is not an raf . notice , however that this example violates condition ( a3 ) . we have seen from the last section that applying @xmath188 , even infinitely often , does not seem to provide a way to determine whether a crs possesses an raf . however , in most applications , the main interest will generally be in finite rafs . from the earlier theory it is clear that if @xmath202 is finite for some integer @xmath196 then any rafs that may exist for @xmath59 are necessarily finite , and finite in number . moreover , if @xmath205 and this set is finite , then @xmath202 is the unique ( and necessarily finite ) maximal raf for @xmath59 . however , it is also quite possible that a crs might contain both finite and infinite rafs , and in this section we describe a characterisation of when an raf contains a finite raf . given a crs @xmath59 define a sequence @xmath206 of subsets of @xmath1 as follows : @xmath207 @xmath208 in words , @xmath209 is the set of reactions that have all their reactants in @xmath3 , and for @xmath80 @xmath94 is the set of reactions for which each reactant is either an element of @xmath3 or products of some reaction in @xmath210 for @xmath211 . [ finiteraf ] suppose a crs @xmath59 satisfies ( a1)(a3 ) . let @xmath212 for all @xmath135 , where @xmath213 is as defined above . then : * @xmath214 is a nested increasing sequence of finite sets . * @xmath59 has a finite raf if and only if @xmath215 for some @xmath135 . * if @xmath216 for some @xmath98 , then @xmath217 is a finite raf for @xmath59 for all @xmath218 . * every finite raf for @xmath59 is contained in @xmath219 for some @xmath220 . by ( a1 ) and ( a3 ) , it follows that @xmath221 is finite , and , by induction , that @xmath213 is finite for all @xmath80 . moreover , if @xmath222 then @xmath223 and so @xmath224 ( i.e. @xmath225 ) and so the sets @xmath226 form an increasing nested sequence . this establishes ( i ) . for parts ( ii ) and ( iii ) , suppose that @xmath59 contains a finite raf @xmath32 . since ( a1 ) and ( a2 ) hold , we can apply lemma [ lem1 ] to deduce that every reaction @xmath49 is an element of @xmath213 for some @xmath98 . thus , since @xmath32 is finite , and the sequence @xmath213 is a nested increasing sequence of finite sets , it follows that @xmath227 for some fixed @xmath203 , in which case @xmath228 . conversely , if @xmath216 , then it is clear from the definitions that @xmath229 is an finite raf for @xmath59 ; moreover , so also is @xmath217 for all @xmath230 . part ( iv ) also follows easily from the definitions , since if @xmath32 is a finite raf for @xmath59 then @xmath231 for some @xmath232 , and since @xmath32 is finite we have @xmath233 and so @xmath234 . this completes the proof . @xmath56 theorem [ finiteraf ] provides an algorithm to search for finite rafs in any infinite crs that satisfies ( a1)(a3 ) . given @xmath59 , construct @xmath221 and run the ( standard ) raf algorithm @xcite and @xcite on @xmath221 . if it fails to find an raf , then construct @xmath235 and run the algorithm on this set , and continue in the same manner . if @xmath59 contains a finite raf , then this process is guaranteed to find it , however , there is no assurance in advance of how long this might take ( if not constraint is placed on the size of the how large the smallest finite raf might be ) . finally , we show how proposition [ infpro ] can be reformulated more abstractly in order to makes clear the underlying mathematical principles ; the added generality may also be useful for settings beyond chemical reaction systems . this uses the notion of `` @xmath236-compatibility '' from @xcite , which we now explain . suppose we have an arbitrary set @xmath30 and an arbitrary partially ordered set @xmath34 , together with some functions @xmath237 consider the function @xmath238 , where @xmath239 we are interested in the non - empty subsets of @xmath30 fixed points of @xmath240 , particularly , when @xmath104 is _ monotonic _ ( i.e. , where @xmath241 ) . a subset @xmath19 of @xmath30 is said to be _ @xmath236-compatible _ if @xmath19 is non - empty and @xmath242 . the notion of an raf can be captured in this general setting as follows . given a crs @xmath243 satisfying ( a2 ) , take @xmath244 and @xmath245 ( partially ordered by set inclusion ) , and define @xmath246 as follows : @xmath247}(f ) \mbox { and } g((x , r ) ) = \{x\ } \cup \rho(r),\ ] ] where , as earlier , @xmath248 $ ] is the set of reactions @xmath27 for which there is some @xmath249 with @xmath250 . notice that @xmath104 is monotonic and when @xmath59 is finite , the set @xmath251 can be computed in polynomial time in the size of @xmath59 . [ lemcom ] suppose we have a crs @xmath59 satisfying ( a2 ) , and with @xmath104 and @xmath252 defined as in ( [ fgeq ] ) . if @xmath19 is @xmath236-compatible , then @xmath248 $ ] is an raf for @xmath59 . conversely , if @xmath32 is an raf for @xmath59 , then a @xmath236-compatible set @xmath19 exists with @xmath248 = { { \mathcal r}}'$ ] . in particular , @xmath59 has an raf if and only if @xmath30 contains a @xmath236-compatible set . if @xmath19 is @xmath236-compatible subset of @xmath30 , then for @xmath253 $ ] , each reaction @xmath27 has at least one molecule type @xmath167 for which @xmath254 . @xmath236-compatibility ensures that @xmath255 , in other words , @xmath256 for some catalyst @xmath7 of @xmath8 . this holds for every @xmath49 , so @xmath32 is an raf for @xmath59 . conversely , if @xmath32 is an raf , then for each reaction @xmath49 , we can choose an associated catalyst @xmath173 so that @xmath257 . then @xmath258 is a @xmath236-compatible subset of @xmath30 , with @xmath248 = { { \mathcal r}}'$ ] . @xmath56 the problem of finding a @xmath236-compatible set ( if one exists ) in a general setting ( arbitrary @xmath30 , and @xmath34 , not necessarily related to chemical reaction networks ) can be solved in general polynomial time when @xmath30 is finite and @xmath104 is monotonic and computable in finite time . this provides a natural generalization of the classical raf algorithm . in @xcite , we showed how other problems ( including a toy problem in economics ) could by formulated within this more general framework . however , if we allow the set @xmath30 to be infinite , then monotonicity of @xmath104 needs to be supplemented with a further condition on @xmath104 . we will consider a condition ( ` @xmath201-continuity ' ) , which generalizes ( a4)@xmath70 , and that applies automatically when @xmath30 is finite we say that @xmath259 is ( weakly ) @xmath201-continuous if , for any nested descending chain @xmath260 of sets , we have : @xmath261 recall that an element in a partially ordered set need not have a greatest lower bound ( glb ) ; but if it does , it has a unique one . notice that when @xmath30 is finite , this property holds trivially , since then @xmath262 for the last set @xmath263 in the ( finite ) nested chain . for a subset @xmath19 of @xmath30 and @xmath264 , define @xmath265 to be the result of applying function @xmath240 iteratively @xmath203 times starting with @xmath19 . thus @xmath266 and for @xmath264 , @xmath267 . taking the particular interpretation of @xmath104 and @xmath252 in ( [ fgeq ] ) , the sequence @xmath268 is nothing more than the sequence @xmath269 from ( [ cieqx ] ) . notice that the sequence @xmath270 is a nested decreasing sequence of subsets of @xmath30 , and so we may define the set : @xmath271 which is a ( possibly empty ) subset of @xmath30 ( in the setting of proposition [ infpro ] , @xmath272 ) . given ( finite or infinite ) sets @xmath273 , where @xmath34 is partially ordered , together with functions @xmath274 , it is routine to verify that the following properties hold : * the @xmath236-compatible subsets of @xmath30 are precisely the non - empty subsets of @xmath30 that are fixed points of @xmath240 ; * if @xmath104 is monotonic then @xmath275 contains all @xmath236-compatible subsets of @xmath30 ; in particular , if @xmath276 , then there is no @xmath236compatible subset of @xmath30 . * if @xmath104 is @xmath201-continuous then @xmath275 is @xmath236-compatible , provided it is non - empty ; in particular , if @xmath104 is monotonic and @xmath201-continuous then ( by ( ii ) ) there a @xmath236-compatible subset of @xmath30 exists if and only if @xmath275 is nonempty . * without the assumption that @xmath104 is weakly @xmath201-continuous in part ( iii ) , it is possible for @xmath275 to fail to be @xmath236-compatible when @xmath30 is infinite , even if @xmath104 is monotone . the proof of parts ( i)(iii ) proceeds exactly as in @xcite , with the addition of one extra step required to justify part ( iii ) , assuming @xmath201-continuity . namely , condition ( [ glbeq ] ) ensures that @xmath277 is also @xmath201-continuous in the sense that for any nested descending chain @xmath260 of sets , we have : @xmath278 and so @xmath279 . the proof of ( [ psieq ] ) from ( [ glbeq ] ) is straightforward : firstly , @xmath280 holds for _ any _ function @xmath240 , while if @xmath281 , then , by definition of @xmath240 , @xmath282 for all @xmath98 and @xmath283 for all @xmath82 and so @xmath284 , and @xmath283 for all @xmath82 . now , since @xmath285 is a glb of @xmath286 , we have @xmath287 for all @xmath98 ( i.e. @xmath288 ) and so @xmath289 . part ( iv ) follows directly from parts ( ii ) and ( iii ) . for part ( vi ) , consider the infinite crs @xmath117 in example 2 . as above , take @xmath290 and , for @xmath291 , with @xmath104 and @xmath252 defined as in ( [ fgeq ] ) . then @xmath292 , where @xmath293 however , @xmath19 is not @xmath236-compatible , since @xmath294 and @xmath295 but this is not a subset of @xmath296}(f ) = { { \mathcal f}}$ ] since @xmath297 . in this example , @xmath104 fails to be weakly @xmath201-continuous , and the argument is analogous to where we showed earlier that @xmath129 fails to satisfy ( a4)@xmath70 . more precisely , for each @xmath82 , let @xmath298 , where @xmath94 is defined in ( [ r1eqx ] ) and where , for each reaction @xmath299 , @xmath300 is the unique catalyst of @xmath8 . then @xmath301 and so @xmath302 . however , @xmath303 and so @xmath304 , which differs from the glb of @xmath305 , namely @xmath302 . the examples in this paper are particularly simple indeed mostly we took the food set to consist of just a single molecule , and reactions often had only one possible catalyst . in reality more ` realistic ' examples can be constructed , based on polymer models over an alphabet , however the details of those examples tends to obscure the underlying principles so we have kept with our somewhat ` toy ' examples in order that the reader can readily verify certain statements . section [ finitesec ] describes a process for determining whether an arbitrary infinite crs ( satisfying ( a1)(a3 ) ) contains a finite raf . however , from an algorithmic point of view , proposition [ finiteraf ] is somewhat limited , since the process described is not guaranteed to terminate in any given number of steps . if no further restriction is placed on the ( infinite ) crs , then it would seem difficult to hope for any sort of meaningful algorithm ; however , if the crs has a ` finite description ' ( as do our main examples above ) , then the question of the algorithmic decidability of the existence of an raf or of a finite raf arises . more precisely , suppose an infinite crs @xmath306 consists of ( i ) a countable set of molecule types @xmath307 , where we may assume ( in line with ( a1 ) ) that @xmath308 , for some finite value @xmath309 , and ( ii ) a countable set @xmath310 of reactions , where @xmath64 has a finite set @xmath311 of reactants , a finite set @xmath312 of products , and a finite or countable set @xmath313 of catalysts , where @xmath314 and @xmath315 are computable ( i.e. partial recursive ) set - valued functions defined on the positive integers . given this setting , a possible question for further investigation is whether ( and under what conditions ) there exists an algorithm to determine whether or not @xmath59 contains an raf , or more specifically a finite raf ( i.e. when is this question decidable ? ) . the author thanks the allan wilson centre for funding support , and wim hordijk for some useful comments on an earlier version of this manuscript . i also thank marco stenico ( personal communication ) for pointing out that @xmath201-consistency is required for part ( iii ) of the @xmath236-compatibility result above when @xmath30 is infinite , and for a reference to a related fixed - point result in domain theory ( theorem 2.3 in @xcite ) , from which this result can also be derived . p. dittrich , p. speroni di fenizio , chemical organisation theory . bull . math . biol . * 69 * , 11991231 ( 2007 ) p. g. higgs , n. lehman , the rna world : molecular cooperation at the origins of life . genet . * 16*(1 ) , 717 ( 2015 ) w. hordijk , m. steel , autocatalytic sets extended : dynamics , inhibition , and a generalization . . chem . * 3*:5 ( 2012 ) w. hordijk , m. steel , autocatalytic sets and boundaries . j. syst . 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given any finite and closed chemical reaction system , it is possible to efficiently determine whether or not it contains a ` self - sustaining and collectively autocatalytic ' subset of reactions , and to find such subsets when they exist . however , for systems that are potentially open - ended ( for example , when no prescribed upper bound is placed on the complexity or size / length of molecules types ) , the theory developed for the finite case breaks down . we investigate a number of subtleties that arise in such systems that are absent in the finite setting , and present several new results .

the statistical mechanics of systems with long - range interactions has recently attracted a lot of attention @xcite . typical systems with long - range interactions include self - gravitating systems @xcite , two - dimensional vortices @xcite , non - neutral plasmas @xcite , free electrons lasers @xcite and toy models such as the hamiltonian mean field ( hmf ) model @xcite . unusual properties of systems with long - range interaction such as negative specific heats or ensembles inequivalence have been evidenced and linked with lack of additivity @xcite . in addition , a striking property of these systems is the rapid formation of quasi stationary self - organized states ( coherent structures ) such as galaxies in the universe @xcite , large scale vortices in geophysical and astrophysical flows @xcite or quasi - stationary states in the hmf model @xcite . these qsss can be explained in terms of statistical mechanics using the theory developed by lynden - bell @xcite for the vlasov equation or by miller @xcite and robert & sommeria @xcite for the 2d euler equation . two - dimensional vortices interact via a logarithmic potential . interaction of vortices in 3d turbulence is weaker than in 2d turbulence , but still long - range . due to dissipative anomaly and vortex stretching , statistical mechanics of 3d turbulence has so far eluded theories . recent progress was recently made considering 3d inviscid axisymmetric flows @xcite that are intermediate between 2d and 3d flows : they are subject to vortex stretching like in 3d turbulence , but locally conserve a scalar quantity in the ideal limit , like in 2d turbulence . it is therefore interesting to study whether these systems obey the peculiarities observed in other systems with long - range interactions such as violent relaxation , existence of long - lived quasi - stationary states , negative specific heats and ensembles inequivalence . the general study of the stability of axisymmetric flows , and the possible occurrence of phase transitions , is difficult due to the presence of an infinite number of casimir invariants linked with the axisymmetry of the flow . in a previous paper @xcite , hereafter paper i , we have considered a simplified axisymmetric euler system characterized by only three conserved quantities : the fine - grained energy @xmath0 , the helicity @xmath1 and the angular momentum @xmath2 . we have developed the corresponding statistical mechanics and shown that equilibrium states of this system have the form of beltrami mean flows on which are superimposed gaussian fluctuations . we have shown that the maximization of entropy @xmath3 at fixed helicity @xmath1 , angular momentum @xmath2 and microscopic energy @xmath0 ( microcanonical ensemble ) is equivalent to the maximization of free energy @xmath4 at fixed helicity @xmath1 and angular momentum @xmath2 ( canonical ensemble ) . these variational principles are also equivalent to the minimization of macroscopic energy @xmath5 at fixed helicity @xmath1 and angular momentum @xmath2 . this provides a justification of the minimum energy principle ( selective decay ) from statistical mechanics . we have furthermore discussed the analogy with the simplified thermodynamical approach of 2d turbulence developed in @xcite based on only three conserved quantities : the fine - grained enstrophy @xmath6 , the energy @xmath7 and the circulation @xmath8 . we have shown that equilibrium states of this system have the form of beltrami mean flows ( linear vorticity - stream function relationship ) on which are superimposed gaussian fluctuations . we have shown that the maximization of entropy @xmath3 at fixed energy @xmath7 , circulation @xmath8 and microscopic enstrophy @xmath6 ( microcanonical ensemble ) is equivalent to the maximization of grand potential @xmath9 at fixed energy @xmath7 and circulation @xmath8 ( grand microcanonical ensemble ) . these variational principles are also equivalent to the minimization of macroscopic enstrophy @xmath10 at fixed energy @xmath7 and circulation @xmath8 . this provides a justification of a minimum enstrophy principle ( selective decay ) from statistical mechanics . in the analogy between 2d turbulence and 3d axisymmetric turbulence , the energy plays the role of the enstrophy . in the present paper , we study more closely the equilibrium states of axisymmetric flows and explore their stability . we show that all critical points of macroscopic energy at fixed helicity and angular momentum are _ saddle points _ , so that they are unstable in a strict sense . indeed , there is no minimum ( macroscopic ) energy state at fixed helicity and angular momentum ( either globally or locally ) because we can always decrease the energy by considering a perturbation at smaller scales . this is reminiscent of the richardson energy cascade in 3d turbulence . inversely , in 2d turbulence , there exists minimum enstrophy states that develop at large scales ( inverse cascade ) . therefore , our system is intermediate between 2d and 3d turbulence : there exists equilibrium states in the form of coherent structures ( that are solutions of a mean field differential equation ) like in 2d turbulence , but they are saddle points of macroscopic energy and are expected to cascade towards smaller and smaller scales like in 3d turbulence . however , we give arguments showing that saddle points can be robust in practice and play a role in the dynamics . indeed , they are unstable only for some particular ( optimal ) perturbations and can persist for a long time if the system does not spontaneously generate these perturbations . therefore , these large - scale coherent structures can play a role in the dynamics and they have indeed been observed in experiments of von krmn flows @xcite . in order to make this idea more precise , we have explored their stability numerically using phenomenological relaxation equations derived in @xcite . we have found some domains of robustness in the parameter space . in particular , the one cell structure is highly robust for large values of the angular momentum @xmath11 and becomes weakly robust for low values of the angular momentum . in that case , we expect a phase transition ( bifurcation ) from the one - cell structure to the two - cells structure . we have also found that the value of the critical angular momentum @xmath12 changes depending whether we use relaxation equations associated with a canonical ( fixed temperature ) or microcanonical ( fixed microscopic energy ) description . at low temperatures @xmath13 , we have evidence a new kind of `` ensembles inequivalence '' characterizing the robustness of saddle points with respect to random perturbations . the paper is organized as follows : in sec . [ setup ] , we set - up the various notations and hypotheses we are going to use . the computation and characterization of equilibrium states is done in sec . [ computation ] . the stability analysis of these equilibrium states is performed in sec . [ analytics ] where we show analytically that all states are unstable with respect to large wavenumber perturbations . we evidence a process of energy condensation at small scales that is reminiscent of the richardson cascade . we explore numerically the robustness of the equilibria in both canonical and microcanonical ensembles in sec . [ stability ] . our numerical method is probabilistic and rather involved . a discussion of our results is done in sec . [ discussion ] where a bifurcation scenario relevant to the turbulent experimental von krmn flow is suggested . we consider a system with a cylindrical geometry enclosed in the volume delimited above and below by surfaces @xmath14 and @xmath15 , and radially by @xmath16 . like in paper i , we consider an axisymmetric euler - beltrami system characterized by a velocity field @xmath17 , with axisymmetric time averaged @xmath18 . we furthermore assume that the only relevant invariants of the axisymmetric euler equations for our problem are the averaged energy @xmath19 , the averaged helicity @xmath20 and the averaged angular momentum @xmath21 where @xmath22 . we introduce the potential vorticity @xmath23 and the stream function @xmath24 such that @xmath25 and @xmath26 . they are related to each other by the generalized laplacian operator @xmath27 in actual turbulent von krmn experiments , we have been able to observe that the largest part of the kinetic energy is contained in the toroidal motions . it is therefore natural , as a first elementary step , to consider a model in which only toroidal fluctuations are considered , and suppose that the fluctuations in the other ( poloidal ) directions are simply frozen . with such an assumption , poloidal vorticity fluctuations are allowed , but toroidal vorticity fluctuations are excluded . we therefore only include a fraction of the vorticity fluctuations , that presumably become predominant at small scale , due to the existence of vortex stretching . as shown below and in the next paper @xcite , this simplification however still allows for vortex stretching and energy cascades towards smaller scales , and leads to predictions that are in good agreement with experiments . moreover , our hypotheses lead to a model that is self - contained and analytically tractable . according to our hypotheses , neither @xmath28 nor @xmath24 fluctuates in time : @xmath29 and @xmath30 . in that case , the conserved quantities can be rewritten @xmath31 @xmath32 @xmath33 where @xmath34 denotes the spatial average . ] @xmath35 the helicity and the angular momentum are _ robust constraints _ because they can be expressed in terms of coarse - grained quantities @xmath36 and @xmath37 . by contrast , the energy is a _ fragile constraint _ because it can not be expressed in terms of coarse - grained quantities . indeed , it involves the fluctuations of angular momentum @xmath38 . to emphasize that point , we have introduced the notation @xmath39 to designate the fine - grained ( microscopic ) energy . splitting @xmath40 into a mean part @xmath37 and a fluctuating part @xmath41 , we define the coarse - grained ( macroscopic ) energy by @xmath42 then , the energy contained in the fluctuations is simply @xmath43 where @xmath44 is the local centered variance of angular momentum . we stress that the microscopic energy @xmath45 is conserved while the macroscopic energy @xmath5 is _ not _ conserved and is likely to decrease ( see below ) . in paper i , we have developed a simplified thermodynamic approach of axisymmetric flows under the above - mentioned hypothesis . let @xmath46 denote the pdf of @xmath40 and let us recall the expression of the entropy @xmath47 we have proven the equivalence between the _ microcanonical _ ensemble @xmath48\ , | \ , e^{f.g . } , \ , h,\ , i , \ , \int \rho d\eta=1 \rbrace , \label{bes1}\ ] ] and the _ canonical _ ensemble @xmath49=s-\beta e^{f.g.}\ , | \ , h,\ , i , \ , \int \rho d\eta=1 \rbrace . \label{bes2}\ ] ] in each ensemble , the critical points are determined by the first order condition @xmath50 . the equilibrium distribution is gaussian @xmath51 the mean flow is a beltrami state @xmath52 @xmath53 and the centered variance of angular momentum is @xmath54 these equations determine _ critical points _ of the variational problems ( [ bes1 ] ) and ( [ bes2 ] ) that cancel the first order variations of the thermodynamical potential . clearly , ( [ bes1 ] ) and ( [ bes2 ] ) have the same critical points . furthermore , it is shown in paper i that ( [ bes1 ] ) and ( [ bes2 ] ) are equivalent for the _ maximization _ problem linked with the sign of the second order variations of the thermodynamical potential : a critical point determined by eqs . ( [ vp1])-([vp4 ] ) is a maximum of @xmath3 at fixed microscopic energy , helicity and angular momentum iff it is a maximum of @xmath55 at fixed helicity and angular momentum . this equivalence is not generic . we always have the implication ( [ bes2 ] ) @xmath56 ( [ bes1 ] ) but the reciprocal may be wrong . here , the microcanonical and canonical ensembles are equivalent due to the quadratic nature of the microscopic energy @xmath57 . we note that , according to eq . ( [ vp4 ] ) , @xmath58 is positive . in the canonical ensemble , @xmath58 is prescribed . in the microcanonical ensemble , @xmath58 is a lagrange multiplier that must be related to the energy @xmath45 . according to eqs . ( [ fluct ] ) and ( [ vp4 ] ) , we find that @xmath59 is determined by the condition @xmath60 this relation shows that @xmath61 plays the role of a temperature associated with the fluctuations of angular momentum . in 3d is the counterpart of the chemical potential @xmath62 associated with the conservation of the fine - grained enstrophy in 2d ( see introduction ) . ] finally , we have proven in paper i that the two variational problems ( [ bes1 ] ) and ( [ bes2 ] ) are equivalent to @xmath63=-\beta e^{c.g.}\ , | \ , h,\ , i \rbrace,\label{res13}\ ] ] or equivalently @xmath64\ , | \ , h,\ , i \rbrace,\label{res13b}\ ] ] in the sense that the solution of ( [ bes1 ] ) or ( [ bes2 ] ) is given by eq . ( [ vp1 ] ) where @xmath65 are the solutions of ( [ res13 ] ) or ( [ res13b ] ) . this justifies a _ selective decay principle _ from statistical mechanics . indeed , it is often argued that an axisymmetric turbulent flow should evolve so as to minimize energy at fixed helicity and angular momentum . in general , this phenomenological principle is motivated by viscosity or other dissipative processes . in our approach , it is justified by the maximum entropy principle ( [ bes1 ] ) of statistical mechanics when a coarse - graining is introduced . in the sequel , we shall study the maximization problem ( [ res13b ] ) since it is simpler than ( [ bes1 ] ) or ( [ bes2 ] ) , albeit equivalent . remark : _ although the variational problems ( [ bes1 ] ) and ( [ bes2 ] ) determining equilibrium states are equivalent , this does not mean that the relaxation equations associated with these variational problems are equivalent . to take an analogy , the boltzmann ( microcanonical ) and the kramers ( canonical ) equations have the same equilibrium states -the maxwell distribution- but a different dynamics . in the following , we will show that the equilibrium variational problems ( [ bes1 ] ) and ( [ bes2 ] ) have no solution . indeed , there is no maximum of entropy at fixed @xmath0 , @xmath1 and @xmath2 and no minimum of free energy at fixed @xmath1 and @xmath2 . all the critical points of ( [ bes1 ] ) and ( [ bes2 ] ) are saddle points of the thermodynamical potentials . then , the idea is to consider the out - of - equilibrium problem , introduce relaxation equations and study the robustness of saddle points with respect to random perturbations . for what concerns the out - of - equilibrium problem , the microcanonical and canonical ensembles may be inequivalent . we will see that they are indeed inequivalent . in this section , we shall study the minimization problem @xmath64\ , | \ , h,\ , i \rbrace.\label{res13c}\ ] ] the critical points of macroscopic energy at fixed helicity and angular momentum are determined by the condition @xmath66 where @xmath67 ( helical potential ) and @xmath68 ( chemical potential ) are lagrange multipliers . introducing the notations @xmath69 and @xmath70 , the variations on @xmath71 and @xmath72 lead to @xmath73 @xmath74 which are equivalent to eqs . ( [ vp2])-([vp3 ] ) up to a change of notations . in the following , it will be convenient to work with the new field @xmath75 . it is easy to check that @xmath76 where @xmath77 is the usual laplacian . therefore , eq . ( [ su1 ] ) becomes @xmath78 and the previous equations can be rewritten @xmath79 @xmath80 where @xmath81 is solution of @xmath82 with @xmath83 on the boundary . this is the fundamental differential equation of the problem . note that a particular solution of this differential equation is @xmath84 but it does not satisfy the boundary conditions . using eqs . ( [ dh ] ) and ( [ di ] ) , the helicity and the angular momentum are given by @xmath85 @xmath86 these equations are relationships between @xmath87 and @xmath88 . _ remark : _ we have not taken into account the conservation of circulation @xmath89 because this would lead to a term @xmath90 in the r.h.s . of eq . ( [ phi ] ) that diverges as @xmath91 . to construct the different solutions of eq . ( [ phi ] ) and study their stability , we shall follow the general procedure developed by chavanis & sommeria @xcite for the 2d euler equation . we first introduce an eigenmode decomposition to compute all critical points of ( [ res13c ] ) . then , we investigate their stability by determining whether they are ( local ) minima of macroscopic energy or saddle points . we first assume that @xmath92 in that case , the differential equation ( [ phi ] ) becomes @xmath93 with @xmath83 on the domain boundary . we introduce the eigenfunctions @xmath94 of the operator @xmath95 . they are defined by @xmath96 with @xmath97 on the domain boundary . it is easy to show that the eigenvalues @xmath98 of @xmath99 are positive ( hence the notation @xmath100 ) . indeed , we have @xmath101 and @xmath102 , which proves the result . it is also easy to show that the eigenfunctions are orthogonal with respect to the scalar product @xmath103 finally , we normalize them so that @xmath104 . the eigenvalues and eigenfunctions of the operator @xmath99 can be determined analytically . the differential equation ( [ e3 ] ) can be rewritten @xmath105 we look for solutions in the form @xmath106 . this yields @xmath107 where the sign of the constant has been chosen in order to satisfy the boundary condition @xmath83 in @xmath14 and @xmath15 . the differential equation for @xmath108 is readily solved and we obtain @xmath109 with @xmath110 where @xmath111 is a strictly positive integer . on the other hand , the differential equation for @xmath112 is @xmath113 if we define @xmath114 and @xmath115 , the foregoing equation can be rewritten @xmath116 this is a bessel equation whose solution is @xmath117 now , the boundary condition @xmath118 implies @xmath119 so that @xmath120 where @xmath121 is the @xmath122-th zero of bessel function @xmath123 . in conclusion , the eigenvalues are @xmath124 and the eigenfunctions are @xmath125 with the normalization constant @xmath126 the mode @xmath127 corresponds to @xmath122 cells in the @xmath128-direction and @xmath111 cells in the @xmath129-direction . we shall distinguish two kinds of modes , according to their properties regarding the symmetry @xmath130 with respect to the plane @xmath131 . the _ odd eigenmodes _ denoted @xmath132 are such that @xmath133 and correspond to @xmath111 even . they have zero mean value in the @xmath129 direction ( @xmath134 ) . for example , the mode @xmath135 is a two - cells solution in the vertical direction . the _ even eigenmodes _ denoted @xmath136 are such that @xmath137 and correspond to @xmath111 odd . they have non zero mean value in the vertical direction ( @xmath138 ) . in particular , the mode @xmath139 is a one - cell solution . returning to eq . ( [ e2 ] ) , this differential equation has solutions only for quantized values of @xmath140 ( eigenvalues ) and the corresponding solutions ( eigenfunctions ) are @xmath141 where we have used the helicity constraint ( [ de20 ] ) to determine the normalization constant . note that eq . ( [ de20 ] ) implies that @xmath142 and @xmath1 have the same sign , so that the square root is always defined . substituting this result in eq . ( [ de21 ] ) , and introducing the control parameter @xmath143 we find that these solutions exist only for @xmath144 with @xmath145 for the odd eigenmodes @xmath132 , we have @xmath146 and for the even eigenmodes @xmath136 , we have @xmath147 . we now assume that @xmath148 and define @xmath149 in that case , the fundamental differential equation ( [ phi ] ) becomes @xmath150 with @xmath151 on the domain boundary . we also assume that @xmath152 . in that case , eq . ( [ c2 ] ) admits a unique solution that can be obtained by expanding @xmath153 on the eigenmodes . using the identity @xmath154 we get @xmath155 of course , @xmath153 can also be obtained by solving the differential equation ( [ phi ] ) numerically . note that this solution is even since only the even modes are `` excited '' . substituting eq . ( [ c1 ] ) in eq . ( [ de21 ] ) , we obtain @xmath156 then , substituting eqs . ( [ c1 ] ) and ( [ c4 ] ) in eq . ( [ de20 ] ) , we get @xmath157 this equation gives a relationship between @xmath158 and @xmath159 . then , @xmath160 is determined by eq . ( [ c4 ] ) . these equations can therefore be viewed as the equations of state of the system . they determine the branch formed by the solutions of the continuum . using @xmath161 and @xmath162 we obtain @xmath163 this implies that @xmath158 is of the same sign as @xmath159 , hence @xmath1 . furthermore , @xmath159 is an odd function of @xmath158 . in the sequel , we shall consider only cases with @xmath164 , i.e. @xmath165 and @xmath166 for illustration and figures . note that eq . ( [ c5 ] ) involves the important function @xmath167 for @xmath146 , the inverse helical potential is @xmath168 or @xmath169 where @xmath170 is any zero of @xmath171 , i.e. @xmath172 for simplicity , we shall call @xmath173 the first zero of @xmath171 . this first zero is always between the first and the second even eigenmodes ( see appendix a ) . its location with respect to the first odd eigenmode @xmath174 depends on the aspect ratio of the cylinder : for @xmath175 , we have @xmath176 ( case l - for large aspect ratio ) while for @xmath177 , @xmath178 ( case s - for small aspect ratio ) . we now consider the case where @xmath148 and @xmath140 . for @xmath179 , we recover the eigenfunction @xmath180 as a limit case . therefore , the even eigenmodes are limit points of the main branch . on the other hand , for @xmath181 , the solution of eq . ( [ c2 ] ) is not unique . indeed , we can always add to the solution ( [ c3 ] ) an eigenmode @xmath182 . this leads to the mixed solution @xmath183 the `` proportion '' @xmath184 of the eigenmode present in the mixed solution is determined by the control parameter @xmath159 . taking the norm of @xmath185 and its scalar product with @xmath128 , we get @xmath186 substituting these results in eqs . ( [ de20 ] ) and ( [ de21 ] ) , we find that @xmath184 is determined by @xmath159 according to @xmath187 with @xmath181 . these mixed solutions exist in the range @xmath188 and they form a plateau at constant @xmath181 . for @xmath189 , we recover the odd eigenmode @xmath132 at @xmath146 and for @xmath190 , the plateau connects the branch of continuum solutions . the mixed solutions are therefore symmetry breaking solutions . they can be seen as a mixture of a continuum solution and an eigenmode solution , like in situations with different phase coexistence . in this section , we plot @xmath158 as a function of @xmath159 . for given @xmath191 , this curve determines the inverse helical potential @xmath192 as a function of the inverse helicity @xmath193 ( conjugate variables ) . it is represented in figs . [ blaml ] and [ blams ] for the cases l and s respectively . one sees that , for a given value of the control parameter @xmath159 , there exists multiple solutions with different values of @xmath158 . we will see in sec . [ coarsegrainedenergy ] . that , for a given value of @xmath159 , the macroscopic energy @xmath5 decreases as @xmath158 increases . therefore , low values of @xmath158 correspond to high energies states and high values of @xmath158 correspond to low energies states . as a function of @xmath159 for case l ( we have taken @xmath194 and @xmath195 ) . for a given value of @xmath159 ( we have taken @xmath196 ) , the solutions of the continuum are denoted by red circles and the mixed solutions by green circles . the mixed solution branches are drawn using dotted lines . one observes multiplicity of solutions : at given @xmath159 correspond several solutions with different @xmath158 . ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] of the four first solutions for @xmath197 . from left to right : @xmath198 ( direct monopole ) , @xmath199 ( vertical dipole ) , @xmath200 ( reversed monopole ) and @xmath201 . increasing values from blue to red . by convention , we call direct ( resp . reversed ) monopole the one - cell solution with maximal ( resp . minimal ) inner stream function - see above . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202 . , title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] + along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] + along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] along the three branches of solution at @xmath203 , @xmath204 and @xmath205 from left to right for each branch . top= branch 1 , direct monopole ; middle : mixed branch ( vertical dipole ) ; bottom : branch 2,reversed monopole . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] _ case l : _ in this case @xmath176 and the curve @xmath206 looks typically like in fig . [ blaml ] . for a given value of @xmath159 , we have different solutions as represented in fig . [ fig : sel_ex2 ] . the highest energy solution is a one cell solution ( continuum branch ) , that we choose to call `` direct monopole '' . the second one is a two vertical cells solution ( mixed branch ) . the cells are symmetric for @xmath146 but one of the two cells grows for increasing @xmath159 . the third highest energy solution is another one - cell solution ( continuum branch ) rotating in a direction opposite to that of the highest energy solution . we therefore call it a `` reversed monopole '' . we call these three respective branches of solutions `` branch 1 '' and `` branch 2 '' for the continuum solutions , and `` mixed branch '' for the mixed solutions . the branches 1 and 2 connect each other at @xmath207 , the location of the first even eigenmode . a typical sequence of variation of the stream function with increasing @xmath159 on these three branches is given in fig . [ fig : branches ] . one sees that , as we increase @xmath159 on the mixed branch , the two cells solution , with a mixing layer at @xmath131 continuously transforms itself into a one cell solution , via a continuous shift of the mixing layer towards the vertical boundary . _ case s : _ in this case @xmath178 and the curve @xmath206 looks typically like in fig . [ blams ] . the highest energy solution is a one cell solution ( continuum branch ) , ( direct monopole ) . the second solution is another one - cell solution ( continuum branch ) rotating in the opposite direction ( reversed monopole ) . the third solution is a two horizontal cells solutions ( continuum branch ) . some stream functions are represented in fig . [ bifu04 ] . as a function of @xmath159 for case s ( here @xmath194 and @xmath208 ) . the solutions of the continuum are denoted by red circles and the mixed solutions by green circles . the mixed solution branches are drawn using dotted lines . ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] of the four first solutions at @xmath209 for case s. from left to right : @xmath210 ( direct monopole ) , @xmath211 ( reversed monopole ) , @xmath212 and @xmath213 . for simplicity , we show at each point only one solution , corresponding to a given sign of @xmath2 . the solution corresponding to opposite sign of @xmath2 can be found by a change @xmath202.,title="fig : " ] _ remark : _ there is a maximum value of @xmath214 above which there is no critical point of energy at fixed helicity and angular momentum . in that case , the system is expected to cascade towards smaller and smaller scales since there is no possibility to be blocked in a `` saddle point '' . this is a bit similar to the antonov instability in stellar dynamics due to the absence of critical point of entropy at fixed mass and energy below a critical value of energy @xcite . in that case , the system is expected to collapse ( gravothermal catastrophe ) . it is not yet clear whether a similar process can be achieved in experiments of turbulent axisymmetric flows . for @xmath215 , the system could become non - axisymmetric ruling out the theoretical analysis . in the previous section , we have found several solutions with different values of @xmath158 for each value of the control parameter @xmath216 . according to the variational principle ( [ res13c ] ) , we should select the solution with the minimum macroscopic energy . combining eqs . ( [ dh ] ) , ( [ di ] ) , ( [ e_cg ] ) , ( [ bel1 ] ) and ( [ bel2 ] ) , we obtain the relation @xmath217 for the eigenmodes ( @xmath218 ) , we find that @xmath219 let us consider the odd eigenmodes @xmath132 that exist for @xmath146 only . they are in competition with each other . we see that there is no minimum energy state since the energy decreases when @xmath127 increase , i.e. when the eigenmodes develop smaller and smaller scales . therefore , the minimum energy state corresponds to the structure concentrated at the smallest accessible scale . for the solutions of the continuum , using eq . ( [ c4 ] ) , the macroscopic energy is @xmath220 we can easily plot it as a function of @xmath158 ( see figs . [ fig : sel_ex ] and [ evsb04 ] ) . combining figs [ blaml ] , [ blams ] , [ fig : sel_ex ] and [ evsb04 ] , we see that , for a given value of @xmath159 , the solution with the smallest macroscopic energy corresponds to the highest @xmath158 , i.e. to small - scale structures . this is in complete opposition to what happens in 2d turbulence . in that case , the counterpart of the macroscopic energy @xmath5 is the macroscopic enstrophy @xmath10 and the minimum enstrophy state corresponds to structures spreading at the largest scale . strikingly , the bifurcation diagram in 2d turbulence @xcite is reversed with respect to the present one . as a function of @xmath158 for case l. the energy of the even eigenmodes are denoted by red circles and the energy of the odd eigenmodes by green circles . ] as a function of @xmath158 for case s. the energy of the even eigenmodes are denoted by red circles and the energy of the odd eigenmodes by green circles . ] in conclusion , there is no global minimum of macroscopic energy at fixed helicity and angular momentum . we can always decrease the macroscopic energy by considering structures at smaller and smaller scales . since ( [ bes1 ] ) , ( [ bes2 ] ) and ( [ res13b ] ) are equivalent , we also conclude that there is no global maximum of entropy at fixed microscopic energy , helicity and angular momentum . we may note a similar fact in astrophysics . it is well - known that a stellar system has no global entropy maximum at fixed mass and energy @xcite . this is associated to gravitational collapse ( called the gravothermal catastrophe in the microcanonical ensemble ) leading to the formation of binary stars . however , in the astrophysical problem , there exists local entropy maxima ( metastable states ) at fixed mass and energy if the energy is sufficiently high ( above the antonov energy ) . similarly , we could investigate the existence of metastable states in the present problem . however , we will show in sec . [ analytics ] that there is no local minimum of macroscopic energy at fixed helicity and angular momentum . all the critical points ( [ bel1])-([bel2 ] ) of the variational problem ( [ res13c ] ) are saddle points ! in our system , the chemical potential is @xmath221 . for given @xmath1 , we have to plot @xmath68 as a function of @xmath2 ( conjugate variables ) . the chemical potential is zero for the eigenmodes . using the equation of state ( [ c4 ] ) , we can express @xmath68 for the continuum solutions as @xmath222 where @xmath159 is expressed as a function of @xmath158 by eq . ( [ c5 ] ) . therefore , eq . ( [ chemical ] ) gives @xmath223 as a function of @xmath158 . eliminating @xmath158 between eqs . ( [ chemical ] ) and ( [ c5 ] ) , we obtain @xmath223 as a function of @xmath159 for the continuum . for the mixed solutions , we have @xmath224 corresponding to straight lines as a function of @xmath225 . the chemical potential curve @xmath223 as a function of @xmath225 is represented in fig . [ csursqrth - casel ] for case l and in fig . [ csursqrth - cases ] for case s. for fixed @xmath1 , this gives @xmath68 as a function of @xmath2 . if we come back to the initial variational problem ( [ bes1 ] ) , the caloric curve should give @xmath58 as a function of the microscopic energy @xmath45 ( conjugate variables ) for fixed values of @xmath1 and @xmath2 . now , the temperature is determined by the expression @xmath226 for given @xmath1 and @xmath2 , we can determine the _ discrete _ values of @xmath227 and the corresponding _ discrete _ values of @xmath228 as explained previously . then , for each discrete value , the temperature is related to the energy by eq . ( [ et ] ) . therefore , the mean flow ( beltrami state ) is fully determined by @xmath1 and @xmath2 and , for a given mean flow , the variance of the fluctuations ( temperature ) is determined by the energy @xmath45 according to @xmath229 in conclusion , the caloric curve @xmath230 , or more properly the series of equilibria , is formed by a a discrete number of straight lines with value at the origin @xmath231 and with constant specific heats @xmath232 . the specific heat is positive since the microcanonical and canonical ensembles are equivalent in our problem . in this section , we prove that the critical points of macroscopic energy at fixed helicity and angular momentum are all saddle points . a critical point of macroscopic energy at fixed helicity and angular momentum is a minimum ( resp . maximum ) iff the second order variations @xmath233 are definite positive ( resp . definite negative ) for all perturbations that conserve helicity and angular momentum at first order , i.e. @xmath234 and @xmath235 . adapting the procedure of chavanis & sommeria @xcite to the present context , we shall determine sufficient conditions of _ \(i ) let us prove that there is no local maximum of macroscopic energy at fixed angular momentum and helicity . consider first the even solutions , including the continuum solutions and the even eigenmodes . we choose a perturbation such that @xmath236 is odd and @xmath237 . for symmetry reason , this perturbation does not change @xmath2 nor @xmath1 at first order . on the other hand , for this perturbation @xmath238 . consider now the odd eigenmodes . we choose a perturbation of the form @xmath237 and @xmath239 , where @xmath240 is the first continuum solution such that @xmath241 . for this perturbation , we have @xmath242 , @xmath243 and @xmath244 since @xmath132 is orthogonal to @xmath240 . therefore , this perturbation does not change the helicity and the angular momentum at first order . on the other hand , for this perturbation @xmath238 . as a result , the critical points of macroscopic energy at fixed helicity and angular momentum can not be energy maxima since we can always find particular perturbations that increase the energy while conserving the constraints . \(ii ) let us prove that there is no local minimum of macroscopic energy at fixed angular momentum and helicity . to that purpose , we consider perturbations of the form @xmath245 and @xmath246 . the corresponding stream function is @xmath247 . consider first the even solutions , including the continuum solutions and the even eigenmodes . in that case , we have @xmath248 , @xmath249 and @xmath250 since @xmath251 is orthogonal to @xmath153 . the preceding relations remain valid for the odd eigenmodes @xmath127 provided that @xmath252 . therefore , these perturbations do not change the helicity and the angular momentum at first order . on the other hand , for these perturbations , we have @xmath253 thus , for given @xmath158 and @xmath254 sufficiently large , if the critical point is an odd eigenmode . ] i.e. @xmath255 , we have @xmath256 . as a result , the critical points of macroscopic energy at fixed helicity and angular momentum can not be energy minima since we can always find particular perturbations that decrease the energy while conserving the constraints . in conclusion , the critical points of macroscopic energy at fixed helicity and angular momentum are saddle points since we can find perturbations making @xmath257 positive and perturbations making @xmath257 negative . this analysis shows that all beltrami solutions are unstable . however , saddle points may be characterized by very long lifetimes as long as the system does not explore dangerous perturbations that destabilize them . this motivates the numerical stability analysis of sec . [ stability ] . _ remark : _ let us consider the odd eigenmode @xmath135 . we have seen that it can be destabilized by a perturbation @xmath258 or by a perturbation @xmath251 at smaller scale . let us now consider the effect of a perturbation of the form @xmath259 and @xmath239 , where @xmath240 is the first continuum mode such that @xmath241 . the corresponding stream function is @xmath260 . for this perturbation , we have @xmath242 , @xmath261 and @xmath262 since @xmath263 is orthogonal to @xmath240 . therefore , this perturbation does not change the helicity and the angular momentum at first order . for this perturbation , we have in addition @xmath264 this quantity is negative when @xmath265 corresponding to case l. this implies that the eigenmode @xmath135 is also destabilized by the perturbation @xmath260 which is at larger scale than the perturbations @xmath258 . the stability analysis performed in sec . [ analytics ] has shown that all the critical points of entropy at fixed microscopic energy , helicity and angular momentum are saddle points . we shall now investigate their robustness by using the relaxation equations derived in paper i ( for a review of relaxation equations in the context of 2d hydrodynamics , see @xcite ) . these relaxation equations can serve as numerical algorithms to compute maximum entropy states or minimum energy states with relevant constraints . their study is interesting in its own right since these equations constitute non trivial dynamical systems leading to rich bifurcations . although these relaxation equations do not provide a parametrization of turbulence ( we have no rigorous argument for that ) , they may however give an idea of the true dynamical evolution of the flow . in that respect , it would be interesting to compare these relaxation equations with navier - stokes simulations . this will , however , not be attempted in the present paper . by construction , the relaxation equations monotonically increase entropy , or decrease energy , with relevant constraints . different generic evolutions are possible : ( i ) they can relax towards a fully stable state ( global maximum of entropy or global minimum of energy ) ; ( ii ) they can relax towards a metastable state ( local maximum of entropy or local minimum of energy ) ; ( iii ) they do not relax towards a steady state and develop structures at smaller and smaller scales . in the present situation , we have seen that there are no stable and metastable states . therefore , the stability analysis of sec . [ analytics ] predicts that the system should cascade towards smaller and smaller scales without limit ( except the one fixed by the finite resolution of the simulations ) . this is a possible regime ( see top of fig . [ fig : ex_relaxinst ] ) but this is not what is generically observed in the experiments where long - lived structures at large scales are found ( like at the bottom of fig . [ fig : ex_relaxinst ] ) . here , we explore the possibility that these long - lived structures are saddle points of entropy or energy with relevant constraints . these saddle points are steady states of the relaxation equations . although they are unstable ( strictly speaking ) , we argue that these saddle points can be long - lived and relatively robust ( this idea was previously developed for 2d flows in @xcite ) . indeed , they are unstable only for certain ( dangerous ) perturbations , but not for all perturbations . therefore , they can be stable as long as the system does not explore dangerous perturbations that destabilize them . of course , the rigorous characterization of this form of stability is extremely complex . in order to test this idea in a simple manner , we shall use the relaxation equations and study the robustness of the saddle points with respect to them . our stability analysis is based on the numerical integration of the relaxation equations @xmath266 where @xmath267 and @xmath268 are given functions of @xmath128 and @xmath129 , and @xmath58 , @xmath67 and @xmath68 evolve in time ( see below ) so as to guarantee the conservation of the invariants . in the canonical ensemble , the temperature @xmath58 is fixed and the conserved quantities are the helicity and the angular momentum . the lagrange multipliers @xmath269 and @xmath270 are computed at each time so as to guarantee the conservation of @xmath1 and @xmath2 . one may check that they are solutions of the system of algebraic equations ( see paper i ) @xmath271 these relaxation equations are associated with the maximization problem ( [ bes2 ] ) provided that , at any given time , the distribution of angular momentum is given by eq . ( [ vp1 ] ) with constant @xmath58 ( see paper i for details ) . by properly redefining the lagrange multipliers , they are also associated with the minimization problem ( [ res13b ] ) . in the microcanonical ensemble , the conserved quantities are @xmath7 , @xmath1 and @xmath2 . in the sequel , it will be convenient to fix the time dependence of @xmath58 by imposing @xmath272 at each time . taking into account the two other invariants , one may check that @xmath273 , @xmath270 and @xmath274 are solution of the system of algebraic equations @xmath275 these relaxation equations are associated with the maximization problem ( [ bes1 ] ) provided that , at any given time , the distribution of angular momentum is given by eq . ( [ vp1 ] ) with @xmath276 ( see paper i for details ) . in the sequel we focus on the special case @xmath277 and @xmath278 , where @xmath279 and @xmath280 are constants , that allows a simple numerical treatement of the relaxation equations by projection along the beltrami eigenmodes : @xmath281 where @xmath282 , @xmath283 , @xmath284 label the modes and @xmath285 is the number of modes . in that case , eqs . ( [ rel1 ] ) and ( [ eq : canorelax ] ) can be transformed into a set of @xmath286 odes : @xmath287 , \label{ode1 } \\ & & \dot{x_{\bf n } } = -\chi _ * \left [ \beta p_{\bf n } + \mu s_{\bf n}\right],\label{ode2}\\ & & p_{\bf n } = b_{\bf n}^{-2 } x_{\bf n } , \label{eq : relax_simp_proj}\end{aligned}\ ] ] where @xmath288 is such that @xmath289 . note that the constraints couple eqs . ( [ ode1])-([eq : relax_simp_proj ] ) through the parameters @xmath58 , @xmath68 and @xmath67 . to investigate the robustness of a given stationary solution , we first perturb it with a suitable perturbation ( see below ) , and then follow its dynamics thanks to the relaxation equations . two typical time evolutions are provided in fig [ fig : ex_relaxinst ] : if the solution is fragile with respect to the perturbation , it will cascade to another solution ( usually the solution of smallest scale permitted by our resolution ) ; if the solution is robust with respect to this perturbation , it will eventually return to its initial unperturbed state . to quantify the robustness of a given solution , we define a probabilistic stability criterion by computing the probability for the solution `` to escape '' from its basin of attraction . to that purpose , we select a threshold @xmath290 and compute at each time the probability of escape @xmath291,\ ] ] using @xmath292 realizations with perturbations drawn at random at @xmath293 from a suitable ensemble ( see below ) . this allows us to define `` statistically fragile '' solutions as those for which @xmath294 when @xmath295 , the others being referred to as `` statistically robust '' . in practice , the limit @xmath295 is not accessible . we thus generalize this notion to a `` finite time '' , by considering the asymptotic value of @xmath296 reached at the largest time of the simulation , @xmath297 . in addition , the asymptotic value of @xmath298 provides a mean to quantify the degree of robustness of a solution . examples are given in fig . [ fig : criteres ] , for a fragile and for a robust solution . as can be seen , the fragile solution is fragile whatever the threshold @xmath290 . however , the degree of robustness of a solution depends on the threshold @xmath290 . quite naturally , the larger the threshold , the more robust the solution . made with 200 perturbations around two beltrami states . top : fragile solution ; bottom : robust solution . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] made with 200 perturbations around two beltrami states . top : fragile solution ; bottom : robust solution . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] _ remark _ : although the variational problems ( [ bes1 ] ) , ( [ bes2 ] ) and ( [ res13b ] ) are equivalent , and all lead to the absence of stable equilibrium state , the corresponding relaxation equations described previously are different . therefore , the robustness of the saddle points will be different in the canonical and microcanonical settings . this can be viewed as a form of `` ensembles inequivalence '' for an out - of - equilibrium situation . the stability must be investigated using perturbations that rigorously conserve the integral constraints . this puts some conditions regarding the shape of the possible perturbations that we can use . in the canonical ensemble , the integral constraints are @xmath1 and @xmath2 . given an initial stationary solution @xmath301 , the perturbations @xmath302 must obey @xmath303 one can check that this set of constraints is satisfied by any perturbation of the form @xmath304\\ & & \delta \xi = { \epsilon } r^{-1 } { s_{\bf i_2}^\star}^{-1}\left[\displaystyle\frac{\langle r\phi_{\bf i_1}\rangle } { \langle r\phi_{\bf i_0}\rangle } x^\star_{\bf i_0}-x^\star_{\bf i_1}\right]\phi_{\bf i_2}\\ & & \delta \psi = { \epsilon } r { s_{\bf i_2}^\star}^{-1 } b_{\bf i_2}^{-2}\left[\displaystyle\frac{\langle r\phi_{\bf i_1}\rangle } { \langle r\phi_{\bf i_0}\rangle } x^\star_{\bf i_0}-x^\star_{\bf i_1}\right]\phi_{\bf i_2 } \label{eq : perturb}\end{aligned}\ ] ] where @xmath305 is the amplitude of the perturbation , @xmath306 labels an even mode while @xmath307 and @xmath308 label two different modes different from @xmath306 such that @xmath309 . following eqs . ( [ dec1 ] ) and ( [ dec2 ] ) , we have set @xmath310 and @xmath311 . in the sequel , we fix the amplitude of the perturbation @xmath305 through the norm @xmath312 by imposing @xmath313^{-\frac{1}{2}}.\ ] ] the modes @xmath306 , @xmath307 and @xmath308 are chosen randomly according to the following procedure : i ) we draw @xmath306 following a uniform law among the @xmath314 even modes ; ii ) we draw @xmath308 following a uniform law among the @xmath314 or @xmath315 modes of the set of allowed @xmath316 , excluding @xmath306 . this mode is therefore necessarily even for solution of continuum , and often even for mixed solutions ; iii ) we draw @xmath307 following a uniform law among the @xmath317 even and odd modes , excluding @xmath306 and @xmath308 . this choice allows the generation of @xmath292 random perturbations with the same amplitude @xmath318 . in the microcanonical ensemble , the relaxation equations conserve in addition the energy . to satisfy this additional constraint , we choose the perturbations according to the same procedure as in the canonical case , and then determine the initial value of the temperature @xmath319 in order to guarantee the conservation of the energy . and angular momentum @xmath2 determine the mean flow while the energy @xmath7 determines the temperature . ] as explained previously , this amounts to taking @xmath320 in the following , we shall group the perturbations into subclasses such that perturbations of the same class have the same temperature @xmath321 or , equivalently , the same macroscopic energy @xmath322 . note that the initial temperature of the perturbation differs from the temperature of the equilibrium state which is given by @xmath323 in the sequel , we focus on the stability analysis in the case l , for the first three branches of solutions , relevant for comparison with experiments , see paper iii @xcite . our parameters are as follows : * the number of modes is @xmath324 with @xmath325 ( radial modes ) and @xmath326 ( vertical modes ) corresponding to 120 even modes and 120 odd modes . the radial and vertical lengths are @xmath327 and @xmath328 . * the amplitude of the perturbations is @xmath329 . we consider @xmath330 realizations for each given stationary solution . * the parameters @xmath279 and @xmath280 are both taken equal to @xmath331 . the relaxation equations are integrated using an implicit heun scheme . the time step is empirically chosen proportional to @xmath332 . for @xmath333 , the time step is 0.02 . we have checked that this time step is small enough to guarantee the numerical conservation of @xmath2 and @xmath1 ( canonical case ) or @xmath7 , @xmath2 and @xmath1 ( microcanonical case ) . for any value of @xmath159 on a given branch of solutions , we proceed as follows : [ [ canonical - ensemble ] ] canonical ensemble * we fix the value of the temperature @xmath58 ( it remains constant during the evolution ) . in the sequel , we focus on five arbitrary values , @xmath334 , chosen so as to span a wide range . * we compute the beltrami solution @xmath335 corresponding to a prescribed value of @xmath159 on the given branch . * we generate @xmath292 perturbed initial conditions leaving unchanged the helicity and the angular momentum of @xmath335 . * we evolve the perturbed initial conditions through eqs . ( [ ode1])-([eq : relax_simp_proj ] ) and eq . ( [ eq : coefrelaxcano1],[eq : coefrelaxcano2 ] ) for a certain amount of time @xmath297 . [ [ microcanonical - ensemble ] ] microcanonical ensemble * we fix the value of the energy @xmath7 ( it remains constant during the evolution ) . in the sequel , it is fixed after arbitrary choice of five values of the temperature , @xmath336 , chosen so as to span a wide range . once @xmath321 has been fixed , the total energy is then fixed . it can vary from one realization to the other , but does not vary along the evolution . * we compute the stationary beltrami solution @xmath335 corresponding to a prescribed value of @xmath159 on the given branch . * we generate @xmath292 perturbed initial conditions leaving unchanged the helicity and the angular momentum of @xmath335 . * we group together the perturbations that have the same initial temperature @xmath337 measuring the initial energy of the fluctuations ( equivalently , these perturbations have the same value of macroscopic energy @xmath5 ) . * we evolve the perturbed initial condition through eqs . ( [ ode1])-([eq : relax_simp_proj ] ) and ( [ eq : coefrelaxmicro1]-[eq : coefrelaxmicro3 ] ) for a certain amount of time @xmath297 . on the three branches , we computed the value @xmath338 ( computed at @xmath297 ) as a function of @xmath159 for different temperatures @xmath339 . the results are displayed on fig . [ fig : inequivalence_cano ] . the mixed branch ( vertical dipoles ) and branch 2 ( reversed monopoles ) are found very robust for high threshold , and still retain a certain degree of robustness for a small threshold , with a 40 per cent probability of escape . there is no clear dependence on the temperature . this is natural , since temperature can be eliminated by a suitable rescaling of time ( or of coefficients @xmath267 and @xmath268 ) and redefinition of lagrange parameters . the behavior on branch 1 ( direct monopoles ) is more contrasted and provides a very clear transition around the critical value @xmath340 . for @xmath341 , the probability to escape is close to 1 , meaning large fragility of the branch . for @xmath342 , the branch becomes much more robust , reaching a larger degree of robustness than the two other branches for high threshold , while reaching the same robustness for small threshold . the different behaviors are summarized on fig . [ fig : stabcano ] . as function of @xmath159 on the three branches ( left : branch 1 , middle : branch 2 , right : mixed branch ) at different temperatures @xmath343 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig:"][fig : micro_branche1 ] as function of @xmath159 on the three branches ( left : branch 1 , middle : branch 2 , right : mixed branch ) at different temperatures @xmath343 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig:"][fig : micro_branche2 ] as function of @xmath159 on the three branches ( left : branch 1 , middle : branch 2 , right : mixed branch ) at different temperatures @xmath343 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig:"][fig : micro_mixte ] . the lines are increasingly fat with increasing @xmath338 , i.e. robustness . note that the value of @xmath349 increases with increasing @xmath297 . ] note that the value @xmath350 is somewhat arbitrary . indeed , increasing @xmath297 further , we observed the same qualitative scenario , with an increased value of @xmath349 . we also observed that over sufficiently long time , the branch 2 tends to become unstable , past a value of the order @xmath351 . on the three branches , we computed the value @xmath338 ( computed at @xmath297 ) as a function of @xmath159 for classes of perturbations with different initial temperature @xmath352 . note that the initial temperature fixes the amplitude of the velocity fluctuations . the results are displayed on fig . [ fig : inequivalence_micro ] . for the mixed branch ( vertical dipoles ) and branch 2 ( reversed monopoles ) , the microcanonical results do not noticeably differ from the canonical results : the two branches are found very robust for large threshold , and still retain a certain degree of robustness for a small threshold , with a 40 per cent probability of escape . there is no clear dependence on the initial temperature . there is therefore no ensembles inequivalence for these two branches . this is not true anymore for branch 1 ( direct monopoles ) . indeed , one still observes a transition from robustness to fragility around a critical value @xmath349 but this quantity depends on the initial temperature @xmath353 : it takes a value @xmath340 at large initial temperatures ( large velocity fluctuations ) and then decreases to @xmath354 for small initial temperatures ( small velocity fluctuations ) . the difference of robustness observed between the two ensembles may be seen as a kind of inequivalence of ensembles at small initial temperatures . the different behaviors are summarized on fig . [ fig : stabmicro ] . like in the canonical case , we checked that an increase of @xmath297 results in a larger fragility of the branch 1 and 2 towards small @xmath159 , at a given temperature . _ remark : _ note that perturbations with small initial temperature have large macroscopic energies corresponding to perturbations at large scales . according to sec . [ analytics ] such perturbations are less destabilizing than perturbations at small scales ( associated with small macroscopic energies hence large temperatures ) . this may explain the numerical results . as function of @xmath159 on the three branches ( left : branch 1 ; middle : branch 2 ; right : mixed branch ) at different initial temperatures @xmath355 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] as function of @xmath159 on the three branches ( left : branch 1 ; middle : branch 2 ; right : mixed branch ) at different initial temperatures @xmath355 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] as function of @xmath159 on the three branches ( left : branch 1 ; middle : branch 2 ; right : mixed branch ) at different initial temperatures @xmath355 : @xmath344 , @xmath345 , @xmath346 , @xmath347 , and @xmath348 . two different thresholds are used : @xmath299 ( continuous line ) @xmath300 ( dotted line).,title="fig : " ] . left : for a low initial temperature ; right : for a high initial temperature . the lines are increasingly fat with increasing @xmath338 , i.e. robustness . note that the value of @xmath349 increases with increasing @xmath297.,title="fig:"][fig : stab0 ] . left : for a low initial temperature ; right : for a high initial temperature . the lines are increasingly fat with increasing @xmath338 , i.e. robustness . note that the value of @xmath349 increases with increasing @xmath297.,title="fig:"][fig : stab1 ] we have studied the thermodynamics of axisymmetric euler - beltrami flows and proved the coexistence of several equilibrium states for the same values of the control parameters . all these states are saddle points of entropy but they can have very long lifetime as long as the system does not spontaneously develop dangerous perturbations . we have numerically explored the robustness of some of these states by using relaxation equations in the canonical and microcanonical ensembles . the dipoles ( mixed branch ) and the reversed monopoles ( branch 2 ) were found to be rather robust in both ensembles . furthermore , in the microcanonical ensemble there is no dependence on the initial temperature on these branches . by contrast , the direct monopoles ( branch 1 ) display a sharp transition around a critical value @xmath349 . the value of @xmath349 increases with increasing integration time . in the microcanonical ensemble , this value also decreases with decreasing initial temperature , resulting in a difference of robustness in the canonical and microcanonical ensembles . this difference may be seen as a kind of `` ensembles inequivalence '' . this is , however , a very unconventional terminology since it concerns here the robustness of _ saddle points _ with respect to random perturbations that keep the energy or the temperature fixed , over a finite amount of time . the simulations have shown that the dipole ( two - cells solution ) is relatively robust for any value of the angular momentum . on the other hand , the direct monopole ( one - cell solution ) is very fragile at low angular momentum but becomes robust at high angular momentum . in that case , it is even more robust than the dipole . therefore , increasing the total angular momentum of the flow , one expects to observe a transition from the two - cells solution ( antisymmetric with respect to the middle plane ) to the one - cell solution ( symmetric with respect to the middle plane ) . this bifurcation scenario is sketched in fig . [ fig : scenarhyst ] . it is reminiscent of the turbulent transition reported in the von krmn flow @xcite in which the initial two - cells flow observed at zero global rotation suddenly bifurcates when the rotation is large enough . once the bifurcation has taken place , the level of fluctuation is experimentally observed to decrease strongly , resulting in a decrease of the statistical temperature . in our scenario , this means that the monopole branch is suddenly stabilized with respect to redecrease of the total angular momentum of the flow , resulting in a hysteresis that has also been observed experimentally . it would therefore be interesting to investigate more closely the relevance of our scenario to the experimental system . this is done in the next paper @xcite . , on the mixed branch , in a two - cells topology ( vertical dipole ) ; middle : increasing @xmath159 up to @xmath356 ( red arrow ) , the system follows the mixed branch . one cell grows at the expense of the other , resulting in a shift of the mixing layer upwards ; right : at @xmath349 , the system bifurcates towards branch 1 ( more stable ) , resulting in a one - cell topology ( direct monopole ) . in this new state , the fluctuations are much milder ( empirical fact from experiments ) , thereby allowing a stabilization of the branch towards lower @xmath159 . therefore , the monopole subsists beyond this point , even after a redecreasing of @xmath159 , depicted by the black arrow.,title="fig : " ] , on the mixed branch , in a two - cells topology ( vertical dipole ) ; middle : increasing @xmath159 up to @xmath356 ( red arrow ) , the system follows the mixed branch . one cell grows at the expense of the other , resulting in a shift of the mixing layer upwards ; right : at @xmath349 , the system bifurcates towards branch 1 ( more stable ) , resulting in a one - cell topology ( direct monopole ) . in this new state , the fluctuations are much milder ( empirical fact from experiments ) , thereby allowing a stabilization of the branch towards lower @xmath159 . therefore , the monopole subsists beyond this point , even after a redecreasing of @xmath159 , depicted by the black arrow.,title="fig : " ] , on the mixed branch , in a two - cells topology ( vertical dipole ) ; middle : increasing @xmath159 up to @xmath356 ( red arrow ) , the system follows the mixed branch . one cell grows at the expense of the other , resulting in a shift of the mixing layer upwards ; right : at @xmath349 , the system bifurcates towards branch 1 ( more stable ) , resulting in a one - cell topology ( direct monopole ) . in this new state , the fluctuations are much milder ( empirical fact from experiments ) , thereby allowing a stabilization of the branch towards lower @xmath159 . therefore , the monopole subsists beyond this point , even after a redecreasing of @xmath159 , depicted by the black arrow.,title="fig : " ] an interesting outcome of our study lies in the fate of the solutions when they are destabilized by a dangerous perturbation : due to the energy minimization principle , the unstable solution tends to `` cascade '' towards a higher wavenumber solution in a way reminiscent to the richardson energy cascade of 3d turbulence ( see fig . [ fig : ex_relaxinst ] ) . the cascade stops when the largest available wavenumber is reached , since dangerous perturbations are necessarily at smaller scale than the achieved state . this form of energy condensation at the smallest scale may be seen as an interesting counterpart ( in the opposite sense ) of the large scale energy condensation observed in 2d turbulence via the inverse energy cascade process . this is the signature of the 2d and a half nature of our system , intermediate between 2d and 3d turbulence . we have characterized the thermodynamical equilibrium states of axisymmetric euler - beltrami flows and proved the coexistence of several equilibrium states for a given value of the control parameter like in 2d turbulence @xcite . we further showed that all states are saddle points of entropy and can , in principle , be destabilized by a perturbation with a larger wavenumber , resulting in a structure at the smallest available scale . this mechanism is therefore reminiscent of the 3d richardson energy cascade towards smaller and smaller scales . therefore , our system is truly intermediate between 2d turbulence ( coherent structures ) and 3d turbulence ( energy cascade ) . through a numerical exploration of the robustness of the equilibrium states with respect to random perturbations using a relaxation algorithm in both canonical and microcanonical ensembles , we showed however that these saddle points of entropy can be very robust and therefore play a role in the dynamics . we evidenced differences in the robustness of the solutions in the canonical and microcanonical ensembles leading to a theoretical scenario of bifurcation between two different equilibria ( with one or two cells ) that resembles a recent observation of a turbulent bifurcation in a von krmn experiment @xcite . this work was supported by european contract wallturb . we show that the first zero of @xmath357 , denoted @xmath358 , is always between the first @xmath359 and the second @xmath360 even eigenmode . to that purpose , we note that if @xmath361 then @xmath362 for any @xmath127 so that @xmath363 . there is no discontinuity of @xmath171 in the interval @xmath364 so that there is no zero in that interval . consider now the interval @xmath365b''_1,b''_2[$ ] . in that interval , @xmath171 is also continuous and increasing since @xmath366 moreover , for @xmath367 , @xmath368 . similarly , @xmath369 when @xmath370 . therefore , there exists a unique value of @xmath358 in the range @xmath365b''_1,b''_2[$ ] , such that @xmath371 . this shows that the first zero of @xmath171 lies in between the first two even eigenmodes . this property remains true for the successive values of @xmath170 and the successive even eigenmodes .

we characterize the thermodynamical equilibrium states of axisymmetric euler - beltrami flows . they have the form of coherent structures presenting one or several cells . we find the relevant control parameters and derive the corresponding equations of state . we prove the coexistence of several equilibrium states for a given value of the control parameter like in 2d turbulence [ chavanis & sommeria , j. fluid mech . * 314 * , 267 ( 1996 ) ] . we explore the stability of these equilibrium states and show that all states are saddle points of entropy and can , in principle , be destabilized by a perturbation with a larger wavenumber , resulting in a structure at the smallest available scale . this mechanism is therefore reminiscent of the 3d richardson energy cascade towards smaller and smaller scales . therefore , our system is truly intermediate between 2d turbulence ( coherent structures ) and 3d turbulence ( energy cascade ) . we further explore numerically the robustness of the equilibrium states with respect to random perturbations using a relaxation algorithm in both canonical and microcanonical ensembles . we show that saddle points of entropy can be very robust and therefore play a role in the dynamics . we evidence differences in the robustness of the solutions in the canonical and microcanonical ensembles . a scenario of bifurcation between two different equilibria ( with one or two cells ) is proposed and discussed in connection with a recent observation of a turbulent bifurcation in a von krmn experiment [ ravelet _ et al . _ , phys . rev . lett . * 93 * , 164501 ( 2004 ) ] .

the formation and evolution of galaxies is one of the fundamental problems in astrophysics . the recent deep imaging of very faint galaxies made with _ hubble space telescope _ ( williams et al . 1996 ) and the detection of co emission from a high-@xmath7 quasar br 1202@xmath40725 at @xmath7 = 4.69 ( ohta et al . 1996 ; omont et al . 1996 ) have encouraged us to study the problem mentioned above . since the galaxies should form from gaseous system , it is important to investigate the major epoch of star formation in the gas system and to study how stars have been made during the course of galaxy evolution . when we study evolution of galaxies , we usually use stellar lights as the tracer of evolution ( cf . tinsley 1980 ; arimoto & yoshii 1986 , 1987 ; bruzual & charlot 1993 ) . however , much data of interstellar medium ( ism ) of galaxies from x - ray emitting hot gas through warm hi gas to cold molecular gas and dust have been accumulated for these decades ( cf . wiklind & henkel 1989 ; lees et al . 1991 ; fabbiano , kim , & trinchier 1992 ; kim , fabbiano , & trinchier 1992 ; wang , kenney , & ishizuki 1992 ) . therefore , the time is ripe to begin the study of evolution of ism of galaxies from the epoch of galaxy formation to the present day for both elliptical and disk galaxies . in this _ paper _ , appreciating the recent detection of co emission from the high-@xmath7 quasar br 1202@xmath40725 ( ohta et al . 1996 ; omont et al . 1996 ) , we discuss the evolution of molecular gas content in galaxies . since active galactic nuclei ( agn ) are associated with their host galaxies , the co luminosity of agn depends on the gaseous content of their host galaxies . therefore , any observations of molecular - line emission from high-@xmath7 objects are very useful in studying evolution of molecular gas content in galaxies.0725 , there are two more successful detections of co from high-@xmath7 objects ; 1 ) the hyperluminous infrared galaxy iras f10214 + 4725 at @xmath7 = 2.286 ( brown & vanden bout 1991 ; solomon , downes , & radford 1992 ; tsuboi & nakai 1992 ; radford et al . 1996 ) , and 2 ) the cloverleaf quasar h1413 + 135 at @xmath7 2.556 ( barvanis et al . 1994 ) . since , however , these two sources are gravitationally amplified ones ( elston , thompson , & hill 1994 ; soifer et al . 1995 ; trentham 1995 ; graham & liu 1995 ; broadhurst & lehr 1995 ; serjeant et al . 1995 ; close et al . 1995 ) , we do not use these data in this study because there may be uncertainty in the amplification factor . ] in spite of the successful co detection from br 1202@xmath40725 , evans et al . ( 1996 ) reported the negative co detection from 11 high-@xmath7 ( @xmath6 ) powerful radio galaxies ( prgs ) and gave the upper limits of order @xmath8 ( hereafter @xmath9 = 50 km s@xmath10 mpc@xmath10 and @xmath11 ) , being comparable to or larger than that of br 1202@xmath40725 ( ohta et al . 1996 ; omont et al . 1996 ) . here a question arises as why co emission was detected from the high-@xmath7 quasar at @xmath12 while not detected from the high-@xmath7 ( @xmath6 ) prgs . there may be two alternative answers : 1 ) the host galaxies are different between quasars and prgs in terms of molecular gas contents . or , 2 ) although the host galaxies are basically similar between quasars and prgs , their evolutionary stages are different and thus the molecular gas contents are systematically different between the two classes . provided that the current unified model for quasars and radio galaxies ( barthel 1989 ) is also applicable to high-@xmath7 populations , it is unlikely that their host galaxies are significantly different . since it is usually considered that luminous agns like quasars and prgs are associated with either massive ellipticals or bulges of disk galaxies as well as merger nuclei , the evolution of ism would be rapidly proceeded during the era of spheroidal component formation therefore , we investigate the latter possibility ( i.e. , evolutionary effect ) and discuss some implications on the evolution of ism in galaxies . assuming that the luminous agns are harbored in giant elliptical galaxies and/or in bulges of spiral galaxies , we investigate the evolution of co luminosity based on a galactic wind model for elliptical galaxies proposed by arimoto & yoshii ( 1987 ) and a bulge - disk model for spiral galaxies by arimoto & jablonka ( 1991 ) . the so - called _ infall _ model of galaxy chemical evolution is adopted for both spheroidals and disks and time variations of gas mass and gas metallicity , in particular @xmath13 , are calculated numerically by integrating usual differential equations for chemical evolution without introducing the instantaneous recycling approximation for stellar lifetime . model parameters , such as star formation rate ( sfr ) @xmath14 , a slope of initial mass function ( imf ) @xmath15 , and gas accretion rate ( acr ) @xmath16 , are taken from arimoto & yoshii ( 1987 ) and arimoto & jablonka ( 1991 ) . the lower and upper stellar mass limits are set to be @xmath17 m@xmath18 and @xmath19 m@xmath18 , respectively . according to jablonka , martin , & arimoto ( 1996 ) , who found that the @xmath20 relation of bulges are exactly identical to that of elliptical galaxies , we consider that bulges are small ellipticals of equivalent luminosity and that both spheroidal systems share the similar history of star formation . thus , for ellipticals and bulges , we assume that the remaining gas is expelled completely after the onset of galactic wind , which takes place once the thermal energy released from supernovae exceeds the binding energy of the gas . the wind times , @xmath21 gyr for giant ellipticals ( m@xmath22 m@xmath18 ) and @xmath23 gyr for bulges ( m@xmath24 m@xmath18 ) , are taken from arimoto & yoshii ( 1987 ) . for spiral galaxies , assuming that the bulge and disk evolve independently , we construct a model by combining the bulge and disk models with m@xmath24 m@xmath18 and @xmath25 m@xmath18 , respectively . this model gives @xmath26 mag for the bulge and @xmath27 mag for the disk at the age of 15 gyr old ( arimoto & jablonka 1991 ) . the bulge - to - disk light ratio in v - band is @xmath28 , nearly twice of typical values for early type spirals ( simien & de vaucouleurs 1986 ) . the @xmath29 refers to that of co(@xmath30=1 - 0 ) . note that the @xmath29 of high-@xmath7 galaxies are measured by using much higher transitions such as @xmath30=3 - 2 , 4 - 3 , and so on . however , it is known that that local co - rich galactic nuclei and starburst nuclei have @xmath29(@xmath30=3 - 2)@xmath31(@xmath30=1 - 0 ) @xmath32 ( devereux et al . 1994 ; israel & van der werf 1996 ) . therefore high-@xmath7 analogs may have the similar properties . in fact , two high-@xmath7 objects iras f10214 + 4724 and h1413 + 117 have @xmath29(@xmath30=4 - 3)@xmath31(@xmath30=3 - 2 ) @xmath32 and @xmath29(@xmath30=6 - 5)@xmath31(@xmath30=3 - 2 ) @xmath33 ( see table 1 of israel & van der werf ) . thus , the uncertainty due to use of higher transition data may be 50 percent at most , when we compare model @xmath29(@xmath30=1 - 0 ) and the observed @xmath29 at higher transitions . ] of a model galaxy can be calculated from molecular hydrogen mass by using the empirical co to h@xmath34 conversion factor ( @xmath35 ) . arimoto , sofue & tsujimoto ( 1995 ) showed that @xmath35 strongly depends on the gas metallicity and derived the following relationship valid for nearby spirals and irregular galaxies : @xmath36 where @xmath37 h@xmath34/ k km s@xmath10 = @xmath38 and o / h is the oxygen abundance of hii regions . we introduce a fractional mass of hydrogen molecule to that of atomic hydrogen , @xmath39 , and write the co luminosity in k km s@xmath10 pc@xmath40 as follows : @xmath41 where @xmath42 and @xmath43 are in @xmath44 . chemical evolution model gives @xmath45 and o / h as a function of time and the co luminosity evolution can be traced with a help of eq.(2 ) provided that @xmath46 is known _ a priori_. we assume time invariant @xmath46 throughout the course of galaxy evolution . in principle , @xmath46 itself should evolve as well , since the hydrogen molecule is newly produced on the surface of dust ejected from evolving stars and/or formed in expanding shells of supernovae remnants while at the same time a part of molecules are dissociated by uv photons emitted from young hot stars . the mass of dust and the number of uv photons should also evolve as a result of galactic chemical evolution ( honma , sofue , & arimoto 1995 ) . detailed evolution of @xmath46 will be shown in our subsequent paper ( ikuta et al . 1997 ) , instead in this _ paper _ we assume @xmath47 . recent studies of nearby ellipticals suggest @xmath48 ( wiklind & rydbeck 1986 ; sage & wrobel 1989 ; lees et al . 1991 ; eckart , cameron , & genzel 1991 ) . the contribution of helium to the gas mass is entirely ignored for simplicity , but our conclusions change little even if the evolution of helium gas is precisely taken into account . the formation epoch of galaxies is assumed to be @xmath49 . although the choice of @xmath50 is rather arbitrary , @xmath51 has some supports from recent studies on the metallicity of broad emission - line regions of high-@xmath7 quasars ( hamann & ferland 1992 , 1993 ; kawara et al . 1996 ; taniguchi et al . 1997 ) . figure 1 shows the result for elliptical galaxies . the thick solid line represents the galactic wind model , and the dotted line a model with continuous star formation ( the wind is suppressed even after the wind criterion is satisfied ) . the dashed line shows a case for a wind model , but the gas ejected from evolving giants after the wind is bound and accumulated in the galaxy to form neutral gas ( bound - wind model ; arimoto 1989 ) . the co luminosity , @xmath52 , of elliptical galaxies increases prominently soon after their birth , and attains the maximum at an epoch of about 0.85 gyr since the birth , or at @xmath53 . then , it suddenly decreases when the galactic wind has expelled the ism from the galaxy . the extremely luminous phase in co observed for the high @xmath7 quasar br 1202@xmath40725 ( ohta et al . 1996 ; omont et al . 1996 ) can be well explained , if it is in the star forming phase of the whole elliptical system . moreover , the non - detection of the smaller redshift galaxies as observed by evans et al . ( 1996 ) and van ojik et al . ( 1997 ) is also naturally understood by the present model : it is because of the fact that elliptical galaxies at @xmath54 contains little ism . in the figure , we also superpose co observational data for lower redshift elliptical galaxies ( wiklind & rydbeck 1986 ; sage & wrobel 1989 ; wiklind & henkel 1989 ; eckart et al . 1991 ; lees et al . 1991 ; sage & galletta 1993 ; sofue & wakamatsu 1993 ; wiklind , combes , & henkel 1995 ) . the theoretical curve for the bound - wind model is clearly inconsistent with the observations for galaxies at @xmath55 . this suggests that the gas has been expelled continuously after the galactic wind ( @xmath7 @xmath56 4 ) and has not been bound to the system . this , in turn , is consistent with the idea that the intracluster hot gas with high metallicity , as observed in x - rays , may have been supplied by the winds from early type galaxies ( ishimaru & arimoto 1997 ) . although it is not clarified how the gas has been expelled out of the galaxies , without being bound to the system , recent studies suggest that it is probably due to the energy supply from either the type ia supernovae ( renzini et al . 1994 ) or the intermittent agn activities ( ciotti & ostriker 1997 ) . figure 2 shows the result for a spiral galaxy , where the initial masses of bulge and disk are taken to be @xmath57 and @xmath58 , respectively . the co luminosity of the bulge evolves in almost the same fashion as an elliptical galaxy as above : @xmath52 increases rapidly after the birth , attains the maximum within 0.36 gyr , and , then , suddenly decreases because of the strong wind from the star - forming bulge . the co luminosity of the thus - calculated forming bulge seems insufficient to be detected as the observed luminosity of br 1202@xmath40725 , unless the bulge is much heavier than @xmath57 . moreover , we emphasize that the duration of this bright phase in @xmath52 is shorter than that obtained for ellipticals by a factor of two , and therefore , the probability to detect such co - bright phase for a bulge would be much smaller than that for elliptical galaxies . on the other hand , formation of the gaseous disk due to gas infall and star formation then proceeds mildly , and , therefore , the metal pollution of ism in the disk is slower , which results in a slower increase of the co luminosity . as a consequence , the co luminosity increases gradually and monotonically until today . also , the less - luminous phase due to the disk , following the wind phase of the bulge , is in agreement with the upper - limit observations of evans et al . ( 1996 ) and van ojik et al . ( 1997 ) . we also plot co observations for more other nearby spiral galaxies , as plotted by filled circles ( braine et al . the evolution of the co luminosity of these galaxies can be traced back by adjusting the present - day luminosity of the calculated track . the most luminous nearby spirals in co is ngc 4565 ( @xmath59 k km s@xmath10 pc@xmath40 ) . it is interesting to mention that , if the model is normalized to this galaxy , the peak co luminosity corresponding to the forming bulge phase can be still sufficient to explain the luminosity of br 1202@xmath40725 . the present study has shown that the current radio telescope facilities are capable of detecting co emission from high - redshift galaxies which experience their initial starbursts if the following two conditions are satisfied ; 1 ) the masses of systems should exceed @xmath60 , and 2 ) their evolutionary phases should be prior to the galactic wind . therefore , the detectability of co emission from high-@xmath7 galaxies is severely limited by the above two conditions . our study suggests that co emission can be hardly detected from galaxies with redshift @xmath3 without an amplification either by galaxy mergers and/or by gravitational lensing . this prescription is consistent with the observations ; co emission was detected from the high - redshift quasar br 1202@xmath40725 at @xmath5 ( ohta et al . 1996 ; omont et al . 1996 ) while not detected from the radio galaxies with @xmath6 ( evans et al . 1996 and van ojik et al . 1997 ) and quasars with redshift @xmath0 2 ( takahara et al . further , the two convincing detections of co emission from the high-@xmath7 objects at @xmath61 , iras f10214 + 4724 ( cf . radford et al . 1996 ) and the cloverleaf quasar h1413 + 135 ( barvanis et al . 1994 ) , are actually gravitationally amplified sources . the striking non - detection of high-@xmath7 galaxies in co at @xmath6 implies that most elliptical galaxies and bulges of spiral galaxies were formed before @xmath62 , or high-@xmath7 galaxies with @xmath6 observed in the optical and infrared studies may be galaxies after the epoch of galactic wind . this implication is consistent with the formation epoch ( @xmath63 ) of high-@xmath7 quasars studied by chemical properties of the broad emission - line regions ( hamann & ferland 1992 , 1993 ; hill , thompson , & elston 1993 ; elston et al . 1994 ; kawara et al . 1996 ; taniguchi et al . therefore it is strongly suggested that most host galaxies of high-@xmath7 agn were formed before @xmath62 . according to our model , it would be worth noting that quasar nuclei are hidden by the dusty clouds unless the galactic wind could expel them from the host galaxies . we also mention that any quasar nuclei are not necessarily to associate with gas - rich circumnuclear environment though this implication is in contradiction to what suggested for low-@xmath7 agn ( yamada 1994 ) . therefore , it seems very lucky that the co emission was detected from br 1202@xmath40725 at @xmath12 . finally , we revisit the important question : what is br 1202@xmath40725 ? as shown in section 3 , the unambiguous co detection from br 1202@xmath40725 is interpreted as an initial starburst galaxy which is forming either an elliptical or a bulge with mass larger than @xmath64 . the elongated ( ohta et al . 1996 ) or the double - peaked ( omont et al . 1996 ) co distribution may be understood as possible evidence for galactic wind in terms of our scenario . if it is an elliptical galaxy , its formation epoch is estimated to be @xmath65 . however , if it were a bulge former , the mass of bulge should be comparable with that of typical ellipticals . since such massive bulges are rarer by two orders of magnitude than elliptical with similar masses ( e.g. , woltier 1990 ) , the host of br 1202@xmath40725 may be an elliptical from a statistical ground . we gratefully acknowledge t. kodama and o. nakamura for kindly providing us chemical evolution program packages . our special thanks to k. ohta , t. yamada , and r. mcmahon for fruitful discussions . we also thank t. hasegawa and m. honma for useful comments . this work was financially supported in part by a grant - in - aid for the scientific research by the japanese ministry of education , culture , sports and science ( nos . 07044054 and 09640311 ) .

we present co luminosity evolution of both elliptical and spiral galaxies based on a galactic wind model and a bulge - disk model , respectively . we have found that the co luminosity peaks around the epoch of galactic wind caused by collective supernovae @xmath0 0.85 gyr after the birth of the elliptical with @xmath1 while @xmath0 0.36 gyr after the birth of the bulge with @xmath2 . after these epochs , the co luminosity decreases abruptly because the majority of molecular gas was expelled from the galaxy system as the wind . taking account of typical masses of elliptical galaxies and bulges of spiral galaxies , we suggest that co emission can be hardly detected from galaxies with redshift @xmath3 unless some amplification either by galaxy mergers and/or by gravitational lensing is working . therefore , our study explains reasonably why co emission was detected from the high - redshift quasar br 1202@xmath40725 at @xmath5 while not detected from the powerful radio galaxies with @xmath6 .

an important diagnostic of the physical state of the interstellar medium is its large - scale velocity dispersion . this parameter is however very difficult to derive , since it is in general dominated by the contribution of the systematic velocity gradients in the beam , which are not well - known . exactly face - on galaxies are ideal objects for this study , since the line - width can be attributed almost entirely to the z - velocity dispersion @xmath0 . indeed , the systematic gradients perpendicular to the plane are expected negligible ; for instance no systematic pattern associated to spiral arms have been observed in face - on galaxies ( e.g. shostak & van der kruit 1984 , dickey et al 1990 ) , implying that the z - streaming motions at the arm crossing are not predominant . in an inclined galaxy on the contrary , it is very difficult to obtain the true velocity dispersion , since the systematic motions in the plane @xmath1 ( rotation , arm streaming motions ) widen the spectra due to the finite spatial resolution of the observations ( e.g. garcia - burillo et al 1993 , vogel et al 1994 ) . nearly face - on galaxies have already been extensively studied in the atomic gas component , in order to derive the true hi velocity dispersion ( van der kruit & shostak 1982 , 1984 , shostak & van der kruit 1984 , dickey et al 1990 ) . the evolution of @xmath0 as a function of radius was derived : the velocity dispersion is remarkably constant all over the galaxy @xmath0 = 6 = @xmath2 , and only in the inner parts it increases up to 12 . the constancy of @xmath0 in the plane , and in particular in the outer parts of the galaxy disk , is not yet well understood ; it might be related to the large - scale gas stability and to the linear flaring of the plane , as is observed in the milky - way ( merrifield 1992 ) and m31 ( brinks & burton 1984 ) . in the isothermal sheet model of a thin plane , where the z - velocity dispersion @xmath3 is independent of z , the height @xmath4(r ) of the gaseous plane , if assumed self - gravitating , is @xmath5 where @xmath6 is the gas velocity dispersion , and @xmath7 the gas surface density . the density profile is then a sech@xmath8 law . but to have the gas self - gravitating , we have to assume that either there is no dark matter component , or the gas is the dark matter itself ( e.g. pfenniger et al 1994 ) . since in general the hi surface density decreases as 1/r in the outer parts of galaxies ( e.g. bosma 1981 ) , a linear flaring ( @xmath9 ) corresponds to a constant velocity dispersion with radius . on the contrary hypothesis of the gas plane embedded in an external potential of larger scale height , where the gravitational acceleration close to the plane can be approximated by @xmath10 , the z - density profile is then a gaussian : @xmath11 and the characteristic height , or gaussian scale height of the gas is : @xmath12 and @xmath13 is @xmath14 , where @xmath15 is the density in the plane of the total matter , stellar component plus dark matter component , in which the gas is embedded . if the dark component is assumed spherical , the density in the plane is dominated by the stellar component , which is distributed in an exponential disk . this hypothesis would predict an exponential flare in the gas , while the gas flares appear more linear than exponential ( e.g. merrifield 1992 , brinks & burton 1983 ) . the knowledge of their true shape is however hampered by the presence of warps . also , the flattening of the dark matter component , and its participation to the density @xmath16 in the plane , is unknown . as for the stability arguments , let us assume here the z - velocity dispersion comparable to the radial velocity dispersion , or at least their ratio constant with radius . the velocity dispersion of the gas component is self - regulated by dynamical instabilities . if the toomre q parameter for the gas @xmath17 is lower than 1 , instabilities set in , heat the medium and increase @xmath18 until @xmath19 is 1 . the critical velocity dispersion @xmath20 depends on the epicyclic frequency @xmath21 and on the gas surface density @xmath7 ; assuming again an hi surface density decreasing as 1/r in the outer parts and a flat rotation curve , where @xmath21 also varies as 1/r , then @xmath20 is constant . to maintain @xmath22 all over the outer parts , @xmath6 should also remain constant . however , the gas density gradient appears often steeper than @xmath23 and the @xmath19 parameter is increasing towards the outer parts . this has been noticed by kennicutt ( 1989 ) , who concluded that there exists some radius in every galaxy where the gas density reaches the threshold of global instability ( @xmath24 ) ; he identifies this radius to the onset of star formation in the disk . in fact , this threshold does not occur exactly at @xmath19 = 1 , but at a slightly higher value , around 1.4 , which could be due to the fact that the @xmath25 criterion is a single - fluid one , which does not take into account the coupling between gas and stars . the determination of the z - velocity dispersion in the molecular component has not yet been done . it could bring complementary insight to the hi results , since in general the center of galaxies is much better sampled through co emission ( a central hi depletion is frequent ) , and also the thickness of the h@xmath26 plane can be lower by a factor 3 or 4 than the hi layer ( case of mw , m31 , boulanger et al 1981 ) . in the case of m51 , an almost face - on galaxy ( i=20@xmath27 ) , the estimated @xmath0 determined from the co lines is surprisingly large ( up to @xmath0 = 25 in the southern arm ) once the rotation field , and even streaming - motions are taken into account , at the beam scale . an interpretation could be that the co lines are broadened by macroscopic opacity , i.e. cloud overlapping ( garcia - burillo et al 1993 ) , since such large line - widths are not observed in galaxies with less co emission . however , one could also suspect turbulent motions , generated at large - scale by gravitational instabilities or viscous shear . the level of star formation could be another factor : as for turbulence , it generally affects the molecular component more than the hi , except for very violent events like sne . but the finite inclination ( 20@xmath27 ) of m51 makes the discrimination between in - plane and z - dispersion very delicate . it is therefore necessary to investigate in more details this problem in exactly face - on galaxies , and determine whether there exist spatial variations of @xmath0 over the galaxy plane . in this paper we report molecular gas observations of two face - on galaxies ngc 628 ( m74 ) and ngc 3938 , in the co(1 - 0 ) , co(2 - 1 ) and @xmath28co lines , using the iram 30m telescope . after a brief description of the galaxy parameters in section 2 , and the observational parameters in section 3 , we derive the amplitude and the spatial variations of @xmath0 perpendicular to the plane in ngc 628 and ngc 3938 . section 5 summarises and discusses the physical interpretations . from these mass models , we have derived the epicyclic frequency as a function of radius ( this does not depend on the precise model used , as long as the rotation curve is fitted ) , and the critical velocity dispersion required for axisymmetric stability , for the stellar and gaseous components ( figures [ vrot628 ] and [ vrot3938 ] ) . the comparison with the observed vertical velocity dispersions for hi and co is clear : the observed values are most of the time larger , in particular for ngc 3938 . this means that , if the gas velocity dispersion can be considered isotropic , the toomre stability parameter in the galaxy plane is always @xmath31 , and most of the time @xmath32 2 - 3 , for ngc 3938 . for ngc 628 , @xmath19 is near 1 between 3 and 20kpc , and the threshold for star formation , @xmath33 according to kennicutt ( 1989 ) is reached at 23 kpc . this is far in the outer parts of the galaxy , since r@xmath30 = 15.5 kpc . if the vertical dispersion is lower than in the plane , as could be the case ( e.g. olling 1995 ) , than @xmath19 is even larger . the gas appears then to be quite stable , unless the coupling gas - stars has a very large effect . figures [ vrot628 ] and [ vrot3938 ] also plot the critical velocity dispersion for the stellar component , together with a fit to the observed stellar velocity dispersions , from van der kruit & freeman ( 1984 ) for ngc 628 and from bottema ( 1988 , 1993 ) for ngc 3938 . from a sample of 12 galaxies where such data are available , bottema ( 1993 ) concludes that the stellar velocity dispersion is declining exponentially as @xmath34 , as expected for an exponential disk of scale - length @xmath35 and constant thickness , as found by van der kruit & searle ( 1981 ) . since mostly the vertical stellar dispersion @xmath36 is measured , it is assumed that there is a constant ratio between the radial dispersion @xmath37 , comparable to that observed in the solar neighbourhood @xmath38 = 0.6 . this is already well above the minimum ratio required for vertical stability , i.e. @xmath38 = 0.3 ( araki 1985 , merritt & sellwood 1994 ) . within these assumptions , it can be derived that the toomre parameter for the stars @xmath39 is about constant with radius , within the optical disk ; it depends of course on the mass - to - light ratio adopted for the luminous component , and is in the range @xmath40 1 for m(stars)/l@xmath41 = 3 . figures [ vrot628 ] and [ vrot3938 ] confirm the result of almost constant @xmath39 , but with low values , especially for ngc 3938 . this could be explained , if the vertical dispersion is indeed much lower than the radial one . the minimum value for the ratio @xmath38 is 0.3 ( for stability reasons ) , so that the derived @xmath39 values displayed in figures [ vrot628 ] and [ vrot3938 ] could be multiplied by @xmath42 2 . the idea of stellar velocity dispersion regulated by gravitational instabilities appears therefore supported by the data , within the uncertainties . the most intriguing result is the large gas vertical dispersion observed for ngc 3938 , and its distribution with radius . the large corresponding @xmath19 values , that will mean comfortable stability , are difficult to reconcile with the observed large and small - scales gas instabilities : clear spiral arms are usually observed in the outer hi disks , with small - scale structure as well ( see e.g. van der hulst & sancisi 1988 , richter & sancisi 1994 ) . this is also the case here for ngc 628 showing all signs of gravitational instabilities in its outer hi disk ( kamphuis & briggs 1992 ) , and for ngc 3938 ( van der kruit & shostak 1982 ) . a possibility to reduce @xmath19 is that also the gas dispersion is anisotropic , this time the vertical one being larger than in the plane . however we will see , through comparison with gas dispersion in the plane of the galaxy ( cf next section ) that the anisotropy of gas dispersion does not appear so large . another explanation could be that the present rough calculations of the @xmath25-parameter concern only a simplified one - component stability analysis , and could be significantly modified by multi - components analysis . it has been shown ( jog & solomon 1984 , romeo 1992 , jog 1992 & 1996 ) that the coupling between several components de - stabilises every dynamical component . the apparent stability ( @xmath43 ) of the gas component might therefore not be incompatible with an instability - regulated velocity dispersion for the gas . but then , in the vertical direction , the dispersion is much higher than the minimum required for vertical stability . could this large velocity dispersion be powered by star formation ? this is not likely , at least for the majority of the hi gas well outside the optical disk , where no stellar activity is observed . a possible explanation would be to suppose that the hi is tracing a much larger amount of gas , in the form of molecular clouds , which will then be self - gravitating , with @xmath44 ( pfenniger et al 1994 ; pfenniger & combes 1994 ) . with a flat rotation curve , and a gas surface density decreasing as @xmath23 , the critical dispersion would then be constant with radius . another puzzle is the similarity of the co and hi vertical velocity dispersions . if the gas layers are indeed isothermal in z , we can deduce that both atomic and molecular layers have also similar heights . this means that the atomic and molecular components can be considered as a unique dynamical component , which can be observed under two phases , according to the local physical conditions ( density , excitation temperature , etc .. ) . the amplitudes of z - oscillations of the molecular and atomic gas are the same , only we see the gas as molecular when it is at heights lower than @xmath42 50pc . at these heights , the molecular fraction is @xmath45 ( imamura & sofue 1997 ) , which means that almost all clouds are molecular , taking into account their atomic envelope . in fact it is not clear whether we see the co or h@xmath26 formation and destruction , since we can rely only on the co tracer . also , it is possible that the density of clouds at high altitude is not enough to excite the co molecule , which means that the limit for observing co will not be coinciding with the limit for molecular presence itself . the latter is strongly suggested by the observed vertical density profiles of the h@xmath26 and hi number density : there is a sharp boundary where the apparent @xmath46 falls to zero , while we expect a smoother profile for a unique dynamical gas component . that the gas can change phase from molecular to atomic and vice - versa several times in one z - oscillation is not unexpected , since the time - scale of molecular formation and destruction is smaller than the z - oscillation period , of @xmath42 10@xmath47 yrs at the optical radius : the chemical time - scale is of the order of 10@xmath48 yrs ( leung et al 1984 , langer & graedel 1989 ) . morever , as discussed in the previous section ( _ 5.1 _ ) , the key factor controlling the presence of molecules is photodestruction , which explains why there is a column density threshold above which the gas phase turns to molecular ( elmegreen 1993 ) . this threshold could be reached at some particular height above the plane . should we expect the existence of several layers of gas at different tmperatures , and therefore different thicknesses , in galaxy planes ? in the very simple model of a diffuse and homogeneous gas , unperturbed by star - formation , we can compute the mixing time - scale of two layers at different temperatures , through atomic or molecule collisions : this is of the order of the collisional time - scale , @xmath42 10@xmath49 yrs for an average volumic density of 1 @xmath50 , and a thermal velocity of 0.3 . this is very short with respect to the z - oscillation time scale of @xmath42 10@xmath47 yrs , and therefore mixing should occur , if differential dissipation or gravitational heating is not taken into account . this simple model is of course very far from realistic . we know that the interstellar medium , atomic as well as molecular , is distributed in a hierachical ensemble of clouds , similar to a fractal . let us then consider another simple modelisation of an ideal gas where the particles are in fact the interstellar clouds , undergoing collisions ( cf oort 1954 , cowie 1980 ) . for typical clouds of 1pc size , and 10@xmath51 @xmath50 volumic density , the collisional time - scale is of the order of 10@xmath47 yrs , comparable with the vertical oscillations time - scale . this figure should not be taken too seriously , given the rough simplifications , but it corresponds to what has been known for a long time , i.e. the ensemble of clouds can not be considered as a fluid in equilibrium , since the collisional time - scale is comparable to the dynamical time , like the spiral - arm crossing time ( cf bash 1979 , kwan 1979 , casoli & combes 1982 , combes & gerin 1985 ) . if the collisions were able to redistribute the kinetic energy completely , there should be equipartition , i.e. the velocity dispersion would decrease with the mass @xmath52 of the clouds like @xmath53 . in fact the cloud - cloud relative velocities are roughly constant with mass ( between clouds of masses 100 m@xmath54 and gmcs of 10@xmath55 m@xmath54 , a ratio of 100 would be expected in velocity dispersions , which is not observed , stark 1979 ) . towards the galactic anticenter , where streaming motions should be minimised , the one - dimensional dispersion for the low - mass and giant clouds are found to be about 9.1 and 6.6 respectively , with near constancy over several orders of magnitude , and therefore no equipartition of energy ( stark 1984 ) . the almost constancy of velocity dispersions with mass requires to find other mechanisms responsible for the heating . if relatively small clouds can be heated by star - formation , supernovae , etc ... (e.g . chize & lazareff 1980 ) , the largest clouds could be heated by gravitational scattering ( jog & ostriker 1988 , gammie et al 1991 ) . in the latter mechanism , encounters between clouds with impact parameters of the order of their tidal radius in a differentially rotating disk are equivalent to a gravitational viscosity that pumps the rotational energy into random cloud kinetic energy . a 1d velocity dispersion of 5 - 7is the predicted result , independent of mass . this value is still slightly lower than the observed 1d dispersion of clouds observed in the milky way . stark & brand ( 1989 ) find 7.8from a study within 3 kpc of the sun . but collective effects , gravitational instabilities forming structures like spiral arms , etc ... have not yet been taken into account . given the high degree of structure and apparent permanent instability of the gas , they must play a major role in the heating , the source of energy being also the global rotational energy . dissipation lowering the gas dispersion continuously maintains the gas at the limit of instability , closing the feedback loop of the self - regulation ( lin & pringle 1987 , bertin & romeo 1988 ) . in the external parts of galaxies , where there is no star formation , gravitational instabilities are certainly the essential heating mechanism this again will tend to an isothermal , or more exactly isovelocity , ensemble of clouds , since the gravitational mechanism does not depend on the particle mass . the molecular or atomic gas are equivalent in this process , and should reach the same equilibrium dispersion . in the milky way , although the kinematics of gas is much complicated due to our embedded perspective , we have also the same puzzle . the velocity dispersion has been estimated through several methods , with intrinsic biases for each method , but essentially the dispersion has been estimated in the plane . only with high - latitude molecular clouds , can we have an idea of the local vertical velocity dispersion . magnani et al ( 1996 ) have recently made a compilation of more than 100 of these high - latitude clouds . the velocity dispersion of the ensemble is 5.8if seven intermediate velocity objects are excluded , and 9.9 otherwise . this is interestingly close to the values we find for ngc 628 ( 6 ) and ngc 3938 ( 8.5 ) . unfortunately there is always some doubt in the galaxy that all molecular clouds are taken into account , due to many selection effects , while the measurement is much more direct at large scale in external face - on galaxies . in fact , it has been noticed by magnani et al ( 1996 ) that there were an inconsistency between the local measured scale - height of molecular clouds ( about 60pc ) and the vertical velocity dispersion . however , they conclude in terms of a different population for the local high - latitude clouds ( hlc ) . indeed , the total mass of observed hlc is still a small fraction of the molecular surface density at the solar radius . the local gaussian scale height of the molecular component has been derived to be 58pc ( at r@xmath54 = 8.5kpc ) through a detailed data modelling by malhotra ( 1994 ) ; this is also compatible with all previous values ( dame et al 1987 , clemens et al 1988 ) . the local hi scale height is 220pc ( malhotra 1995 ) . we therefore would have expected a ratio of 3.8 between the dispersions of the h@xmath26 and hi gas , but these are very similar , within the uncertainties , which come mainly from the clumpiness of the clouds for the h@xmath26 component . if we believe the more easily determined hi dispersion of 9(malhotra 1995 ) , then the h@xmath26 dispersion is expected to be 2.4 , clearly outside of the error bars or intrinsic scatter : the value at the solar radius is estimated at 7.8by malhotra ( 1994 ) . of course , all this discussion is hampered by the fact that we discuss mainly horizontal dispersions in the case of the milky way , while the gas dispersions could well be anisotropic . this is why the present results on external face - on galaxies are more promising . the vertical gas velocity dispersion in spiral galaxies is an important parameter required to determine the flattening of the dark matter component , combined with the observation of the gas layer thickness ( cf olling 1995 , becquaert & combes 1997 ) . we have shown here that the gas dispersion does not appear very anisotropic , in the sense that the vertical dispersion is not much smaller that what has been derived in the plane of our galaxy ( for instance by the terminal velocity method , burton 1992 , malhotra 1994 ) . such vertical dispersion data should be obtained in much larger samples , to consolidate statistically this result . adler d.s . , liszt h.s . : 1989 , ap.j . 339 , 836 araki s. : 1985 , phd thesis , massachussetts institute of technology bash f.h . : 1979 , apj 233 , 524 becquaert j - f . , combes f. : 1997 , a&a in press bertin g. , romeo a. : 1988 , a&a 195 , 105 binney , j. & tremaine , s. 1987 , `` galactic dynamics '' , princeton university press , princeton , new jersey bosma a. : 1981 , aj 86 , 1971 bottema r. : 1988 , a&a 197 , 105 bottema r. : 1993 , a&a 275 , 16 boulanger f. , stark a.a . , combes f. : 1981 , a&a 93 , l1 braine j. , combes f. , casoli f. et al : 1993 a&as 97 , 887 brinks e. , burton w.b . : 1984 , a&a 141 , 195 briggs f.h . , wolfe a.m. , krumm n. , salpeter e.e . : 1980 , apj 238 , 510 burton w.b . : 1992 , in `` the galactic interstellar medium '' , saas - 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we present co(1 - 0 ) and co(2 - 1 ) observations of the two nearly face - on galaxies ngc 628 and ngc 3938 , in particular cuts along the major and minor axis . the contribution of the beam - smeared in - plane velocity gradients to the observed velocity width is quite small in the outer parts of the galaxies . this allows us to derive the velocity dispersion of the molecular gas perpendicular to the plane . we find that this dispersion is remarkably constant with radius , 6 for ngc 628 and 8.5 for ngc 3938 , and of the same order as the hi dispersion . the constancy of the value is interpreted in terms of a feedback mechanism involving gravitational instabilities and gas dissipation . the similarity of the co and hi dispersions suggests that the two components are well mixed , and are only two different phases of the same kinematical gas component . the gas can be transformed from the atomic phase to the molecular phase and vice - versa several times during a z - oscillation . psfig = + 2.0 cm

over the past decade or so two separate developments have occurred in computer science whose intersection promises to open a vast new area of research , an area extending far beyond the current boundaries of computer science . the first of these developments is the growing realization of how useful it would be to be able to control distributed systems that have little ( if any ) centralized communication , and to do so `` adaptively '' , with minimal reliance on detailed knowledge of the system s small - scale dynamical behavior . the second development is the maturing of the discipline of reinforcement learning ( rl ) . this is the branch of machine learning that is concerned with an agent who periodically receives `` reward '' signals from the environment that partially reflect the value of that agent s private utility function . the goal of an rl algorithm is to determine how , using those reward signals , the agent should update its action policy to maximize its utility @xcite . ( until our detailed discussions below , we will use the term `` reinforcement learning '' broadly , to include any algorithm of this sort , including ones that rely on detailed bayesian modeling of underlying markov processes @xcite . intuitively , one might hope that rl would help us solve the distributed control problem , since rl is adaptive , and , in particular , since it is not restricted to domains having sufficient breadths of communication . however , by itself , conventional single - agent rl does not provide a means for controlling large , distributed systems . this is true even if the system @xmath0 have centralized communication . the problem is that the space of possible action policies for such systems is too big to be searched . we might imagine as a variant using a large set of agents , each controlling only part of the system . since the individual action spaces of such agents would be relatively small , we could realistically deploy conventional rl on each one . however , now we face the central question of how to map the world utility function concerning the overall system into private utility functions for each of the agents . in particular , how should we design those private utility functions so that each agent can realistically hope to optimize its function , and at the same time the collective behavior of the agents will optimize the world utility ? we use the term `` collective intelligence '' ( coin ) to refer to any pair of a large , distributed collection of interacting computational processes among which there is little to no centralized communication or control , together with a ` world utility ' function that rates the possible dynamic histories of the collection . the central coin design problem we consider arises when the computational processes run rl algorithms : how , without any detailed modeling of the overall system , can one set the utility functions for the rl algorithms in a coin to have the overall dynamics reliably and robustly achieve large values of the provided world utility ? the benefits of an answer to this question would extend beyond the many branches of computer science , having major ramifications for many other sciences as well . section [ sec : back ] discusses some of those benefits . section [ sec : lit ] reviews previous work that has bearing on the coin design problem . section [ sec : math ] section constitutes the core of this chapter . it presents a quick outline of a promising mathematical framework for addressing this problem in its most general form , and then experimental illustrations of the prescriptions of that framework . throughout , we will use italics for emphasis , single quotes for informally defined terms , and double quotes to delineate colloquial terminology . there are many design problems that involve distributed computational systems where there are strong restrictions on centralized communication ( `` we ca nt all talk '' ) ; or there is communication with a central processor , but that processor is not sufficiently powerful to determine how to control the entire system ( `` we are nt smart enough '' ) ; or the processor is powerful enough in principle , but it is not clear what algorithm it could run by itself that would effectively control the entire system ( `` we do nt know what to think '' ) . just a few of the potential examples include : \i ) designing a control system for constellations of communication satellites or for constellations of planetary exploration vehicles ( world utility in the latter case being some measure of quality of scientific data collected ) ; \ii ) designing a control system for routing over a communication network ( world utility being some aggregate quality of service measure ) \iii ) construction of parallel algorithms for solving numerical optimization problems ( the optimization problem itself constituting the world utility ) ; \iv ) vehicular traffic control , _ e.g. _ , air traffic control , or high - occupancy toll - lanes for automobiles . ( in these problems the individual agents are humans and the associated utility functions must be of a constrained form , reflecting the relatively inflexible kinds of preferences humans possess . ) ; \v ) routing over a power grid ; \vi ) control of a large , distributed chemical plant ; \vii ) control of the elements of an amorphous computer ; \viii ) control of the elements of a ` noisy ' phased array radar ; \ix ) compute - serving over an information grid . such systems may be best controlled with an artificial coin . however , the potential usefulness of deeper understanding of how to tackle the coin design problem extends far beyond such engineering concerns . that s because the coin design problem is an inverse problem , whereas essentially all of the scientific fields that are concerned with naturally - occurring distributed systems analyze them purely as a `` forward problem . '' that is , those fields analyze what global behavior would arise from provided local dynamical laws , rather than grapple with the inverse problem of how to configure those laws to induce desired global behavior . ( indeed , the coin design problem could almost be defined as decentralized adaptive control theory for massively distributed stochastic environments . ) it seems plausible that the insights garnered from understanding the inverse problem would provide a trenchant novel perspective on those fields . just as tackling the inverse problem in the design of steam engines led to the first true understanding of the macroscopic properties of physical bodes ( aka thermodynamics ) , so may the cracking of the coin design problem may improve our understanding of many naturally - occurring coins . in addition , although the focuses of those other fields are not on the coin design problem , in that they are related to the coin design problem , that problem may be able to serve as a `` touchstone '' for all those fields . this may then reveal novel connections between the fields . as an example of how understanding the coin design problem may provide a novel perspective on other fields , consider countries with capitalist human economies . although there is no intrinsic world utility in such systems , they can still be viewed from the perspective of coins , as naturally occurring coins . for example , one can declare world utility to be a time average of the gross domestic product ( gdp ) of the country in question . ( world utility per se is not a construction internal to a human economy , but rather something defined from the outside . ) the reward functions for the human agents in this example could then be the achievements of their personal goals ( usually involving personal wealth to some degree ) . now in general , to achieve high world utility in a coin it is necessary to avoid having the agents work at cross - purposes . otherwise the system is vulnerable to economic phenomena like the tragedy of the commons ( toc ) , in which individual avarice works to lower world utility @xcite , or the liquidity trap , where behavior that helps the entire system when employed by some agents results in poor global behavior when employed by all agents @xcite . one way to avoid such phenomena is by modifying the agents utility functions . in the context of capitalist economies , this kind of effect can be achieved via punitive legislation that modifies the rewards the agents receive for engaging in certain kinds of activity . a real world example of an attempt to make just such a modification was the creation of anti - trust regulations designed to prevent monopolistic practices . in designing a coin we usually have more freedom than anti - trust regulators though , in that there is no base - line `` organic '' private utility function over which we must superimpose legislation - like incentives . rather , the entire `` psychology '' of the individual agents is at our disposal when designing a coin . this obviates the need for honesty - elicitation ( ` incentive compatible ' ) mechanisms , like auctions , which form a central component of conventional economics . accordingly , coins can differ in certain crucial respects from human economies . the precise differences the subject of current research seem likely to present many insights into the functioning of economic structures like anti - trust regulators . to continue with this example , consider the usefulness , as far as the world utility is concerned , of having ( commodity , or especially fiat ) money in the coin . formally , from a coin perspective , the use of ` money ' for trading between agents constitutes a particular class of couplings between the states and utility functions of the various agents . for example , if one agent s ` bank account ' variable goes up in a ` trade ' with another agent , then a corresponding ` bank account ' variable in that other agent must decrease to compensate . in addition to this coupling between the agents states , there is also a coupling between their utilities , if one assume that both agents will prefer to have more money rather than less , everything else being equal . however one might formally define such a ` money ' structure , we can consider what happens if it does ( or does not ) obtain for an arbitrary dynamical system , in the context of an arbitrary world utility . for some such dynamical systems and world utilities , a money structure will improve the value of that world utility . but for the same dynamics , the use of a money structure will simultaneously induce _ low levels _ of other world utilities ( a trivial example being a world utility that equals the negative of the first one ) . this raises a host of questions , like how to formally specify the most general set of world utilities that benefits significantly from using money - based private utility functions . if one is provided a world utility that is not a member of that set , then an `` economics - like '' configuration of the system is likely to result in poor performance . such a characterization of how and when money helps improve world utilities of various sorts might have important implications for conventional human economics , especially when one chooses world utility to be one of the more popular choices for social welfare function . ( see @xcite and references therein for some of the standard economics work that is most relevant to this issue . ) there are many other scientific fields that are currently under investigation from a coin - design perspective . some of them are , like economics , part of ( or at least closely related to ) the social sciences . these fields typically involve rl algorithms under the guise of human agents . an example of such a field is game theory , especially game theory of bounded rational players . as illustrated in our money example , viewing such systems from the perspective of a non - endogenous world utility , _ i.e. _ , from a coin - design perspective , holds the potential for providing novel insight into them . ( in the case of game theory , it holds the potential for leading to deeper understanding of many - player inverse stochastic game theory . ) however there are other scientific fields that might benefit from a coin - design perspective even though they study systems that do nt even involve rl algorithms . the idea here is that if we viewed such systems from an `` artificial '' teleological perspective , both in concentrating on a non - endogenous world utility and in casting the nodal elements of the system as rl algorithms , we could learn a lot about the form of the ` design space ' in which such systems live . ( just as in economics , where the individual nodal elements _ are _ rl algorithms , investigating the system using an externally imposed world utility might lead to insight . ) examples here are ecosystems ( individual genes , individuals , or species being the nodal elements ) and cells ( individual organelles in eukaryotes being the nodal elements ) . in both cases , the world utility could involve robustness of the desired equilibrium against external perturbation , efficient exploitation of free energy in the environment , etc . the following list elaborates what we mean by a coin : \1 ) there are many processors running concurrently , performing actions that affect one another s behavior . \2 ) there is little to no centralized personalized communication , _ i.e. _ , little to no behavior in which a small subset of the processors not only communicates with all the other processors , but communicates differently with each one of those other processors . any single processor s `` broadcasting '' the same information to all other processors is not precluded . \3 ) there is little to no centralized personalized control , _ i.e. _ , little to no behavior in which a small subset of the processors not only controls all the other processors , but controls each one of those other processors differently . `` broadcasting '' the same control signal to all other processors is not precluded . \4 ) there is a well - specified task , typically in the form of extremizing a utility function , that concerns the behavior of the entire distributed system . so we are confronted with the inverse problem of how to configure the system to achieve the task . the following elements characterize the sorts of approaches to coin design we are concerned with here : \5 ) the approach for tackling ( 4 ) is scalable to very large numbers of processors . \6 ) the approach for tackling ( 4 ) is very broadly applicable . in particular , it can work when little ( if any ) `` broadcasting '' as in ( 2 ) and ( 3 ) is possible . \7 ) the approach for tackling ( 4 ) involves little to no hand - tailoring . \8 ) the approach for tackling ( 4 ) is robust and adaptive , with minimal need to `` get the details exactly right or else , '' as far as the stochastic dynamics of the system is concerned . \9 ) the individual processors are running rl algorithms . unlike the other elements of this list , this one is not an _ a priori _ engineering necessity . rather , it is a reflection of the fact that rl algorithms are currently the best - understood and most mature technology for addressing the points ( 8) and ( 9 ) . there are many approaches to coin design that do not have every one of those features . these approaches constitute part of the overall field of coin design . as discussed below though , not having every feature in our list , no single one of those approaches can be extended to cover the entire breadth of the field of coin design . ( this is not too surprising , since those approaches are parts of fields whose focus is not the coin design problem per se . ) the rest of this section consists of brief presentations of some of these approaches , and in particular characterizes them in terms of our list of nine characteristics of coins and of our desiredata for their design . of the approaches we discuss , at present it is probably the ones in artificial intelligence and machine learning that are most directly applicable to coin design . however it is fairly clear how to exploit those approaches for coin design , and in that sense relatively little needs to be said about them . in contrast , as currently employed , the toolsets in the social sciences are not as immediately applicable to coin design . however , it seems likely that there is more yet to be discovered about how to exploit them for coin design . accordingly , we devote more space to those social science - based approaches here . we present an approach that holds promise for covering all nine of our desired features in section [ sec : math ] . there is an extensive body of work in ai and machine learning that is related to coin design . indeed , one of the most famous speculative works in the field can be viewed as an argument that ai should be approached as a coin design problem @xcite . much work of a more concrete nature is also closely related to the problem of coin design . as discussed in the introduction , the maturing field of reinforcement learning provides a much needed tool for the types of problems addressed by coins . because rl generally provides model - free and `` online '' learning features , it is ideally suited for the distributed environment where a `` teacher '' is not available and the agents need to learn successful strategies based on `` rewards '' and `` penalties '' they receive from the overall system at various intervals . it is even possible for the learners to use those rewards to modify _ how _ they learn @xcite . although work on rl dates back to samuel s checker player @xcite , relatively recent theoretical @xcite and empirical results @xcite have made rl one of the most active areas in machine learning . many problems ranging from controlling a robot s gait to controlling a chemical plant to allocating constrained resource have been addressed with considerable success using rl @xcite . in particular , the rl algorithms @xmath1 ( which rates potential states based on a _ value function _ ) @xcite and @xmath2learning ( which rates action - state pairs ) @xcite have been investigated extensively . a detailed investigation of rl is available in @xcite . although powerful and widely applicable , solitary rl algorithms will not perform well on large distributed heterogeneous problems in general . this is due to the very big size of the action - policy space for such problems . in addition , without centralized communication and control , how a solitary rl algorithm could run the full system at all , poorly or well , becomes a major concern . for these reasons , it is natural to consider deploying many rl algorithms rather than a single one for these large distributed problems . we will discuss the coordination issues such an approach raises in conjunction with multi - agent systems in section [ sec : mas ] and with learnability in coins in section [ sec : math ] . the field of distributed artificial intelligence ( dai ) has arisen as more and more traditional artificial intelligence ( ai ) tasks have migrated toward parallel implementation . the most direct approach to such implementations is to directly parallelize ai production systems or the underlying programming languages @xcite . an alternative and more challenging approach is to use distributed computing , where not only are the individual reasoning , planning and scheduling ai tasks parallelized , but there are _ different modules _ with different such tasks , concurrently working toward a common goal @xcite . in a dai , one needs to ensure that the task has been modularized in a way that improves efficiency . unfortunately , this usually requires a central controller whose purpose is to allocate tasks and process the associated results . moreover , designing that controller in a traditional ai fashion often results in brittle solutions . accordingly , recently there has been a move toward both more autonomous modules and fewer restrictions on the interactions among the modules @xcite . despite this evolution , dai maintains the traditional ai concern with a pre - fixed set of _ particular _ aspects of intelligent behavior ( _ e.g. _ reasoning , understanding , learning etc . ) rather than on their _ cumulative _ character . as the idea that intelligence may have more to do with the interaction among components started to take shape @xcite , focus shifted to concepts ( _ e.g. _ , multi - agent systems ) that better incorporated that idea @xcite . the field of multi - agent systems ( mas ) is concerned with the interactions among the members of such a set of agents @xcite , as well as the inner workings of each agent in such a set ( _ e.g. _ , their learning algorithms ) @xcite . as in computational ecologies and computational markets ( see below ) , a well - designed mas is one that achieves a global task through the actions of its components . the associated design steps involve @xcite : 1 . decomposing a global task into distributable subcomponents , yielding tractable tasks for each agent ; 2 . establishing communication channels that provide sufficient information to each of the agents for it to achieve its task , but are not too unwieldly for the overall system to sustain ; and 3 . coordinating the agents in a way that ensures that they cooperate on the global task , or at the very least does not allow them to pursue conflicting strategies in trying to achieve their tasks . step ( 3 ) is rarely trivial ; one of the main difficulties encountered in mas design is that agents act selfishly and artificial cooperation structures have to be imposed on their behavior to enforce cooperation @xcite . an active area of research , which holds promise for addressing parts the coin design problem , is to determine how selfish agents `` incentives '' have to be engineered in order to avoid the tragedy of the commons ( toc ) @xcite . ( this work draws on the economics literature , which we review separately below . ) when simply providing the right incentives is not sufficient , one can resort to strategies that actively induce agents to cooperate rather than act selfishly . in such cases coordination @xcite , negotiations @xcite , coalition formation @xcite or contracting @xcite among agents may be needed to ensure that they do not work at cross purposes . unfortunately , all of these approaches share with dai and its offshoots the problem of relying excessively on hand - tailoring , and therefore being difficult to scale and often nonrobust . in addition , except as noted in the next subsection , they involve no rl , and therefore the constituent computational elements are usually not as adaptive and robust as we would like . because it neither requires explicit modeling of the environment nor having a `` teacher '' that provides the `` correct '' actions , the approach of having the individual agents in a mas use rl is well - suited for mas s deployed in domains where one has little knowledge about the environment and/or other agents . there are two main approaches to designing such mas s : + ( i ) one has ` solipsistic agents ' that do nt know about each other and whose rl rewards are given by the performance of the entire system ( so the joint actions of all other agents form an `` inanimate background '' contributing to the reward signal each agent receives ) ; + ( ii ) one has ` social agents ' that explicitly model each other and take each others actions into account . both ( i ) and ( ii ) can be viewed as ways to ( try to ) coordinate the agents in a mas in a robust fashion . * solipsistic agents : * mas s with solipsistic agents have been successfully applied to a multitude of problems @xcite . generally , these schemes use rl algorithms similar to those discussed in section [ sec : control ] . however much of this work lacks a well - defined global task or broad applicability ( _ e.g. _ , @xcite ) . more generally , none of the work with solipsistic agents scales well . ( as illustrated in our experiments on the `` bar problem '' , recounted below . ) the problem is that each agent must be able to discern the effect of its actions on the overall performance of the system , since that performance constitutes its reward signal . as the number of agents increases though , the effects of any one agent s actions ( signal ) will be swamped by the effects of other agents ( noise ) , making the agent unable to learn well , if at all . ( see the discussion below on learnability . ) in addition , of course , solipsistic agents can not be used in situations lacking centralized calculation and broadcast of the single global reward signal . * social agents : * mas s whose agents take the actions of other agents into account synthesize rl with game theoretic concepts ( _ e.g. _ , nash equilibrium ) . they do this to try to ensure that the overall system both moves toward achieving the overall global goal and avoids often deleterious oscillatory behavior @xcite . to that end , the agents incorporate internal mechanisms that actively model the behavior of other agents . in section [ sec : bar ] , we discuss a situation where such modeling is necessarily self - defeating . more generally , this approach usually involves extensive hand - tailoring for the problem at hand . some human economies provides examples of naturally occurring systems that can be viewed as a ( more or less ) well - performing coin . the field of economics provides much more though . both empirical economics ( _ e.g. _ , economic history , experimental economics ) and theoretical economics ( _ e.g. _ , general equilibrium theory @xcite , theory of optimal taxation @xcite ) provide a rich literature on strategic situations where many parties interact . in fact , much of the entire field of economics can be viewed as concerning how to maximize certain constrained kinds of world utilities , when there are certain ( very strong ) restrictions on the individual agents and their interactions , and in particular when we have limited freedom in setting either the utility functions of those agents or modifying their rl algorithms in any other way . in this section we summarize just two economic concepts , both of which are very closely related to coins , in that they deal with how a large number of interacting agents can function in a stable and efficient manner : general equilibrium theory and mechanism design . we then discuss general attempts to apply those concepts to distributed computational problems . we follow this with a discussion of game theory , and then present a particular celebrated toy - world problem that involves many of these issues . often the first version of `` equilibrium '' that one encounters in economics is that of supply and demand in single markets : the price of the market s good is determined by where the supply and demand curves for that good intersect . in cases where there is interaction among multiple markets however , even when there is no production but only trading , one can not simply determine the price of each market s good individually , as both the supply and demand for each good depends on the supply / demand of other goods . considering the price fluctuations across markets leads to the concept of ` general equilibrium ' , where prices for each good are determined in such a way to ensure that all markets ` clear ' @xcite . intuitively , this means that prices are set so the total supply of each good is equal to the demand for that good . the existence of such an equilibrium , proven in @xcite , was first postulated by leon walras @xcite . a mechanism that calculates the equilibrium ( _ i.e. _ , ` market - clearing ' ) prices now bears his name : the walrasian auctioner . in general , for an arbitrary goal for the overall system , there is no reason to believe that having markets clear achieves that goal . in other words , there is no _ a priori _ reason why the general equilibrium point should maximize one s provided world utility function . however , consider the case where one s goal for the overall system is in fact that the markets clear . in such a context , examine the case where the interactions of real - world agents will induce the overall system to adopt the general equilibrium point , so long as certain broad conditions hold . then if we can impose those conditions , we can cause the overall system to behave in the manner we wish . however general equilibrium theory is not sufficient to establish those `` broad conditions '' , since it says little about real - world agents . in particular , general equilibrium theory suffers from having no temporal aspect ( _ i.e. _ , no dynamics ) and from assuming that all the agents are perfectly rational . another shortcoming of general equilibrium theory as a model of real - world systems is that despite its concerning prices , it does not readily accommodate the full concept of money @xcite . of the three main roles money plays in an economy ( medium of exchange in trades , store of value for future trades , and unit of account ) none are essential in a general equilibrium setting . the unit of account aspect is not needed as the bookkeeping is performed by the walrasian auctioner . since the supplies and demands are matched directly there is no need to facilitate trades , and thus no role for money as a medium of exchange . and finally , as the system reaches an equilibrium in one step , through the auctioner , there is no need to store value for future trading rounds @xcite . the reason that money is not needed can be traced to the fact that there is an `` overseer '' with global information who guides the system . if we remove the centralized communication and control exerted by this overseer , then ( as in a real economy ) agents will no longer know the exact details of the overall economy . they will be forced to makes guesses as in any learning system , and the differences in those guesses will lead to differences in their actions @xcite . such a decentralized learning - based system more closely resembles a coin than does a conventional general equilibrium system . in contrast to general equilibrium systems , the three main roles money plays in a human economy are crucial to the dynamics of such a decentralized system @xcite . this comports with the important effects in coins of having the agents utility functions involve money ( see background section above ) . even if there exists centralized communication so that we are nt considering a full - blown coin , if there is no centralized walras - like control , it is usually highly non - trivial to induce the overall system to adopt the general equilibrium point . one way to try to do so is via an auction . ( this is the approach usually employed in computational markets see below . ) along with optimal taxation and public good theory @xcite , the design of auctions is the subject of the field of mechanism design . more generally , mechanism design is concerned with the incentives that must be applied to any set of agents that interact and exchange goods @xcite in order to get those agents to exhibit desired behavior . usually that desired behavior concerns pre - specified utility functions of some sort for each of the individual agents . in particular , mechanism design is usually concerned with incentive schemes which induce ` ( pareto ) efficient ' ( or ` pareto optimal ' ) allocations in which no agent can be made better off without hurting another agent @xcite . one particularly important type of such an incentive scheme is an auction . when many agents interact in a common environment often there needs to be a structure that supports the exchange of goods or information among those agents . auctions provide one such ( centralized ) structure for managing exchanges of goods . for example , in the english auction all the agents come together and ` bid ' for a good , and the price of the good is increased until only one bidder remains , who gets the good in exchange for the resource bid . as another example , in the dutch auction the price of a good is decreased until one buyer is willing to pay the current price . all auctions perform the same task : match supply and demand . as such , auctions are one of the ways in which price equilibration among a set of interacting agents ( perhaps an equilibration approximating general equilibrium , perhaps not ) can be achieved . however , an auction mechanism that induces pareto efficiency does not necessarily maximize some other world utility . for example , in a transaction in an english auction both the seller and the buyer benefit . they may even have arrived at an allocation which is efficient . however , in that the winner may well have been willing to pay more for the good , such an outcome may confound the goal of the market designer , if that designer s goal is to maximize revenue . this point is returned to below , in the context of computational economics . ` computational economies ' are schemes inspired by economics , and more specifically by general equilibrium theory and mechanism design theory , for managing the components of a distributed computational system . they work by having a ` computational market ' , akin to an auction , guide the interactions among those components . such a market is defined as any structure that allows the components of the system to exchange information on relative valuation of resources ( as in an auction ) , establish equilibrium states ( _ e.g. _ , determine market clearing prices ) and exchange resources ( _ i.e. _ , engage in trades ) . such computational economies can be used to investigate real economies and biological systems @xcite . they can also be used to design distributed computational systems . for example , such computational economies are well - suited to some distributed resource allocation problems , where each component of the system can either directly produce the `` goods '' it needs or acquire them through trades with other components . computational markets often allow for far more heterogeneity in the components than do conventional resource allocation schemes . furthermore , there is both theoretical and empirical evidence suggesting that such markets are often able to settle to equilibrium states . for example , auctions find prices that satisfy both the seller and the buyer which results in an increase in the utility of both ( else one or the other would not have agreed to the sale ) . assuming that all parties are free to pursue trading opportunities , such mechanisms move the system to a point where all possible bilateral trades that could improve the utility of both parties are exhausted . now restrict attention to the case , implicit in much of computational market work , with the following characteristics : first , world utility can be expressed as a monotonically increasing function @xmath3 where each argument @xmath4 of @xmath3 can in turn be interpreted as the value of a pre - specified utility function @xmath5 for agent @xmath4 . second , each of those @xmath5 is a function of an @xmath4-indexed ` goods vector ' @xmath6 of the non - perishable goods `` owned '' by agent @xmath4 . the components of that vector are @xmath7 , and the overall system dynamics is restricted to conserve the vector @xmath8 . ( there are also some other , more technical conditions . ) as an example , the resource allocation problem can be viewed as concerning such vectors of `` owned '' goods . due to the second of our two conditions , one can integrate a market - clearing mechanism into any system of this sort . due to the first condition , since in a market equilibrium with non - perishable goods no ( rational ) agent ends up with a value of its utility function lower than the one it started with , the value of the world utility function must be higher at equilibrium than it was initially . in fact , so long as the individual agents are smart enough to avoid all trades in which they do not benefit , any computational market can only improve this kind of world utility , even if it does not achieve the market equilibrium . ( see the discussion of `` weak triviality '' below . ) this line of reasoning provides one of the main reasons to use computational markets when they can be applied . conversely , it underscores one of the major limitations of such markets : starting with an arbitrary world utility function with arbitrary dynamical restrictions , it may be quite difficult to cast that function as a monotonically increasing @xmath3 taking as arguments a set of agents goods - vector - based utilities @xmath5 , if we require that those @xmath5 be well - enough behaved that we can reasonably expect the agents to optimize them in a market setting . one example of a computational economy being used for resource allocation is huberman and clearwater s use of a double blind auction to solve the complex task of controlling the temperature of a building . in this case , each agent ( individual temperature controller ) bids to buy or sell cool or warm air . this market mechanism leads to an equitable temperature distribution in the system @xcite . other domains where market mechanisms were successfully applied include purchasing memory in an operating systems @xcite , allocating virtual circuits @xcite , `` stealing '' unused cpu cycles in a network of computers @xcite , predicting option futures in financial markets @xcite , and numerous scheduling and distributed resource allocation problems @xcite . computational economics can also be used for tasks not tightly coupled to resource allocation . for example , following the work of maes @xcite and ferber @xcite , baum shows how by using computational markets a large number of agents can interact and cooperate to solve a variant of the blocks world problem @xcite . viewed as candidate coins , all market - based computational economics fall short in relying on both centralized communication and centralized control to some degree . often that reliance is extreme . for example , the systems investigated by baum not only have the centralized control of a market , but in addition have centralized control of all other non - market aspects of the system . ( indeed , the market is secondary , in that it is only used to decide which single expert among a set of candidate experts gets to exert that centralized control at any given moment ) . there has also been doubt cast on how well computational economies perform in practice @xcite , and they also often require extensive hand - tailoring in practice . finally , return to consideration of a world utility function that is a monotonically increasing function @xmath9 whose arguments are the utilities of the agents . in general , the maximum of such a world utility function will be a pareto optimal point . so given the utility functions of the agents , by considering all such @xmath9 we map out an infinite set @xmath10 of pareto optimal points that maximize _ some _ such world utility function . ( @xmath10 is usually infinite even if we only consider maximizing those world utilities subject to an overall conservation of goods constraint . ) now the market equilibrium is a pareto optimal point , and therefore lies in @xmath10 . but it is only one element of @xmath10 . moreover , it is usually set in full by the utilities of the agents , in concert with the agents initial endowments . in particular , it is independent of the world utility . in general then , given the utilities of the agents and a world utility @xmath9 , there is no _ a priori _ reason to believe that the particular element in @xmath10 picked out by the auction is the point that maximizes that particular world utility . this subtlety is rarely addressed in the work on using computational markets to achieve a global goal . it need not be uncircumventable however . for example , one obvious idea would be to to try to distort the agents _ perceptions _ of their utility functions and/or initial endowments so that the resultant market equilibrium has a higher value of the world utility at hand . game theory is the branch of mathematics concerned with formalized versions of `` games '' , in the sense of chess , poker , nuclear arms races , and the like @xcite . it is perhaps easiest to describe it by loosely defining some of its terminology , which we do here and in the next subsection . the simplest form of a game is that of ` non - cooperative single - stage extensive - form ' game , which involves the following situation : there are two or more agents ( called ` players ' in the literature ) , each of which has a pre - specified set of possible actions that it can follow . ( a ` finite ' game has finite sets of possible actions for all the players . ) in addition , each agent @xmath4 has a utility function ( also called a ` payoff matrix ' for finite games ) . this maps any ` profile ' of the action choices of all agents to an associated utility value for agent @xmath4 . ( in a ` zero - sum ' game , for every profile , the sum of the payoffs to all the agents is zero . ) the agents choose their actions in a sequence , one after the other . the structure determining what each agent knows concerning the action choices of the preceding agents is known as the ` information set . ' games in which each agent knows exactly what the preceding ( ` leader ' ) agent did are known as ` stackelberg games ' . ( a variant of such a game is considered in our experiments below . see also @xcite . ) in a ` multi - stage ' game , after all the agents choose their first action , each agent is provided some information concerning what the other agents did . the agent uses this information to choose its next action . in the usual formulation , each agent gets its payoff at the end of all of the game s stages . an agent s ` strategy ' is the rule it elects to follow mapping the information it has at each stage of a game to its associated action . it is a ` pure strategy ' if it is a deterministic rule . if instead the agent s action is chosen by randomly sampling from a distribution , that distribution is known a ` mixed strategy ' . note that an agent s strategy concerns @xmath11 possible sequences of provided information , even any that can not arise due to the strategies of the other agents . any multi - stage extensive - form game can be converted into a ` normal form ' game , which is a single - stage game in which each agent is ignorant of the actions of the other agents , so that all agents choose their actions `` simultaneously '' . this conversion is acieved by having the `` actions '' of each agent in the normal form game correspond to an entire strategy in the associated multi - stage extensive - form game . the payoffs to all the agents in the normal form game for a particular strategy profile is then given by the associated payoff matrices of the multi - stage extensive form - game . a ` solution ' to a game , or an ` equilibrium ' , is a profile in which every agent behaves `` rationally '' . this means that every agent s choice of strategy optimizes its utility subject to a pre - specified set of conditions . in conventional game theory those conditions involve , at a minimum , perfect knowledge of the payoff matrices of all other players , and often also involve specification of what strategies the other agents adopted and the like . in particular , a ` nash equilibrium ' is a a profile where each agent has chosen the best strategy it can , _ given the choices of the other agents_. a game may have no nash equilibria , one equilibrium , or many equilibria in the space of pure strategies . a beautiful and seminal theorem due to nash proves that every game has at least one nash equilibrium in the space of mixed strategies @xcite . there are several different reasons one might expect a game to result in a nash equilibrium . one is that it is the point that perfectly rational bayesian agents would adopt , assuming the probability distributions they used to calculate expected payoffs were consistent with one another @xcite . a related reason , arising even in a non - bayesian setting , is that a nash equilibrium equilibrium provides `` consistent '' predictions , in that if all parties predict that the game will converge to a nash equilibrium , no one will benefit by changing strategies . having a consistent prediction does not ensure that all agents payoffs are maximized though . the study of small perturbations around nash equilibria from a stochastic dynamics perspective is just one example of a ` refinement ' of nash equilibrium , that is a criterion for selecting a single equilibrium state when more than one is present @xcite . in cooperative game theory the agents are able to enter binding contracts with one another , and thereby coordinate their strategies . this allows the agents to avoid being `` stuck '' in nash equilibria that are pareto inefficient , that is being stuck at equilibrium profiles in which all agents would benefit if only they could agree to all adopt different strategies , with no possibility of betrayal . characteristic function _ of a game involves subsets ( ` coalitions ' ) of agents playing the game . for each such subset , it gives the sum of the payoffs of the agents in that subset that those agents can guarantee if they coordinate their strategies . an @xmath12 is a division of such a guaranteed sum among the members of the coalition . it is often the case that for a subset of the agents in a coalition one imputation @xmath13 another , meaning that under threat of leaving the coalition that subset of agents can demand the first imputation rather than the second . so the problem each agent @xmath4 is confronted with in a cooperative game is which set of other agents to form a coalition with , given the characteristic function of the game and the associated imputations @xmath4 can demand of its partners . there are several different kinds of solution for cooperative games that have received detailed study , varying in how the agents address this problem of who to form a coalition with . some of the more popular are the ` core ' , the ` shapley value ' , the ` stable set solution ' , and the ` nucleolus ' . in the real world , the actual underlying game the agents are playing does not only involve the actions considered in cooperative game theory s analysis of coalitions and imputations . the strategies of that underlying game also involve bargaining behavior , considerations of trying to cheat on a given contract , bluffing and threats , and the like . in many respects , by concentrating on solutions for coalition formation and their relation with the characteristic function , cooperative game theory abstracts away these details of the true underlying game . conversely though , progress has recently been made in understanding how cooperative games can arise from non - cooperative games , as they must in the real world @xcite . not surprisingly , game theory has come to play a large role in the field of multi - agent systems . in addition , due to darwinian natural selection , one might expect game theory to be quite important in population biology , in which the `` utility functions '' of the individual agents can be taken to be their reproductive fitness . as it turns out , there is an entire subfield of game theory concerned with this connection with population biology , called ` evolutionary game theory ' @xcite . to introduce evolutionary game theory , consider a game in which all players share the same space of possible strategies , and there is an additional space of possible ` attribute vectors ' that characterize an agent , along with a probability distribution @xmath14 across that new space . ( examples of attributes in the physical world could be things like size , speed , etc . ) we select a set of agents to play a game by randomly sampling @xmath14 . those agents attribute vectors jointly determine the payoff matrices of each of the individual agents . ( intuitively , what benefit accrues to an agent for taking a particular action depends on its attributes and those of the other agents . ) however each agent @xmath4 has limited information concerning both its attribute vector and that of the other players in the game , information encapsulated in an ` information structure ' . the information structure specifies how much each agent knows concerning the game it is playing . in this context , we enlarge the meaning of the term `` strategy '' to not just be a mapping from information sets and the like to actions , but from entire information structures to actions . in addition to the distribution @xmath14 over attribute vectors , we also have a distribution over strategies , @xmath15 . a strategy @xmath16 is a ` population strategy ' if @xmath15 is a delta function about @xmath16 . intuitively , we have a population strategy when each animal in a population `` follows the same behavioral rules '' , rules that take as input what the animal is able to discern about its strengths and weakness relative to those other members of the population , and produce as output how the animal will act in the presence of such animals . given @xmath14 , a population strategy centered about @xmath16 , and its own attribute vector , any player @xmath4 in the support of @xmath14 has an expected payoff for any strategy it might adopt . when @xmath4 s payoff could not improve if it were to adopt any strategy other than @xmath16 , we say that @xmath16 is ` evolutionary stable ' . intuitively , an evolutionary stable strategy is one that is stable with respect to the introduction of mutants into the population . now consider a sequence of such evolutionary games . interpret the payoff that any agent receives after being involved in such a game as the ` reproductive fitness ' of that agent , in the biological sense . so the higher the payoff the agent receives , in comparison to the fitnesses of the other agents , the more `` offspring '' it has that get propagated to the next game . in the continuum - time limit , where games are indexed by the real number @xmath17 , this can be formalized by a differential equation . this equation specifies the derivative of @xmath18 evaluated for each agent @xmath4 s attribute vector , as a montonically increasing function of the relative difference between the payoff of @xmath4 and the average payoff of all the agents . ( we also have such an equation for @xmath15 . ) the resulting dynamics is known as ` replicator dynamics ' , with an evolutionary stable population strategy , if it exists , being one particular fixed point of the dynamics . now consider removing the reproductive aspect of evolutionary game theory , and instead have each agent propagate to the next game , with `` memory '' of the events of the preceding game . furthermore , allow each agent to modify its strategy from one game to the next by `` learning '' from its memory of past games , in a bounded rational manner . the field of learning in games is concerned with exactly such situations @xcite . most of the formal work in this field involves simple models for the learning process of the agents . for example , in ` ficticious play ' @xcite , in each successive game , each agent @xmath4 adopts what would be its best strategy if its opponents chose their strategies according to the empirical frequency distribution of such strategies that @xmath4 has encountered in the past . more sophisticated versions of this work employ simple bayesian learning algorithms , or re - inventions of some of the techniques of the rl community @xcite . typically in learning in games one defines a payoff to the agent for a sequence of games , for example as a discounted sum of the payoffs in each of the constituent games . within this framework one can study the long term effects of strategies such as cooperation and see if they arise naturally and if so , under what circumstances . many aspects of real world games that do not occur very naturally otherwise arise spontaneously in these kinds of games . for example , when the number of games to be played is not pre - fixed , it may behoove a particular agent @xmath4 to treat its opponent better than it would otherwise , since @xmath4 @xmath19 have to rely on that other agent s treating it well in the future , if they end up playing each other again . this framework also allows us to investigate the dependence of evolving strategies on the amount of information available to the agents @xcite ; the effect of communication on the evolution of cooperation @xcite ; and the parallels between auctions and economic theory @xcite . in many respects , learning in games is even more relevant to the study of coins than is traditional game theory . however it suffers from the same major shortcoming ; it is almost exclusively focused on the forward problem rather than the inverse problem . in essence , coin design is the problem of @xmath20 game theory . the `` el farol '' bar problem and its variants provide a clean and simple testbed for investigating certain kinds of interactions among agents @xcite . in the original version of the problem , which arose in economics , at each time step ( each `` night '' ) , each agent needs to decide whether to attend a particular bar . the goal of the agent in making this decision depends on the total attendance at the bar on that night . if the total attendance is below a preset capacity then the agent should have attended . conversely , if the bar is overcrowded on the given night , then the agent should not attend . ( because of this structure , the bar problem with capacity set to @xmath21 of the total number of agents is also known as the ` minority game ' ; each agent selects one of two groups at each time step , and those that are in the minority have made the right choice ) . the agents make their choices by predicting ahead of time whether the attendance on the current night will exceed the capacity and then taking the appropriate course of action . what makes this problem particularly interesting is that it is impossible for each agent to be perfectly `` rational '' , in the sense of correctly predicting the attendance on any given night . this is because if most agents predict that the attendance will be low ( and therefore decide to attend ) , the attendance will actually high , while if they predict the attendance will be high ( and therefore decide not to attend ) the attendance will be low . ( in the language of game theory , this essentially amounts to the property that there are no pure strategy nash equilibria @xcite . ) alternatively , viewing the overall system as a coin , it has a prisoner s dilemma - like nature , in that `` rational '' behavior by all the individual agents thwarts the global goal of maximizing total enjoyment ( defined as the sum of all agents enjoyment and maximized when the bar is exactly at capacity ) . this frustration effect is similar to what occurs in spin glasses in physics , and makes the bar problem closely related to the physics of emergent behavior in distributed systems @xcite . researchers have also studied the dynamics of the bar problem to investigate economic properties like competition , cooperation and collective behavior and especially their relationship to market efficiency @xcite . properly speaking , biological systems do not involve utility functions and searches across them with rl algorithms . however it has long been appreciated that there are many ways in which viewing biological systems as involving searches over such functions can lead to deeper understanding of them @xcite . conversely , some have argued that the mechanism underlying biological systems can be used to help design search algorithms @xcite . these kinds of reasoning which relate utility functions and biological systems have traditionally focussed on the case of a single biological system operating in some external environment . if we extend this kind of reasoning , to a set of biological systems that are co - evolving with one another , then we have essentially arrived at biologically - based coins . this section discusses some of how previous work in the literature bears on this relationship between coins and biology . the fields of population biology and ecological modeling are concerned with the large - scale `` emergent '' processes that govern the systems that consist of many ( relatively ) simple entities interacting with one another @xcite . as usually cast , the `` simple entities '' are members of one or more species , and the interactions are some mathematical abstraction of the process of natural selection as it occurs in biological systems ( involving processes like genetic reproduction of various sorts , genotype - phenotype mappings , inter and intra - species competitions for resources , etc . ) . population biology and ecological modeling in this context addresses questions concerning the dynamics of the resultant ecosystem , and in particular how its long - term behavior depends on the details of the interactions between the constituent entities . broadly construed , the paradigm of ecological modeling can even be broadened to study how natural selection and self - regulating feedback creates a stable planet - wide ecological environment gaia @xcite . the underlying mathematical models of other fields can often be usefully modified to apply to the kinds of systems population biology is interested in @xcite . ( see also the discussion in the game theory subsection above . ) conversely , the underlying mathematical models of population biology and ecological modeling can be applied to other non - biological systems . in particular , those models shed light on social issues such as the emergence of language or culture , warfare , and economic competition @xcite . they also can be used to investigate more abstract issues concerning the behavior of large complex systems with many interacting components @xcite . going a bit further afield , an approach that is related in spirit to ecological modeling is ` computational ecologies ' . these are large distributed systems where each component of the system s acting ( seemingly ) independently results in complex global behavior . those components are viewed as constituting an `` ecology '' in an abstract sense ( although much of the mathematics is not derived from the traditional field of ecological modeling ) . in particular , one can investigate how the dynamics of the ecology is influenced by the information available to each component and how cooperation and communication among the components affects that dynamics @xcite . although in some ways the most closely related to coins of the current ecology - inspired research , the field of computational ecologies has some significant shortcomings if one tries to view it as a full science of coins . in particular , it suffers from not being designed to solve the inverse problem of how to configure the system so as to arrive at a particular desired dynamics . this is a difficulty endemic to the general program of equating ecological modeling and population biology with the science of coins . these fields are primarily concerned with the `` forward problem '' of determining the dynamics that arises from certain choices of the underlying system . unless one s desired dynamics is sufficiently close to some dynamics that was previously catalogued ( during one s investigation of the forward problem ) , one has very little information on how to set up the components and their interactions to achieve that desired dynamics . in addition , most of the work in these fields does not involve rl algorithms , and viewed as a context in which to design coins suffers from a need for hand - tailoring , and potentially lack of robustness and scalability . the field of ` swarm intelligence ' is concerned with systems that are modeled after social insect colonies , so that the different components of the system are queen , worker , soldier , etc . it can be viewed as ecological modeling in which the individual entities have extremely limited computing capacity and/or action sets , and in which there are very few types of entities . the premise of the field is that the rich behavior of social insect colonies arises not from the sophistication of any individual entity in the colony , but from the interaction among those entities . the objective of current research is to uncover kinds of interactions among the entity types that lead to pre - specified behavior of some sort . more speculatively , the study of social insect colonies may also provide insight into how to achieve learning in large distributed systems . this is because at the level of the individual insect in a colony , very little ( or no ) learning takes place . however across evolutionary time - scales the social insect species as a whole functions as if the various individual types in a colony had `` learned '' their specific functions . the `` learning '' is the direct result of natural selection . ( see the discussion on this topic in the subsection on ecological modeling . ) swarm intelligences have been used to adaptively allocate tasks in a mail company @xcite , solve the traveling salesman problem @xcite and route data efficiently in dynamic networks @xcite among others . despite this , such intelligences do not really constitute a general approach to designing coins . there is no general framework for adapting swarm intelligences to maximize particular world utility functions . accordingly , such intelligences generally need to be hand - tailored for each application . and after such tailoring , it is often quite a stretch to view the system as `` biological '' in any sense , rather than just a simple and _ a priori _ reasonable modification of some previously deployed system . the two main objectives of artificial life , closely related to one another , are understanding the abstract functioning and especially the origin of terrestrial life , and creating organisms that can meaningfully be called `` alive '' @xcite . the first objective involves formalizing and abstracting the mechanical processes underpinning terrestrial life . in particular , much of this work involves various degrees of abstraction of the process of self - replication @xcite . some of the more real - world - oriented work on this topic involves investigating how lipids assemble into more complex structures such as vesicles and membranes , which is one of the fundamental questions concerning the origin of life @xcite . many computer models have been proposed to simulate this process , though most suffer from overly simplifying the molecular morphology . more generally , work concerned with the origin of life can constitute an investigation of the functional self - organization that gives rise to life @xcite . in this regard , an important early work on functional self - organization is the _ lambda calculus _ , which provides an elegant framework ( recursively defined functions , lack of distinction between object and function , lack of architectural restrictions ) for studying computational systems @xcite . this framework can be used to develop an artificial chemistry `` function gas '' that displays complex cooperative properties @xcite . the second objective of the field of artificial life is less concerned with understanding the details of terrestrial life per se than of using terrestrial life as inspiration for how to design living systems . for example , motivated by the existence ( and persistence ) of computer viruses , several workers have tried to design an immune system for computers that will develop `` antibodies '' and handle viruses both more rapidly and more efficiently than other algorithms @xcite . more generally , because we only have one sampling point ( life on earth ) , it is very difficult to precisely formulate the process by which life emerged . by creating an artificial world inside a computer however , it is possible to study far more general forms of life @xcite . see also @xcite where the argument is presented that the richest way of approaching the issue of defining `` life '' is phenomenologically , in terms of self-@xmath22similar scaling properties of the system . cellular automata can be viewed as digital abstractions of physical gases @xcite . formally , they are discrete - time recurrent neural nets where the neurons live on a grid , each neuron has a finite number of potential states , and inter - neuron connections are ( usually ) purely local . ( see below for a discussion of recurrent neural nets . ) so the state update rule of each neuron is fixed and local , the next state of a neuron being a function of the current states of it and of its neighboring elements . the state update rule of ( all the neurons making up ) any particular cellular automaton specifies the mapping taking the initial configuration of the states of all of its neurons to the final , equilibrium ( perhaps strange ) attractor configuration of all those neurons . so consider the situation where we have a desired such mapping , and want to know an update rule that induces that mapping . this is a search problem , and can be viewed as similar to the inverse problem of how to design a coin to achieve a pre - specified global goal , albeit a `` coin '' whose nodal elements do not use rl algorithms . genetic algorithms are a special kind of search algorithm , based on analogy with the biological process of natural selection via recombination and mutation of a genome @xcite . although genetic algorithms ( and ` evolutionary computation ' in general ) have been studied quite extensively , there is no formal theory justifying genetic algorithms as search algorithms @xcite and few empirical comparisons with other search techniques . one example of a well - studied application of genetic algorithms is to ( try to ) solve the inverse problem of finding update rules for a cellular automaton that induce a pre - specified mapping from its initial configuration to its attractor configuration . to date , they have used this way only for extremely simple configuration mappings , mappings which can be trivially learned by other kinds of systems . despite the simplicity of these mappings , the use of genetic algorithms to try to train cellular automata to exhibit them has achieved little success @xcite . equilibrium statistical physics is concerned with the stable state character of large numbers of very simple physical objects , interacting according to well - specified local deterministic laws , with probabilistic noise processes superimposed @xcite . typically there is no sense in which such systems can be said to have centralized control , since all particles contribute comparably to the overall dynamics . aside from mesoscopic statistical physics , the numbers of particles considered are usually huge ( _ e.g. _ , @xmath23 ) , and the particles themselves are extraordinarily simple , typically having only a few degrees of freedom . moreover , the noise processes usually considered are highly restricted , being those that are formed by `` baths '' , of heat , particles , and the like . similarly , almost all of the field restricts itself to deterministic laws that are readily encapsulated in hamilton s equations ( schrodinger s equation and its field - theoretic variants for quantum statistical physics ) . in fact , much of equilibrium statistical physics is nt even concerned with the dynamic laws by themselves ( as for example is stochastic markov processes ) . rather it is concerned with invariants of those laws ( _ e.g. _ , energy ) , invariants that relate the states of all of the particles . trivially then , deterministic laws without such readily - discoverable invariants are outside of the purview of much of statistical physics . one potential use of statistical physics for coins involves taking the systems that statistical physics analyzes , especially those analyzed in its condensed matter variant ( _ e.g. _ , spin glasses @xcite ) , as simplified models of a class of coins . this approach is used in some of the analysis of the bar problem ( see above ) . it is used more overtly in ( for example ) the work of galam @xcite , in which the equilibrium coalitions of a set of `` countries '' are modeled in terms of spin glasses . this approach can not provide a general coin framework though . in addition to the restrictions listed above on the kinds of systems it considers , this is due to its not providing a general solution to arbitrary coin inversion problems , and to its not employing rl algorithms . another contribution that statistical physics can make is with the mathematical techniques it has developed for its own purposes , like mean field theory , self - averaging approximations , phase transitions , monte carlo techniques , the replica trick , and tools to analyze the thermodynamic limit in which the number of particles goes to infinity . although such techniques have not yet been applied to coins , they have been successfully applied to related fields . this is exemplified by the use of the replica trick to analyze two - player zero - sum games with random payoff matrices in the thermodynamic limit of the number of strategies in @xcite . other examples are the numeric investigation of iterated prisoner s dilemma played on a lattice @xcite , the analysis of stochastic games by expressing of deviation from rationality in the form of a `` heat bath '' @xcite , and the use of topological entropy to quantify the complexity of a voting system studied in @xcite . other quite recent work in the statistical physics literature is formally identical to that in other fields , but presents it from a novel perspective . a good example of this is @xcite , which is concerned with the problem of controlling a spatially extended system with a single controller , by using an algorithm that is identical to a simple - minded proportional rl algorithm ( in essence , a rediscovery of rl ) . much of the theory of physics can be cast as solving for the extremization of an actional , which is a functional of the worldline of an entire ( potentially many - component ) system across all time . the solution to that extremization problem constitutes the actual worldline followed by the system . in this way the calculus of variations can be used to solve for the worldline of a dynamic system . as an example , simple newtonian dynamics can be cast as solving for the worldline of the system that extremizes a quantity called the ` lagrangian ' , which is a function of that worldline and of certain parameters ( _ e.g. _ , the ` potential energy ' ) governing the system at hand . in this instance , the calculus of variations simply results in newton s laws . if we take the dynamic system to be a coin , we are assured that its worldline automatically optimizes a `` global goal '' consisting of the value of the associated actional . if we change physical aspects of the system that determine the functional form of the actional ( _ e.g. _ , change the system s potential energy function ) , then we change the global goal , and we are assured that our coin optimizes that new global goal . counter - intuitive physical systems , like those that exhibit braess paradox @xcite , are simply systems for which the `` world utility '' implicit in our human intuition is extremized at a point different from the one that extremizes the system s actional . the challenge in exploiting this to solve the coin design problem is in translating an arbitrary provided global goal for the coin into a parameterized actional . note that that actional must govern the dynamics of the physical coin , and the parameters of the actional must be physical variables in the coin , variables whose values we can modify . the field of active walker models @xcite is concerned with modeling `` walkers '' ( be they human walkers or instead simple physical objects ) crossing fields along trajectories , where those trajectories are a function of several factors , including in particular the trails already worn into the field . often the kind of trajectories considered are those that can be cast as solutions to actional extremization problems so that the walkers can be explicitly viewed as agents optimizing a private utility . one of the primary concerns with the field of active walker models is how the trails worn in the field change with time to reach a final equilibrium state . the problem of how to design the cement pathways in the field ( and other physical features of the field ) so that the final paths actually followed by the walkers will have certain desirable characteristics is then one of solving for parameters of the actional that will result in the desired worldline . this is a special instance of the inverse problem of how to design a coin . using active walker models this way to design coins , like action extremization in general , probably has limited applicability . also , it is not clear how robust such a design approach might be , or whether it would be scalable and exempt from the need for hand - tailoring . this subsection presents a `` catch - all '' of other fields that have little in common with one another except that they bear some relation to coins . an extremely well - researched body of work concerns the mathematical and numeric behavior of systems for which the probability distribution over possible future states conditioned on preceding states is explicitly provided . this work involves many aspects of monte carlo numerical algorithms @xcite , all of markov chains @xcite , and especially markov fields , a topic that encompasses the chapman - kolmogorov equations @xcite and its variants : liouville s equation , the fokker - plank equation , and the detailed - balance equation in particular . non - linear dynamics is also related to this body of work ( see the synopsis of iterated function systems below and the synopsis of cellular automata above ) , as is markov competitive decision processes ( see the synopsis of game theory above ) . formally , one can cast the problem of designing a coin as how to fix each of the conditional transition probability distributions of the individual elements of a stochastic field so that the aggregate behavior of the overall system is of a desired form . unfortunately , almost all that is known in this area instead concerns the forward problem , of inferring aggregate behavior from a provided set of conditional distributions . although such knowledge provides many `` bits and pieces '' of information about how to tackle the inverse problem , those pieces collectively cover only a very small subset of the entire space of tasks we might want the coin to perform . in particular , they tell us very little about the case where the conditional distribution encapsulates rl algorithms . the technique of iterated function systems @xcite grew out of the field of nonlinear dynamics @xcite . in such systems a function is repeatedly and recursively applied to itself . the most famous example is the logistic map , @xmath24 for some @xmath25 between 0 and 4 ( so that @xmath26 stays between 0 and 1 ) . more generally the function along with its arguments can be vector - valued . in particular , we can construct such functions out of affine transformations of points in a euclidean plane . iterated functions systems have been applied to image data . in this case the successive iteration of the function generically generates a fractal , one whose precise character is determined by the initial iteration-1 image . since fractals are ubiquitous in natural images , a natural idea is to try to encode natural images as sets of iterated function systems spread across the plane , thereby potentially garnering significant image compression . the trick is to manage the inverse step of starting with the image to be compressed , and determining what iteration-1 image(s ) and iterating function(s ) will generate an accurate approximation of that image . in the language of nonlinear dynamics , we have a dynamic system that consists of a set of iterating functions , together with a desired attractor ( the image to be compressed ) . our goal is to determine what values to set certain parameters of our dynamic system to so that the system will have that desired attractor . the potential relationship with coins arises from this inverse nature of the problem tackled by iterated function systems . if the goal for a coin can be cast as its relaxing to a particular attractor , and if the distributed computational elements are isomorphic to iterated functions , then the tricks used in iterated functions theory could be of use . although the techniques of iterated function systems might prove of use in designing coins , they are unlikely to serve as a generally applicable approach to designing coins . in addition , they do not involve rl algorithms , and often involve extensive hand - tuning . a recurrent neural net consists of a finite set of `` neurons '' each of which has a real - valued state at each moment in time . each neuron s state is updated at each moment in time based on its current state and that of some of the other neurons in the system . the topology of such dependencies constitute the `` inter - neuronal connections '' of the net , and the associated parameters are often called the `` weights '' of the net . the dynamics can be either discrete or continuous ( _ i.e. _ , given by difference or differential equations ) . recurrent nets have been investigated for many purposes @xcite . one of the more famous of these is associative memories . the idea is that given a pre - specified pattern for the ( states of the neurons in the ) net , there may exist inter - neuronal weights which result in a basin of attraction focussed on that pattern . if this is the case , then the net is equivalent to an associative memory , in that a complete pre - specified pattern across all neurons will emerge under the net s dynamics from any initial pattern that partially matches the full pre - specified pattern . in practice , one wishes the net to simultaneously possess many such pre - specified associative memories . there are many schemes for `` training '' a recurrent net to have this property , including schemes based on spin glasses @xcite and schemes based on gradient descent @xcite . as can the fields of cellular automata and iterated function systems , the field of recurrent neural nets can be viewed as concerning certain variants of coins . also like those other fields though , recurrent neural nets has shortcomings if one tries to view it as a general approach to a science of coins . in particular , recurrent neural nets do not involve rl algorithms , and training them often suffers from scaling problems . more generally , in practice they can be hard to train well without hand - tailoring . packet routing in a data network @xcite presents a particularly interesting domain for the investigation of coins . in particular , with such routing : + ( i ) the problem is inherently distributed ; + ( ii ) for all but the most trivial networks it is impossible to employ global control ; + ( iii ) the routers have only access to local information ( routing tables ) ; + ( iv ) it constitutes a relatively clean and easily modified experimental testbed ; and + ( v ) there are potentially major bottlenecks induced by ` greedy ' behavior on the part of the individual routers , which behavior constitutes a readily investigated instance of the tragedy of the commons ( toc ) . many of the approaches to packet routing incorporate a variant on rl @xcite . q routing is perhaps the best known such approach and is based on routers using reinforcement learning to select the best path @xcite . although generally successful , q routing is not a general scheme for inverting a global task . this is even true if one restricts attention to the problem of routing in data networks there exists a global task in such problems , but that task is directly used to construct the algorithm . a particular version of the general packet routing problem that is acquiring increased attention is the quality of service ( qos ) problem , where different communication packets ( voice , video , data ) share the same bandwidth resource but have widely varying importances both to the user and ( via revenue ) to the bandwidth provider . determining which packet has precedence over which other packets in such cases is not only based on priority in arrival time but more generally on the potential effects on the income of the bandwidth provider . in this context , rl algorithms have been used to determine routing policy , control call admission and maximize revenue by allocating the available bandwidth efficiently @xcite . many researchers have exploited the noncooperative game theoretic understanding of the toc in order to explain the bottleneck character of empirical data networks behavior and suggest potential alternatives to current routing schemes @xcite . closely related is work on various `` pricing''-based resource allocation strategies in congestable data networks @xcite . this work is at least partially based upon current understanding of pricing in toll lanes , and traffic flow in general ( see below ) . all of these approaches are particularly of interest when combined with the rl - based schemes mentioned just above . due to these factors , much of the current research on a general framework for coins is directed toward the packet - routing domain ( see next section ) . traffic congestion typifies the toc public good problem : everyone wants to use the same resource , and all parties greedily trying to optimize their use of that resource not only worsens global behavior , but also worsens _ their own _ private utility ( _ e.g. _ , if everyone disobeys traffic lights , everyone gets stuck in traffic jams ) . indeed , in the well - known braess paradox @xcite , keeping everything else constant including the number and destinations of the drivers but opening a new traffic path can _ increase _ everyone s time to get to their destination . ( viewing the overall system as an instance of the prisoner s dilemma , this paradox in essence arises through the creation of a novel ` defect - defect ' option for the overall system . ) greedy behavior on the part of individuals also results in very rich global dynamic patterns , such as stop and go waves and clusters @xcite . much of traffic theory employs and investigates tools that have previously been applied in statistical physics @xcite ( see subsection above ) . in particular , the spontaneous formation of traffic jams provides a rich testbed for studying the emergence of complex activity from seemingly chaotic states @xcite . furthermore , the dynamics of traffic flow is particular amenable to the application and testing of many novel numerical methods in a controlled environment @xcite . many experimental studies have confirmed the usefulness of applying insights gleaned from such work to real world traffic scenarios @xcite . finally , there are a number of other fields that , while either still nascent or not extremely closely related to coins , are of interest in coin design : * amorphous computing : * amorphous computing grew out of the idea of replacing traditional computer design , with its requirements for high reliability of the components of the computer , with a novel approach in which widespread unreliability of those components would not interfere with the computation @xcite . some of its more speculative aspects are concerned with `` how to program '' a massively distributed , noisy system of components which may consist in part of biochemical and/or biomechanical components @xcite . work here has tended to focus on schemes for how to robustly induce desired geometric dynamics across the physical body of the amorphous computer issue that are closely related to morphogenesis , and thereby lend credence to the idea that biochemical components are a promising approach . especially in its limit of computers with very small constituent components , amorphous computing also is closely related to the fields of nanotechnology @xcite and control of smart matter ( see below ) . * control of smart matter:*. as the prospect of nanotechnology - driven mechanical systems gets more concrete , the daunting problem of how to robustly control , power , and sustain protean systems made up of extremely large sets of nano - scale devices looms more important @xcite . if this problem were to be solved one would in essence have `` smart matter '' . for example , one would be able to `` paint '' an airplane wing with such matter and have it improve drag and lift properties significantly . * morphogenesis : * how does a leopard embryo get its spots , or a zebra embryo its stripes ? more generally , what are the processes underlying morphogenesis , in which a body plan develops among a growing set of initially undifferentiated cells ? these questions , related to control of the dynamics of chemical reaction waves , are essentially special cases of the more general question of how ontogeny works , of how the genotype - phenotype mapping is carried out in development . the answers involve homeobox ( as well as many other ) genes @xcite . under the presumption that the functioning of such genes is at least in part designed to facilitate genetic changes that increase a species fitness , that functioning facilitates solution of the inverse problem , of finding small - scale changes ( to dna ) that will result in `` desired '' large scale effects ( to body plan ) when propagated across a growing distributed system . * self organizing systems * the concept of self - organization and self - organized criticality @xcite was originally developed to help understand why many distributed physical systems are attracted to critical states that possess long - range dynamic correlations in the large - scale characteristics of the system . it provides a powerful framework for analyzing both biological and economic systems . for example , natural selection ( particularly punctuated equilibrium @xcite ) can be likened to self - organizing dynamical system , and some have argued it shares many the properties ( _ e.g. _ , scale invariance ) of such systems @xcite . similarly , one can view the economic order that results from the actions of human agents as a case of self - organization @xcite . the relationship between complexity and self - organization is a particularly important one , in that it provides the potential laws that allow order to arise from chaos @xcite . * small worlds ( 6 degrees of separation ) : * in many distributed systems where each component can interact with a small number of `` neighbors '' , an important problem is how to propagate information across the system quickly and with minimal overhead . on the one extreme the neighborhood topology of such systems can exist on a completely regular grid - like structure . on the other , the topology can be totally random . in either case , certain nodes may be effectively ` cut - off ' from other nodes if the information pathways between them are too long . recent work has investigated `` small worlds '' networks ( sometimes called 6 degrees of separation ) in which underlying grid - like topologies are `` doped '' with a scattering of long - range , random connections . it turns out that very little such doping is necessary to allow for the system to effectively circumvent the information propagation problem @xcite . * control theory : * adaptive control @xcite , and in particular adaptive control involving locally weighted rl algorithms @xcite , constitute a broadly applicable framework for controlling small , potentially inexactly modeled systems . augmented by techniques in the control of chaotic systems @xcite , they constitute a very successful way of solving the `` inverse problem '' for such systems . unfortunately , it is not clear how one could even attempt to scale such techniques up to the massively distributed systems of interest in coins . the next section discusses in detail some of the underlying reasons why the purely model - based versions of these approaches are inappropriate as a framework for coins . summarizing the discussion to this point , it is hard to see how any already extant scientific field can be modified to encompass systems meeting all of the requirements of coins listed at the beginning of section [ sec : lit ] . this is not too surprising , since none of those fields were explicitly designed to analyze coins . this section first motivates in general terms a framework that is explicitly designed for analyzing coins . it then presents the formal nomenclature of that framework . this is followed by derivations of some of the central theorems of that framework . finally , we present experiments that illustrate the power the framework provides for ensuring large world utility in a coin . what mathematics might one employ to understand and design coins ? perhaps the most natural approach , related to the stochastic fields work reviewed above , involves the following three steps : \1 ) first one constructs a detailed stochastic model of the coin s dynamics , a model parameterized by a vector @xmath27 . as an example , @xmath27 could fix the utility functions of the individual agents of the coin , aspects of their rl algorithms , which agents communicate with each other and how , etc . \2 ) next we solve for the function @xmath28 which maps the parameters of the model to the resulting stochastic dynamics . \3 ) cast our goal for the system as a whole as achieving a high expected value of some `` world utility '' . then as our final step we would have to solve the inverse problem : we would have to search for a @xmath27 which , via @xmath9 , results in a high value of e(world utility @xmath29 ) . let s examine in turn some of the challenges each of these three steps entrain : \i ) we are primarily interested in very large , very complex systems , which are noisy , faulty , and often operate in a non - stationary environment . moreover , our `` very complex system '' consists of many rl algorithms , all potentially quite complicated , all running simultaneously . clearly coming up with a detailed model that captures the dynamics of all of this in an accurate manner will often be extraordinarily difficult . moreover , unfortunately , given that the modeling is highly detailed , often the level of verisimilitude required of the model will be quite high . for example , unless the modeling of the faulty aspects of the system were quite accurate , the model would likely be `` brittle '' , and overly sensitive to which elements of the coin were and were not operating properly at any given time . \ii ) even for models much simpler than the ones called for in ( i ) , solving explicitly for the function @xmath9 can be extremely difficult . for example , much of markov chain theory is an attempt to broadly characterize such mappings . however as a practical matter , usually it can only produce potentially useful characterizations when the underlying models are quite inaccurate simplifications of the kinds of models produced in step ( i ) . \iii ) even if one can write down an @xmath9 , solving the associated inverse problem is often impossible in practice . \iv ) in addition to these difficulties , there is a more general problem with the model - based approach . we wish to perform our analysis on a `` high level '' . our thesis is that due to the robust and adaptive nature of the individual agents rl algorithms , there will be very broad , easily identifiable regions of @xmath27 space all of which result in excellent e(world utility @xmath29 ) , and that these regions will not depend on the precise learning algorithms used to achieve the low - level tasks ( cf . the list at the beginning of section [ sec : lit ] ) . to fully capitalize on this , one would want to be able to slot in and out different learning algorithms for achieving the low - level tasks without having to redo our entire analysis each time . however in general this would be possible with a model - based analysis only for very carefully designed models ( if at all ) . the problem is that the result of step ( 3 ) , the solution to the inverse problem , would have to concern aspects of the coin that are ( at least approximately ) invariant with respect to the precise low - level learning algorithms used . coming up with a model that has this property while still avoiding problems ( i - iii ) is usually an extremely daunting challenge . fortunately , there is an alternative approach which avoids the difficulties of detailed modeling . little modeling of any sort ever is used in this alternative , and what modeling does arise has little to do with dynamics . in addition , any such modeling is extremely high - level , intented to serve as a decent approximation to almost any system having `` reasonable '' rl algorithms , rather than as an accurate model of one particular system . we call any framework based on this alternative a * descriptive framework*. in such a framework one identifies certain * salient characteristics * of coins , which are characteristics of a coin s entire worldline that one strongly expects to find in coins that have large world utility . under this expectation , one makes the assumption that if a coin is explicitly modified to have the salient characteristics ( for example in response to observations of its run - time behavior ) , then its world utility will benefit . so long as the salient characteristics are ( relatively ) easy to induce in a coin , then this assumption provides a ready indirect way to cause that coin to have large world utility . an assumption of this nature is the central leverage point that a descriptive framework employs to circumvent detailed modeling . under it , if the salient characteristics can be induced with little or no modeling ( e.g. , via heuristics that are nt rigorously and formally justified ) , then they provide an indirect way to improve world utility without recourse to detailed modeling . in fact , since one does not use detailed modeling in a descriptive framework , it may even be that one does not have a fully rigorous mathematical proof that the central assumption holds in a particular system for one s choice of salient characteristics . one may have to be content with reasonableness arguments not only to justify one s scheme for inducing the salient characteristics , but for making the assumption that characteristics are correlated with large world utility in the first place . of course , the trick in the descriptive framework is to choose salient characteristics that both have a beneficial relationship with world utility and that one expects to be able to induce with relatively little detailed modeling of the system s dynamics . there exist many ways one might try to design a descriptive framework . in this subsection we present nomenclature needed for a ( very ) cursory overview of one of them . ( see @xcite for a more detailed exposition , including formal proofs . ) this overview concentrates on the four salient characteristics of intelligence , learnability , factoredness , and the wonderful life utility , all defined below . intelligence is a quantification of how well an rl algorithm performs . we want to do whatever we can to help those algorithms achieve high values of their utility functions . learnability is a characteristic of a utility function that one would expect to be well - correlated with how well an rl algorithm can learn to optimize it . a utility function is also factored if whenever its value increases , the overall system benefits . finally , wonderful life utility is an example of a utility function that is both learnable and factored . after the preliminary definitions below , this section formalizes these four salient characteristics , derives several theorems relating them , and illustrates in some computer experiments how those theorems can be used to help the system achieve high world utility . * 1 ) * we refer to an rl algorithm by which an individual component of the coin modifies its behavior as a * microlearning * algorithm . we refer to the initial construction of the coin , potentially based upon salient characteristics , as the coin * initialization*. we use the phrase * macrolearning * to refer to externally imposed run - time modifications to the coin which are based on statistical inference concerning salient characteristics of the running coin . * 2 ) * for convenience , we take time , @xmath17 , to be discrete and confined to the integers , _ z_. when referring to coin initialization , we implicitly have a lower bound on @xmath17 , which without loss of generality we take to be less than or equal to @xmath30 . * 3 ) * all variables that have any effect on the coin are identified as components of euclidean - vector - valued * states * of various discrete * nodes*. as an important example , if our coin consists in part of a computational `` agent '' running a microlearning algorithm , the precise configuration of that agent at any time @xmath17 , including all variables in its learning algorithm , all actions directly visible to the outside world , all internal parameters , all values observed by its probes of the surrounding environment , etc . , all constitute the state vector of a node representing that agent . we define @xmath31 to be a vector in the euclidean vector space @xmath32 , where the components of @xmath31 give the state of node @xmath33 at time @xmath17 . the @xmath4th component of that vector is indicated by @xmath34 . * observation 3.1 : * in practice , many coins will involve variables that are most naturally viewed as discrete and symbolic . in such cases , we must exercise some care in how we choose to represent those variables as components of euclidean vectors . there is nothing new in this ; the same issue arises in modern work on applying neural nets to inherently symbolic problems . in our coin framework , we will usually employ the same resolution of this issue employed in neural nets , namely representing the possible values of the discrete variable with a unary representation in a euclidean space . just as with neural nets , values of such vectors that do not lie on the vertices of the unit hypercube are not meaningful , strictly speaking . fortunately though , just as with neural nets , there is almost always a most natural way to extend the definitions of any function of interest ( like world utility ) so that it is well - defined even for vectors not lying on those vertices . this allows us to meaningfully define partial derivatives of such functions with respect to the components of @xmath35 , partial derivatives that we will evaluate at the corners of the unit hypercube . * 4 ) * for notational convenience , we define @xmath36 to be the vector of the states of all nodes at time @xmath17 ; @xmath37 to be the vector of the states of all nodes other than @xmath33 at time @xmath17 ; and @xmath38 to be the entire vector of the states of all nodes at all times . @xmath39 is infinite - dimensional in general , and usually assumed to be a hilbert space . we will often assume that all spaces @xmath40 over all times @xmath17 are isomorphic to a space @xmath41 , i.e. , @xmath39 is a cartesian product of copies of @xmath41 . also for notational convenience , we define gradients using @xmath42-shorthand . so for example , @xmath43 is the vector of the partial derivative of @xmath44 with respect to the components of @xmath45 . also , we will sometimes treat the symbol `` @xmath17 '' specially , as delineating a range of components of @xmath35 . so for example an expression like `` @xmath46 '' refers to all components @xmath45 with @xmath47 . * 5 ) * to avoid confusion with the other uses of the comma operator , we will often use @xmath48 rather than @xmath49 to indicate the vector formed by concatenating the two ordered sets of vector components @xmath50 and @xmath51 . for example , @xmath52 refers to the vector formed by concatenating those components of the worldline @xmath35 involving node @xmath33 for times less than 0 with those components involving node @xmath53 that have times greater than 0 . * 6 ) * we take the universe in which our coin operates to be completely deterministic . this is certainly the case for any coin that operates in a digital system , even a system that emulates analog and/or stochastic processes ( _ e.g. _ , with a pseudo - random number generator ) . more generally , this determinism reflects the fact that since the real world obeys ( deterministic ) physics , @xmath54 real - world system , be it a coin or something else , is , ultimately , embedded in a deterministic system . the perspective to be kept in mind here is that of nonlinear time - series analysis . a physical time series typically reflects a few degrees of freedom that are projected out of the underlying space in which the full system is deterministically evolving , an underlying space that is actually extremely high - dimensional . this projection typically results in an illusion of stochasticity in the time series . * 7 ) * formally , to reflect this determinism , first we bundle all variables we are not directly considering but which nonetheless affect the dynamics of the system as components of some catch - all * environment node*. so for example any `` noise processes '' and the like affecting the coin s dynamics are taken to be inputs from a deterministic , very high - dimensional environment that is potentially chaotic and is never directly observed @xcite . given such an environment node , we then stipulate that for all @xmath55 such that @xmath56 , @xmath45 sets @xmath57 uniquely . * observation 7.1 : * when nodes are `` computational devices '' , often we must be careful to specify the physical extent of those devices . such a node may just be the associated cpu , or it may be that cpu together with the main ram , or it may include an external storage device . almost always , the border of the device @xmath33 will end before any external system that @xmath33 is `` observing '' begins . this means that since at time @xmath17 @xmath33 only knows the value of @xmath31 , its `` observational knowledge '' of that external system is indirect . that knowledge reflects a coupling between @xmath31 and @xmath58 , a coupling that is induced by the dynamical evolution of the system from preceding moments up to the time @xmath17 . if the dynamics does not force such a coupling , then @xmath33 has no observational knowledge of the outside world . * 8) * we express the dynamics of our system by writing @xmath59 . ( in this paper there will be no need to be more precise and specify the precise dependency of @xmath60 on @xmath17 and/or @xmath61 . ) we define @xmath62 to be a set of constraint equations enforcing that dynamics , and also , more generally , fixing the entire manifold @xmath63 of vectors @xmath64 that we consider to be ` allowed ' . so @xmath63 is a subset of the set of all @xmath65 that are consistent with the deterministic laws governing the coin , _ i.e. _ , that obey @xmath66 . we generalize this notation in the obvious way , so that ( for example ) @xmath67 is the manifold consisting of all vectors @xmath68 that are projections of a vector in @xmath63 . * observation 8.1 : * note that @xmath67 is parameterized by @xmath69 , due to determinism . note also that whereas @xmath60 is defined for any argument of the form @xmath70 for some @xmath17 ( _ i.e. _ , we can evolve any point forward in time ) , in general not all @xmath71 lie in @xmath72 . in particular , there may be extra restrictions constraining the possible states of the system beyond those arising from its need to obey the relevant dynamical laws of physics . finally , whenever trying to express a coin in terms of the framework presented here , it is a good rule to try to write out the constraint equations explicitly to check that what one has identified as the space @xmath40 contains all quantities needed to uniquely fix the future state of the system . * observation 8.2 : * we do not want to have @xmath73 be the phase space of every particle in the system . we will instead usually have @xmath73 consist of variables that , although still evolving deterministically , exist at a larger scale of granularity than that of individual particles ( _ e.g. _ , thermodynamic variables in the thermodynamic limit ) . however we will often be concerned with physical systems obeying entropy - driven dynamic processes that are contractive at this high level of granularity . examples are any of the many - to - one mappings that can occur in digital computers , and , at a finer level of granularity , any of the error - correcting processes in the electronics of such a computer that allow it to operate in a digital fashion . accordingly , although the dynamics of our system will always be deterministic , it need not be invertible . * observation 8.3 : * intuitively , in our mathematics , all behavior across time is pre - fixed . the coin is a single fixed worldline through @xmath39 , with no `` unfolding of the future '' as the die underlying a stochastic dynamics get cast . this is consistent with the fact that we want the formalism to be purely descriptive , relating different properties of any single , fixed coin s history . we will often informally refer to `` changing a node s state at a particular time '' , or to a microlearner s `` choosing from a set of options '' , and the like . formally , in all such phrases we are really comparing different worldlines , with the indicated modification distinguishing those worldlines . * observation 8.4 : * since the dynamics of any real - world coin is deterministic , so is the dynamics of any component of the coin , and in particular so is any learning algorithm running in the coin , ultimately . however that does not mean that those deterministic components of the coin are not allowed to be `` based on '' , or `` motivated by '' stochastic concepts . the _ motivation _ behind the algorithms run by the components of the coin does not change their underlying nature . indeed , in our experiments below , we explicitly have the reinforcement learning algorithms that are trying to maximize private utility operate in a ( pseudo- ) probabilistic fashion , with pseudo - random number generators and the like . more generally , the deterministic nature of our framework does not preclude our superimposing probabilistic elements on top of that framework , and thereby generating a stochastic extension of our framework . exactly as in statistical physics , a stochastic nature can be superimposed on our space of deterministic worldlines , potentially by adopting a degree of belief perspective on `` what probability means '' @xcite . indeed , the macrolearning algorithms we investigate below implicitly involve such a superimposing ; they implicitly assume a probabilistic coupling between the ( statistical estimate of the ) correlation coefficient connecting the states of a pair of nodes and whether those nodes are in the one another s `` effect set '' . similarly , while it does not salient characteristics that involve probability distributions , the descriptive framework does not preclude such characteristics either . as an example , the `` intelligence '' of an agent s particular action , formally defined below , measures the fraction of alternative actions an agent could have taken that would have resulted in a lower utility value . to define such a fraction requires a measure across the space of such alternative actions , even if only implicitly . accordingly , intelligence can be viewed as involving a probability distribution across the space of potential actions . in this paper though , we concentrate on the mathematics that obtains before such probabilistic concerns are superimposed . whereas the deterministic analysis presented here is related to game - theoretic structures like nash equilibria , a full - blown stochastic extension would in some ways be more related to structures like correlated equilibria @xcite . * 9 ) * formally , there is a lot of freedom in setting the boundary between what we call `` the coin '' , whose dynamics is determined by @xmath63 , and what we call `` macrolearning '' , which constitutes perturbations to the coin instigated from `` outside the coin '' , and which therefore is @xmath74 reflected in @xmath63 . as an example , in much of this paper , we have clearly specified microlearners which are provided fixed private utility functions that they are trying to maximize . in such cases usually we will implicitly take @xmath63 to be the dynamics of the system , microlearning and all , _ for fixed private utilities _ that are specified in @xmath35 . for example , @xmath35 could contain , for each microlearner , the bits in an associated computer specifying the subroutine that that microlearner can call to evaluate what its private utility would be for some full worldline @xmath35 . macrolearning overrides @xmath63 , and in this situation it refers ( for example ) to any statistical inference process that modifies the private utilities at run - time to try to induce the desired salient characteristics . concretely , this would involve modifications to the bits \{@xmath75 } specifying each microlearner @xmath4 s private utility , modifications that are @xmath74 accounted for in @xmath63 , and that are potentially based on variables that are not reflected in @xmath39 . since @xmath63 does not reflect such macrolearning , when trying to ascertain @xmath63 based on empirical observation ( as for example when determining how best to modify the private utilities ) , we have to take care to distinguish which part of the system s observed dynamics is due to @xmath63 and which part instead reflects externally imposed modifications to the private utilities . more generally though , other boundaries between the coin and macrolearning - based perturbations to it are possible , reflecting other definitions of @xmath39 , and other interpretations of the elements of each @xmath64 . for example , say that under the perspective presented in the previous paragraph , the private utility is a function of some components @xmath16 of @xmath35 , components that do not include the \{@xmath75}. now modify this perspective so that in addition to the dynamics of other bits , @xmath63 also encapsulates the dynamics of the bits \{@xmath75}. having done this , we could still view each private utility as being fixed , but rather than take the bits \{@xmath75 } as `` encoding '' the subroutine that specifies the private utility of microlearner @xmath4 , we would treat them as `` parameters '' specifying the functional dependence of the ( fixed ) private utility on the components of @xmath35 . in other words , formally , they constitute an extra set of arguments to @xmath4 s private utility , in addition to the arguments @xmath16 . alternatively , we could simply say that in this situation our private utilities are time - indexed , with @xmath4 s private utility at time @xmath17 determined by \{@xmath76 } , which in turn is determined by evolution under @xmath63 . under either interpretation of private utility , any modification under @xmath63 to the bits specifying @xmath4 s utility - evaluation subroutine constitutes dynamical laws by which the parameters of @xmath4 s microlearner evolves in time . in this case , macrolearning would refer to some further removed process that modifies the evolution of the system in a way not encapsulated in @xmath63 . for such alternative definitions of @xmath63/@xmath39 , we have a different boundary between the coin and macrolearning , and we must scrutinize different aspects of the coin s dynamics to infer @xmath63 . whatever the boundary , the mathematics of the descriptive framework , including the mathematics concerning the salient characteristics , is restricted to a system evolving according to @xmath63 , and explicitly does not account for macrolearning . this is why the strategy of trying to improve world utility by using macrolearning to try to induce salient characteristics is almost always ultimately based on an assumption rather than a proof . * 10 ) * we are provided with some von neumann * world utility * @xmath77 that ranks the various conceivable worldlines of the coin . note that since the environment node is never directly observed , we implicitly assume that the world utility is not directly ( ! ) a function of its state . our mathematics will not involve @xmath78 alone , but rather the relationship between @xmath78 and various sets of * personal utilities * @xmath79 . intuitively , as discussed below , for many purposes such personal utilities are equivalent to arbitrary `` virtual '' versions of the private utilities mentioned above . in particular , it is only private utilities that will occur within any microlearning computer algorithms that may be running in the coin as manifested in @xmath63 . personal utilities are external mathematical constructions that the coin framework employs to analyze the behavior of the system . they can be involved in learning processes , but only as tools that are employed outside of the coin s evolution under @xmath63 , _ i.e. _ , only in macrolearning . ( for example , analysis of them can be used to modify the private utilities . ) * observation 10.1 : * these utility definitions are very broad . in particular , they do not require casting of the utilities as discounted sums . note also that our world utility is not indexed by @xmath17 . again reflecting the descriptive , worldline character of the formalism , we simply assign a single value to an entire worldline of the system , implicitly assuming that one can always say which of two candidate worldlines are preferable . so given some `` present time '' @xmath80 , issues like which of two `` potential futures '' @xmath81 , @xmath82 is preferable are resolved by evaluating the relevant utility at two associated points @xmath35 and @xmath83 , where the @xmath84 components of those points are the futures indicated , and the two points share the same ( usually implicit ) @xmath85 `` past '' components . this time - independence of @xmath78 automatically avoids formal problems that can occur with general ( _ i.e. _ , not necessarily discounted sum ) time - indexed utilities , problems like having what s optimal at one moment in time conflict with what s optimal at other moments in time .. the effects of the actions by the nodes , adn therefore whether those actions are `` optimal '' or not , depends on the future actions of the nodes . however if they too are to be `` optimal '' , according to their world - utility , those future actions will depend on _ their _ futures . so we have a potentially infinite regress of differing stipulations of what `` optimal '' actions at time @xmath17 entails . ] for personal utilities such formal problems are often irrelevant however . before we begin our work , we as coin designers must be able to rank all possible worldlines of the system at hand , to have a well - defined design task . that is why world utility can not be time - indexed . however if a particular microlearner s goal keeps changing in an inconsistent way , that simply means that that microlearner will grow `` confused '' . from our perspective as coin designers , there is nothing _ a priori _ unacceptable about such confusion . it may even result in better performance of the system as a whole , in whic case we would actually want to induce it . nonetheless , for simplicity , in most of this paper we will have all @xmath86 be independent of @xmath17 , just like world utility . world utility is defined as that function that we are ultimately interested in optimizing . in conventional rl it is a discounted sum , with the sum starting at time @xmath17 . in other words , conventional rl has a time - indexed world utility . it might seem that in this at least , conventional rl considers a case that has more generality than that of the coin framework presented here . ( it obviously has less generality in that its world utility is restricted to be a discounted sum . ) in fact though , the apparent time - indexing of conventional rl is illusory , and the time - dependent discounted sum world utilty of conventional rl is actually a special case of the non - time - indexed world utility of our coin framework . to see this formally , consider any ( time - independent ) world utility @xmath87 that equals @xmath88 for some function @xmath89 and some positive constant @xmath90 with magnitude less than 1 . then for any @xmath91 and any @xmath83 and @xmath92 where @xmath93 , @xmath94 = sgn[\sum_{t=0}^{\infty } \gamma^t r(\underline{\zeta}'_{,t } ) - \sum_{t=0}^{\infty } \gamma^t r(\underline{\zeta''}_{,t})]$ ] . conventional rl merely expresses this in terms of time - dependent utilities @xmath95 by writing @xmath94 = sgn[u_{t'}(\underline{\zeta } ' ) - u_{t'}(\underline{\zeta}'')]$ ] for all @xmath61 . since utility functions are , by definition , only unique up to the relative orderings they impose on potential values of their arguments , we see that conventional rl s use of a time - dependent discounted sum world utility @xmath96 is identical to use of a particular time - independent world utility in our coin framework . * 11 ) * as mentioned above , there may be variables in each node s state which , under one particular interpretation , represent the `` utility functions '' that the associated microlearner s computer program is trying to extremize . when there are such components of @xmath35 , we refer to the utilities they represent as * private utilities*. however even when there are private utilities , formally we allow the personal utilities to differ from them . the personal utility functions \{@xmath97 } do not exist `` inside the coin '' ; they are not specified by components of @xmath35 . this separating of the private utilities from the \{@xmath97 } will allow us to avoid the teleological problem that one may not always be able to explicitly identify `` the '' private utility function reflected in @xmath35 such that a particular computational device can be said to be a microlearner `` trying to increase the value of its private utility '' . to the degree that we can couch the theorems purely in terms of personal rather than private utilities , we will have successfully adopted a purely behaviorist approach , without any need to interpret what a computational device is `` trying to do '' . despite this formal distinction though , often we will implicitly have in mind deploying the personal utilities onto the microlearners as their private utilities , in which case the terms can usually be used interchangeably . the context should make it clear when this is the case . we will need to quantify how well the entire system performs in terms of @xmath78 . to do this requires a measure of the performance of an arbitrary worldline @xmath35 , for an arbitrary utility function , under arbitrary dynamic laws @xmath63 . formally , such a measure is a mapping from three arguments to @xmath98 . such a measure will also allow us to quantify how well each microlearner performs in purely behavioral terms , in terms of its personal utility . ( in our behaviorist approach , we do not try to make specious distinctions between whether a microlearner s performance is due to its level of `` innate sophistication '' , or rather due to dumb luck all that matters is the quality of its behavior as reflected in its utility value for the system s worldline . ) this behaviorism in turn will allow us to avoid having private utilities explicitly arise in our theorems ( although they still arise frequently in pedagogical discussion ) . even when private utilities exist , there will be no formal need to explicitly identify some components of @xmath35 as such utilities . assuming a node s microlearner is competent , the fact that it is trying to optimize some particular private utility @xmath99 will be manifested in our performance measure s having a large value at @xmath35 for @xmath63 for that utility @xmath99 . the problem of how to formally define such a performance measure is essentially equivalent to the problem of how to quantify bounded rationality in game theory . some of the relevant work in game theory , for example that involving ` trembling hand equilibria ' or ` @xmath100 equilibria ' @xcite is concerned with refinements or modifications of nash equilibria ( see also @xcite ) . rather than a behaviorist approach , such work adopts a strongly teleological perspective on rationality . in general , such work is only applicable to those situations where the rationality is bounded due to the precise causal mechanisms investigated in that work . most of the other game - theoretic work first models ( ! ) the microlearner , as some extremely simple computational device ( _ e.g. _ , a deterministic finite automaton ( dfa ) . one then assumes that the microlearner performs perfectly for that device , so that one can measure that learner s performance in terms of some computational capacity measure of the model ( _ e.g. _ , for a dfa , the number of states of that dfa ) @xcite . however , if taken as renditions of real - world computer - based microlearners never mind human microlearners the models in this approach are often extremely abstracted , with many important characteristics of the real learners absent or distorted . in addition , there is little reason to believe that any results arising from this approach would not be highly dependent on the model choice and on the associated representation of computational capacity . yet another disadvantage is that this approach concentrates on perfect , fully rational behavior of the microlearners , within their computational restrictions . we would prefer a less model - dependent approach , especially given our wish that the performance measure be based solely on the utility function at hand , @xmath35 , and @xmath63 . now we do nt want our performance measure to be a `` raw '' utility value like @xmath101 , since that is not invariant with respect to monotonic transformations of @xmath102 . similarly , we do nt want to penalize the microlearner for not achieving a certain utility value if that value was impossible to achieve not due to the microlearner s shortcomings , but rather due to @xmath63 and the actions of other nodes . a natural way to address these concerns is to generalize the game - theoretic concept of `` best - response strategy '' and consider the problem of how well @xmath33 performs _ given the actions of the other nodes_. such a measure would compare the utility ultimately induced by each of the possible states of @xmath33 at some particular time , which without loss of generality we can take to be 0 , to that induced by the actual state @xmath103 . in other words , we would compare the utility of the actual worldline @xmath35 to those of a set of alternative worldlines @xmath83 , where @xmath104 , and use those comparisons to quantify the quality of @xmath33 s performance . now we are only concerned with comparing the effects of replacing @xmath35 with @xmath83 on @xmath105 contributions to the utility . but if we allow arbitrary @xmath106 , then in and of themselves the difference between those past components of @xmath83 and those of @xmath35 can modify the value of the utility , regardless of the effects of any difference in the future components . our presumption is that for many coins of interest we can avoid this conundrum by restricting attention to those @xmath83 where @xmath106 differs from @xmath107 only in the internal parameters of @xmath33 s microlearner , differences that only at times @xmath108 manifest themselves in a form the utility is concerned with . ( in game - theoretic terms , such `` internal parameters '' encode full extensive - form strategies , and we only consider changes to the vertices at or below the @xmath109 level in the tree of an extensive - form strategy . ) although this solution to our conundrum is fine when we can apply it , we do nt want to restrict the formalism so that it can only concern systems having computational algorithms which involve a clearly pre - specified set of extensive strategy `` internal parameters '' and the like . so instead , we formalize our presumption behaviorally , even for computational algorithms that do not have explicit extensive strategy internal parameters . since changing the internal parameters does nt affect the @xmath110 components of @xmath111 _ that the utility is concerned with _ , and since we are only concerned with changes to @xmath35 that affect the utility , we simply elect to not change the @xmath110 values of the internal parameters of @xmath111 at all . in other words , we leave @xmath112unchanged . the advantage of this stipulation is that we can apply it just as easily whether @xmath33 does or does nt have any `` internal parameters '' in the first place . so in quantifying the performance of @xmath33 for behavior given by @xmath35 we compare @xmath35 to a set of @xmath83 , a set restricted to those @xmath83 sharing @xmath35 s past : @xmath113 , @xmath114 , and @xmath115 . since @xmath116 is free to vary ( reflecting the possible changes in the state of @xmath33 at time 0 ) while @xmath106 is not , @xmath117 , in general . we may even wish to allow @xmath118 in certain circumstances . ( recall that @xmath63 may reflect other restrictions imposed on allowed worldlines besides adherence to the underlying dynamical laws , so simply obeying those laws does not suffice to ensure that a worldline lies on @xmath63 . ) in general though , our presumption is that as far as utility values are concerned , considering these dynamically impossible @xmath83 is equivalent to considering a more restricted set of @xmath83 with `` modified internal parameters '' , all of which are @xmath119 . we now present a formalization of this performance measure . given @xmath63 and a measure @xmath120 demarcating what points in @xmath121 we are interested in , we define the ( @xmath122 ) * intelligence * for node @xmath33 of a point @xmath35 with respect to a utility @xmath99 as follows : @xmath123 \cdot \delta(\underline{\zeta}'_{\;\hat{}\eta,0 } - \underline{\zeta}_{\;\hat{}\eta,0})\ ] ] where @xmath124 is the heaviside theta function which equals 0 if its argument is below 0 and equals 1 otherwise , @xmath125 is the dirac delta function , and we assume that @xmath126 . intuitively , @xmath127 measures the fraction of alternative states of @xmath33 which , if @xmath33 had been in those states at time 0 , would either degrade or not improve @xmath33 s performance ( as measured by @xmath99 ) . sometimes in practice we will only want to consider changes in those components of @xmath103 that we consider as `` free to vary '' , which means in particular that those changes are consistent with @xmath63 and the state of the external world , @xmath128 . ( this consistency ensures that @xmath33 s observational information concerning the external world is correct ; see observation 7.1 above . ) such a restriction means that even though @xmath129 may not be consistent with @xmath63 and @xmath107 , by itself it is still consistent with @xmath63 ; in quantifying the quality of a particular @xmath103 . so we do nt compare our point to other @xmath129 that are physically impossible , no matter what the past is . any such restrictions on what changes we are considering are reflected implicitly in intelligence , in the measure @xmath130 . as an example of intelligence , consider the situation where for each player @xmath33 , the support of the measure @xmath131 extends over all possible actions that @xmath33 could take that affect the ultimate value of its personal utility , @xmath97 . in this situation we recover conventional full rationality game theory involving nash equilibria , as the analysis of scenarios in which the intelligence of each player @xmath33 with respect to @xmath97 equals 1 . whose components need not all equal 1 . many of the theorems of conventional game theory can be directly carried over to such bounded - rational games @xcite by redefining the utility functions of the players . in other words , much of conventional full rationality game theory applies even to games with bounded rationality , under the appropriate transformation . this result has strong implications for the legitimacy of the common criticism of modern economic theory that its assumption of full rationality does not hold in the real world , implications that extend significantly beyond the sonnenschein - mantel - debreu theorem equilibrium aggregate demand theorem @xcite . ] as an alternative , we could for each @xmath33 restrict @xmath131 to some limited `` set of actions that @xmath33 actively considers '' . this provides us with an `` effective nash equilibrium '' at the point @xmath35 where each @xmath132 equals 1 , in the sense that _ as far it s concerned _ , each player @xmath33 has played a best possible action at such a point . as yet another alternative , we could restrict each @xmath131 to some infinitesimal neighborhood about @xmath129 , and thereby define a `` local nash equilibrium '' by having @xmath133 for each player @xmath33 . in general , competent greedy pursuit of private utility @xmath99 by the microlearner controlling node @xmath33 means that the intelligence of @xmath33 for personal utility @xmath99 , @xmath127 , is close to 1 . accordingly , we will often refer interchangeably to a capable microlearner s `` pursuing private utility @xmath99 '' , and to its having high intelligence for personal utility @xmath99 . alternatively , if the microlearner for node @xmath33 is incompetent , then it may even be that `` by luck '' its intelligence for some personal utility \{@xmath102 } exceeds its intelligence for the different private utility that it s actually trying to maximize , @xmath134 . say that we expect that a particular microlearner is `` smart '' , in that it is more likely to have high rather than low intelligence . we can model this by saying that given a particular @xmath135 , the conditional probability that @xmath136 is a monotonically increasing function of @xmath137 . since for a given @xmath135 the intelligence @xmath138 is a monotonically increasing function of @xmath102 , this modelling assumption means that the probability that @xmath136 is a monotonically increasing function of @xmath139 . an alternative weaker model is to only stipulate that the probability of having a particular pair @xmath140 with @xmath138 equal to @xmath141 is a monotonically increasing function of @xmath141 . ( this probability is an integral over a joint distribution , rather than a conditional distribution , as in the original model . ) in either case , the `` better '' the microlearner , the more tightly peaked the associated probability distribution over intelligence values is . any two utility functions that are related by a monotonically increasing transformation reflect the same preference ordering over the possible arguments of those functions . since it is only that ordering that we are ever concerned with , we would like to remove this degeneracy by `` normalizing '' all utility functions . in other words , we would like to reduce any equivalence set of utility functions that are monotonic transformations of one another to a canonical member of that set . to see what this means in the coin context , fix @xmath142 . viewed as a function from @xmath143 , @xmath144 is itself a utility function , one that is a monotonically increasing function of @xmath99 . ( it says how well @xmath33 would have performed for all vectors @xmath111 . ) accordingly , the integral transform taking @xmath99 to @xmath144 is a ( contractive , non - invertible ) mapping from utilities to utilities . applied to any member of a utility in @xmath99 s equivalence set , this mapping produces the same image utility , one that is also in that equivalence set . it can be proven that any mapping from utilities to utilities that has this and certain other simple properties must be such an integral transform . in this , intelligence is the unique way of `` normalizing '' von neumann utility functions . for those conversant with game theory , it is worth noting some of the interesting aspects that ensue from this normalizing nature of intelligences . at any point @xmath35 that is a nash equilibrium in the set of personal utilities \{@xmath97 } , all intelligences @xmath132 must equal 1 . since that is the maximal value any intelligence can take on , a nash equilibrium in the \{@xmath97 } is a pareto optimal point in the associated intelligences ( for the simple reason that no deviation from such a @xmath35 can raise any of the intelligences ) . conversely , if there exists at least one nash equilibrium in the \{@xmath97 } , then there is not a pareto optimal point in the \{@xmath132 } that is not a nash equilibrium . now restrict attention to systems with only a single instant of time , _ i.e. _ , single - stage games . also have each of the ( real - valued ) components of each @xmath145 be a mixing component of an associated one of @xmath33 s potential strategies for some underlying finite game . then have @xmath146 be the associated expected payoff to @xmath33 . ( so the payoff to @xmath33 of the underlying pure strategies is given by the values of @xmath146 when @xmath35 is a unit vector in the space @xmath147 of @xmath33 s possible states . ) then we know that there must exist at least one nash equilibrium in the \{@xmath97}. accordingly , in this situation the set of nash equilibria in the \{@xmath97 } is identical to the set of points that are pareto optimal in the associated intelligences . ( see eq . 5 in the discussion of factored systems below . ) intelligence can be a difficult quantity to work with , unfortunately . as an example , fix @xmath33 , and consider any ( small region centered about some ) @xmath35 along with some utility @xmath99 , where @xmath35 is not a local maximum of @xmath99 . then by increasing the values @xmath99 takes on in that small region we will increase the intelligence @xmath127 . however in doing this we will also necessarily @xmath148 the intelligence at points outside that region . so intelligence has a non - local character , a character that prevents us from directly modifying it to ensure that it is simultaneously high for any and all @xmath35 . a second , more general problem is that without specifying the details of a microlearner , it can be extremely difficult to predict which of two private utilities the microlearner will be better able to learn . indeed , even @xmath149 the details , making that prediction can be nearly impossible . so it can be extremely difficult to determine what private utility intelligence values will accrue to various choices of those private utilities . in other words , macrolearning that involves modifying the private utilities to try to directly increase intelligence with respect to those utilities can be quite difficult . fortunately , we can circumvent many of these difficulties by using a proxy for ( private utility ) intelligence . although we expect its value usually to be correlated with that of intelligence in practice , this proxy does not share intelligence s non - local nature . in addition , the proxy does not depend heavily on the details of the microlearning algorithms used , _ i.e. _ , it is fairly independent of those aspects of @xmath63 . intuitively , this proxy can be viewed as a `` salient characteristic '' for intelligence . we motivate this proxy by considering having @xmath150 for all @xmath33 . if we try to actually use these \{@xmath102 } as the microlearners private utilities , particularly if the coin is large , we will invariably encounter a very bad signal - to - noise problem . for this choice of utilities , the effects of the actions taken by node @xmath33 on its utility may be `` swamped '' and effectively invisible , since there are so many other processes going into determining @xmath78 s value . this makes it hard for @xmath33 to discern the echo of its actions and learn how to improve its private utility . it also means that @xmath33 will find it difficult to decide how best to act once learning has completed , since so much of what s important to @xmath33 is decided by processes outside of @xmath33s immediate purview . in such a scenario , there is nothing that @xmath33 s microlearner can do to reliably achieve high intelligence . in addition to this `` observation - driven '' signal / noise problem , there is an `` action - driven '' one . for reasons discussed in observation 7.1 above , we can define a distribution @xmath151 reflecting what @xmath33 does / doesnt know concerning the actual state of the outside world @xmath152 at time 0 . if the node @xmath33 chooses its actions in a bayes - optimal manner , then @xmath153 $ ] , where @xmath141 runs over the allowed action components of @xmath33 at time 0 . since this will differ from @xmath154 $ ] in general , this bayes - optimal node s intelligence will be less than 1 for the particular @xmath35 at hand , in general . moreover , the less @xmath99 s ultimate value ( after the application of @xmath63 , etc . ) depends on @xmath135 , the smaller the difference in these two argmax - based @xmath141 s , and therefore the higher the intelligence of @xmath33 , in general.s ultimate value to not depend on @xmath135 . ] we would like a measure of @xmath99 that captures these efects , but without depending on function maximization or any other detailed aspects of how the node determines its actions . one natural way to do this is via the * ( utility ) learnability * : given a measure @xmath131 restricted to a manifold @xmath63 , the ( @xmath155 ) utility learnability of a utility @xmath99 for a node @xmath33 at @xmath35 is : @xmath156 * intelligence learnability * is defined the same way , with @xmath157 replaced by @xmath158 . note that any affine transformation of @xmath99 has no effect on either the utility learnability @xmath159 or the associated intelligence learnability , @xmath160 . the integrand in the numerator of the definition of learnability reflects how much of the change in @xmath99 that results from replacing @xmath129 with @xmath161 is due to the change in @xmath33 s @xmath122 state ( the `` signal '' ) . the denominator reflects how much of the change in @xmath99 that results from replacing @xmath35 with @xmath83 is due to the change in the @xmath162 states of nodes other than @xmath33 ( the `` noise '' ) . so learnability quantifies how easy it is for the microlearner to discern the `` echo '' of its behavior in the utility function @xmath99 . our presumption is that the microlearning algorithm will achieve higher intelligence if provided with a more learnable private utility . intuitively , the ( utility ) * differential learnability * of @xmath99 at a point @xmath35 is the learnability with @xmath130 restricted to an infinitesimal ball about @xmath35 . we formalize it as the following ratio of magnitudes of a pair of gradients , one involving @xmath33 , and one involving @xmath152 : @xmath163 note that a particular value of differential utility learnability , by itself , has no significance . simply rescaling the units of @xmath103 will change that value . rather what is important is the ratio of differential learnabilities , at the same @xmath35 , for different @xmath99 s . such a ratio quantifies the relative preferability of those @xmath99 s . one nice feature of differential learnability is that unlike learnability , it does not depend on choice of some measure @xmath164 . this independence can lead to trouble if one is not careful however , and in particular if one uses learnability for purposes other than choosing between utility functions . for example , in some situations , the coin designer will have the option of enlarging the set of variables from the rest of the coin that are `` input '' to some node @xmath33 at @xmath162 and that therefore can be used by @xmath33 to decide what action to take . intuitively , doing so will not affect the rl `` signal '' for @xmath33 s microlearner ( the magnitude of the potential `` echo '' of @xmath33 s actions are not modified by changing some aspect of how it chooses among those actions ) . however it _ will _ reduce the `` noise '' , in that @xmath33 s microlearner now knows more about the state of the rest of the system . in the full integral version of learnability , this effect can be captured by having the support of @xmath164 restricted to reflect the fact that the extra inputs to @xmath33 at @xmath122 are correlated with the @xmath122 state of the external system . in differential learnability however this is not possible , precisely because no measure @xmath164 occurs in its definition . so we must capture the reduction in noise in some other fashion . occurring in the definition of differential learnability with something more nuanced . for example , one may wish to replace it with the maximum of the dot product of @xmath165 with any @xmath39 vector @xmath166 , subject not only to the restrictions that @xmath167 and @xmath168 * 0 * , but also subject to the restriction that @xmath166 must lie in the tangent plane of @xmath63 at @xmath35 . the first two restrictions , in concert with the extra restriction that @xmath169 * 0 * , give the original definition of the noise term . if they are instead joined with the third , new restriction , they will enforce any applicable coupling between the state of @xmath33 at time 0 and the rest of the system at time 0 . solving with lagrange multipliers , we get @xmath170 , where @xmath171 is the normal to @xmath63 at @xmath35 , @xmath172 , and @xmath173 while @xmath174 . as a practical matter though , it is often simplest to assume that the @xmath135 can vary arbitrarily , independent of @xmath103 , so that the noise term takes the form in eq . 3 . ] alternatively , if the extra variables are being input to @xmath33 for all @xmath108 , not just at @xmath162 , and if @xmath33 `` pays attention '' to those variables for all @xmath108 , then by incorporating those changes into our system @xmath63 itself has changed , @xmath175 . hypothesize that at those @xmath17 the node @xmath33 is capable of modifying its actions to `` compensate '' for what ( due to our augmentation of @xmath33 s inputs ) @xmath33 now knows to be going on outside of it . under this hypothesis , those changes in those external events will have less of an effect on the ultimate value of @xmath97 than they would if we had not made our modification . in this situation , the noise term has been reduced , so that the differential learnabiliity properly captures the effect of @xmath33 s having more inputs . another potential danger to bear in mind concerning differential learnability is that it is usually best to consider its average over a region , in particular over points with less than maximal intelligence . it is really designed for such points ; in fact , at the intelligence - maximizing @xmath35 , @xmath176 . whether in its differential form or not , and whether referring to utilities or intelligence , learnability is not meant to capture all factors that will affect how high an intelligence value a particular microlearner will achieve . such an all - inclusive definition is not possible , if for no other reason the fact that there are many such factors that are idiosyncratic to the particular microlearner used . beyond this though , certain more general factors that affect most popular learning algorithms , like the curse of dimensionality , are also not ( explicitly ) designed into learnability . learnability is not meant to provide a full characterization of performance that is what intelligence is designed to do . rather ( relative ) learnability is ony meant to provide a _ guide _ for how to improve performance . a system that has infinite ( differential , intelligence ) learnability for all its personal utilities is said to be `` perfectly '' ( differential , intelligence ) learnable . it is straight - forward to prove that a system is perfectly learnable @xmath177 iff @xmath178 can be written as @xmath179 for some function @xmath180 . ( see the discussion below on the general condition for a system s being perfectly factored . ) with these definitions in hand , we can now present ( a portion of ) one descriptive framework for coins . in this subsection , after discussing salient characteristics in general , we present some theorems concerning the relationship between personal utilities and the salient characteristic we choose to concentrate on . we then discus how to use these theorems to induce that salient characteristic in a coin . the starting point with a descriptive framework is the identification of `` salient characteristics of a coin which one strongly expects to be associated with its having large world utility '' . in this chapter we will focus on salient characteristics that concern the relationship between personal and world utilities . these characteristics are formalizations of the intuition that we want coins in which the competent greedy pursuit of their private utilities by the microlearners results in large world utility , without any bottlenecks , toc , `` frustration '' ( in the spin glass sense ) or the like . one natural candidate for such a characteristic , related to pareto optimality @xcite , is * weak triviality*. it is defined by considering any two worldlines @xmath35 and @xmath83 both of which are consistent with the system s dynamics ( _ i.e. _ , both of which lie on @xmath63 ) , where for every node @xmath33 , @xmath181 . , and require only that both of the `` partial vectors '' @xmath182 and @xmath183 obey the relevant dynamical laws , and therefore lie in @xmath184 . ] if for any such pair of worldlines where one `` pareto dominates '' the other it is necessarily true that @xmath185 , we say that the system is weakly trivial . we might expect that systems that are weakly trivial for the microlearners private utilities are configured correctly for inducing large world utility . after all , for such systems , if the microlearners collectively change @xmath35 in a way that ends up helping all of them , then necessarily the world utility also rises . more formally , for a weakly trivial system , the maxima of @xmath78 are pareto - optimal points for the personal utilities ( although the reverse need not be true ) . as it turns out though , weakly trivial systems can readily evolve to a world utility @xmath186 , one that often involves toc . to see this , consider automobile traffic in the absence of any traffic control system . let each node be a different driver , and say their private utilities are how quickly they each individually get to their destination . identify world utility as the sum of private utilities . then by simple additivity , for all @xmath35 and @xmath83 , whether they lie on @xmath63 or not , if @xmath187 it follows that @xmath185 ; the system is weakly trivial . however as any driver on a rush - hour freeway with no carpool lanes or metering lights can attest , every driver s pursuing their own goal definitely does not result in acceptable throughput for the system as a whole ; modifications to private utility functions ( like fines for violating carpool lanes or metering lights ) would result in far better global behavior . a system s being weakly trivial provides no assurances regarding world utility . this does not mean weak triviality is never of use . for example , say that for a set of weakly trivial personal utilities each agent can guarantee that _ regardless of what the other agents do _ , its utility is above a certain level . assume further that , being risk - averse , each agent chooses an action with such a guarantee . say it is also true that the agents are provided with a relatively large set of candidate guaranteed values of their utilities . under these circumstances , the system s being weakly trivial provides some assurances that world utility is not too low . moreover , if the overhead in enforcing such a future - guaranteeing scheme is small , and having a sizable set of guaranteed candidate actions provided to each of the agents does not require an excessively centralized infrastructure , we can actually employ this kind of scheme in practice . indeed , in the extreme case , one can imagine that every agent is guaranteed exactly what its utility would be for every one of its candidate actions . ( see the discussion on general equilibrium in the background section above . ) in this situation , nash equilibria and pareto optimal points are identical , which due to weak triviality means that the point maximizing @xmath78 is a nash equilibrium . however in any less extreme situation , the system may not achieve a value of world utility that is close to optimal . this is because even for weakly trivial systems a pareto optimal point may have poor world utility , in general . situations where one has guarantees of lower bounds on one s utility are not too common , but they do arise . one important example is a round of trades in a computational market ( see the background section above ) . in that scenario , there is an agent - indexed set of functions \{@xmath188 } and the personal utility of each agent @xmath189 is given by @xmath190 , where @xmath191 is the end of the round of trades . there is also a function @xmath192 @xmath193 that is a monotonically increasing function of its arguments , and world utility @xmath78 is given by @xmath194 . so the system is weakly trivial . in turn , each @xmath195 is determined solely by the `` allotment of goods '' possessed by @xmath33 , as specified in the appropriate components of @xmath196 . to be able to remove uncertainty about its future value of @xmath197 in this kind of system , in determining its trading actions each agent @xmath33 must employ some scheme like inter - agent contracts . this is because without such a scheme , no agent can be assured that if it agrees to a proposed trade with another agent that the full proposed transaction of that trade actually occurs . given such a scheme , if in each trade round @xmath17 each agent @xmath33 myopically only considers those trades that are assured of increasing the corresponding value of @xmath197 , then we are guaranteed that the value of the world utility is not less than the initial value @xmath198 . the problem with using weak triviality as a general salient characteristic is precisely the fact that the individual microlearners @xmath199 greedy . in a coin , there is no system - wide incentive to replace @xmath35 with a different worldline that would improve everybody s private utility , as in the definition of weak triviality . rather the incentives apply to each microlearner individually and motivate the learners to behave in a way that may well hurt some of them . so weak triviality is , upon examination , a poor choice for the salient characteristic of a coin . one alternative to weak triviality follows from consideration of the stricture that we must ` expect ' a salient characteristic to be coupled to large world utility in a running real - world coin . what can we reasonably expect about a running real - world coin ? we can not assume that all the private utilities will have large values witness the traffic example . but we @xmath200 assume that if the microlearners are well - designed , each of them will be doing close to as well it can _ given the behavior of the other nodes_. in other words , within broad limits we can assume that the system is more likely to be in @xmath35 than @xmath83 if for all @xmath33 , @xmath201 . we define a system to be * coordinated * iff for any such @xmath35 and @xmath83 lying on @xmath63 , @xmath185 . ( again , an obvious variant is to restrict @xmath202 , and require only that both @xmath183 and @xmath182 lie in @xmath184 . ) traffic systems are @xmath74 coordinated , in general . this is evident from the simple fact that if all drivers acted as though there were metering lights when in fact there were nt any , they would each be behaving with lower intelligence given the actions of the other drivers ( each driver would benefit greatly by changing its behavior by no longer pretending there were metering lights , etc . ) . but nonetheless , world utility would be higher . like weak triviality , coordination is intimately related to the economics concept of pareto optimality . unfortunately , there is not room in this chapter to present the mathematics associated with coordination and its variants . we will instead discuss a third candidate salient characteristic of coins , one which like coordination ( and unlike weak triviality ) we can reasonably expect to be associated with large world utility this alternative fixes weak triviality not by replacing the personal utilities \{@xmath97 } with the intelligences \{@xmath203 } as coordination does , but rather by only considering worldlines whose difference at time 0 involves a single node . this results in this alternative s being related to nash equilibria rather than pareto optimality . say that our coin s worldline is @xmath35 . let @xmath83 be any other worldline where @xmath204 , and where @xmath205 . now restrict attention to those @xmath83 where at @xmath162 @xmath35 and @xmath83 differ only for node @xmath33 . if for all such @xmath83 @xmath206 = sgn[g(\underline{\zeta } ) - g(\underline{\zeta}_{,t<0 } \bullet c(\underline{\zeta}'_{,0 } ) ) ] \ ; , \ ] ] and if this is true for all nodes @xmath33 , then we say that the coin is * factored * for all those utilities \{@xmath102 } ( at @xmath35 , with respect to time 0 and the utility @xmath78 ) . for a factored system , for any node @xmath33 , _ given the rest of the system _ , if the node s state at @xmath162 changes in a way that improves that node s utility over the rest of time , then it necessarily also improves world utility . colloquially , for a system that is factored for a particular microlearner s private utility , if that learner does something that improves that personal utility , then everything else being equal , it has also done something that improves world utility . of two potential microlearners for controlling node @xmath33 ( _ i.e. _ , two potential @xmath145 ) whose behavior until @xmath122 is identical but which differ there , the microlearner that is smarter with respect to @xmath14 will always result in a larger @xmath14 , by definition of intelligence . accordingly , for a factored system , the smarter microlearner is also the one that results in better @xmath78 . so as long as we have deployed a sufficiently smart microlearner on @xmath33 , we have assured a good @xmath78 ( given the rest of the system ) . formally , this is expressed in the fact @xcite that for a factored system , for all nodes @xmath33 , @xmath207 one can also prove that nash equilibria of a factored system are local maxima of world utility . note that in keeping with our behaviorist perspective , nothing in the definition of factored requires the existence of private utilities . indeed , it may well be that a system having private utilities \{@xmath134 } is factored , but for personal utilities \{@xmath102 } that differ from the \{@xmath134}. a system s being factored does @xmath74 mean that a change to @xmath103 that improves @xmath146 can not also hurt @xmath208 for some @xmath209 . intuitively , for a factored system , the side effects on the rest of the system of @xmath33 s increasing its own utility do not end up decreasing world utility but can have arbitrarily adverse effects on other private utilities . ( in the language of economics , no stipulation is made that @xmath33 s `` costs are endogenized . '' ) for factored systems , the separate microlearners successfully pursuing their separate goals do not frustrate each other _ as far as world utility is concerned_. in addition , if @xmath210 is factored with respect to @xmath78 , then a change to @xmath211 that improves @xmath212 improves @xmath213 . but it may @xmath214 some @xmath215 and/or @xmath216 . ( this is even true for a discounted sum of rewards personal utility , so long as @xmath217 . ) an example of this would be an economic system cast as a single individual , @xmath33 , together with an environment node , where @xmath78 is a steeply discounted sum of rewards @xmath33 receives over his / her lifetime , @xmath217 , and @xmath218 , @xmath219 . for such a situation , it may be appropriate for @xmath33 to live extravagantly at the time @xmath61 , and `` pay for it '' later . as an instructive example of the ramifications of eq . 5 , say node @xmath33 is a conventional computer . we want @xmath220 to be as high as possible , i.e. , given the state of the rest of the system at time 0 , we want computer @xmath33 s state then to be the best possible , as far as the resultant value of @xmath78 is concerned . now a computer s `` state '' consists of the values of all its bits , including its code segment , i.e. , including the program it is running . so for a factored personal utility @xmath102 , if the program running on the computer is better than most others as far as @xmath102 is concerned , then it is also better than most other programs as far as @xmath78 is concerned . our task as coin designers engaged in coin initialization or macrolearning is to find such a program and such an associated @xmath102 . one way to approach this task is to restrict attention to programs that consist of rl algorithms with private utility specified in the bits \{@xmath75 } of @xmath33 . this reduces the task to one of finding a private utility \{@xmath75 } ( and thereby fully specifying @xmath103 ) such that our rl algorithm working with that private utility has high @xmath138 , i.e. , such that that algorithm outperforms most other programs as far as the personal utility @xmath97 is concerned . perhaps the simplest way to address this reduced task is to exploit the fact that for a good enough rl algorithm @xmath221 will be large , and therefore adopt such an rl algorithm and fix the private utility to equal @xmath97 . in this way we further reduce the original task , which was to search over all personal utilities @xmath97 and all programs @xmath222 to find a pair such that both @xmath97 is factored with respect to @xmath78 and there are relatively few programs that outperform @xmath222 , as far as @xmath97 . the task is now instead to search over all private utilities \{@xmath75 } such that both \{@xmath75 } is factored with respect to @xmath78 and such that there are few programs ( _ of any sort _ , rl - based or not ) that outperform our rl algorithm working on \{@xmath75 } , as far as that self - same private utility is concerned . the crucial assumption being leveraged in this approach is that our rl algorithm is `` good enough '' , and the reason we want learnable \{@xmath75 } is to help effect this assumption . in general though , we ca nt have both perfect learnability and perfect factoredness . as an example , say that @xmath223 , and that the dynamics is the identity operator : @xmath218 , @xmath224 . then if @xmath225 and the system is perfectly learnable , it is not perfectly factored . this is because perfect learnability requires that @xmath226 for some function @xmath180 . however any change to @xmath103 that improves such a @xmath97 will either help or @xmath214 @xmath87 , depending on the sign of @xmath135 . for the `` wrong '' sign of @xmath135 , this means the system is actually `` anti - factored '' . due to such incompatibility between perfect factoredness and perfect learnability , we must usually be content with having high degree of factoredness and high learnability . in such situations , the emphasis of the macrolearning process should be more and more on having high degree of factoredness as we get closer and closer to a nash equilibrium . this way the system wo nt relax to an incorrect local maximum . in practice of course , a coin will often not be perfectly factored . nor in practice are we always interested only in whether the system is factored at one particular point ( rather than across a region say ) . these issues are discussed in @xcite , where in particular a formal definition of of the * degree of factoredness * of a system is presented . if a system is factored for utilities @xmath227 , then it is also factored for any utilities @xmath228 where for each @xmath33 @xmath229 is a monotonically increasing function of @xmath97 . more generally , the following result characterizes the set of all factored personal utilities : * theorem 1 : * a system is factored at all @xmath230 iff for all those @xmath35 , @xmath231 , we can write @xmath232 for some function @xmath233 such that @xmath234 for all @xmath230 and associated @xmath78 values . ( the form of the \{@xmath97 } off of @xmath63 is arbitrary . ) * proof : * for fixed @xmath103 and @xmath107 , any change to @xmath103 which keeps @xmath235 on @xmath63 and which at the same time increases @xmath236 must increase @xmath237 , due to the restriction on @xmath238 . this establishes the backwards direction of the proof . for the forward direction , write @xmath239 . define this formulation of @xmath97 as @xmath240 , which we can re - express as @xmath241 . now since the system is factored , @xmath242 , @xmath243 @xmath244 @xmath245 so consider any situation where the system is factored , and the values of @xmath78 , @xmath246 , and @xmath135 are specified . then we can find _ any _ @xmath103 consistent with those values ( _ i.e. _ , such that our provided value of @xmath78 equals @xmath247 ) , evaluate the resulting value of @xmath248 , and know that we would have gotten the same value if we had found a different consistent @xmath103 . this is true for all @xmath249 . therefore the mapping @xmath250 is single - valued , and we can write @xmath251 . * qed . * by thm . 1 , we can ensure that the system is factored without any concern for @xmath63 , by having each @xmath252 . alternatively , by only requiring that @xmath253 does @xmath254 ( _ i.e. _ , does @xmath255 ) , we can access a broader class of factored utilities , a class that @xmath0 depend on @xmath63 . loosely speaking , for those utilities , we only need the projection of @xmath256 onto @xmath257 to be parallel to the projection of @xmath258 onto @xmath257 . given @xmath78 and @xmath63 , there are infinitely many @xmath259 having this projection ( the set of such @xmath260 form a linear subspace of @xmath39 ) . the partial differential equations expressing the precise relationship are discussed in @xcite . as an example of the foregoing , consider a ` team game ' ( also known as an ` exact potential game ' @xcite ) in which @xmath261 for all @xmath33 . such coins are factored , trivially , regardless of @xmath63 ; if @xmath97 rises , then @xmath78 must as well , by definition . ( alternatively , to confirm that team games are factored just take @xmath262 in thm . 1 . ) on the other hand , as discussed below , coins with ` wonderful life ' personal utilities are also factored , but the definition of such utilities depends on @xmath63 . due to their often having poor learnability and requiring centralized communication ( among other infelicities ) , in practice team game utilities often are poor choices for personal utilities . accordingly , it is often preferable to use some other set of factored utilities . to present an important example , first define the ( @xmath122 ) * effect set * of node @xmath33 at @xmath35 , @xmath263 , as the set of all components @xmath264 for which @xmath265 . define the effect set @xmath266 with no specification of @xmath35 as @xmath267 . ( we take this latter definition to be the default meaning of `` effect set '' . ) we will also find it useful to define @xmath268 as the set of components of the space @xmath39 that are not in @xmath266 . intuitively , @xmath33 s effect set is the set of all components @xmath264 which would be affected by a change in the state of node @xmath33 at time 0 . ( they may or may not be affected by changes in the @xmath122 states of the other nodes . ) note that the effect sets of different nodes may overlap . the extension of the definition of effect sets for times other than 0 is immediate . so is the modification to have effect sets only consist of those components @xmath34 that vary with with the state of node @xmath33 at time 0 , rather than consist of the full vectors @xmath31 possessing such a component . these modifications will be skipped here , to minimize the number of variables we must keep track of . next for any set @xmath269 of components ( @xmath270 ) , define @xmath271 as the `` virtual '' vector formed by clamping the @xmath269-components of @xmath35 to an arbitrary fixed value . ( in this paper , we take that fixed value to be @xmath272 for all components listed in @xmath269 . ) consider in particular a * wonderful life * set @xmath269 . the value of the * wonderful life utility * ( wlu for short ) for @xmath269 at @xmath35 is defined as : @xmath273 in particular , the wlu for the effect set of node @xmath33 is @xmath274 , which for @xmath249 can be written as @xmath275 . we can view @xmath33 s effect set wlu as analogous to the change in world utility that would have arisen if node @xmath33 `` had never existed '' . ( hence the name of this utility - cf . the frank capra movie . ) note however , that @xmath276 is a purely `` fictional '' , counter - factual operation , in the sense that it produces a new @xmath35 without taking into account the system s dynamics . indeed , no assumption is even being made that @xmath277 is consistent with the dynamics of the system . the sequence of states the node @xmath33 is clamped to in the definition of the wlu need not be consistent with the dynamical laws embodied in @xmath63 . this dynamics - independence is a crucial strength of the wlu . it means that to evaluate the wlu we do _ not _ try to infer how the system would have evolved if node @xmath33 s state were set to 0 at time 0 and the system evolved from there . so long as we know @xmath35 extending over all time , and so long as we know @xmath78 , we know the value of wlu . this is true even if we know nothing of the dynamics of the system . an important example is effect set wonderful life utilities when the set of all nodes is partitioned into ` subworlds ' in such a way that all nodes in the same subworld @xmath278 share substantially the same effect set . in such a situation , all nodes in the same subworld @xmath278 will have essentially the same personal utilities , exactly as they would if they used team game utilities with a `` world '' given by @xmath278 . when all such nodes have large intelligence values , this sharing of the personal utility will mean that all nodes in the same subworld are acting in a coordinated fashion , loosely speaking . the importance of the wlu arises from the following results : * theorem 2 : * i ) a system is factored at all @xmath230 iff for all those @xmath35 , @xmath231 , we can write @xmath279 for some function @xmath280 such that @xmath281 for all @xmath230 and associated @xmath78 values . ( the form of the \{@xmath97 } off of @xmath63 is arbitrary . ) \ii ) in particular , a coin is factored for personal utilities set equal to the associated effect set wonderful life utilities . * proof : * to prove ( i ) , first write @xmath282 . for all @xmath249 , @xmath283 is independent of @xmath103 , and so by definition of @xmath60 it is a single - valued function of @xmath135 for such @xmath35 . therefore @xmath284 for some function @xmath285 . accordingly , by thm . 1 , for \{@xmath97 } of the form stipulated in ( i ) , the system is factored . going the other way , if the system is factored , then by thm . 1 it can be written as @xmath237 . since both @xmath107 and @xmath286 , we can rewrite this as @xmath287_{,t<0 } , [ \hat{}c^{eff}_{\eta}]_{\;\hat{}\eta,0 } , g(\underline{\zeta}))$ ] . * qed . * part ( ii ) of the theorem follows immediately from part ( i ) . for pedagogical value though , here we instead derive it directly . first , since @xmath288 is independent of @xmath264 for all @xmath289 , so is the @xmath290 vector @xmath291 , _ i.e. _ , @xmath292_{\eta',t } = \vec{0 } \;\ ; \forall ( \eta ' , t ) \in c^{eff}_\eta$ ] . this means that viewed as a @xmath107-parameterized function from @xmath293 to @xmath290 , @xmath294 is a single - valued function of the @xmath295 components . therefore @xmath296 can only depend on @xmath107 and the non-@xmath33 components of @xmath129 . accordingly , the wlu for @xmath266 is just @xmath78 minus a term that is a function of @xmath107 and @xmath295 . by choosing @xmath233 in thm . 1 to be that difference , we see that @xmath33 s effect set wlu is of the form necessary for the system to be factored . * qed . * as a generalization of ( ii ) , the system is factored if each node @xmath33 s personal utility is ( a monotonically increasing function of ) the wlu for a set @xmath297 that contains @xmath266 . for conciseness , except where explicitly needed , for the remainder of this subsection we will suppress the argument `` @xmath107 '' , taking it to be implicit . the next result concerning the practical importance of effect set wlu is the following : * theorem 3 : * let @xmath269 be a set containing @xmath298 . then @xmath299 * proof : * writing it out , @xmath300 the second term in the numerator equals 0 , by definition of effect set . dividing by the similar expression for @xmath301 then gives the result claimed . * qed . * so if we expect that ratio of magnitudes of gradients to be large , effect set wlu has much higher learnability than team game utility while still being factored , like team game utility . as an example , consider the case where the coin is a very large system , with @xmath33 being only a relatively minor part of the system ( _ e.g. _ , a large human economy with @xmath33 being a `` typical john doe living in peoria illinois '' ) . often in such a system , for the vast majority of nodes @xmath302 , how @xmath78 varies with @xmath303 will be essentially independent of the value @xmath103 . ( for example , how gdp of the us economy varies with the actions of our john doe from peoria , illinois will be independent of the state of some jane smith living in los angeles , california . ) in such circumstances , thm . 3 tells us that the effect set wonderful life utility for @xmath33 will have a far larger learnability than does the world utility . for any fixed @xmath269 , if we change the clamping operation ( _ i.e. _ , change the choice of the `` arbitrary fixed value '' we clamp each component to ) , then we change the mapping @xmath304 , and therefore change the mapping @xmath305 . accordingly , changing the clamping operation can affect the value of @xmath306 evaluated at some point @xmath129 . therefore , by thm . 3 , changing the clamping operation can affect @xmath307 . so properly speaking , for any choice of @xmath269 , if we are going to use @xmath308 , we should set the clamping operation so as to maximize learnability . for simplicity though , in this paper we will ignore this phenomenon , and simply set the clamping operation to the more or less `` natural '' choice of * 0 * , as mentioned above . next consider the case where , for some node @xmath33 , we can write @xmath309 as @xmath310 . say it is also true that @xmath33 s effect set is a small fraction of the set of all components . in this case it often true that the values of @xmath311 are much larger than those of @xmath312 , which means that partial derivatives of @xmath311 are much larger than those of @xmath312 . in such situations the effect set wlu is far more learnable than the world utility , due to the following results : * theorem 4 : * if for some node @xmath33 there is a set @xmath269 containing @xmath313 , a function @xmath314 , and a function @xmath315 , such that @xmath316 , then @xmath317 * proof : * for brevity , write @xmath318 and @xmath319 both as functions of full @xmath320 , just such functions that are only allowed to depend on the components of @xmath35 that lie in @xmath269 and those components that do not lie in @xmath269 , respectively . then the @xmath269 wlu for node @xmath33 is just @xmath321 . since in that second term we are clamping all the components of @xmath35 that @xmath312 cares about , for this personal utility @xmath322 . so in particular @xmath323 . now by definition of effect set , @xmath324 , since @xmath325 does not contain @xmath298 . so @xmath326 . * qed . * the obvious extensions of thm.s 3 and 4 to effect sets with respect to times other than 0 can also be proven @xcite . an important special case of thm . 4 is the following : * corollary 1 : * if for some node @xmath33 we can write \i ) @xmath327_{t\ge0 } ) + g_3(\underline{\zeta}_{,t<0})$ ] for some set @xmath269 containing @xmath313 , and if \ii ) @xmath328_{\sigma } ) ||$ ] , then @xmath329 . in practice , to assure that condition ( i ) of this corollary is met might require that @xmath269 be a proper superset of @xmath266 . countervailingly , to assure that condition ( ii ) is met will usually force us to keep @xmath269 as small as possible . one can often remove elements from an effect set and still have the results of this section hold . most obviously , if ( @xmath53 , t ) @xmath330 but @xmath331 = * 0 * , we can remove ( @xmath53 , t ) from @xmath266 without invalidating our results . more generally , if there is a set @xmath332 such that for each component ( @xmath333 the chain rule term @xmath334 \;\cdot \ ; [ \partial_{\underline{\zeta}_{\eta,0;i } } [ c(\underline{\zeta}_{,0})]_{\eta',t}]$ ] = 0 , then the effects on @xmath78 of changes to @xmath103 that are `` mediated '' by the members of @xmath335 cancel each other out . in this case we can usually remove the elements of @xmath335 from @xmath266 with no ill effects . usually the mathematics of a descriptive framework a formal investigation of the salient characteristics will not provide theorems of the sort , `` if you modify the coin the following way at time @xmath17 , the value of the world utility will increase . '' rather it provides theorems that relate a coin s salient characteristics with the general properties of the coin s entire history , and in particular with those properties embodied in @xmath63 . in particular , the salient characteristic that we are concerned with in this chapter is that the system be highly intelligent for personal utilities for which it is factored , and our mathematics concerns the relationship between factoredness , intelligence , personal utilities , effect sets , and the like . more formally , the desideratum associated with our salient characteristic is that we want the coin to be at a @xmath35 for which there is some set of \{@xmath97 } ( not necessarily consisting of private utilities ) such that ( a ) @xmath35 is factored for the \{@xmath97 } , and ( b ) @xmath132 is large for all @xmath33 . now there are several ways one might try to induce the coin to be at such a point . one approach is to have each algorithm controlling @xmath33 explicitly try to `` steer '' the worldline towards such a point . in this approach @xmath33 need nt even have a private utility in the usual sense . ( the overt `` goal '' of the algorithm controlling @xmath33 involves finding a @xmath35 with a good associated extremum over the class of all possible @xmath97 , independent of any private utilities . ) now initialization of the coin , _ i.e. _ , fixing of @xmath129 , involves setting the algorithm controlling @xmath33 , in this case to the steering algorithm . accordingly , in this approach to initialization , we fix @xmath129 to a point for which there is some special @xmath97 such that both @xmath336 is factored for @xmath97 , and @xmath337 is large . there is nothing peculiar about this . what is odd though is that in this approach we do not know what that `` special '' @xmath97 is when we do that initialization ; it s to be determined , by the unfolding of the system . in this chapter we concentrate on a different approach , which can involve either initialization or macrolearning . in this alternative we deploy the \{@xmath97 } as the microlearners private utilities at some @xmath338 , in a process not captured in @xmath63 , so as to induce a factored coin that is as intelligent as possible . ( it is with that `` deploying of the \{@xmath97 } '' that we are trying to induce our salient characteristic in the coin . ) since in this approach we are using private utilities , we can replace intelligence with its surrogate , learnability . so our task is to choose \{@xmath97 } which are as learnable as possible while still being factored . solving for such utilities can be expressed as solving a set of coupled partial differential equations . those equations involve the tangent plane to the manifold @xmath63 , a functional trading off ( the differential versions of ) degree of factoredness and learnability , and any communication constraints on the nodes we must respect . while there is not space in the current chapter to present those equations , we can note that they are highly dependent on the correlations among the components of @xmath31 . so in this approach , in coin initialization we use some preliminary guesses as to those correlations to set the initial \{@xmath97}. for example , the effect set of a node constitutes all components @xmath339 that have non - zero correlation with @xmath103 . furthermore , by thm . 2 the system is factored for effect set wlu personal utilities . and by coroll . 1 , for small effect sets , the effect set wlu has much greater differential utility learnability than does @xmath78 . extending the reasoning behind this result to all @xmath35 ( or at least all likely @xmath35 ) , we see that for this scenario , the descriptive framework advises us to use wonderful life private utilities based on ( guesses for ) the associated effect sets rather than the team game private utilities , @xmath340 . in macrolearning we must instead run - time estimate an approximate solution to our partial differential equations , based on statistical inference . as an example , we might start with an initial guess as to @xmath33 s effect set , and set its private utility to the associated wlu . but then as we watch the system run and observe the correlations among the components of @xmath35 , we might modify which components we think comprise @xmath33 s effect set , and modify @xmath33 s personal utility accordingly . as implied above , often one can perform reasonable coin initialization and/or macrolearning without writing down the partial differential equations governing our salient characteristic explicitly . simply `` hacking '' one s way to the goal of maximizing both degree of factoredness and intelligibility , for example by estimating effect sets , often results in dramatic improvement in performance . this is illustrated in the experiments recounted in the next two subsections . even if we do nt exactly know the effect set of each node @xmath33 , often we will be able to make a reasonable guess about which components of @xmath35 comprise the `` preponderance '' of @xmath33 s effect set . we call such a set a * guessed effect set*. as an example , often the primary effects of changes to @xmath33 s state will be on the future state of @xmath33 , with only relatively minor effects on the future states of other nodes . in such situations , we would expect to still get good results if we approximated the effect set wlu of each node @xmath33 with a wlu based on the guessed effect set @xmath341 . in other words , we would expect to be able to replace wlu@xmath342 with wlu@xmath343 and still get good performance . this phenomenon was borne out in the experiments recounted in @xcite that used coin initialization for distributed control of network packet routing . in a conventional approach to packet routing , each router runs what it believes ( based on the information available to it ) to be a shortest path algorithm ( spa ) , _ i.e. _ , each router sends its packets in the way that it surmises will get those packets to their destinations most quickly . unlike with an approach based on our coin framework , with spa - based routing the routers have no concern for the possible deleterious side - effects of their routing decisions on the global performance ( _ e.g. _ , they have no concern for whether they induce bottlenecks ) . we performed simulations in which we compared such a coin - based routing system to an spa - based system . for the coin - based system @xmath78 was global throughput and no macrolearning was used . the coin initialization was to have each router s private utility be a wlu based on an associated guessed effect set generated _ a priori_. in addition , the coin - based system was realistic in that each router s reinforcement algorithm had imperfect knowledge of the state of the system . on the other hand , the spa was an idealized `` best - possible '' system , in which each router knew exactly what the shortest paths were at any given time . despite the handicap that this disparity imposed on the coin - based system , it achieved significantly better global throughput in our experiments than did the perfect - knowledge spa - based system , and in particular , avoided the braess paradox that was built - in to some of those systems @xcite . the experiments in @xcite were primarily concerned with the application of packet - routing . to concentrate more precisely on the issue of coin initialization , we ran subsequent experiments on variants of arthur s famous `` el farol bar problem '' ( see section [ sec : lit ] ) . to facilitate the analysis we modified arthur s original problem to be more general , and since we were not interested in directly comparing our results to those in the literature , we used a more conventional ( and arguably `` dumber '' ) machine learning algorithm than the ones investigated in @xcite . in this formulation of the bar problem @xcite , there are @xmath344 agents , each of whom picks one of seven nights to attend a bar the following week , a process that is then repeated . in each week , each agent s pick is determined by its predictions of the associated rewards it would receive . these predictions in turn are based solely upon the rewards received by the agent in preceding weeks . an agent s `` pick '' at week @xmath17 ( _ i.e. _ , its node s state at that week ) is represented as a unary seven - dimensional vector . ( see the discussion in the definitions subsection of our representing discrete variables as euclidean variables . ) so @xmath33 s zeroing its state in some week , as in the cl@xmath345 operation , essentially means it elects not to attend any night that week . the world utility is @xmath346 where : @xmath347 ; @xmath348 is the total attendance on night @xmath349 at week @xmath17 ; @xmath350 ; and @xmath351 and each of the \{@xmath352 } are real - valued parameters . intuitively , the `` world reward '' @xmath222 is the sum of the global `` rewards '' for each night in each week . it reflects the effects in the bar as the attendance profile of agents changes . when there are too few agents attending some night , the bar suffers from lack of activity and therefore the global reward for that night is low . conversely , when there are too many agents the bar is overcrowded and the reward for that night is again low . note that @xmath353 reaches its maximum when its argument equals @xmath351 . in these experiments we investigate two different @xmath354 s . one treats all nights equally ; @xmath355 $ ] . the other is only concerned with one night ; @xmath356 $ ] . in our experiments , @xmath357 and @xmath344 is chosen to be 4 times larger than the number of agents necessary to have @xmath351 agents attend the bar on each of the seven nights , _ i.e. _ , there are @xmath358 agents ( this ensures that there are no trivial solutions and that for the world utility to be maximized , the agents have to `` cooperate '' ) . as explained below , our microlearning algorithms worked by providing a real - valued `` reward '' signal to each agent at each week @xmath17 . each agent s reward function is a surrogate for an associated utility function for that agent . the difference between the two functions is that the reward function only reflects the state of the system at one moment in time ( and therefore is potentially observable ) , whereas the utility function reflects the agent s ultimate goal , and therefore can depend on the full history of that agent across time . we investigated three agent reward functions . one was based on effect set wlu . the other two were `` natural '' rewards included for comparison purposes . with @xmath359 the night selected by @xmath33 , the three rewards are : @xmath360 \mbox{global ( g ) : } \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; & r_{\eta}(\underline{\zeta}_{,t } ) & \equiv r(\underline{\zeta}_{,t } ) = \sum_{k=1}^7 \gamma_k(x_k(\underline{\zeta}_{,t } ) ) \\ [ -.05 in ] \mbox{wonderful life ( wl ) : } \ ; \ ; \ ; & r_{\eta}(\underline{\zeta}_{,t } ) & \equiv r(\underline{\zeta}_{,t } ) - r(\mbox{cl}_{\underline{\zeta}_{\eta , t}}(\underline{\zeta}_{,t } ) ) \\ [ -.05 in ] & & = \gamma_{d_\eta } ( x_{d_\eta } ( \underline{\zeta}_{,t } ) ) - \gamma_{d_\eta } ( x_{d_\eta } ( \mbox{cl}_{\underline{\zeta}_{\eta , t } } ( \underline{\zeta}_{,t})))\end{aligned}\ ] ] the conventional ud reward is a natural `` naive '' choice for the agents reward ; the total reward on each night gets uniformly divided among the agents attending that night . if we take @xmath361 ( _ i.e. _ , @xmath33 s utility is an undiscounted sum of its rewards ) , then for the ud reward @xmath362 , so that the system is weakly trivial . the original version of the bar problem in the physics literature @xcite is the special case where ud reward is used but there are only two `` nights '' in the week ( one of which corresponds to `` staying at home '' ) ; @xmath354 is uniform ; and @xmath363 for some vector @xmath364 , taken to equal ( .6 , .4 ) in the very original papers . so the reward to agent @xmath33 is 1 if it attends the bar and attendance is below capacity , or if it stays at home and the bar is over capacity . reward is 0 otherwise . ( in addition , unlike in our coin - based systems , in the original work on the bar problem the microlearners work by explicitly predicting the bar attendance , rather than by directly modifying behavior to try to increase a reward signal . ) in contrast to the ud reward , providing the g reward at time @xmath17 to each agent results in all agents receiving the same reward . this is the team game reward function , investigated for example in @xcite . for this reward function , the system is automatically factored if we define @xmath365 . however , evaluation of this reward function requires centralized communication concerning all seven nights . furthermore , given that there are 168 agents , g is likely to have poor learnability as a reward for any individual agent . this latter problem is obviated by using the wl reward , where the subtraction of the clamped term removes some of the `` noise '' of the activity of all other agents , leaving only the underlying `` signal '' of how the agent in question affects the utility . so one would expect that with the wl reward the agents can readily discern the effects of their actions on their rewards . even though the conditions in coroll . 1 do nt hold elements of @xmath366 are just @xmath367 , but the contributions of @xmath368 to @xmath78 can not be written as a sum of a @xmath367 contribution and a @xmath369 contribution . ] , this reasoning accords with the implicit advice of coroll . 1 under the approximation of the @xmath162 effect set as @xmath370 . in other words , it agrees with that corollary s implicit advice under the identification of @xmath371 as @xmath33 s @xmath122 guessed effect set . in fact , in this very simple system , we can explicitly calculate the ratio of the wl reward s learnability to that of the g reward , by recasting the system as existing for only a single instant so that @xmath372 exactly and then applying thm . so for example , say that all @xmath373 , and that the number of nodes @xmath344 is evenly divided among the seven nights . the numerator term in thm . 3 is a vector whose components are some of the partials of g evaluated when @xmath374 . this vector is @xmath375 dimensional , one dimension for each of the 7 components of ( the unary vector comprising ) each node in @xmath152 . for any particular @xmath302 and night @xmath4 , the associated partial derivative is @xmath376 $ ] , where as usual `` @xmath377 '' indicates the @xmath4th component of the unary vector @xmath378 . since @xmath379 , for any fixed @xmath4 and @xmath53 , this sum just equals @xmath380 . since there are @xmath375 such terms , after taking the norm we obtain @xmath381 \ ; \sqrt{7(n-1})|$ ] . the denominator term in thm . 3 is the difference between the gradients of the global reward and the clamped reward . these differ on only @xmath382 terms , one term for that component of each node @xmath302 corresponding to the night @xmath33 attends . ( the other @xmath383 terms are identical in the two partials and therefore cancel . ) this yields @xmath381 \ ; [ 1 - e^{1/c } ( 1 - \frac{7}{n-7c } ) ] \ ; \sqrt{n-1}$ ] . combining with the result of the previous paragraph , our ratio is @xmath384 . in addition to this learnability advantage of the wl reward , to evaluate its wl reward each agent only needs to know the total attendance on the night it attended , so no centralized communication is required . finally , although the system wo nt be perfectly factored for this reward ( since in fact the effect set of @xmath33 s action at @xmath17 would be expected to extend a bit beyond @xmath31 ) , one might expect that it is close enough to being factored to result in large world utility . each agent keeps a seven dimensional euclidean vector representing its estimate of the reward for attending each night of the week . at the end of each week , the component of this vector corresponding to the night just attended is proportionally adjusted towards the actual reward just received . at the beginning of the succeeding week , the agent picks the night to attend using a boltzmann distribution with energies given by the components of the vector of estimated rewards , where the temperature in the boltzmann distribution decays in time . ( this learning algorithm is equivalent to claus and boutilier s @xcite independent learner algorithm for multi - agent reinforcement learning . ) we used the same parameters ( learning rate , boltzmann temperature , decay rates , etc . ) for all three reward functions . ( this is an _ extremely _ primitive rl algorithm which we only chose for its pedagogical value ; more sophisticated rl algorithms are crucial for eliciting high intelligence levels when one is confronted with more complicated learning problems . ) figure [ fig : barfig ] presents world reward values as a function of time , averaged over 50 separate runs , for all three reward functions , for both @xmath355 $ ] and @xmath356 $ ] . the behavior with the g reward eventually converges to the global optimum . this is in agreement with the results obtained by crites @xcite for the bank of elevators control problem . systems using the wl reward also converged to optimal performance . this indicates that for the bar problem our approximations of effects sets are sufficiently accurate , _ i.e. _ , that ignoring the effects one agent s actions will have on future actions of other agents does not significantly diminish performance . this reflects the fact that the only interactions between agents occurs indirectly , via their affecting each others reward values . however since the wl reward is more learnable than than the g reward , convergence with the wl reward should be far quicker than with the g reward . indeed , when @xmath356 $ ] , systems using the g reward converge in 1250 weeks , which is 5 times worse than the systems using wl reward . when @xmath355 $ ] systems take 6500 weeks to converge with the g reward , which is more than _ 30 times _ worse than the time with the wl reward . in contrast to the behavior for reward functions based on our coin framework , use of the conventional ud reward results in very poor world reward values , values that deteriorated as the learning progressed . this is an instance of the toc . for example , for the case where @xmath356 $ ] , it is in every agent s interest to attend the same night but their doing so shrinks the world reward `` pie '' that must be divided among all agents . a similar toc occurs when @xmath354 is uniform . this is illustrated in fig . [ fig : attend ] which shows a typical example of daily attendance figures ( \{@xmath385 } ) for each of the three reward functions for @xmath386 . in this example optimal performance ( achieved with the wl reward ) has 6 agents each on 6 separate nights , ( thus maximizing the reward on 6 nights ) , and the remaining 132 agents on one night . figure [ fig : numagents ] shows how @xmath387 performance scales with @xmath344 for each of the reward signals for @xmath356 $ ] . systems using the ud reward perform poorly regardless of @xmath344 . systems using the g reward perform well when @xmath344 is low . as @xmath344 increases however , it becomes increasingly difficult for the agents to extract the information they need from the g reward . ( this problem is significantly worse for uniform @xmath354 . ) because of their superior learnability , systems using the wl reward overcome this signal - to - noise problem ( _ i.e. _ , because the wl reward is based on the _ difference _ between the actual state and the state where one agent is clamped , it is much less affected by the total number of agents ) . in the experiments recounted above , the agents were sufficiently independent that assuming they did not affect each other s actions ( when forming guesses for effect sets ) allowed the resultant wl reward signals to result in optimal performance . in this section we investigate the contrasting situation where we have initial guesses of effect sets that are quite poor and that therefore result in bad global performance when used with wl rewards . in particular , we investigate the use of macrolearning to correct those guessed effect sets at run - time , so that with the corrected guessed effect sets wl rewards will instead give optimal performance . this models real - world scenarios where the system designer s initial guessed effect sets are poor approximations of the actual associated effect sets and need to be corrected adaptively . in these experiments the bar problem is significantly modified to incorporate constraints designed to result in poor @xmath78 when the wl reward is used with certain initial guessed effect sets . to do this we forced the nights actually attended by some of the agents ( followers ) to agree with those attended by other agents ( leaders ) , regardless of what night those followers `` picked '' via their microlearning algorithms . ( for leaders , picked and actually attended nights were always the same . ) we then had the world utility be the sum , over all leaders , of the values of a triply - indexed reward matrix whose indices are the nights that each leader - follower set attends : @xmath388 where @xmath389 is the night the @xmath390 leader attends in week @xmath17 , and @xmath391 and @xmath392 are the nights attended by the followers of leader @xmath4 , in week @xmath17 ( in this study , each leader has two followers ) . we also had the states of each node be one of the integers \{0 , 1 , ... , 6 } rather than ( as in the bar problem ) a unary seven - dimensional vector . this was a bit of a contrivance , since constructions like @xmath393 are nt meaningful for such essentially symbolic interpretations of the possible states @xmath103 . as elaborated below , though , it was helpful for constructing a scenario in which guessed effect set wlu results in poor performance , _ i.e. _ , a scenario in which we can explore the application of macrolearning . to see how this setup can result in poor world utility , first note that the system s dynamics is what restricts all the members of each triple @xmath394 to equal the night picked by leader @xmath4 for week so @xmath395 and @xmath392 are both in leader @xmath4 s actual effect set at week @xmath17 whereas the initial guess for @xmath4 s effect set may or may not contain nodes other than @xmath389 . ( for example , in the bar problem experiments , the guessed effect set does not contain any nodes beyond @xmath389 . ) on the other hand , @xmath78 and @xmath222 are defined for all possible triples ( @xmath396 ) . so in particular , @xmath222 is defined for the dynamically unrealizable triples that can arise in the clamping operation . this fact , combined with the leader - follower dynamics , means that for certain @xmath222 s there exist guessed effect sets such that the dynamics assures poor world utility when the associated wl rewards are used . this is precisely the type of problem that macrolearning is designed to correct . as an example , say each week only contains two nights , 0 and 1 . set @xmath397 and @xmath398 . so the contribution to @xmath78 when a leader picks night 1 is 1 , and when that leader picks night 0 it is 0 , independent of the picks of that leader s followers ( since the actual nights they attend are determined by their leader s picks ) . accordingly , we want to have a private utility for each leader that will induce that leader to pick night 1 . now if a leader s guessed effect set includes both of its followers ( in addition to the leader itself ) , then clamping all elements in its effect set to 0 results in an @xmath222 value of @xmath398 . therefore the associated guessed effect set wlu will reward the leader for choosing night 1 , which is what we want . ( for this case wl reward equals @xmath399 if the leader picks night 1 , compared to reward @xmath400 for picking night 0 . ) however consider having two leaders , @xmath401 and @xmath402 , where @xmath401 s guessed effect set consists of @xmath401 itself together with the two followers of @xmath402 ( rather than together with the two followers of @xmath401 itself ) . so neither of leader @xmath401 s followers are in its guessed effect set , while @xmath401 itself is . accordingly , the three indices to @xmath401 s @xmath222 need not have the same value . similarly , clamping the nodes in its guessed effect set wo nt affect the values of the second and third indices to @xmath401 s @xmath222 , since the values of those indices are set by @xmath401 s followers . so for example , if @xmath402 and its two followers go to night 0 in week 0 , and @xmath401 and its two followers go to night 1 in that week , then the associated guessed effect set wonderful life reward for @xmath401 for week 0 is @xmath403 $ ] . this equals @xmath404 . simply by setting @xmath405 we can ensure that this is negative . conversely , if leader @xmath401 had gone to night 0 , its guessed effect wlu would have been 0 . so in this situation leader @xmath401 will get a greater reward for going to night 0 than for going to night 1 . in this situation , leader @xmath401 s using its guessed effect set wlu will lead it to make the wrong pick . to investigate the efficacy of the macrolearning , two sets of separate experiments were conducted . in the first one the reward matrix @xmath222 was chosen so that if each leader is maximizing its wl reward , but for guessed effect sets that contain none of its followers , then the system evolves to @xmath406 world reward . so if a leader incorrectly guesses that some @xmath269 is its effect set even though @xmath269 does nt contain both of that leader s followers , and if this is true for all leaders , then we are assured of worst possible performance . in the second set of experiments , we investigated the efficacy of macrolearning for a broader spectrum of reward matrices by generating those matrices randomly . we call these two kinds of reward matrices _ worst - case _ and _ random _ reward matrices , respectively . in both cases , if it can modify the initial guessed effect sets of the leaders to include their followers , then macrolearning will induce the system to be factored . the microlearning in these experiments was the same as in the bar problem . all experiments used the wl personal reward with some ( initially random ) guessed effect set . when macrolearning was used , it was implemented starting after the microlearning had run for a specified number of weeks . the macrolearner worked by estimating the correlations between the agents selections of which nights to attend . it did this by examining the attendances of the agents over the preceding weeks . given those estimates , for each agent @xmath33 the two agents whose attendances were estimated to be the most correlated with those of agent @xmath33 were put into agent @xmath33 s guessed effect set . of course , none of this macrolearning had any effect on global performance when applied to follower agents , but the macrolearning algorithm can not know that ahead of time ; it applied this procedure to each and every agent in the system . figure [ fig : worstreward ] presents averages over 50 runs of world reward as a function of weeks using the worst - case reward matrix . for comparison purposes , in both plots the top curve represents the case where the followers are in their leader s guessed effect sets . the bottom curve in both plots represents the other extreme where no leader s guessed effect set contains either of its followers . in both plots , the middle curve is performance when the leaders guessed effect sets are initially random , both with ( right ) and without ( left ) macrolearning turned on at week 500 . the performance for random guessed effect sets differs only slightly from that of having leaders guessed effect sets contain none of their followers ; both start with poor values of world reward that deteriorates with time . however , when macrolearning is performed on systems with initially random guessed effect sets , the system quickly rectifies itself and converges to optimal performance . this is reflected by the sudden vertical jump through the middle of the right plot at 500 weeks , the point at which macrolearning changed the guessed effect sets . by changing those guessed effect sets macrolearning results in a system that is factored for the associated wl reward function , so that those reward functions quickly induced the maximal possible world reward . figure [ fig : randomreward ] presents performance averaged over 50 runs for world reward as a function of weeks using a spectrum of reward matrices selected at random . the ordering of the plots is exactly as in figure [ fig : worstreward ] . macrolearning is applied at 2000 weeks , in the right plot . the simulations in figure [ fig : randomreward ] were lengthened from those in figure [ fig : worstreward ] because the convergence time of the full spectrum of reward matrices case was longer . in figure [ fig : randomreward ] the macrolearning resulted in a transient degradation in performance at 2000 weeks followed by convergence to the optimal . without macrolearning the system s performance no longer varied after 2000 weeks . combined with the results presented in figure [ fig : worstreward ] , these experiments demonstrate that macrolearning induces optimal performance by aligning the agents guessed effect sets with those agents that they actually do influence the most . many distributed computational tasks can not be addressed by direct modeling of the underlying dynamics , or are at best poorly addressed that way due to robustness and scalability concerns . such tasks should instead be addressed by model - independent machine learning techniques . in particular , reinforcement learning ( rl ) techniques are often a natural choice for how to address such tasks . when as is often the case we can not rely on centralized control and communication , such rl algorithms have to be deployed locally , throughout the system . this raises the important and profound question of how to configure those algorithms , and especially their associated utility functions , so as to achieve the ( global ) computational task . in particular we must ensure that the rl algorithms do not `` work at cross - purposes '' as far as the global task is concerned , lest phenomena like tragedy of the commons occur . how to initialize a system to do this is a novel kind of inverse problem , and how to adapt a system at run - time to better achieve such a global task is a novel kind of learning problem . we call any distributed computational system analyzed from the perspective of such an inverse problem a collective intelligence ( coin ) . as discussed in the literature review section of this chapter , there are many approaches / fields that address aspects of coins . these range from multi - agent systems through conventional economics and on to computational economics . ( human economies are a canonical model of a functional coin . ) they range onward to game theory , various aspects of distributed biological systems , and on through physics , active walker models , and recurrent neural nets . unfortunately , none of these fields seems appropriate as a general approach to understanding coins . after this literature review we present a mathematical theory for coins . we then present experiments on two test problems that validate the predictions of that theory for how best to design a coin to achieve a global computational task . the first set of experiments involves a variant of arthur s famous el farol bar problem . the second set instead considers a leader - follower problem that is hand - designed to cause maximal difficulty for the advice of our theory on how to initialize a coin . this second set of experiments is therefore a test of the on - line learning aspect of our approach to coins . in both experiments the procedures derived from our theory , procedures using only local information , vastly outperformed natural alternative approaches , even such approaches that exploited global information . indeed , in both problems , following the theory summarized in this chapter provides good solutions even when the exact conditions required by the associated theorems hold only approximately . there are many directions in which future work on coins will proceed ; it is a vast and rich area of research . we are already successfully applying our current understanding of coins , tentative as it is , to internet packet routing problems . we are also investigating coins in a more general optimization context where economics - 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this paper surveys the emerging science of how to design a `` collective intelligence '' ( coin ) . a coin is a large multi - agent system where : \i ) there is little to no centralized communication or control . \ii ) there is a provided world utility function that rates the possible histories of the full system . in particular , we are interested in coins in which each agent runs a reinforcement learning ( rl ) algorithm . the conventional approach to designing large distributed systems to optimize a world utility does not use agents running rl algorithms . rather , that approach begins with explicit modeling of the dynamics of the overall system , followed by detailed hand - tuning of the interactions between the components to ensure that they `` cooperate '' as far as the world utility is concerned . this approach is labor - intensive , often results in highly nonrobust systems , and usually results in design techniques that have limited applicability . in contrast , we wish to solve the coin design problem implicitly , via the `` adaptive '' character of the rl algorithms of each of the agents . this approach introduces an entirely new , profound design problem : assuming the rl algorithms are able to achieve high rewards , what reward functions for the individual agents will , when pursued by those agents , result in high world utility ? in other words , what reward functions will best ensure that we do not have phenomena like the tragedy of the commons , braess s paradox , or the liquidity trap ? although still very young , research specifically concentrating on the coin design problem has already resulted in successes in artificial domains , in particular in packet - routing , the leader - follower problem , and in variants of arthur s el farol bar problem . it is expected that as it matures and draws upon other disciplines related to coins , this research will greatly expand the range of tasks addressable by human engineers . moreover , in addition to drawing on them , such a fully developed science of coin design may provide much insight into other already established scientific fields , such as economics , game theory , and population biology . 0.2 in 5.8 in -0.5 in 9.0 in 0.2 in

the discovery of gliese 229b ( oppenheimer et al . 1995 ) and the successes of the 2mass ( reid 1994 ; stiening , skrutskie , and capps 1995 ) , sloan ( strauss et al . 1999 ) , and denis ( delfosse et al . 1997 ) surveys have collectively opened up a new chapter in stellar astronomy . the l and t dwarfs ( kirkpatrick et al . 1999,2000 ; martn et al . 1999 ; burgasser et al . 1999 , 2000a , b , c ) that have thereby been discovered and characterized comprise the first new stellar " types to be added to the stellar zoo in nearly 100 years . the lower edge of the solar - metallicity main sequence is an l dwarf not an m dwarf , with a near 1700 kelvin ( k ) , and more than 200 l dwarfs spanning a range from @xmath02200 k to @xmath01300 k are now inventoried . the coolest l dwarfs are also brown dwarfs , objects too light ( @xmath4 m@xmath5 ) to ignite hydrogen stably on the main sequence ( burrows et al . similarly , to date approximately 40 t dwarfs have been discovered spanning the range from @xmath01200 k to @xmath0750 k. these are all brown dwarfs and are the coldest stars " currently known . however , the edge of the stellar " mass function in the field , in the solar neighborhood , or in star clusters has not yet been reached and it is strongly suspected that in the wide mass and gap between the currently known t dwarfs and jovian - like planets there resides a population of very cool ( @xmath6 k ) brown dwarfs . such objects could be too dim in the optical and near - infrared to have been seen with current technology , but might be discovered in the not - too - distant future by the ngss / wise infrared space survey ( wright et al . 2001 ) , sirtf ( space infrared telescope facility ; werner and fanson 1995 ) , and/or jwst ( james webb space telescope ; mather and stockman 2000 ) . in this paper , we calculate the spectra and colors of such a population in order to provide a theoretical underpinning for the future study of these coolest of brown dwarfs . dwelling as they do at beyond those of the currently - known t dwarfs , these stars " emit strongly in the near- and mid - infrared . consequently , we highlight their fluxes from 1 to 30 microns and compare these fluxes with the putative sensitivities of instruments on sirtf and planned instruments on jwst . we include the effects of water clouds that form in the coolest of these objects . the presence of clouds of any sort emphasizes the kinship of this transitional class with solar system planets , in which clouds play a prominent role . ( note , however , that on jupiter itself water clouds are too deep below the ammonia cloud layer to have been unambiguously detected . ) since we focus on isolated free - floaters or wide binary brown dwarfs , we do not include external irradiation by companions . the of this model set ( @xmath7 k ) are such that silicate and iron clouds are deeply buried . hence , unlike for l dwarfs and early t dwarfs ( marley et al . 2002 ; burrows et al . 2002 ) , the effect of such refractory clouds on emergent spectra can be ignored . in [ approach ] , we discuss our numerical approaches and inputs . we go on in [ models ] to describe our mass - age model set and our use of the burrows et al . ( 1997 ) evolutionary calculations to provide the mapping between ( , gravity [ @xmath8 ) and ( mass , age ) . in [ profiles ] , we present a representative sample of derived atmospheric temperature(t)/pressure(p ) profiles and their systematics . this leads in [ sense ] to a short discussion of the sirtf and jwst point - source sensitivities . section [ spectra ] concerns the derived spectra and is the central section of the paper . in it , we discuss prominent spectral features from the optical to 30 microns , trends as a function of age , , and mass , diagnostics of particular atmospheric constituents , and detectability with instruments on sirtf and jwst . we find that these platforms can in principle detect brown dwarfs cooler than the current t dwarfs out to large distances . we also explore the evolution of @xmath9 and its eventual return to the red " ( marley et al . 2002 ; stephens , marley , and noll 2001 ) , reversing the blueward trend with decreasing that roughly characterizes the known t dwarfs . furthermore , we make suggestions for filter sets that may optimize their study with nircam on jwst . finally , we present physical reasons for anticipating the emergence of a new stellar type beyond the t dwarfs . in [ conclusion ] , we summarize what we have determined about this coolest - dwarf family and the potential for their detection . to calculate model atmospheres of cool brown dwarfs requires 1 ) a method to solve the radiative transfer , radiative equilibrium , and hydrostatic equilibrium equations , 2 ) a convective algorithm , 3 ) an equation of state that also provides the molecular and atomic compositions , 4 ) a method to model clouds that may form , and 5 ) an extensive opacity database for the constituents that arise in low - temperature , high - pressure atmospheres . the computer program we use to solve the atmosphere and spectrum problem in a fully self - consistent fashion is an updated version of the planar code tlusty ( hubeny 1988 ; hubeny & lanz 1995 ) , which uses a hybrid of complete linearization and accelerated lambda iteration ( hubeny 1992 ) . to handle convection , we use mixing - length theory ( with a mixing length equal to one pressure - scale height ) . the equation of state we use to find the p / t / density(@xmath10 ) relation is that of saumon , chabrier , and van horn ( 1995 ) and the molecular compositions are calculated using a significantly updated version of the code solgasmix ( burrows and sharp 1999 ) . the latter incorporates a rainout algorithm for refractory silicates and iron ( burrows et al . the most important molecules are h@xmath1 , h@xmath1o , ch@xmath2 , co , n@xmath1 , and nh@xmath3 and the most important atoms are na and k. we determine when water condenses by comparing the water ice condensation curve ( the total pressure at which the partial pressure of water is at saturation ) with the object s t / p profile . for pressures lower than that near the associated intercept , we deplete the vapor phase through the expected rainout and embed an absorbing / scattering water - ice cloud with a thickness of one pressure - scale - height in the region above . note that the total gas pressures at which the partial pressure of water is at the triple - point pressure of water are generally higher than the intercept pressures we find hence , the water gas to water ice ( solid ) transition is the more relevant . note also that the optical properties of water ice and water droplets are not very different . the ice particles are assumed to be spherical and their modal particle radii are derived using the theory of cooper et al . they vary in size from @xmath020 ( higher-@xmath11/lower- ) to @xmath0150 ( lower-@xmath11/higher- ) and we assume that the particle size is independent of altitude . a canonical super - saturation factor ( cooper et al . 2003 ; ackermann and marley 2001 ) of 0.01 ( 1.0% ) is used . curiously , with such large particles and such a small super - saturation , the absorptive opacity of our baseline water - ice clouds , when they do form , is not large . in fact , the consequences for the emergent spectrum of the associated drying of the upper atmosphere , and the corresponding diminution of the water vapor abundance there , are comparable to the effects on the spectrum of the clouds themselves . without an external flux source , and the scattering of that flux back into space by clouds , water - ice clouds seem to have only a secondary influence on the spectra of the coolest isolated brown dwarfs . we use the constantly - updated opacity database described in burrows et al . ( 1997,2001,2002 ) . this includes rayleigh scattering , collision - induced absorption ( cia ) for h@xmath1 ( borysow and frommhold 1990 ; borysow , jrgensen , and zheng 1997 ) , and t / p - dependent absorptive opacities from 0.3 to 300 for h@xmath1o , ch@xmath2 , co , and nh@xmath3 . the opacities of the alkali metal atoms are taken from burrows , marley , and sharp ( 2000 ) , which are similar in the line cores and near wings to those found in burrows and volobuyev ( 2003 ) . the opacities are tabulated in t/@xmath10/frequency space using the abundances derived for a solar - metallicity elemental abundance pattern ( anders and grevesse 1989 ; grevesse and sauval 1998 ; allende - prieto , lambert , and asplund 2002 ) . during the tlusty iterations , the opacity at any thermodynamic point and for any wavelength is obtained by interpolation . the absorptive opacities for the ice particles are derived using mie theory with the frequency - dependent spectrum of the complex index of refraction of water ice . ammonia clouds form in the upper atmospheres of the coldest exemplars of the late brown dwarf family ( @xmath12 160 k ; [ profiles ] ) . nevertheless , since the scattering of incident radiation that gives them their true importance in the jovian context is absent , we ignore them here . we have chosen for this study a set of models with the masses and ages given in table 1 . also shown in table 1 are the corresponding gravities and . these models span an effective temperature range from @xmath0800 k to @xmath0150 k that allows us to probe the realm between the known t dwarfs and the known jovian planets . to establish the mapping between mass / age pairs and the /@xmath11 pairs that are needed for atmospheric calculations , we use the evolutionary models of burrows et al . ( 1997 ) . while this procedure does not ensure that the atmospheres we calculate are fully consisitent with those evolutionary tracks , the errors are not large . figure [ fig:1 ] depicts evolutionary trajectories and isochrones in /@xmath11 space for models in the realm beyond the t dwarfs . the depicted isochrones span the range from 10@xmath13 to @xmath14 years and the masses cover the range from 0.5 to 25 . the large dots denote the models found in table 1 for which we have calculated spectra and atmospheres . for contrast , the approximate region in which the currently known t dwarfs reside is also shown . in addition , we provide the demarcation lines that separate ( in a rough sense ) the cloud - free models from those with water clouds and ammonia clouds . the clouds form to the left of the corresponding condensation lines . figure [ fig:1 ] emphasizes the transitional and as - yet - unstudied character of this family of objects . it also provides at a glance a global summary of family properties . figure [ fig:2 ] is a companion figure to fig . [ fig:1 ] , but shows iso - lines in mass / age space . for a given mass , fig . [ fig:2 ] allows one to determine the evolution of and at what age a given is achieved . it also makes easy the determination of the combination of mass and age for which clouds form , as well as the minimum mass for which a given is reached after approximately the galactic disk s or the sun s age ( @xmath15 and @xmath16 years , respectively ) . for instance , fig . [ fig:2 ] shows that it takes @xmath17 myr for a 2-object to reach a of 400 k , that it takes the same object 1 gyr to reach a of @xmath0250 k , and that in the age of the solar system a 2-object can reach the nh@xmath3 condensation line . similarly , fig . [ fig:2 ] indicates that a 10-object takes @xmath01 gyr to reach a of @xmath0400 k , and that it has water - ice clouds in its upper atmosphere . figures [ fig:1 ] and [ fig:2 ] are , therefore , useful maps of the model domain to which the reader may want often to return . to calculate absolute fluxes at 10 parsecs one needs the radius of the object . we determine this for each model in table 1 by using a fit to the results of burrows et al . ( 1997 ) that works reasonably well below @xmath025 and after deuterium burning has ended : @xmath18 where @xmath19 is jupiter s radius ( @xmath20 cm ) . shown in fig . [ fig:3 ] are representative temperature - pressure profiles at 300 myr ( blue ) for models with masses of 1 , 2 , 5 , 7 , and 10 and at 5 gyr ( red ) for models with masses of 2 , 5 , 7 , 10 , 15 , 20 , and 25 . superposed are the water ice and ammonia condensation lines at solar metallicity . the radiative - convective boundary pressures are near 0.1 - 1.0 bars for the lowest - mass , oldest models and are near 10 - 30 bars for the youngest , most massive models . at a given temperature , lower - mass objects have higher pressures ( at a given age ) . similarly , an object with a given mass evolves to higher and higher pressures at a given temperature . this trend is made clear in fig . [ fig:4 ] , in which the evolving t / p profiles for 1-and 5-models are depicted , and is not unexpected ( marley et al . 1996,2002 ; burrows et al . note that fig . [ fig:4 ] implies that a 5-object takes @xmath0300 myr to form water clouds , but that a 1-object takes only @xmath0100 myr . after @xmath01 gyr , a 1-object forms ammonia clouds , signature features of jupiter itself . these numbers echo the information also found in fig . [ fig:2 ] . the appearance of a water - ice cloud manifests itself in figs . [ fig:3 ] and [ fig:4 ] by the kink in the t / p profile near the intercept with the associated condensation line . generally , the higher the intercept of the t / p profile with the condensation line ( the lower the intercept pressure ) the smaller the droplet size ( cooper et al . 2003 ) . note that after an age of @xmath0300 myr a 7-object is expected to form water clouds high up in its atmosphere and that after @xmath05 gyr even a 25-object will do so . the higher the atmospheric pressure at which the cloud forms the greater the column thickness of the cloud . this results in a stronger cloud signature for the lower - mass models than for the higher - mass models . however , given the generally large ice particle sizes derived with the cooper et al . ( 2003 ) model , the low assumed supersaturation ( [ approach ] ) , the tendency for larger particle radii to form for larger intercept pressures , and the modest to low imaginary part of the index of refraction for pure water ice , the effect of water clouds in our model set is not large . this translates into a small cloud effect on the corresponding flux spectra ( [ spectra ] ) . figure [ fig:5 ] portrays the evolution of the t / p profiles for 10-and 20-objects . this figure is provided to show , among other things , the position of the forsterite ( mg@xmath1sio@xmath2 ) condensation line relative to that of water ice . mg@xmath1sio@xmath2 clouds exist in these brown dwarfs , but at significantly higher pressures and temperatures and are , therefore , buried from view . hence , unlike in l dwarfs , such clouds have very little effect on the emergent spectra of the coolest brown dwarfs that are the subject of this paper . finally , the high pressures achieved at low temperatures for the lowest mass , oldest objects shown in figs . [ fig:3 ] , [ fig:4 ] , and [ fig:5 ] suggest that the cia ( pressure - induced ) opacity of h@xmath1 might for them be important . this is indeed the case at longer wavelengths and is discussed in [ spectra ] . we mention this because cia opacity is yet another characteristic signature of the jovian planets in our own solar system and to emphasize yet again that our cold brown dwarf model suite is a bridge between the realms of the planets and the stars . " before we present and describe our model spectra , we discuss the anticipated point - source sensitivities of the instruments on board the sirtf and jwst space telescopes . sirtf has a 0.84-meter aperture and is to be launched in mid - april of 2003 . jwst is planned to have a collecting area of @xmath025 square meters over a segmented 6-meter diameter mirror and is to be launched at the beginning of the next decade . while sirtf is the last of the great observatories , " and will view the sky with unprecedented infrared sensitivity , jwst will in turn provide a two- to four - order - of - magnitude gain in sensitivity through much of the mid - infrared up to 27 microns . while their fields of view are limited and missions like wise ( formerly ngss ; wright et al . 2001 ) are more appropriate for large - area surveys , the extreme sensitivity of both sirtf and jwst will bring the coolest brown dwarfs and isolated giant planets into the realm of detectability and study . sirtf / irac has four channels centered at 3.63 , 4.53 , 5.78 , and 8.0 that are thought to have 5-@xmath21 point - source sensitivities for 200-second integrations of @xmath02.5 , @xmath04.5 , @xmath015.5 , and @xmath025.0 microjanskys , respectively . hst / nicmos achieves a bit better than one microjansky sensitivity at 2.2 , but does not extend as far into the near ir . the short - wavelength , low - spectral resolution module ( short - low " ) of sirtf / irs extends from @xmath05.0 to @xmath014.0 and has a 5-@xmath21 point - source sensitivity for a 500-second integration of @xmath0100 microjanskys . the other three modules on irs cover other mid - ir wavelength regimes at either low- or high - spectral resolution , but will have smaller brown dwarf detection ranges . the @xmath020.5 to @xmath026 channel on sirtf / mips is the most relevant channel on mips for brown dwarf studies and has a suggested 1-@xmath21 point - source sensitivity at @xmath024 of @xmath070 microjanskys . this is @xmath01000 times better in imaging mode than for the pioneering iras . all these sirtf sensitivities are derived from various sirtf web pages and are pre - launch estimates ( ` http://sirtf.caltech.edu ` ) . furthermore , for all three sirtf instruments , one can estimate the point - source sensitivities for different values of the signal - to - noise and integration times . these signals - to - noise and integration times are the nominal combinations for each instrument and the quoted sensitivities serve to guide our assessment of sirtf s capabilities for cool brown dwarf studies in advance of real on - orbit calibrations and measurements . the capabilities of jwst are even more provisional , but the design goals for its instruments are impressive ( ` http://ngst.gsfc.nasa.gov ` ) . jwst / nircam is to span @xmath00.6 to @xmath05.0 in various wavelength channels / filters , though the final design has not been frozen . the seven so - called b " filters have widths of 0.51.0 microns centered at @xmath00.71 , @xmath01.1 , @xmath01.5 , @xmath02.0 , @xmath02.7 , @xmath03.6 , and @xmath04.4 microns and are expected to have 5-@xmath21 point - source sensitivities in imaging mode , for an assumed exposure time of @xmath22 seconds , of @xmath01.6 , @xmath00.95 , @xmath01.0 , @xmath01.2 , @xmath00.95 , @xmath01.05 , and @xmath01.5 nanojanskys ( nj ) , respectively . in addition , a set of so - called i " filters , with about half to one quarter the spectral width of the b filters , and sensitivites comparable to that of the b filters , are available in the 1.55.0 region . furthermore , jwst / nircam may have a tunable filter to examine selected spectral regions beyond 2.5 at a resolution ( @xmath23 ) of @xmath0100 , though at the time of this writing the availability of such a capability remained uncertain . hence , with jwst / nircam we enter the world of _ nano_jansky sensitivity . this is greater than one hundred times more sensitive than hst / nicmos at 2.2 and enables one to probe deeply in space , as well as broadly in wavelength . jwst / miri spans the mid - ir wavelength range from @xmath05.0 to @xmath027.0 and will have in imaging mode a 10-@xmath21 point - source sensitivity for a 10@xmath24-second integration of from @xmath063 nj at the shortest wavelength to @xmath010 microjanskys at the longest . this is orders of magnitude more sensitive than any previous mid - ir telescope in imaging mode . ( in spectral mode with an @xmath25 near 1000 , jwst / miri will be @xmath0100 times less sensitive than in imaging mode . ) given the importance of the mid - ir for understanding those brown dwarfs that may exist in relative abundance at cooler than those of the currently known t dwarfs , miri provides what is perhaps a transformational capability . as with sirtf , the quoted jwst sensitivities are taken from the associated web pages and , hence , should be considered tentative . we now turn to a discussion of the spectra , spectral evolution , defining features , systematics , and diagnostics for the cool brown dwarf models listed in table 1 and embedded in figs . [ fig:1 ] and [ fig:2 ] . on each of figs . [ fig:6 ] to [ fig:11 ] in [ spectra ] , we plot for the sirtf ( red ) and jwst ( blue ) instruments the broadband sensitivities we have summarized in this section . using the numerical tools and data referred to in [ approach ] , and the mapping between /@xmath11 and mass / age found in table 1 , we have generated a grid of spectral and atmospheric models for cool brown dwarfs that reside in the low - sector of /@xmath11 space ( fig . [ fig:1 ] ) . some of the associated t / p profiles were given in figs . [ fig:3 ] , [ fig:4 ] , and [ fig:5 ] . in figs . [ fig:6 ] to [ fig:11 ] , we plot theoretical flux spectra ( f@xmath26 , in millijanskys ) from the optical to 30 at a distance of 10 parsecs . these figures constitute the major results of our paper . for comparision , superposed on each figure are the estimated point - source sensitivities of the instruments on board sirtf and jwst ( [ sense ] ) . in addition , included at the top of figs . [ fig:8 ] through [ fig:11 ] are the rough positions of the major atmospheric absorption features . ( the full model set is available from the first author upon request . ) figures [ fig:6 ] and [ fig:7 ] portray the mass dependence of a cool brown dwarf s flux spectrum at 10 parsecs for ages of one and five gyr , respectively . the model masses are 25 , 20 , 15 , 10 , 7 , 5 , 2 , and 1 . the top panels depict the most massive four , while the bottom panels depict the least massive four ( three for fig . [ fig:7 ] ) . together they show the monotonic diminution of flux with object mass at a given age that parallels the associated decrease in with mass ( from @xmath0800 k to @xmath0130 k ) seen in table 1 and fig . [ fig:2 ] . figures [ fig:6 ] through [ fig:11 ] show the peaks due to enhanced flux through the water vapor absorption bands that define the classical terrestrial photometric bands ( @xmath27 , @xmath28 , @xmath29 , @xmath30 , and @xmath31 ) and that have come to characterize brown dwarfs since the discovery of gliese 229b ( oppenheimer et al . 1995 ; marley et al . 1996 ) . for the more massive models , the near - ir fluxes are significantly above black - body values . at @xmath0800 k , the 25-/1-gyr model shown in fig . [ fig:6 ] could represent the known late t dwarfs , but all other models in this model set are later " and , hence , represent as yet undetected objects . apart from the distinctive water troughs , generic features are the hump at 4 - 5 microns ( @xmath31 band ) , the broad hump near 10 microns , the methane features at 2.2 , 3.3 , 7.8 , and in the optical ( particularly at 0.89 ) , the ammonia features at @xmath01.5 , @xmath01.95 , @xmath02.95 , and @xmath010.5 , and the na - d and k i resonance lines at 0.589 and 0.77 , respectively . however , as figs . [ fig:6]-[fig:11 ] indicate , the strengths of each of these features are functions of mass and age . for lower masses or greater ages , the centroid of the @xmath31 band hump shifts from @xmath04.0 to @xmath05.0 . in part , this is due to the swift decrease with at the shorter wavelengths of the wien tail . even after the collapse of the flux in the optical and near - ir after @xmath01 gyr for masses below 5 or after @xmath05 gyr for masses below 10 , the @xmath31 band flux persists as a characteristic marker and will be sirtf s best target . moreover , irac s filters are well - positioned for this task . as one would expect , the relative importance of the mid - ir fluxes , in particular between 10 and 30 microns , grows with decreasing mass and increasing age . since this spectral region is near the linear rayleigh - jeans tail , fluxes here persist despite decreases in from @xmath0800 k to @xmath0130 k. figure [ fig:11 ] depicts this clearly for the older 2-models . the rough periodicity in flux beyond 10 is due predominantly to the presence of pure rotational bands of water and , for cooler models , methane as well . for the coldest models depicted in figs . [ fig:6 ] , [ fig:7 ] , and [ fig:11 ] , this behavior subsides , but is replaced with long - period undulations due to cia absorption by h@xmath1 . such a signature is characteristic of jovian planets and is expected for low - t , high - p atmospheres . its appearance marks yet another transition , seen first in this model set for the old 5-and middle - aged 2-objects , between t - dwarf - like and planet "- like behavior . as figs . [ fig:6]-[fig:11 ] imply , sirtf / mips should be able to detect at 10 parsecs the @xmath024-flux of objects more massive than 2 - 4 at age 1 gyr or more massive than 10 at 5 gyr . methane forms at low temperatures and high pressures and makes its presence felt in older and less massive objects . hence , its features at 0.89 , 2.2 , 3.3 , and 7.8 deepen with age and decreasing mass . an example of such strengthening at 7.8 and 2.2 can be seen in fig . [ fig:10 ] by comparing the 100-myr and 5-gyr models with a mass of 5 . clear indications of the strengthening of the methane absorption feature at 0.89 with decreasing mass can be seen in the upper panel of fig . [ fig:7 ] . this trend is accompanied by a corresponding weakening of the cs i feature on top of it . however , due to its presence in the @xmath32 band at relatively short wavelengths , the methane feature at 0.89 may be difficult to detect for all but the youngest and/or most massive models . the actual strength of the 7.8-feature depends on the t / p profile in the upper layers of the atmosphere , which in turn might be affected by ambient uv ( disfavored for free - floating brown dwarfs ) or processes that could create a stratosphere and a temperature inversion . hence , the filling in or reshaping of the 7.8-feature might signal the presence of a stratosphere . such a temperature inversion could also affect the depths of the water troughs . as can be seen by comparing the top panels of figs . [ fig:8]-[fig:11 ] , the alkali metal features at 0.589 and 0.77 diminish in strength with decreasing mass and increasing age . these features are signatures of the known t dwarfs ( burrows , marley , and sharp 2000 ; burrows et al . 2002 ; tsuji , ohnaka , and aoki 1999 ) , so their decay signals a gradual transformation away from standard t - dwarf behavior . for the 10-model older than 1 gyr and the 2-model older than 100 myr , these alkali resonance features cease to be primary signatures . this happens near a of 450 k. ammonia makes an appearance at even lower temperatures than methane and due to the relatively high abundance of nitrogen its absorption features are generally strong , particularly for the cool objects in our model set . for the higher in the mid - t - dwarf range , ammonia may have been seen , but is weak ( saumon et al . 2000 ) . figs . [ fig:10 ] and [ fig:11 ] evince strong ammonia features in the upper panels at @xmath01.5 , @xmath01.95 , and @xmath02.95 and in figs . [ fig:8]-[fig:11 ] in the lower panels at @xmath010.5 . as figs . [ fig:6]-[fig:11 ] imply , the short - low module on sirtf / irs should be able to study the 10.5-ammonia feature . even for the 25-/1gyr model , the @xmath010.5 feature is prominent . for the more massive objects ( 10 - 25 ) , the strength of the 10.5-feature increases with age . for the lowest mass objects ( 2 - 7 ) , the strength of the 10.5-ammonia feature actually decreases with age , even though the strengths of the other ammonia lines increase . as the more massive objects age , their atmospheric pressures increase , shifting the n@xmath1/nh@xmath3 equilibrium towards nh@xmath3 . for the less massive models , pressured - induced absorption by h@xmath1 grows with increasing atmospheric pressure ( fig . [ fig:3]-[fig:4 ] ) and partially flattens an otherwise strengthening 10.5-ammonia feature . below of @xmath0160 k , figs . [ fig:1 ] and [ fig:2 ] demonstrate that ammonia clouds form . however , given that we are studying isolated objects that have no reflected component ( unlike jupiter and saturn ) , and given that realistic supersaturations are only @xmath01% , we have determined that ammonia clouds do not appreciably affect the emergent spectra . as a consequence , we ignore them in the three relevant models ( fig . [ fig:1 ] ) . as with the known t and l dwarfs , water vapor absorptions dominate and sculpt the flux spectra of the cooler brown dwarfs and these features generally deepen with increasing age and decreasing mass . the latter trend is in part a consequence of the increase with decreasing gravity of the column depth of water above the ( roughly - defined ) photosphere . at below @xmath0400 - 500 [ fig:1]-[fig:5 ] ) , water condenses in brown dwarf atmospheres . the appearance of such water - ice clouds constitutes yet another milestone along the bridge from the known t dwarfs to the giant planets . associated with cloud formation is the depletion of water vapor above the tops of the water cloud , with the concommitant decrease at altitude in the gas - phase abundance of water . within @xmath0100 myr , water clouds form in the atmosphere of an isolated 1-object and within @xmath05 gyr they form in the atmosphere of a 25-object . in fact , approximately two - thirds of the models listed in table 1 incorporate water - ice clouds . however , at supersaturations of 1% and for particle sizes above 10 microns ( [ approach]-[profiles ] ; cooper et al . 2003 ) , such clouds ( and the corresponding water vapor depletions above them ) only marginally affect the calculated emergent spectra . even though we see in figs . [ fig:3]-[fig:5 ] the associated kinks in the t / p profiles , these do not translate into a qualitative change in the emergent spectra at any wavelength . for wavelengths longward of 1 micron , the cloudy spectra differ from the no - cloud spectra by at most a few tens of percent . for a representative 2-model at 300 myr ( @xmath0280 k ) , if we increase the supersturation factor by a factor of ten from 1% to 10% , the flux at 5 microns decreases by approximately a factor of two , while the flux from 10 to 30 microns increases by on average @xmath050% . these are not large changes , given the many orders of magnitude covered by the fluxes in figs . [ fig:6]-[fig:11 ] . the prominence of water features provides a guide to the optimal placement of nircam filters for the detection and characterization of brown dwarfs . for example , the water feature near 0.93 is missed by the b filters , while those features at @xmath01.4 and @xmath01.8 are not centered on the respective adjacent filters and , hence , are diluted by the adjoining continuum . the i filters on nircam would partially overcome these limitations . even so , as fig . [ fig:12 ] shows , the broadband fluxes in the nircam filters provide useful diagnostics of the differences among brown dwarfs and extrasolar giant planets ( here expressed as mass at a given age ) , with particular sensitivity to the large flux differences between the 5-window and the region shortward . a tunable filter could provide even greater diagnostic capability by permitting in and around the 5-window a spectral resolution near 100 to more definitively characterize the effective temperature and , hence , the mass of detected objects ( for a given age and composition ) . nevertheless , fig . [ fig:7 ] indicates that at 10 parsecs even a 7-object at 5 gyr should easily be detected in imaging mode in the @xmath28 and @xmath29 bands . in the @xmath31 band , a 2-object could be seen by nircam out to @xmath0100 parsecs . furthermore , a 25-object at 5 gyr and a distance of 1000 parsecs should be detectable by nircam in a number of its current broadband filters . figure [ fig:13 ] shows predicted spectra of a 20-/5-gyr model in the mid - infrared for the sirtf / irs and jwst / miri instruments . to generate the sirtf / irs curve in fig . [ fig:13 ] , we multiplied the theoretical spectra by the irs response curves for the entire wavelength range , not just the 5 - 14 of the short - low " module . the irs spectral resolution has been assumed to be 100 , while that of jwst / miri is @xmath01000 . we find that the irs spectra are useful at 10 parsecs only for the warmer brown dwarfs ( @xmath33 ) , but for these brown dwarfs even at this modest spectral resolution one can clearly identify the various dominant molecular bands . in its broadband detection ( imaging ) mode , jwst / miri will be @xmath0100 times more capable than sirtf from @xmath05 to @xmath027 ( [ sense ] ) . since the mid - ir is one of the spectral regions of choice for the study of the coolest brown dwarfs , miri will assume for their characterization a role of dramatic importance . at wavelengths longward of 15- , miri will be able to detect objects 10 parsecs away down to 2 or lower . in addition , it could detect an object just 10 times the mass of jupiter with an age of 5 gyr out to a distance of one kiloparsec . furthermore , jwst / miri provides 10 times better spectral resolution than sirtf / irs for objects down to 10 . [ fig:6 ] through [ fig:13 ] collectively summarize the flux spectra and evolution of the cool brown dwarfs yet to be discovered , as well as the extraordinary capabilities of the various instruments on board both sirtf and jwst for the diagnosis and characterization of their atmospheres . these figures highlight the prominent molecular features of h@xmath1o , ch@xmath2 , nh@xmath3 , in particular , that are pivotal in the evolution of the differences between the coolest brown dwarfs and the known t dwarfs , most of which are at higher and gravities . the latest known t dwarf has been typed a t8 ( burgasser et al . 2000a ) , but its effective temperature is near 750 - 800 k ( geballe et al . 2001 ; burrows et al . this does not leave much room for the expansion of the t dwarf subtypes to the lower and masses discussed in this paper , and suggests that yet another spectroscopic class beyond the t dwarfs might be called for . many of the spectral trends described in this paper are gradual , but the near disappearance of the alkali features below = 500 k , the onset of water cloud formation below = 400 - 500 k , the collapse below @xmath0350 k of the optical and near - ir fluxes relative to those longward of @xmath05 , and the growing strengths of the nh@xmath3 features all suggest physical reasons for such a new class . figure [ fig:14 ] depicts isochrones from 100 myr to 5 gyr on the @xmath34 versus @xmath9 color - magnitude diagram and demonstrates that the blueward trend in @xmath9 that so typifies the t dwarfs stops and turns around ( marley et al . 2002 ; stephens , marley , and noll 2001 ) between effective temperatures of 300 and 400 k. this is predominantly due not to the appearance of water clouds , but to the long - expected collapse of flux on the wien tail . note that the at which the @xmath9 color turns around is not the same for all the isochrones . this is because the colors are not functions of just , but of gravity as well . the decrease in , that for the t dwarfs squeezes the @xmath30-band flux more than the @xmath28-band flux , finally does to @xmath9 what people had expected such a decrease to do before the discovery of t dwarfs , i.e. , redden the color . we remind the reader that unlike m dwarfs , the @xmath9 colors of t dwarfs actually get bluer with decreasing ( for a given surface gravity ) . this may be counterintuitive , but it is a result of the increasing role of methane and h@xmath1 collision - induced absorption with decreasing temperature , as well as the positive slope of the opacity / wavelength curve of water and its gradual steepening with decreasing temperature . were it not for the extremely low fluxes at such low shortward of 4 microns , we might have suggested the use of this turnaround to mark the beginning of a new spectroscopic class . moreover , clearly the optical can not be used and with the diminishing utility of the near infrared as drops , that leaves the mid - ir longward of @xmath04 as the most logical part of the spectrum with which to characterize a new spectroscopic class . as is usual , this will be determined observationally , and it might be done arbitrarily to limit the growth of the t sequence . nevertheless , we observe that the region between 300 k and 500 k witnesses a few physical transitions that might provide a natural break between stellar " types . we have generated a new set of brown dwarf spectral models that incorporate state - of - the - art opacities and the effects of water clouds . our focus has been on the low - branch of the brown dwarf tree beyond the known t dwarfs . to this end , we have investigated the range from @xmath0800 k to @xmath0130 k and the low - mass range from 25 to 1 . as fig . [ fig:1 ] indicates , this is mostly unexplored territory . our calculations have been done to provide a theoretical foundation for the new brown dwarf studies that will be enabled by the launch of sirtf and the eventual launch of jwst , as well as for the ongoing ground - based searches for the coolest substellar objects . we provide spectra from @xmath00.4 to 30 , investigate the dependence on age and mass of the strengths of the h@xmath1o , ch@xmath2 , and nh@xmath3 molecular features , address the formation and effect of water clouds , and compare the calculated fluxes with the suggested sensitivities of the instruments on board sirtf and jwst . from the latter , detection ranges can be derived , which for jwst can exceed a kiloparsec . we find that the blueward trend in near - infrared colors so characteristic of the t dwarfs stops near a of 300 - 400 k and we identify a few natural physical transitions in the low - realm which might justify the eventual designation of at least one new spectroscopic type after the t dwarfs . these include the formation of water clouds ( @xmath0400 - 500 k ) , the strengthening of ammonia bands , the eventual collapse in the optical , the shift in the position of the @xmath31 band peak , the turnaround of the @xmath9 color , the near disappearance of the strong na - d and k i resonance lines ( @xmath0500 k ) , and the increasing importance with decreasing of the mid - ir longward of 4 . for these cooler objects , the mid - infrared assumes a new and central importance and first mips and irs on sirtf , then miri on jwst , are destined to play pivotal roles in their future characterization and study . finally , the formation of ammonia clouds below @xmath0160 k suggests yet another natural breakpoint , and a second new stellar " class . therefore , there are reasons to anticipate that perhaps two naturally defined , yet uncharted , spectral types reside beyond the t dwarfs at lower . the current filter set for jwst / nircam from 0.6 to 5.0 is good , but not yet fully optimized for cool brown dwarf detection . placing filters on the derived spectral peaks and troughs ( robustly defined by the water bands ) would improve its already good performance for substellar research . in any case , our theoretical spectra are meant to bridge the gap between the known t dwarfs and those cool , low - mass free - floating brown dwarfs with progressively more planetary features which may inhabit the galaxy in interesting , but as yet unknown , numbers . the authors thank ivan hubeny , bill hubbard , john milsom , christopher sharp , jim liebert , curtis cooper , and jonathan fortney for fruitful conversations and help during the course of this work , as well as nasa for its financial support via grants nag5 - 10760 , nag5 - 10629 , and nag5 - 12459 . ackermann , a. and marley , m.s . 2001 , , 556 , 872 allende - prieto , c. , lambert , d.l . , and asplund , m. 2002 , , 573 , l137 anders , e. and grevesse , n. 1989 , geochim . cosmochim . acta , 53 , 197 bessell , m.s . , and brett , j. m. 1988 , , 100 , 1134 borysow , a. and frommhold , l. 1990 , , 348 , l41 borysow , a. , jrgensen , u.g . , and zheng , c. 1997 , , 324 , 185 burgasser , a.j . , et al . 1999 , , 522 , l65 burgasser , a.j . , et al . 2000a , , 531 , l57 burgasser , a.j . , kirkpatrick , j. d. , reid , i. n. , liebert , j. , gizis , j. e. , & brown , m. e. 2000b , , 120 , 473 burgasser , a.j . , et al . 2000c , , 120 , 1100 burrows , a. , marley m. , hubbard , w.b . lunine , j.i . , guillot , t. , saumon , d. freedman , r. , sudarsky , d. and sharp , c.m . 1997 , , 491 , 856 burrows , a. and sharp , c.m . 1999 , , 512 , 843 burrows , a. , marley , m. s. , and sharp , c. m. 2000 , , 531 , 438 burrows , a. , hubbard , w.b . , lunine , j.i . , and liebert , j. 2001 , rev . phys . , 73 , 719 burrows , a. , burgasser , a.j . , kirkpatrick , j. d. , liebert , j. , milsom , j.a . , sudarsky , d. , and hubeny , i. 2002 , , 573 , 394 burrows , a. and volobuyev , m. 2003 , , 583 , 985 cooper , c.s . , sudarsky , d. , milsom , j.a . , lunine , j.i . , & burrows , a. 2003 , , 586 , 1320 delfosse , x. , tinney , c.g . , forveille , t. , epchtein , n. , bertin , e. , borsenberger , j. , copet , e. , de batz , b. , fouqu , p. , kimeswenger , s. , le bertre , t. , lacombe , f. , rouan , d. , and tiphne , d. 1997 , , 327 , l25 geballe , t.r . , saumon , d. , leggett , s.k . , knapp , g.r . , marley , m.s . , and lodders , k. 2001 , , 556 , 373 grevesse , n. , sauval , a.j . 1998 , space sci . , 85 , 161 hubeny , i. 1988 , computer physics comm . , 52 , 103 hubeny , i. 1992 , in _ the atmospheres of early - 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427 ( sirtf ) wright , e. and the ngss team 2001 , bull . a.a.s . , 198 , 407 @xmath35 & @xmath36 & @xmath37 & @xmath38 + & @xmath39 & @xmath40 & @xmath41 + & @xmath42 & @xmath43 & @xmath44 + @xmath45 & @xmath36 & @xmath46 & @xmath47 + & @xmath39 & @xmath48 & @xmath49 + & @xmath42 & @xmath50 & @xmath51 + & @xmath52 & @xmath53 & @xmath54 + & @xmath55 & @xmath56 & @xmath57 + @xmath58 & @xmath36 & @xmath59 & @xmath60 + & @xmath39 & @xmath61 & @xmath62 + & @xmath42 & @xmath63 & @xmath64 + & @xmath52 & @xmath65 & @xmath66 + & @xmath55 & @xmath67 & @xmath68 + @xmath69 & @xmath36 & @xmath70 & @xmath71 + & @xmath39 & @xmath72 & @xmath73 + & @xmath42 & @xmath74 & @xmath75 + & @xmath52 & @xmath76 & @xmath77 + & @xmath55 & @xmath78 & @xmath79 + @xmath80 & @xmath36 & @xmath81 & @xmath82 + & @xmath39 & @xmath83 & @xmath84 + & @xmath42 & @xmath85 & @xmath86 + & @xmath52 & @xmath87 & @xmath88 + & @xmath55 & @xmath89 & @xmath90 + @xmath91 & @xmath42 & @xmath92 & @xmath93 + & @xmath52 & @xmath94 & @xmath95 + & @xmath55 & @xmath96 & @xmath97 + @xmath98 & @xmath42 & @xmath99 & @xmath100 + & @xmath52 & @xmath101 & @xmath102 + & @xmath55 & @xmath103 & @xmath104 + @xmath105 & @xmath42 & @xmath106 & @xmath107 + & @xmath52 & @xmath108 & @xmath109 + & @xmath55 & @xmath101 & @xmath110 +

we explore the spectral and atmospheric properties of brown dwarfs cooler than the latest known t dwarfs . our focus is on the yet - to - be - discovered free - floating brown dwarfs in the range from @xmath0800 k to @xmath0130 k and with masses from 25 to 1 . this study is in anticipation of the new characterization capabilities enabled by the launch of sirtf and the eventual launch of jwst . in addition , it is in support of the continuing ground - based searches for the coolest substellar objects . we provide spectra from @xmath00.4 to 30 , highlight the evolution and mass dependence of the dominant h@xmath1o , ch@xmath2 , and nh@xmath3 molecular bands , consider the formation and effects of water - ice clouds , and compare our theoretical flux densities with the putative sensitivities of the instruments on board sirtf and jwst . the latter can be used to determine the detection ranges from space of cool brown dwarfs . in the process , we determine the reversal point of the blueward trend in the near - infrared colors with decreasing ( a prominent feature of the hotter t dwarf family ) , the at which water and ammonia clouds appear , the strengths of gas - phase ammonia and methane bands , the masses and ages of the objects for which the neutral alkali metal lines ( signatures of l and t dwarfs ) are muted , and the increasing role as decreases of the mid - infrared fluxes longward of 4 . these changes suggest physical reasons to expect the emergence of at least one new stellar class beyond the t dwarfs . furthermore , studies in the mid - infrared could assume a new , perhaps transformational , importance in the understanding of the coolest brown dwarfs . our spectral models populate , with cooler brown dwarfs having progressively more planet - like features , the theoretical gap between the known t dwarfs and the known giant planets . such objects likely inhabit the galaxy , but their numbers are as yet unknown .

in the standard model ( sm ) , lepton - flavor - violating ( lfv ) decays of charged leptons are forbidden ; even if neutrino mixing is taken into account , they are still highly suppressed . however , lfv is expected to appear in many extensions of the sm . some such models predict branching fractions for @xmath0 lfv decays at the level of @xmath15 @xcite , which can be reached at the present b - factories . observation of lfv will then provide evidence for new physics beyond the sm . in this paper , we report on a search for lfv in @xmath16 decays into neutrinoless final states with one charged lepton @xmath17 and one vector meson @xmath2 : @xmath18 , @xmath19 , @xmath20 , @xmath21 , @xmath22 , @xmath23 , @xmath24 and @xmath25 @xcite . a search for the @xmath26 , @xmath27 and @xmath28 modes was performed for the first time at the cleo detector , where 90% confidence level ( cl ) upper limits ( ul ) for the branching fractions in the range @xmath29 were obtained using a data sample of 4.79 fb@xmath7 @xcite . later we carried out a search for these modes in the belle experiment using 158 fb@xmath7 of data and set upper limits in the range @xmath30 @xcite . here we present results of a new search based on a data sample of 543 fb@xmath7 corresponding to @xmath31 @xmath0-pairs collected with the belle detector @xcite at the kekb asymmetric - energy @xmath32 collider @xcite . the belle detector is a large - solid - angle magnetic spectrometer that consists of a silicon vertex detector , a 50-layer central drift chamber , an array of aerogel threshold cherenkov counters , a barrel - like arrangement of time - of - flight scintillation counters , and an electromagnetic calorimeter comprised of csi(tl ) crystals located inside a superconducting solenoid coil that provides a 1.5 t magnetic field . an iron flux - return located outside the coil is instrumented to detect @xmath33 mesons and identify muons . the detector is described in detail elsewhere @xcite . two inner detector configurations were used . a 2.0 cm radius beam - pipe and a 3-layer silicon vertex detector were used for the first sample of 158 fb@xmath7 , while a 1.5 cm radius beam - pipe , a 4-layer silicon detector and a small - cell inner drift chamber were used to record the remaining 385 fb@xmath7 @xcite . we search for @xmath34 , @xmath10 , @xmath13 and @xmath14 candidates in which one @xmath0 decays into a final state with a @xmath1 , two charged hadrons ( 3-prong decay ) , and the other @xmath0 decays into one charged particle ( 1-prong decay ) , any number of @xmath35 s and missing particle(s ) . we reconstruct @xmath3 candidates from @xmath36 , @xmath4 from @xmath37 , @xmath5 from @xmath38 and @xmath6 from @xmath39 . the selection criteria described below are optimized from studies of monte carlo ( mc ) simulated events and the experimental data in the sideband regions of the @xmath40 and @xmath41 distributions described later . the background ( bg ) mc samples consist of @xmath42 ( 1524 fb@xmath7 ) generated by kkmc @xcite , @xmath43 continuum , and two - photon processes . the signal mc events are generated assuming a phase space distribution for @xmath0 decay . the transverse momentum for a charged track is required to be larger than 0.06 gev/@xmath44 in the barrel region ( @xmath45 , where @xmath46 is the polar angle relative to the direction opposite to that of the incident @xmath47 beam in the laboratory frame ) and 0.1 gev/@xmath44 in the endcap region ( @xmath48 and @xmath49 ) . the energies of photon candidates are required to be larger than 0.1 gev in both regions . to select the signal topology , we require four charged tracks in an event with zero net charge , and a total energy of charged tracks and photons in the center - of - mass ( cm ) frame less than 11 gev . we also require that the missing momentum in the laboratory frame be greater than 0.6 gev/@xmath44 , and that its direction be within the detector acceptance ( @xmath50 ) , where the missing momentum is defined as the difference between the momentum of the initial @xmath8 system , and the sum of the observed momentum vectors . the event is subdivided into 3-prong and 1-prong hemispheres with respect to the thrust axis in the cm frame . these are referred to as the signal and tag side , respectively . we allow at most two photons on the tag side to account for initial state radiation , while requiring at most one photon for the @xmath12 , @xmath13 , @xmath14 modes , and two photons except for @xmath51 daughters for the @xmath10 modes on the signal side to reduce the @xmath52 bg . we require that the muon likelihood ratio @xmath53 be greater than 0.95 for momentum greater than 1.0 gev/@xmath44 and the electron likelihood ratio @xmath54 be greater than 0.9 for momentum greater than 0.5 gev/@xmath44 for the charged lepton - candidate track on the signal side . here @xmath55 is the likelihood ratio for a charged particle of type @xmath56 ( @xmath57 , @xmath58 , @xmath59 or @xmath60 ) , defined as @xmath61 , where @xmath62 is the likelihood for particle type @xmath56 , determined from the responses of the relevant detectors @xcite . the efficiencies for muon and electron identification are 92% for momenta larger than 1.0 gev/@xmath44 and 94% for momenta larger than 0.5 gev/@xmath44 . candidate @xmath3 mesons are selected by requiring the invariant mass of @xmath36 daughters to be in the range @xmath63 . we require that both kaon daughters have kaon likelihood ratios @xmath64 and electron likelihood ratios @xmath65 to reduce the background from @xmath32 conversions . candidate @xmath4 mesons are reconstructed from @xmath37 with the invariant mass requirement @xmath66 . the @xmath51 candidate is selected from @xmath35 pairs with invariant mass in the range , @xmath67 . in order to improve the @xmath4 mass resolution , the @xmath51 mass is constrained to be 135 mev/@xmath68 for the @xmath4 mass reconstruction . candidate @xmath5 and @xmath6 mesons are selected with @xmath69 invariant mass in the range @xmath70 , and requiring that the kaon daughter have @xmath64 and both daughters have @xmath65 . [ fig : vmass](a , b , c ) show the invariant mass distributions of the @xmath3 , @xmath4 and @xmath5 candidates for @xmath71 , @xmath72 and @xmath73 , respectively . the estimated bg distributions agree with the data . the main bg contribution is due to @xmath52 events with @xmath3 mesons for the @xmath74 mode , @xmath75 with the pion misidentified as a lepton for the @xmath76 mode , and @xmath77 with one pion misidentified as a kaon and another misidentified as a lepton for the @xmath78 and @xmath28 modes . for @xmath71 , ( b ) @xmath79 for @xmath72 and ( c ) @xmath80 for @xmath73 after muon identification . the points with error bars are data . the open histogram shows the expected @xmath42 bg mc and the hatched one @xmath52 mc and two - photon mc . the regions between the vertical red lines are selected.,title="fig:",scaledwidth=23.0% ] for @xmath71 , ( b ) @xmath79 for @xmath72 and ( c ) @xmath80 for @xmath73 after muon identification . the points with error bars are data . the open histogram shows the expected @xmath42 bg mc and the hatched one @xmath52 mc and two - photon mc . the regions between the vertical red lines are selected.,title="fig:",scaledwidth=23.0% ] for @xmath71 , ( b ) @xmath79 for @xmath72 and ( c ) @xmath80 for @xmath73 after muon identification . the points with error bars are data . the open histogram shows the expected @xmath42 bg mc and the hatched one @xmath52 mc and two - photon mc . the regions between the vertical red lines are selected.,title="fig:",scaledwidth=23.0% ] to reduce the remaining bg from @xmath42 and @xmath52 , we require the relations between the missing momentum @xmath81 ( gev/@xmath44 ) and missing mass squared @xmath82 ( ( gev/@xmath68)@xmath83 ) summarized in table [ tbl : vcut ] . .selection criteria using @xmath81(gev/@xmath44 ) and @xmath82((gev/@xmath68)@xmath83 ) where @xmath81 is missing momentum and @xmath82 is missing mass squared . [ cols="^,^",options="header " , ] we have searched for lfv decays @xmath74 , @xmath84 , @xmath27 and @xmath28 using a 543 fb@xmath7 data sample from the belle experiment . no evidence for a signal is observed and upper limits on the branching fractions are set in the range @xmath85 at the 90% confidence level . this analysis is the first search for @xmath86 modes . the results for the @xmath74 , @xmath27 and @xmath28 modes are @xmath11 times more restrictive than our previous results obtained using 158 fb@xmath7 of data . the sensitivity improvement includes a factor of 3.4 in data statistics and an optimized analysis with higher efficiency and much improved bg suppression . the improved upper limits can be used to constrain the parameter spaces of various scenarios beyond the sm . we thank the kekb group for the excellent operation of the accelerator , the kek cryogenics group for the efficient operation of the solenoid , and the kek computer group and the national institute of informatics for valuable computing and super - sinet network support . we acknowledge support from the ministry of education , culture , sports , science , and technology of japan and the japan society for the promotion of science ; the australian research council and the australian department of education , science and training ; the national science foundation of china and the knowledge innovation program of the chinese academy of sciences under contract no . 10575109 and ihep - u-503 ; the department of science and technology of india ; the bk21 program of the ministry of education of korea , the chep src program and basic research program ( grant no . r01 - 2005 - 000 - 10089 - 0 ) of the korea science and engineering foundation , and the pure basic research group program of the korea research foundation ; the polish state committee for scientific research ; the ministry of education and science of the russian federation and the russian federal agency for atomic energy ; the slovenian research agency ; the swiss national science foundation ; the national science council and the ministry of education of taiwan ; and the u.s.department of energy .

we have searched for neutrinoless @xmath0 lepton decays into @xmath1 and @xmath2 , where @xmath1 stands for an electron or muon , and @xmath2 for a vector meson ( @xmath3 , @xmath4 , @xmath5 or @xmath6 ) , using 543 fb@xmath7 of data collected with the belle detector at the kekb asymmetric - energy @xmath8 collider . no excess of signal events over the expected background is observed , and we set upper limits on the branching fractions in the range @xmath9 at the 90% confidence level . these upper limits include the first results for @xmath10 as well as new limits that are @xmath11 times more restrictive than our previous results for @xmath12 , @xmath13 and @xmath14 .

high energy particle acceleration and magnetic field amplification appear to be tightly related phenomena in many astrophysical environments . for instance , x - ray observations of synchrotron emission from ultrarelativistic electrons in young supernova remnants ( snrs ) @xcite suggest that electrons are efficiently accelerated in these environments , and that the ambient magnetic field in the dowstream medium of snr forward shocks is amplified by a factor of @xmath5 compared to its typical value in the interstellar medium . also , cosmic rays ( crs ) with energies up to @xmath6ev are believed to originate in snrs . calculations based on the diffusive shock acceleration mechanism @xcite show that such high energies can only be reached if crs are efficiently confined to the remnant @xcite , a condition that would be eased by a substantial magnetic field amplification . while it is supported by direct observations of snrs , the process by which the field amplification takes place is still a mystery . @xcite suggested the possibility of magnetic amplification driven by non - resonant crs propagating in the upstream region of shocks . this instability is different from the alfvn wave amplification due to cyclotron resonance with crs @xcite . instead , it only requires positively charged crs propagating along a background magnetic field , @xmath7 , with larmor radii much larger than the wavelength of the wave . this condition is expected to be satisfied by diffusively shock - accelerated crs in the upstream region of both relativistic and non - relativistic shocks . while individual crs are relativistic , on average they stream with respect to the upstream plasma with a drift velocity , @xmath8 , that depends on the distance from the shock . the lowest energy crs are efficiently confined to the shock vicinity by the upstream turbulence , and drift with the shock with @xmath9 , where @xmath10 is the shock speed . the higher energy crs are less affected by the upstream scattering , so they tend to escape more easily from the shock . thus , as the distance from the shock increases , @xmath8 will increase , asymptotically approaching @xmath11 for non - relativistic shocks ( for relativistic shocks , @xmath12 everywhere ) . also , considering that electrons are more affected by radiative losses than ions , it is reasonable to think that at some distance from the shock crs will be mainly positively charged particles . their drift will drive a constant current , @xmath13 , through the upstream plasma . this current will be compensated by the opposite return current , @xmath14 , provided by the background plasma . if there is a small magnetic field perturbation @xmath15 , perpendicular to @xmath7 , a @xmath16 force will push the background plasma transversely . a force of the same magnitude will push the crs in the opposite direction , so that the net force acting on the background plasma - crs system vanishes . however , given the high rigidity of the crs , only the background plasma will experience a significant transverse motion . for a helical magnetic perturbation @xmath15 , this transverse motion will stretch the magnetic field lines , producing an amplification of @xmath15 . using analytic mhd analysis , @xcite showed that for a right - handed circularly polarized electromagnetic wave , this amplification would be exponential , and faster than the resonant instability and @xmath13 are parallel . in the antiparallel case , the polarization is left - handed ] . the linear dispersion relation of this cosmic ray current - driven ( crcd ) instability has been calculated using both mhd @xcite and kinetic treatments @xcite . these works found that , if @xmath13 is kept constant , the instability will grow at a prefered wavelength @xmath17 , with a growth rate @xmath18 , where @xmath19 is the mass density of the background plasma . the non - linear evolution of the crcd instability has been studied making use of both mhd and particle - in - cell ( pic ) simulations . the mhd studies @xcite have shown a substantial amplification of the ambient magnetic field and the formation of turbulence , which is characterized by prominent density fluctuations in the plasma . they have established a saturation criterion that depends on the wavelength , @xmath20 , of each mode and that is given by @xmath21 . this saturation criterion would imply that , if @xmath22 is constant , the field never stops growing but only migrates into longer wavelengths . this migration would be such that the dominant wavelength , @xmath0 , goes roughly as @xmath23 , where b is the mean value of the magnetic field @xcite . the saturation in mhd simulations with constant cr current is either not observed @xcite or is due to the size of magnetic fluctuations reaching the size of the box @xcite . one important limitation of the mhd simulations is that they can not follow the instability at arbitrarily low densities , which makes it difficult to model large plasma density fluctuations properly . also , the mhd simulations do not include the back - reaction on the crs , which must play a fundamental role in the saturation of the instability . both difficulties can be potentially resolved by fully kinetic pic simulations . a first attempt to use pic simulations for this problem was made by @xcite . even though their results show magnetic amplification , it accurs at a significantly lower rate and through a kind of turbulence that essentially differs from the circularly polarized , growing waves predicted by @xcite . this raised a question about the existence of the crcd instability beyond the mhd approximation . in this work we confirm the existence of the crcd instability using pic simulations and establish the conditions under which it is present . in order to understand the saturation mechanisms of the instability , we separate our study in three parts , presented in [ sec : thewaves ] , [ sec : multidimensional ] , and [ freecrs ] . in [ sec : thewaves ] , we show the main non - linear properties of the crcd waves , focusing on their intrinsic saturation mechanism in the presence of constant cr current , @xmath13 . we do this using an one - dimensional , analytic model , and check our results with one - dimensional pic simulations . we calculate a non - linear dispersion relation that includes the time evolution of the phase velocity of the waves . also , our model quantifies all the plasma motions induced by the waves , which is needed to understand the wave behavior in the multidimensional context . in [ sec : multidimensional ] , we present the multidimensional properties of the instability using two- and three - dimensional simulations , also with constant @xmath13 . first , we determine the conditions under which the crcd instability can grow without being affected by plasma filamentation as in the case of @xcite . then , combining the multidimensional simulations with our results from [ sec : thewaves ] , we determine the main properties of the instability in its non - linear stage . we confirm the generation of turbulence as suggested by previous mhd studies , and reexamine its main properties . we estimate the typical turbulence velocity and length scale as a function of the magnetic amplification , finding a faster migration to longer wavelengths than predicted by mhd simulations . we find that the acceleration of background plasma along the direction of motion of the crs causes the intrinsic saturation of the crcd instability at constant @xmath13 . in [ freecrs ] , we study the effect of the back - reaction on the crs as a second saturation mechanism for the instability . [ conclusions ] presents our conclusions and an application to the case of snr environments . in this section we present the one - dimensional analysis of the crcd waves at constant @xmath13 , i.e. , without considering the back - reaction on the crs . in 2.1 we show an analytic , kinetic model for the crcd waves , valid in the non - linear regime . after that , in 2.2 , we check our model making use of one - dimensional pic simulations . we consider a piece of upstream plasma through which positively charged crs flow , providing the current @xmath13 . we focus on the situation where the initial magnetic field @xmath7 , @xmath13 , and the wave vector of the crcd mode @xmath24 , are parallel . local charge neutrality is assumed as initial condition , so @xmath25 , where @xmath26 , @xmath27 , and @xmath28 correspond to the density of ions , electrons , and crs , respectively . in this section we study the time evolution of crcd waves of different wavevectors @xmath24 , which are characterized by their growth rate @xmath29 and phase velocity @xmath30 , where @xmath31 . the derivation is made for a right - handed polarized wave , and is based on the calculation of the drift velocities of plasma components in the presence of a wave of arbitrary amplitude . if we know these drift velocities and the number densities of the different species , we can calculate the total current provided by the background plasma as a function of space and time . adding this total plasma current to the constant @xmath13 contributed by the crs , the time evolution of the wave can be directly obtained from the ampere s and faraday s laws . the details of the calculation are presented in appendix [ app : analytic ] . here we describe its main results , which are summarized in the dispersion relation given by equation ( [ eq : dispersion ] ) . this dispersion relation assumes a constant @xmath29 and allows @xmath30 to evolve in time . we will see below that @xmath29 is indeed constant as long as @xmath32 , where @xmath2 is the alfvn velocity of the backgroud plasma . this condition not only puts a limit to the validity of the derivation , but also sets a saturation criterion for the crcd waves . also , our derivation is in the low plasma temperature limit and assumes that @xmath33 , where @xmath34 is the initial alfvn velocity of the plasma . using two - dimensional pic simulations , we show below that this second condition is actually a requirement for the crcd waves not to be quenched by weibel - like plasma filamentation . from the real part of equation ( [ eq : dispersion ] ) , we see that the growth rate , @xmath29 , is maximized when the wavenumber @xmath35 , which corresponds to the same wavenumber of maximum growth found in the linear regime @xcite . from the imaginary part , we obtain the following differential equation for @xmath36 as a function of the amplification factor of the waves , @xmath37 ( defined as the ratio between the magnitude of the transverse magnetic field , @xmath38 , and @xmath39 ) , @xmath40 the solution for equation ( [ eq : diferencial ] ) is @xmath41 which , when @xmath42 , can be approximated as @xmath43 . this implies that , although in the linear regime the crcd waves are almost purely growing ( @xmath44 ) , if @xmath45 gets close to @xmath3 , their phase velocity can also become comparable to @xmath3 . taking the real part of equation ( [ eq : dispersion ] ) and evaluating at @xmath46 , we obtain that @xmath47 where we have kept terms only to first order in @xmath48 and @xmath49 . we see that in the regime @xmath50 , the instability grows exponentially with a maximum growth rate , @xmath51 that is constant and has the same value as obtained in the previous linear studies @xcite . even though equation ( [ eq : growthrate ] ) shows that our assumption of constant @xmath52 is only valid when @xmath50 , it also indicates that , as @xmath45 approaches @xmath3 , the growth rate will be substantially reduced , suggesting an intrinsic saturation limit for the crcd waves at @xmath53 . in appendix [ app : analytic ] we also show that the presence of the crcd waves induces bulk motions of the plasma particles both parallel and transverse to @xmath13 . the parallel motion has a velocity @xmath54 , while the transverse motion has a velocity @xmath55 , which always points perpendicular to @xmath15 ( and to @xmath13 ) . the parallel plasma motion implies that , when @xmath53 , the entire plasma will move at a speed close to @xmath3 , which , from the point of view of the plasma , substantially reduces @xmath13 . this reduction in @xmath13 explains the intrinsic saturation of the waves at @xmath53 . the tranverse motion , on the other hand , becomes of the order of the alfvn velocity of the plasma when the crcd waves become non - linear ( @xmath56 ) . we will see below that this increasing transverse velocity is related to turbulence formation in the non - linear regime . we checked our analytical results with one - dimensional pic simulations . we use the pic code tristan - mp @xcite , which can run in one , two , and three dimensions . in these simulations , like in our analytic model , all plasma properties depend only on one spatial direction ( @xmath57 ) , but both the velocities of the particles and the electromagnetic fields keep their three - dimensional components . we set up a periodic box that contains an initially cold background plasma ( with typical particle thermal velocity of @xmath58 ) composed of ions and electrons , and a small population of relativistic ions ( crs ) . the driving current is given by the crs that move along @xmath59 with a mean velocity @xmath3 that we vary between runs . these crs are not allowed to change their velocities , as if they had an infinite lorentz factor , @xmath60 . such locked " crs allow us to study the non - linear evolution of the instability considering a constant @xmath13 , i.e. , eliminating the back - reaction on the crs . we give electrons a small velocity along @xmath59 such that the background plasma carries a current @xmath61 . this way the net current is zero . the initial magnetic field , @xmath7 also points along @xmath59 . since we want to simulate a situation where @xmath62 , having good cr statistics would imply a large number of particles per cell . in order to overcome this difficulty , we have initialized the same number of macroparticles for crs , ions , and electrons , but modified their charges so that @xmath63 and @xmath64 , where @xmath65 . we change the mass of the particles accordingly in order to keep the right charge to mass ratios . particles are initially located randomly in the box such that at the position of each ion we also have an electron and a cr , so the initial charge density is zero . this initialization is also used in our two- and three - dimensional runs . the common numerical parameters for the simulations are @xmath66 @xmath67 ( where @xmath68 is the electron plasma frequency and @xmath67 is the grid cell size ) , ion - electron mass ratio @xmath69 , speed of light @xmath70 @xmath67/@xmath71 ( where @xmath71 is the time step ) , and 12.5 particles per species per cell . here we test the non - linear dispersion relation found in [ sec : analytical ] in the relativistic regime ( @xmath72 ) , which would be appropriate for the upstream medium of a relativistic shock front . we ran one - dimensional simulations in boxes of different sizes @xmath73 , set up to probe the growth rate of different wavelengths , @xmath20 . as an initial condition we used a right - handed , circularly polarized , growing wave of amplitude 0.1@xmath74 , whose fields and particle velocities were determined from our analytic model ( appendix [ app : analytic ] ) . we put only one period of the wave in a box , so that the only other modes that could be excited are shorter or equal to @xmath75 . we choose the density of crs such that the corresponding maximum growth rate of the instability , @xmath52 , is 0.2 @xmath76 , so we are in the regime where the background plasma is well magnetized . simulations were run for @xmath20 equal to 0.5 , 0.75 , 1 , 1.25 , 1.5 , 2 , and 3@xmath77 . we used several values for the initial alfvn velocities , @xmath78 , in the range @xmath79 to @xmath80 . this implies that the initial gyrotime @xmath81 ranges from 189 to 1456 @xmath71 , so it is resolved with about 20 @xmath71 even when @xmath1 , and @xmath82 ranges from 1 to 8 . the results for the cases @xmath83 and @xmath80 are presented in fig . [ fg : figura1 ] . we observe that for @xmath84 there is practically no growth . for @xmath85 1 , 1.25 , and 1.5@xmath77 we obtain nearly the same growth rate , which is very close to the analytic @xmath52 . for longer wavelengths , the growth rate gradually decreases . in all our experiments , for @xmath86 , the exponential growth continues until @xmath45 becomes close to @xmath87 , which confirms our analytical saturation criterion for fixed crs . at later times we see that , depending on @xmath73 , the amplitude of the wave either oscillates or keeps growing but at a much lower rate . while individual crs near a non - relativistic shock move at almost the speed of light , on average they move with respect to the upstream at a drift velocity , @xmath3 , that is less than c. in order to study this case , we ran a series of simulations where , besides not allowing crs to alter their trajectories , we make them drift along @xmath59 at a velocity @xmath88 and @xmath89 . we do this using a box size @xmath90 . in this case we do not seed the instability with a small amplitude , growing wave , as done in [ sec : rel ] . instead , we only put the initial magnetic field , @xmath7 , such that @xmath91 , forming an angle @xmath92 with @xmath93 . we use this set - up to show that the instability can develop from any kind of noise . we tilt @xmath7 by a small angle to inject a small amount of magnetic and kinetic energy perpendicular to @xmath13 , so it acts as an initial seed that does not favor any particular @xmath20 ( experiments with @xmath94 were also run , showing no difference besides requiring a longer initial time for the wave to appear ) . other numerical parameters are the same as in the relativistic experiments . is plotted as a function of time , @xmath95 , for experiments similar to the ones depicted in fig . [ fg : figura1 ] , but for @xmath96 1/10 and a box of @xmath97 , where @xmath77 is the wavelength of the theoretically determined fastest growing mode . the instability is seeded by tilting @xmath7 by @xmath98 with respect to @xmath13 . a series of @xmath3 is tested : @xmath99 ( solid black ) , 0.9 ( solid green ) , 0.8 ( solid red ) , 0.6 ( dotted black ) , 0.4 ( dotted green ) , and 0.2 ( dotted red ) . @xmath52 is the maximum theoretical growth rate for the case @xmath99 . the results are consistent with the theoretical @xmath52 and @xmath77 , and with the intrinsic saturation criterion , @xmath53.,width=292 ] fig . [ fg : lesscurr ] shows the magnetic energy evolution for the six @xmath3 tested . we observe that the growth rate is @xmath100 , where @xmath52 is the maximum growth rate for @xmath101 . the amplitude at which the exponential growth stops is such that @xmath53 , confirming our results from [ sec : analytical ] . ( red line ) and @xmath102(green line ) , are plotted as a function of distance , @xmath57 , at two different times for two of the runs described in fig . [ fg : lesscurr ] ( @xmath99 and 0.6 ) . the black line represents @xmath103 . the two left plots show the case @xmath104 at @xmath105 and 50 . the two right plots show the case @xmath106 at @xmath107 and 50 . the results are consistent with the instability appearing initially as right - handed circularly polarized wave with a preferred wavelength @xmath108 , and with a migration into longer wavelengths as the instability grows.,width=321 ] fig . [ fg : sequenceapj ] shows the different components of the magnetic field as a function of position , @xmath57 , at different times for @xmath101 , and @xmath109 . we can see that in both cases the instability appears as a right - handed polarized wave and at an initial wavelength @xmath110 , where @xmath77 is the wavelength of maximum growth for @xmath101 . after the wave reaches saturation , there is a migration into longer wavelengths . this migration appears because the modes with wavelengths greater than @xmath77 grow more slowly , but still grow and saturate at @xmath53 , as can be seen in fig . [ fg : figura1 ] for the relativistic regime ( @xmath101 ) . it means that , as the instability reaches @xmath53 , the spectrum of the waves gradually receives more contribution from wavelengths longer than @xmath77 . as we saw in [ sec : analytical ] , crcd waves induce plasma motions both parallel and perpendicular to @xmath59 . the left panel in fig . [ parperfig ] shows the mean velocity of plasma particles along @xmath57 ( i.e. , parallel to @xmath13 ) , and the analytic estimate for this velocity , @xmath111 . this velocity comes from the @xmath112 drift of background particles , which in a well magnetized plasma ( @xmath113 ) is much larger than other plasma drifts ( see appendix [ app : analytic ] ) . the velocities in fig . [ parperfig ] are computed for two simulations from the right panel of fig . [ fg : figura1 ] : @xmath114 with @xmath115 and @xmath116 with @xmath117 . these wavelengths correspond to the fastest growing waves for the two @xmath3 . the right panel in fig . [ parperfig ] , shows the mean magnitude of the _ transverse _ velocity of plasma particles for the same simulations , and the analytic estimate , @xmath118 . considering that these two cases keep growing exponentially until @xmath119 and 12 , respectively ( see fig . [ fg : figura1 ] ) , we see that , in the @xmath50 regime , our analytical estimates for both longitudinal and transverse motions are in good agreement with our numerical results . so far we have studied the properties of the crcd waves assuming an ideal one - dimensional geometry and constant cr current . in this section , we relax the first of these conditions and use two- and three - dimensional pic simulations to study the crcd instability , still keeping @xmath13 constant . we identify two main differences with respect to the one - dimensional case . the first has to do with the possibility of plasma filamentation that happens before the crcd instability sets in , as suggested by previous works @xcite . we will see below that this filamentation does not occur if the plasma is sufficiently magneitzed , @xmath120 . the second multidimensional effect is the interference between crcd waves generated in different regions of space . since typically the instability starts from random noise , different regions will give rise to crcd waves which in general are out of phase with each other . during the non - linear stage , this non - coherence makes the transverse plasma motions from adjacent regions interfere with each other , giving rise to density fluctuations and turbulence in the plasma . motivated by previous pic studies by @xcite , we studied the possibility of an initial plasma filamentation that could suppress the formation of the crcd instability . we ran a series of high space resolution ( @xmath121 @xmath67 ) two - dimensional simulations whose numerical parameters and results are described in table [ table : fil2 ] . all our two - dimensional simulations are set up in the @xmath122 plane , with @xmath13 and @xmath7 parallel to the @xmath59 axis . also , as in some of our one - dimensional simulations , there is a small component of @xmath7 pointing along @xmath123 , working as a seed for the instability . we identified three regimes , represented in figs . [ fg : filamentationc ] , [ fg : filamentationb ] and [ fg : filamentationa ] . [ fg : filamentationc ] shows the plasma density and three components of the magnetic field for a simulation with @xmath124 ( run m5 in table [ table : fil2 ] ) at two times @xmath125 and 11 . even though some crcd field is observed , especially in @xmath126 , the dominant instability corresponds to a transverse filamentation that appears initially on the scale of @xmath127 times the electron skin depth . as time goes on , the filaments merge , creating prominent holes in the plasma that preclude the growth of the instability . [ fg : filamentationa ] , on the other hand , shows the same quantities for a simulation where the relative number of crs was decreased by a factor of 10 ( run m7 ) , implying that @xmath128 . in this case , a crcd wave of the size of the box does form ( we have chosen the @xmath57-size of the box to be @xmath129 ) . finally , fig . [ fg : filamentationb ] shows the case @xmath130 ( run m6 ) , in which both the crcd instability and the initial filamentation coexist ( we call these cases transitional " and indicate them with the letter t " in table [ table : fil2 ] ) . these three examples indicate that the crcd instability will develop as long as @xmath120 , which is equivalent to having a well magnetized plasma in the sense that @xmath113 ( see appendix [ app : analytic ] ) . as shown in table [ table : fil2 ] , we tested the dependence of this criterion on both the magnetization of the plasma ( using @xmath131 and 31.5 ) and the mass ratio , @xmath132 ( using @xmath133 and 100 ) . we see no difference in our results except that the runs with @xmath134 require a slightly higher value of the ratio @xmath135 ( a factor of 2 larger ) for the transverse filamentation to dominate , but the qualitative criterion remains the same . also , varying @xmath132 allows us to determine the physical length scale , @xmath136 , at which the transverse filaments appear . this scale shows no dependence on @xmath132 and corresponds to @xmath127 times the electron skin depth . another @xmath69 simulation was run with zero initial magnetic field ( m8 ) , showing that the filaments appear at practically the same scale as in the finite magnetic field case , suggesting a similarity between this filamentation and the weibel instability . finally , these results were also tested for a non - relativistic case , @xmath137 , obtaining the same conclusions . thus , the crcd instability will grow if the condition @xmath120 ( or , equivalently , @xmath113 ) is satisfied . for comparison , the smallest @xmath138 factor used by @xcite is 1.31 , which is close to the regime where the transverse filaments appear . this fact would explain their plasma filamentation , which may have suppressed the appearance of the crcd instability . + + + [ cols= " < , < , < , < , < , < , < , < , < , < , < , < , < " , ] we studied crcd instability with a series of two- and three - dimensional simulations whose numerical parameters are summarized in table [ table : interference ] . as we will see below , when multidimensional effects are considered , the dominant wavelength of the instability , @xmath0 , is initially equal to @xmath77 but then rapidly grows as the field is amplified . this can make @xmath0 equal to the size of the box @xmath73 before the instability reaches saturation , which can make sufficiently large three - dimensional simulations challenging . we discuss here the results of our three - dimensional runs , and check them in appendix [ sec:2d ] with large two - dimensional simulations , for which @xmath0 is always significantly smaller than the size of the box . in this section we present the results of three three - dimensional simulations that test saturation in the non - relativistic and relativistic regimes , and the dependence of the amplification on the initial magnetic field and the cr drift velocity . two of the simulations have the same @xmath139 , but @xmath101 and @xmath11 ( runs i1 and i2 in table [ table : interference ] ) . the third simulation has @xmath91 and @xmath101 ( run i3 in table [ table : interference ] ) . as in all our simulations so far , @xmath13 and @xmath7 point along @xmath59 ( apart from a small component of the magnetic field along @xmath123 of magnitude @xmath140 ) , and the back - reaction on the crs is not included . the rest of the numerical parameters are specified in table [ table : interference ] . , but with overplotted arrows showing the magnetic field projection on the @xmath141 plane . the clock - wise orientation of the magnetic field lines around the plasma holes shows the presence of crcd waves driving the turbulence.,width=302 ] the evolution of the plasma density and the three components of the magnetic field for simulation i1 can be seen in figs . [ fg : cortes1 ] , [ fg : cortes2 ] , [ fg : cortes3 ] , and [ fg : cortes4 ] . these figures show two slices of the simulation box . one is longitudinal and corresponds to the plane @xmath142 ( top panels ) , and the other is transverse and corresponds to @xmath143 ( bottom panels ) , where @xmath144 . the magnetic energy evolution for the same run is depicted in fig . [ fg : departure3d ] . [ fg : cortes1 ] shows the early moments of the instability ( @xmath145 ) . the longitudinal slice shows how the crcd waves form independently in different regions of the box . this is also seen in the transverse slice , which shows how the phases of the waves differ between different points of the plane @xmath143 . [ fg : cortes2 ] shows the beginning of the non - linear regime ( @xmath146 ) , which corresponds to @xmath147 . in this case , the phases of the waves are transversely more correlated compared to fig . [ fg : cortes1 ] , as can be seen in the plots of the transverse slice for @xmath126 and @xmath102 . this increased spatial correlation indicates that , until this moment , the adjacent waves were merging without significantly interfering with each other . also , fig . [ fg : cortes2 ] shows the appearance of prominent density fluctuations ( @xmath148 , where @xmath149 is the plasma density ) . as we saw in [ sec : thewaves ] , when the crcd waves are in the exponential growth regime ( @xmath50 ) , the background plasma will move transversely at @xmath150 . so , when the instability gets non - linear , the transverse velocity of the plasma becomes close to @xmath45 . in a low temperature regime , this velocity corresponds to the magnetosonic sound speed of the plasma . thus , as soon as @xmath151 , the transverse motions will produce moderate shocks , giving rise to significant density fluctuations in the plasma . at this point the transverse plasma motions develop into isotropic turbulence with velocities of the order of @xmath152 . , for which ( @xmath153 ) = ( 1/40,1 ) , ( 1/20,0.5 ) , and ( 1/10,1 ) , respectively . time is normalized in terms of the @xmath52 of each simulation . in all the runs , the departure from exponential growth occurs after @xmath151 , but saturation happens at @xmath1.,width=292 ] what happens after the density fluctuations appear can be seen in fig . [ fg : cortes3 ] ( @xmath154 ) . the longitudinal slice shows how the magnetic fluctuations get distorted and increase rapidly in size . as already mentioned in [ sec : nonrel ] , even in one - dimensional geometry the instability is expected to evolve into wavelengths longer than @xmath77 . however , in a multidimensional set - up , the evolution into magnetic fluctuations of larger size gets accelerated after the appearance of the density fluctuations and turbulence in the plasma . we will quantify this migration in [ sec : migration ] . the transverse slice of fig . [ fg : cortes3 ] shows how the underdense regions ( or holes ) have merged and increased their size with respect to fig . [ fg : cortes2 ] . it is also interesting to see from fig . [ fg : zoomcorte ] how the holes are separated by plasma walls " through which the transverse magnetic field reverses direction . we see that the magnetic field has a clockwise orientation around the holes , which is consistent with the presence of right - handed waves producing the expansion of the holes . finally , fig . [ fg : cortes4 ] ( @xmath155 ) shows essentially no difference with respect to fig . [ fg : cortes3 ] besides the growth of the size of both the magnetic fluctuations and the plasma holes , which at this point are close to @xmath73 . [ fg : departure3d ] shows the magnetic energy evolution for the three - dimensional simulations . we can see that , in the three cases , the departure from the exponential growth occurs shortly after the wave becomes non - linear ( which coincides with the generation of significant density fluctuations and turbulent motions in the plasma ) . we also see that the final saturation satisfies the @xmath53 condition , which suggests that , when multidimensional effects are included , the intrinsic saturation of the crcd instability is still given by the @xmath53 criterion . unfortunately , in our three - dimensional simulations , this saturation happens when the dominant sizes of the holes and magnetic fluctuations have already become close to @xmath156 . saturation still happens at @xmath53 because , when @xmath157 , the three - dimensional simulations behave more like the one - dimensional simulations presented in [ sec : rel ] , in the sense that there is only one dominant mode that saturates at @xmath53 . after @xmath157 , the density fluctuations almost disappear and the turbulent motions transform into more coherent transverse plasma motions . in any case , the @xmath53 saturation criterion is confirmed by two - dimensional simulations presented in appendix [ sec:2d ] for which @xmath0 is always smaller than @xmath73 . fig . [ fg : departure3d ] also shows that , in the three - dimensional simulations , the magnitude of the magnetic component along @xmath59 is comparable to the transverse one , suggesting a rather isotropic orientation of the crcd field . , as a function of the amplification factor , @xmath37 , for three - dimensional simulations i1 ( dot - dashed ) , i2 ( dashed ) , and i3 ( dotted ) . for comparison , our semi - analytical formula , @xmath158 , is shown as solid line.,width=302 ] even though migration to longer wavelengths is already observed in one - dimensional simulations , it becomes faster when multidimensional effects are considered . in this section we propose a semi - analytic model that quantifies this migration in terms of the amplification factor of the field , @xmath37 . as we saw in [ sec : threed ] , the motions associated with the turbulence tend to distort the crcd waves , producing a damping of the shortest wavelength modes . thus , the dominant wavelength , @xmath0 , will correspond to the fastest growing mode that can be amplified without being strongly affected by the turbulence . considering that a crcd wave of wavelength @xmath20 grows in a time scale comparable to the inverse of its growth rate @xmath159 from equation ( [ eq : dispersion ] ) , and that the turbulence will kill it in a time scale comparable to @xmath160 , where @xmath161 is the typical turbulent velocity , then @xmath0 will be such that @xmath162 . since the turbulence is due to the transverse plasma motions produced by non - coherent crcd waves , then @xmath161 must be comparable to the transverse velocity of the waves , which we already determined to be @xmath163 . so , @xmath0 will be such that @xmath164 , where @xmath165 is an unknown constant that quantifies the relative importance of the two time scales . if we get @xmath166 from the real part of equation ( [ eq : dispersion ] ) , we can obtain @xmath165 by fitting the evolution of @xmath0 in our three - dimensional simulations , obtaining @xmath167 . this way we find @xmath0 as a function of @xmath37 , @xmath168/2 , \label{eq : wavelength}\ ] ] which is intended to be valid after the turbulence becomes significant ( @xmath169 ) . in fig . [ fg : wavelength ] we show a comparison between this formula and the evolution of @xmath0 as a function of @xmath37 for our three - dimensional simulations . we computed @xmath0 by performing fourier transforms of @xmath126 and @xmath102 along lines of constant @xmath170 and @xmath171 coordinates , and then finding the mean wavelength of the peak of the fourier transform . as the dominant wavelengths approach @xmath73 ( @xmath172 ) , the determination of @xmath0 becomes quite noisy . due to this reason , we have plotted our simulation results only until @xmath173 . we see from fig . [ fg : wavelength ] that equation ( [ eq : wavelength ] ) appears to provide an acceptable fit for the evolution of @xmath0 . our result shows a growth of @xmath0 substantially faster than the direct proportionality between @xmath0 and @xmath37 suggested by @xcite . we will see below that , when the back - reaction on the crs is considered , this difference has important implications to the saturation of the crcd instability . and @xmath174 normalized in terms of @xmath175 , are plotted as a function of time ( normalized using @xmath52 ) for one- and three - dimensional runs in order to study the effect on the magnetic energy evolution due to the back - reaction on the cr . in all cases crs are monoenergetic and have a semi - isotropic momentum distribution such that @xmath137 . the two one - dimensional simulations have numerical parameters : @xmath176 , @xmath177 , @xmath178 , @xmath179 @xmath67 , @xmath180 , @xmath69 , @xmath181 , @xmath182 @xmath71 , @xmath183 , and @xmath184 . their cr lorentz factor @xmath60 is 20 ( solid , red line ) and 40 ( solid , green line ) , respectively . the three - dimensional simulation , whose transverse magnetic energy is represented by the solid , blue line , has the same parameters as run i2 in table [ table : interference ] , but with @xmath185 . the dotted , blue line represents the longitudinal magnetic energy ( @xmath186 ) . the dashed , black line shows the constant magnetic energy along @xmath59 for the two one - dimensional simulations.,width=302 ] in this section we use one- and three - dimensional simulations to study the effect of the dynamic evolution of the crs on the saturation of the crcd instability . we will concentrate on the case of a beam of monoenergetic crs drifting at half the speed of light ( @xmath187 ) in the @xmath59 direction , which is parallel to @xmath188 ( except for a small magnetic component along @xmath123 ) . this drift is obtained by sampling the cr velocities from an isotropic , monoenergetic momentum distribution , but only keeping the velocities in the positive @xmath57 direction . this choice for the cr momentum distribution has a direct application to the most energetic crs that propagate in the upstream medium of snr shocks ( see [ conclusions ] ) . the red and green curves in fig . [ fg : unlocked ] represent the magnetic energy evolution for two one - dimensional simulations that only differ in their lorentz factors , @xmath60 , taken to be 20 and 40 ( the rest of the numerical parameters are specified in the caption of fig . [ fg : unlocked ] ) . these simulations saturate at @xmath189 and @xmath190 , when the larmor radius of crs is close to the dominant wavelength of the instability ( @xmath191 and @xmath192 for two runs ) . to obtain this result we use that initially @xmath193 , and consider that at saturation the dominant wavelength has grown ( @xmath194 and @xmath195 , respectively ) and crs have lost part of their energy ( the mean @xmath60 of crs is 18.9 and 37.6 , respectively ) . these results are confirmed by a three - dimensional simulation whose magnetic energy evolution is represented by the blue lines in fig . [ fg : unlocked ] . the numerical parameters of this three - dimensional simulation are the same as in run i2 ( see table [ table : interference ] ) but includes the back - reaction on the crs , whose @xmath185 . if we consider that at saturation @xmath196 and the mean @xmath60 of the crs is 27.6 , we obtain that the cr deflection saturates the instability when @xmath197 . note that in this simulation @xmath0 is always a factor of 4 smaller than the size of the box , so the saturation is not affected by box effects . for the semi - isotropic distribution of monoenergetic crs presented here , we find that the saturation due to cr back - reaction will happen when @xmath198 . although this result is valid for our particular choice of cr momentum distribution , we expect that in general the saturation of the crcd instability will be determined either by the intrinsic limit @xmath53 , or by the strong cr deflection when @xmath199 . as will be discussed in [ conclusions ] , achieving @xmath53 requires a very high cr energy density , a condition that is not expected for non - relatistic shocks environments . also , our simulations show that , at saturation , many crs have negative @xmath57 velocity . this suggests that , besides the field amplification , the crcd instability can provide an efficient scattering mechanism for crs upstream of shocks . using fully kinetic pic simulations , we confirmed the existence of the crcd instability predicted by bell ( 2004 ) . combining one- , two- , and three - dimensional simulations with an analytic , kinetic model we studied the non - linear properties of the instability and its possible saturation mechanisms . in the first part , we studied non - linear crcd waves under idealized conditions , namely : _ i _ ) ignoring multidimensional effects , and _ ii _ ) assuming a constant cr current without back - reaction on the crs . we confirm that the crcd waves can grow exponentially at the wavelengths and rates predicted by the analytic dispersion relation @xcite . we find that the exponential growth can continue into the very nonlinear regime , until the alfven velocity in the amplified field is comparable to the cr drift velocity , @xmath53 . this saturation is due to plasma acceleration along the direction of motion of the crs , which reduces the cr current observed by the plasma particles . the plasma moves at the velocity @xmath200 , where @xmath37 is the amplification factor of the field ( @xmath201 ) . at saturation , when @xmath202 , the plasma moves together with crs , decreasing the net driving current . the waves also induce transverse plasma motions with velocities @xmath203 . these motions generate plasma turbulence when multidimensional effects are included . in the second part , we considered more realistic conditions by including the multidimensional effects using two- and three - dimensional simulations with constant @xmath13 . our main results are : _ i ) _ in the linear regime , if the plasma is well magnetized ( @xmath113 , or , equivalently , @xmath120 ) , the crcd waves grow at the rate and preferred wavelength close to the ones obtained in the one - dimensional analysis . if this condition is not met , weibel - like filaments form in the plasma , supressing the appearance of the waves . in this case , the streaming crs can still amplify the magnetic field to non - linear values , but at a rate significantly lower than that of the crcd waves @xcite . this regime , however , might be relevant to the upstream medium of relativistic shocks in grbs , where @xmath28 could exceed @xmath26 @xcite . _ ii ) _ in the non - linear crcd regime , the transverse plasma motions associated with the instability create significant density fluctuations and turbulence in the plasma . these turbulent motions suppress the growth of the shortest crcd waves , producing a fast evolution into longer wavelengths that can be approximated by @xmath204/2 $ ] . also , even though the field will continue to be amplified until @xmath53 , the nonlinear growth will be slower than in the linear regime . in the third part , we include the back - reaction on the crs and find that the cr deflection by the amplified field constitutes another possible saturation mechanism . we tested this effect for a semi - isotropic distribution of monoenergetic crs propagating at @xmath187 with respect to the upstream medium , which would be appropriate for the most energetic crs that escape from snrs . we find that the field is amplified until the larmor radii of the crs becomes approximately equal to the size of the dominant magnetic fluctuations . when that happens , the crs get strongly deflected by the magnetic field , which decreases their current and stops the growth of the field . ignoring the migration to longer wavelengths , for a generic cr momentum distribution with @xmath205 , saturation due to cr deflection will happen when @xmath206 , where @xmath207 is the typical lorentz factor of current - carrying cr . on the other hand , saturation due to plasma acceleration to @xmath3 velocity occurs when @xmath208 . thus , the deflection of crs will dominate if the cr energy density is such that @xmath209 . if the migration to longer wavelengths is included , saturation due to cr deflection would happen at even smaller magnetic amplification . in the upstream medium of snr forward shocks , we expect the cr energy density to be low enough so that the maximum crcd amplification is determined by the back - reaction on the crs . in order to make an estimate of typical magnetic amplification in these environments , let us consider a piece of upstream whose distance from the shock is such that it can only feel the most energetic crs that escape from the remnant . we use only the most energetic particles because they are the only ones whose larmor radii are much larger than the typical wavelength of the crcd waves , which is an essential condition of the instability . also , our estimate is based on the following assumptions . first , all the escaping particles have positive charge , which is reasonable considering the much shorter cooling time of electrons compared to ions , and that ions are presumably more efficiently injected into the acceleration process in shocks . second , we assume that the escaping crs have the same energy , @xmath210 , which is roughly the minimum energy required for them to run away from the remnant . third , there is a fixed ratio , @xmath211 , between the flux of cr energy emitted by the shock , @xmath212 , and the flux of energy coming from the upstream medium as seen from the frame of the shock , @xmath213 , where @xmath214 is the shock velocity and @xmath19 is the mass density of the upstream plasma . finally , we assume a plane geometry and that all the ions are protons . under these conditions , the time scale of growth of the instability , @xmath215 , is @xmath216 and the initial length scale of maximum growth is @xmath217 the ratio @xmath218 is @xmath219 which confirms that in the case of snrs the crcd instability will not be affected by the weibel - like filamentation studied in [ sec : magnetization ] . considering the migration into longer wavelengths given by equation ( [ eq : wavelength ] ) , the amplification factor , @xmath37 , in snrs will satisfy @xmath220 \approx 130\bigg(\frac{10 \textrm{km / sec}}{v_{a,0}}\bigg)^2 \bigg(\frac{\eta_{esc}}{0.05}\bigg)\bigg(\frac{v_{sh}}{10 ^ 4\textrm{km / sec}}\bigg)^3 , \label{eq : estimate}\ ] ] which , for typical parameters would imply @xmath221 . we see that , even though run - away crs can significantly amplify the ambient magnetic field , the upstream amplification alone is not enough to explain the factors of @xmath222 inferred from observations of forward shocks in young snrs @xcite . also , note that , if @xmath223 and @xmath224 , the distance swept by the shock in a time @xmath215 is @xmath225 , which is comparable to the typical size of a snr . this means that the advection of the upstream fluid into the shock may happen faster than the growth of the field , and may put further restrictions on the amplification . it has been suggested that a further crcd amplification could be provided by the current of lower energy crs that are confined closer to the shock and move diffusively at drift velocity @xmath226 @xcite . we believe , however , that this possibility requires a more detailed study . such lower energy crs can be magnetized in the sense that their larmor radii are smaller than the typical size of magnetic fluctuations , @xmath0 , violating the conditions for the crcd instability . although on large scales these crs will still produce a current of magnitude @xmath227 parallel to the shock normal , on scales of the crcd wavelength the local cr current may get significantly affected by the amplified field because of the deflection of crs . field amplification may then proceed in essentially different way . the magnetization of crs could be less of an issue if the wavelength of the instability due to low energy crs is shorter than the cr larmor radius . indeed , since lower energy crs are more numerous than the most energetic ones , their larger current will generate shorter crcd waves ( remember that @xmath17 ) . however , the crcd turbulence generated further upstream by the highest energy crs may modify the condition @xmath228 and may suppress the growth of the small wavelength modes closer to the shock . this suggests that other non - linear mechanisms , such as the cyclotron resonance of crs with alfvn waves @xcite , may still be important components in the amplification of the field . the full effect of the low - energy cr contribution needs to be investigated using a fully kinetic treatment of crs that includes their reacceleration by the shock , the presence of pre - existing turbulence , and the eventual contribution of cr electrons to the cancelation of ion current . although we have applied our results only to the non - relativistic case of snrs ( @xmath205 ) , the crcd instability may also play an important role in relativistic shocks in jets and gamma ray bursts , where @xmath72 . our simulations show that , at constant cr current , the evolution of the instability is the same as in the non - relativistic case . in particular , the intrinsic saturation criterion due to plasma acceleration is valid , implying a maximum magnetic fiel such that @xmath229 . however , if the back - reaction on the crs is considered , the non - linear evolution of the field and the saturation due to cr deflection may be dominated by cr beam filamentation @xcite . also , since in the upstream of grb shocks the density of crs might be close to the density of upstream ions @xcite , the magnetization requirement ( @xmath113 ) may not be satisfied in these environments . in this case , a non - linear magnetic amplification is still expected , but through an instability that is characterized by weibel - like filamentation of the plasma and whose properties may be different to the crcd instability described here @xcite . detailed analysis of the relativistic shock case in application to grbs will be presented elsewhere . in conclusion , we have shown that the crcd instability is a viable mechanism for the non - linear amplification of magnetic field upstream of both non - relativistic and relativistic shocks , and that it can provide an efficient scattering mechanism for crs in these environments . this research is supported by nsf grant ast-0807381 and us - israel binational science foundation grant 2006095 . a.s . acknowledges the support from alfred p. sloan foundation fellowship . we thank yury lyubarsky and ehud nakar for useful discussions . in this appendix we calculate a dispersion relation for the crcd waves for the case where @xmath230 , @xmath13 , and the wave vector of the electromagnetic mode , @xmath24 , are all parallel and point along the @xmath59 axis . we will separate the fields and currents into components that are transverse and parallel to @xmath59 , and will identify them with the subscripts @xmath231 ( standing for transverse " ) and @xmath232 , respectively . thus , the magnetic field perpendicular to @xmath59 will be given by @xmath233 which corresponds to a right - handed polarized wave , where the phase @xmath234 is an unknown function of time , @xmath95 , the growth rate @xmath235 is a constant , and @xmath74 is the magnitude of the initial background field @xmath230 . note that the time is chosen so that the wave is in the linear regime for @xmath236 . then , from the ampere s and faraday s laws , we get that the electric field and the current perpendicular to @xmath59 are given by @xmath237 and @xmath238 where @xmath36 and @xmath239 are the time and second time derivative of @xmath240 , respectively . in order to obtain a dispersion relation , we need another expression connecting @xmath241 with @xmath242 and @xmath15 . we find it by making the following assumptions . first , the thermal velocities of the particles in the background plasma will not give rise to any significant drift velocity . second , we will assume that @xmath29 , @xmath243 and @xmath244/dt^n)/(d^{(n-1)}[\omega]/dt^{(n-1)})$ ] are constant and much smaller than @xmath245 , where @xmath246 , @xmath247 , and @xmath248 are the cyclotron frequency , the charge , and the mass of the @xmath249 species , respectively . ( we will see at the end of this appendix that , in order to satisfy the last three conditions , we need @xmath250 , where @xmath34 is the initial alfvn velocity of the plasma . ) third , the electric and magnetic fields are perpendicular , which is a reasonable assumption in the case of a quasineutral plasma . and finally , @xmath32 , where @xmath2 is the alfvn velocity of the plasma and @xmath3 is the drift velocity of the crs . considering this , given @xmath242 and @xmath15 , we can find @xmath241 as follows . if a particle @xmath251 experiences electric and magnetic fields , @xmath252 and @xmath253 , its velocity perpendicular to @xmath253 has two components , @xmath254 and @xmath255 , that satisfy the equations @xmath256 and @xmath257 . in the case of constant and uniform @xmath252 and @xmath253 , @xmath254 represents the classical gyration around @xmath253 , while @xmath255 corresponds to the drift of the particle , which is @xmath258 . when the fields change both in time and space , we can still decompose the velocity perpendicular to the field into @xmath259 [ again , satisfying @xmath256 and @xmath257 ] . in this case , the space and time variations of @xmath253 can also produce drift velocities due to the @xmath254 motion . the space variations will give rise to a drift due to the curvature of the magnetic field lines ( curvature drift ) . the curvature drift velocity , however , is of the order of the thermal speed of the particles times the ratio between their larmor radii and the curvature radius of the lines . the time variations of the field , on the other hand , can also give rise to drift velocities . to first order , these velocities will also be proportional to the thermal speed of the particles times the ratio between the rate of change of the field ( determined by the quantities @xmath29 and @xmath261 ) and @xmath246 . we will neglect these possible drift velocities using our first assumption that the thermal velocities of the particles are low enough not to produce any important drift velocity . on the other hand , in the case of a non - uniform and time - changing fields , the @xmath255 velocity is given by the series , @xmath262 where @xmath263 for @xmath264 . we see from equations ( [ eq : b ] ) , ( [ eq : e ] ) , and ( [ eq : velocities ] ) that @xmath265 as long as @xmath29 , @xmath243 , and @xmath244/dt^n)/(d^{(n-1)}[\omega]/dt^{(n-1)})$ ] are constant and much smaller than @xmath245 , which is our second assumption . notice that , even if @xmath266 , the currents produced by these two velocities can be comparably important . this is because @xmath267 is independent of @xmath248 and @xmath247 , thus it has the same value for ions and electrons ( so from now we will just drop the subscript @xmath251 and will refer to this velocity as simply @xmath268 ) . thus , when considering both species , the current produced by @xmath269 , @xmath270 , will be due to the tiny excess of electrons in the background required to compensate the crs charge , so it will be proportional to @xmath28 . on the other hand , since @xmath271 is proportional to @xmath272 , it will be much larger for ions than for electrons . so the corresponding @xmath273 will be proportional to the total density of ions in the background , @xmath26 , which is typically much larger than @xmath28 . since @xmath274 for @xmath275 will also affect mainly the ions , their contribution to the current in the plasma will be much smaller than the one of @xmath271 provided that @xmath276 , so we will just neglect them . thus the currents @xmath270 and @xmath273 can be calculated considering equations ( [ eq : b ] ) , ( [ eq : e ] ) , and ( [ eq : velocities ] ) , finding that @xmath277 , \label{eq : jotazero}\ ] ] and @xmath278 , \end{array } \label{eq : jotauno}\end{aligned}\ ] ] where we have defined the field amplification factor @xmath279 . this way we have calculated all the currents in the plasma that are perpendicular to @xmath253 , but we still have to determine the ones that are parallel to @xmath253 . we do that using our third assumption , @xmath280 , which implies that @xmath281 using the ampere s law and the fact that in a one dimensional problem @xmath282 , we have that @xmath283 where @xmath284 is the @xmath57 component of the plasma current parallel to @xmath253 , @xmath285 . given this , the component of @xmath285 perpendicular to @xmath59 is just @xmath286 then , using equations ( [ eq : b ] ) , ( [ eq : e ] ) , ( [ eq : perpendicularity ] ) , ( [ eq : jotapax ] ) and ( [ eq : jotapape ] ) we get that @xmath287 now we have the expressions for all the components of the current perpendicular to @xmath59 , @xmath241 , so we can use equation ( [ eq : jotaperp ] ) to find the dispersion relation , @xmath288 from this derivation we can also obtain an estimate of the plasma velocities due to the crcd waves . we know that the motion of particles perpendicular to @xmath253 is dominated by @xmath289 ( since @xmath268 affects ions and electrons in the same way and @xmath290 ) , and that the motion of particles parallel to @xmath253 is mainly given by electrons moving at @xmath291 . thus , by looking at the expressions for @xmath270 and @xmath285 given by equations ( [ eq : jotazero ] ) , ( [ eq : jotapape ] ) , and ( [ eq : jotapa ] ) , we find that the dominant plasma motion will be given by @xmath268 and will imply a velocity of ions and electrons that can be decomposed into a component along @xmath57 , @xmath54 ( where the subscript @xmath292 stands for @xmath293 ) , and a transverse component , @xmath294 , that is always perpendicular to @xmath15 ( and to @xmath59 ) . in [ sec : analytical ] we use the dispersion relation given by equation ( [ eq : dispersion ] ) to calculate the wavenumber , @xmath295 , and growth rate , @xmath52 , of the fastest growing mode , as well as @xmath36 as a function of the amplitude of the wave . we will use these results here in order to check the consistency of assuming that @xmath29 , @xmath243 , and @xmath244/dt^n)/(d^{(n-1)}[\omega]/dt^{(n-1)})$ ] are constant and much smaller than @xmath76 , which are necessary for neglecting @xmath296 when @xmath297 . we know from equation ( [ eq : phasevelocity ] ) that when @xmath50 , @xmath43 . it means that @xmath298/dt^n)/ ( d^{(n-1)}[\omega]/dt^{(n-1 ) } ) \approx 2\gamma$ ] ( except when @xmath299 and @xmath300 , in which case @xmath298/dt^n)/ ( d^{(n-1)}[\omega]/dt^{(n-1 ) } ) \ll \gamma$ ] ) . so , in order to neglect @xmath296 for @xmath297 , we only require @xmath29 and @xmath243 to be constant and much smaller than @xmath76 . it is possible to show from equation ( [ eq : growthrate ] ) that the first condition , which is equivalent to @xmath113 , where @xmath52 is given by equation ( [ eq : growthrate ] ) , is satified if @xmath250 . from equation ( [ eq : phasevelocity ] ) we see that , if we approximate @xmath301 , @xmath302 becomes approximately equal to @xmath303 , which is also constant and much smaller than @xmath52 . this way we see that our analytical results are valid if the plasma is well magnetized in the sense that @xmath113 , which is equivalent to @xmath304 . using two - dimensional pic simulations , we show in [ sec : magnetization ] that this condition is actually a requirement for the crcd not to be quenched by the weible - like filamentation . we saw in [ sec : threed ] that , when multidimensional effects are considered , the dominant wavelength of the crcd instability , @xmath0 , grows according to equation ( [ eq : wavelength ] ) . this makes it numerically expensive to run three - dimensional simulations that could amplify the field substantially without making @xmath0 too close to the size of the simulation box , @xmath73 . in order to overcome this difficulty , in this section we present the results of two - dimensional simulations whose @xmath73 is always bigger than @xmath0 . despite some artifacts related to the two - dimensional geometry , these simulations help us confirm the main results obtained from the three - dimensional analysis presented above . figures [ fg : shocks1 ] , [ fg : shocks2 ] , and [ fg : shocks3 ] show the results at three different times ( @xmath305 , 9 , and 11 , respectively ) for one of the simulations ( run i4 of table [ table : interference ] ) , which corresponds to crs drifting at the speed of light and without considering their back - reaction . we see that initially the instability is produced independently in different regions of the simulation box ( as seen in fig . [ fg : shocks1 ] ) . in this linear stage of evolution , the waves produced in adjacent regions of space seem to grow without interfering with each other . however , when the waves become non - linear , strong density fluctuations appear on scales of a few @xmath77 ( as shown in fig . [ fg : shocks2 ] ) . the beginning of this stage is shown in fig . [ fg : shocks2 ] . it is also apparent from fig . [ fg : shocks3 ] that the magnetic fluctuations get distorted and evolve into larger scales right after the density fluctuations and turbulence form . the formation of density fluctuations also affects the growth rate of the instability . [ fg : departure ] presents the magnetic energy evolution for the two - dimensional simulations i5 and i6 , whose @xmath91 and @xmath306 , respectively . the rest of their numerical parameters are specified in table [ table : interference ] . we see that , as in the three - dimensional case , the exponential growth stops shortly after @xmath151 . after that , the crcd instability grows at a lower rate , reaching saturation when @xmath53 . this result had already been obtained in the three - dimensional case , but in this case we allow the instability to evolve into larger scales as the magnetic field grows . in two dimensions , the formation of density fluctuations produces a clear differentiation between the @xmath170 and @xmath171 components of the field ( as can be seen in figs . [ fg : shocks3 ] and [ fg : departure ] ) . this is because in the low density regions the plasma can not generate the return current necessary to compensate @xmath13 . thus , the uncompensated cr current produces a toroidal " magnetic field around the underdense regions that , in the two - dimensional case , manifests itself as an amplification of the out of the plane component of the field , @xmath102 . even though , as seen in fig . [ fg : shocks3 ] , both the toroidal " and the crcd field coexist , the two - dimensional simulations can still give us information about the point when the crcd instability stops amplifying the field . , represented by black and red lines , respectively . the @xmath57 , @xmath170 , and @xmath171 components are shown using dotted , dashed , and solid lines , respectively . time is normalized in terms of the @xmath52 of each simulation . the differentiation between the @xmath170 and @xmath171 components of the field as well as the departure from exponential growth after @xmath151 can be seen for both runs . saturation still happens when @xmath53.,width=292 ] axford , w. i. , leer , e. , & skadron , g. , 1977 , 15th int . cosmic ray conf . , 11 , 132 ballet , j. , 2006 , adv . in space res . , 37 , 1902 bell , a. r. , 1978 , , 182 , 147 bell , a. r. , 2004 , , 353 , 550 bell , a. r. , 2005 , , 358 , 181 blandford , r. d. , & ostriker , j. p. , 1978 , , 221 , l29 blasi , p. , & amato , e. , 2008 , arxiv:0806.1223v1 buneman , o. , 1993 , `` computer space plasma physics '' , terra scientific , tokyo , 67 couch , s. , milosavljevi , m. , & nakar , e. , 2008 , arxiv:0807.4117v1 krymsky , g. f. , 1977 , sov . , 23 , 327 kulsrud , r. , & pearce , w. p. , 1969 , , 156 , 445 lagage , p. o. , & cesarsky , c. j. , 1983 , , 125 , 249 mckenzie , j. f. , & volk , h. j. , 1982 , , 116 , 191 niemiec , j. , pohl , m. , stroman , t. , & nishikawa , k. , 2008 , , 684 , 1189 reville , b. , kirk , j. g. , & duffy , p. , 2006 , plasma phys . fusion , 48 , 1741 - 1747 riquelme , m. a. , & spitkovsky , a. , 2008 , int . j. mod d , in press spitkovsky , a. , 2005 , aip conf . proc , 801 , 345 , astro - ph/0603211 uchiyama , y. , aharonian , f. a. , tanaka , t. , takahashi , t. , & maeda , t. , 2007 , nature , 449 volk , h. j. , berezhko , e. g. , & ksenofontov , l. t. , 2005 , 29th int . cosmic ray conf . , 3 , 233 - 236 zirakashvili , v. n. , ptuskin , v. s. , & volk , h. j. , 2008 , , 678 , 255

the cosmic ray current - driven ( crcd ) instability , predicted by @xcite , consists of non - resonant , growing plasma waves driven by the electric current of cosmic rays ( crs ) that stream along the magnetic field ahead of both relativistic and non - relativistic shocks . combining an analytic , kinetic model with one- , two- , and three - dimensional particle - in - cell simulations , we confirm the existence of this instability in the kinetic regime and determine its saturation mechanisms . in the linear regime , we show that , if the background plasma is well magnetized , the crcd waves grow exponentially at the rates and wavelengths predicted by the analytic dispersion relation . the magnetization condition implies that the growth rate of the instability is much smaller than the ion cyclotron frequency . as the instability becomes non - linear , significant turbulence forms in the plasma . this turbulence reduces the growth rate of the field and damps the shortest wavelength modes , making the dominant wavelength , @xmath0 , grow proportional to the square of the field . at constant cr current , we find that plasma acceleration along the motion of crs saturates the instability at the magnetic field level such that @xmath1 , where @xmath2 is the alfvn velocity in the amplified field , and @xmath3 is the drift velocity of crs . the instability can also saturate earlier if crs get strongly deflected by the amplified field , which happens when their larmor radii get close to @xmath0 . we apply these results to the case of crs propagating in the upstream medium of the forward shock in supernova remnants . if we consider only the most energetic crs that escape from the shock , we obtain that the field amplification factor of @xmath4 can be reached . this confirms the crcd instability as a potentially important component of magnetic amplification process in astrophysical shock environments .

low - mass galaxies are arguably the best places to test dark matter ( dm ) models since they are dynamically dominated by the dm haloes they are embedded in well within their inner regions . the kinematical information that is inferred from low surface brightness galaxies ( e.g. * ? ? ? * ) , nearby field dwarf galaxies ( e.g. * ? ? ? * ; * ? ? ? * ) and milky way ( mw ) dwarf spheroidals ( dsphs ) ( e.g. * ? ? ? * ; * ? ? ? * ) , seem to favour the presence of @xmath6 dark matter cores with different degrees of certainty . the former two cases are more strongly established while the latter is still controversial ( e.g. * ? ? ? * ) , which is unfortunate since the mw dsphs have the largest dynamical mass - to - light ratios and are thus particularly relevant to test the dm nature . although not necessarily related to the existence of cores , it has also been pointed out that the population of dark satellites obtained in cdm @xmath7body simulations , are too centrally dense to be consistent with the kinematics of the mw dsphs @xcite . this problem possibly also extends to isolated galaxies @xcite . the increasing evidence of lower than expected central dm densities among dm - dominated systems is a lasting challenge to the prevalent collisionless cold dark matter ( cdm ) paradigm . on the other hand , the low stellar - to - dm content of dwarf galaxies represents a challenge for galaxy formation models since these have to explain the low efficiency of conversion of baryons into stars in dwarf galaxies . it is possible that these two outstanding issues share a common solution rooted in our incomplete knowledge of processes that are key to understand how low - mass galaxies form and evolve : gas cooling , star formation and energetic feedback from supernovae ( sne ) . in particular , episodic high - redshift gas outflows driven by sne have been proposed as a mechanism to suppress subsequent star formation and lower , irreversibly , the central dm densities ( e.g. * ? ? ? * ; * ? ? ? although such mechanism seemingly produces intermediate mass galaxies ( halo mass @xmath8 ) with realistic cores and stellar - to - halo mass ratios @xcite , it is questionable if it is energetically viable for lower mass galaxies @xcite . even though environmental effects such as tidal stripping might alleviate this stringent energetic condition in the case of satellite galaxies @xcite , the issue of low central dm densities seems relevant even for isolated galaxies @xcite . this seems to indicate that sne - driven outflows can only act as a solution to this problem if they occur very early , when the halo progenitors of present - day dwarfs were less massive @xcite . it remains unclear if such systems can avoid regenerating a density cusp once they merge with smaller , cuspier , haloes . it is also far from a consensus that the implementation of strong `` bursty '' star formation recipes in simulations , a key ingredient to reduce central dm densities , is either realistic or required to actually produce consistent stellar - to - halo mass ratios ( e.g. , * ? ? ? * ) , and other observed properties . it is therefore desirable , but challenging , to identify observables that could unambiguously determine whether bursty star formation histories with a strong energy injection efficiency ( into the dm particles ) are realistic or not . an exciting alternative solution to the problems of cdm at the scale of dwarfs is that of self - interacting dark matter ( sidm ) . originally introduced by @xcite , it goes beyond the cdm model by introducing significant self - collisions between dm particles . the currently allowed limit to the self - scattering cross section is imposed more stringently by observations of the shapes and mass distribution of elliptical galaxies and galaxy clusters @xcite , and is set at : @xmath9 . dm particles colliding with roughly this cross section naturally produce an isothermal core with a @xmath6 size in low - mass galaxies , close to what is apparently observed . sidm is well - motivated by particle physics models that introduce new force carriers in a hidden dm sector ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which predict velocity - dependent self - scattering cross sections . in the case of massless bosons for instance , the cross section scales as @xmath10 as in rutherford scattering . the renewed interest in sidm has triggered a new era of high resolution dm - only sidm simulations : velocity - dependent in @xcite ( vzl hereafter ) , and velocity - independent in @xcite , that hint at a solution to the cdm problems in low - mass galaxies . in particular for the mw dsphs , it has been established that the resultant dark satellites of a mw - size halo are consistent with the dynamics of the mw dsphs , have cores of @xmath6 and avoid cluster constraints only if @xmath11 , or if the cross section is velocity - dependent @xcite . recently , simulations of sidm models with new light mediators have shown that is possible to also suppress the abundance of dwarf galaxies due to the modified early - universe power spectrum caused by the interactions of the dm with the dark radiation @xcite . given the recent success of sidm , a natural step is to elevate its status to that of cdm by studying the synergy between baryonic physics and dm collisionality in a suitable galaxy formation model . so far , this has been studied only analytically @xcite , with a focus in more massive galaxies where baryons dominate the central potential . interestingly , in this case , the dm core size is reduced and the central densities are higher compared to sidm simulations without the effect of baryons . in this paper we concentrate on the regime of dwarf galaxies by pioneering cosmological hydrodynamical simulations that include the physics of galaxy formation within a sidm cosmology . we compare them with their counterparts ( under the same initial conditions ) in the cdm model with the main objective of understanding the impact of sidm on the formation and evolution of dwarf galaxies . .dm models considered in this paper . cdm is the standard collisionless model without any self - interaction . sidm10 is a reference model with a constant cross section an order of magnitude larger than allowed by current observational constraints . we note that such a model could still be realized in nature if this large cross section would only hold over a limited relative velocity range . sidm1 is also a model with constant cross section , which is potentially in the allowed range . vdsidma and vdsidmb have a velocity - dependent cross section motivated by the particle physics model presented in @xcite . these two models are allowed by all astrophysical constraints , and solve the `` too big to fail '' problem ( see * ? ? ? * ) as demonstrated in vzl . [ cols="^,^,^,^",options="header " , ] in this case the fit is poor in the inner regions ( @xmath12 ) , and thus , we use only the exponential gas profile instead of the two - component model as in the other cases . the lower right panel of figure [ fig : halo_profiles ] demonstrates that the feedback associated with sne does not alter the dm density distribution in our model . this is not surprising since we do not employ a very bursty star formation model , but a rather smooth star formation prescription . as a consequence , the dm density profile is not affected at all by the formation of the baryonic galaxy and the related feedback processes for the cdm case . the sidm models lead to core formation due to self - interactions of dm particles . such core makes it easier for sne feedback to drive gas outwards , which should cause some effect on the dm distribution . in fact , the lower right panel of figure [ fig : halo_profiles ] demonstrates that the dm density is slightly reduced in the cored region even with a smooth feedback model like ours . however , this effect is rather small and at maximum @xmath13 relative to the sidm10 simulation without baryons . this effect is therefore small compared to the effect of self - interactions , which reduce the central dm density much more significantly . so far we have discussed the relative differences between the different profiles . to quantify the spatial distribution of the dm and the baryons , gas and stars , in more detail , we now find analytical fits to the spherically averaged density distributions . we have found that the different dm models require different density profiles profiles to achieve a reasonable quality of the fits . , are stated for each fit . the fits were performed over the full radial range . all models lead to essentially perfect exponential profiles with no significant bulge components . the largest cross section models ( sidm1 , sidm10 ) produce a stellar core in the center.,scaledwidth=49.0% ] we start with the dm profile for the cdm case . it is well - known that cdm haloes have spherically averaged density profiles that are well described by nfw @xcite or einasto profiles @xcite . we therefore fit the dm profile of the cdm model with the two - parameter nfw profile : @xmath14 on the other hand , the sidm haloes are well fitted by cored - like profiles that vary according to the amplitude of the self - scattering cross section at the typical velocities of the halo . in the case of the strongest cross section , sidm10 , a good fit is obtained with the following three - parameter profile : @xmath15 while for intermediate cross sections , sidm1 and vdsidma , a burkert - like three - parameter formula provides a better fit : @xmath16 finally , for the weakest cross section , vdsidmb , a good fit is given by : @xmath17 next we consider the profiles of the baryonic components . for the stars and the gas , we use a two component density profile : an exponential profile in the outer region , which is a good approximation except for the gas beyond @xmath18 , and a cored profile in the inner region , analogous to eq . ( [ sidm10_rho ] ) : @xmath19 where we find that @xmath20 provides a good fit in all cases except for the gas distribution in the sidm1 case . for each profile ( dm , gas , and stars ) , we find the best fit parameters by minimising the following estimate of the goodness of the fit : @xmath21 where the sum goes over all radial bins . we summarise the best fit parameters for each component in table [ table : fits ] . we stress again that we need distinct parametric density profiles to better describe the spatial dm structure of the halo for the different dm models . for instance , in the case of sidm10 , the value of @xmath22 for the best fit using eq . ( [ sidm10_rho ] ) is @xmath23 , whereas using eqs . ( [ burkert]-[burkert_2 ] ) is @xmath24 and @xmath25 , respectively . on the other hand , for sidm1 , the values of @xmath22 using eqs . ( [ sidm10_rho]-[burkert_2 ] ) , are , respectively : 0.020 , 0.003 , 0.021 . clearly , in this case , eq . ( [ burkert ] ) is the best fit . for the stars we can also inspect the stellar surface density profiles , which are closely related to the measured stellar surface brightness profiles . the stellar surface density profiles of the da dwarfs for the different dm models are shown in figure [ fig : stellar_profiles ] . the exponential scale length , @xmath26 , of the different models is quoted for each model , and the dashed lines show the actual exponential fits for each model . for the cdm case , we find over a large radial range an exponential profile and no significant bulge contribution , similar to what is observed for most dwarfs . we have checked that the surface density profiles do not vary much if the orientation of the galaxy changes . the reason for this is that the dwarfs do not form thin disks , but rather extended puffed up ellipsoidal distributions similar to , for example , the stellar population of the isolated dwarf wlm . the scale length values we find are in reasonable agreement with other recent simulation of dwarf galaxies at this mass scale ( e.g. , * ? ? ? in the case of sidm1 and sidm10 , the presence of a small stellar core is visible in figure [ fig : stellar_profiles ] . the scale length does not change significantly as a function of the underlying dm model . however , it can clearly be seen that dm self - interactions lead to slightly larger exponential scale radii . we note that , contrary to previous studies , we achieve exponential stellar surface density profiles without a bursty star formation model or a high density thresholds for star formation . we therefore find that our quiescent , smooth star formation model leads to non - exponential star formation histories , and to exponential stellar surface density profiles . it has been argued that these characteristics are intimately connected to `` bursty '' star formation rates ( see e.g. * ? ? ? as a corollary , it was argued that the formation of a dm core is then naturally expected . however , we find that this is is not necessarily the case . we should note that @xcite simulated an isolated dwarf of a similar halo mass and stellar mass as our dwarf da but with a considerably bursty star formation model that produced a @xmath27 core . this is in clear contrast to our simulation where baryonic effects are unable to create a dm core despite of the high global efficiency of star formation . the key is then , once more , in the time scales and efficiency of energy injection during sne - driven outflows . it remains to be seen if star formation histories in real dwarf galaxies occur in bursts with a timescale much shorter than the local dm dynamical timescale , and with an effective energy injection into the dm particles that is sufficient to significantly alter the dm distribution . as we have shown above , halo da is in relative isolation and has a quiet merger history . we therefore expect that the final stellar and dm configuration is nearly in equilibrium . in the case of sidm , once the isothermal core forms , further collisions are not relevant anymore in changing the dm phase - space distribution . we can then ignore the collisional term in the boltzmann equation and test the equilibrium hypothesis by solving the jeans equation for the radial velocity dispersion profile using as input the density and anisotropy profiles : @xmath28 where @xmath29 is the total enclosed mass . we solve eq . ( [ jeans_eq ] ) independently for the collisionless components , dm and stars , using the fits to the density profiles with the analytic formulae introduced above . in addition , we also fit the corresponding radial anisotropy profiles for both the dm and the stars with the following five - parameter formula : @xmath30 the best fit parameters for this relation for each dm model are listed in table [ table : fits ] . the result obtained by solving the jeans equation for the cdm and sidm10 cases is seen in figure [ fig : sigma_jeans ] . here we show the predicted dispersion profiles with dashed lines for dm ( thick lines ) and stars ( thin lines ) . the solid lines show the actual simulation results . although the agreement between the velocity dispersion predicted by the jeans analysis and the simulation is not perfect , the comparison still indicates that halo da is roughly in equilibrium and that the spherical approximations assumed above are partially correct . in the sidm10 case , this would suggest that the dark matter core formed in the past and that any subsequent scattering does not affect the final equilibrium configuration once the galaxy forms . this would justify the use of the jeans equation without considering a collisional term . we will consider a more detailed dynamical analysis in a subsequent paper analysing the different sidm cases , having a closer look at the velocity anisotropies , and also investigating departures from spherical symmetry ( zavala & vogelsberger , in prep ) . for dm ( top ) and stellar mass ( bottom ) for halo da . the enclosed dm mass is for all times and for all models significantly larger than the stellar mass , and therefore dynamically dominates the center of the dwarf . the central dm mass is substantially reduced for the sidm1 and sidm10 models , but only slightly for the vdsidm models . similarly , the stellar mass is only reduced for the models with constant cross section , whereas the stellar mass growth of vdsidm closely follows that of the cdm case.,title="fig:",scaledwidth=49.0% ] for dm ( top ) and stellar mass ( bottom ) for halo da . the enclosed dm mass is for all times and for all models significantly larger than the stellar mass , and therefore dynamically dominates the center of the dwarf . the central dm mass is substantially reduced for the sidm1 and sidm10 models , but only slightly for the vdsidm models . similarly , the stellar mass is only reduced for the models with constant cross section , whereas the stellar mass growth of vdsidm closely follows that of the cdm case.,title="fig:",scaledwidth=49.0% ] in this section we study in more detail the matter content and structure of the simulated dwarf da within the central region , @xmath31 , which roughly encloses the dm core size for all models . we start with figure [ fig : histories1000 ] , which shows the mass buildup of dm ( top ) and stars ( bottom ) within @xmath4 as a function of time . in the cases with a constant scattering cross section , it is clear that there is a significant amount of dark matter mass expelled from the central kiloparsec . in the case of sidm1 for example , about @xmath32 have been removed by @xmath2 . for the vdsidm models however , there is only a minimal deviation from the evolution of the base cdm model . in fact , the vdsidmb model mass evolution follows the cdm result very closely and shows a nearly constant central mass after early times @xmath33 . the vdsidma model leads to a small depletion of dm in the central @xmath4 of about @xmath34 . the largest depletion can be seen for the sidm10 model , where the central mass is reduced by nearly a factor @xmath35 . the central stellar mass on the other hand grows steadily with time but it is at all times , and for all dm models , sub - dominant compared to the inner dm mass . for all models the central stellar mass is below @xmath36 at @xmath2 , which is a factor @xmath37 lower than the central dm mass at that time . the stellar mass in sidm1 and sidm10 grows more slowly than in the cdm and vdsidm cases . the vdsidm models behave very similar to the cdm case , where the stellar mass grows nearly linearly with time reaching a mass of about @xmath38 . the stellar mass within @xmath4 grows initially similar sidm10 ( sidm1 ) , however , after @xmath39 ( @xmath40 ) the stellar mass growth is slowed down for sidm10 ( sidm1 ) . after that time the growth is still linear but with a significantly shallower slope compared to the cdm and vdsidm cases . we note that sidm1 is an allowed model , and it is striking how different its stellar mass is growing compared to the other allowed vdsidm models . to quantify this in more detail we present a closer look of the density profiles of dm ( solid lines ) and stars ( dashed lines ) in figure [ fig : rho_inner ] . this reveals a tight correlation between the shape of the dm and stellar density distributions . the stars within the core react to the change in the potential of the dominant dm component due to self - interactions . the size of the stellar core is therefore tied , to certain degree , to the core sizes of the dm distribution . in the cases where the scattering cross section has a velocity dependence , although the creation of a dm core is evident , the impact is minimal in the stellar distribution compared to the models with a constant cross section . this is mainly because even in the cdm case , the stellar distribution forms a core which is roughly the size of the dm core observed in the vdsidm cases . we conclude that self - interactions drive the sizes of the cores in dm and stars to track each other . for sidm1 , the density within the core is a factor of @xmath41 smaller than in cdm . the central distribution of stars can therefore probe the nature of dm and can potentially be used to distinguish different sidm models . the strong correlation between dm and stars that we are finding is similar to the one suggested recently by @xcite using analytical arguments , but the regimes and interpretations are quite different . whereas these authors investigated the response of sidm to a dominant stellar component , we are investigating a system where dm still dominates dynamically . thus in the former , the dm cores sizes are reduced relative to expectations from dm - only simulations due to the formation of the galaxy , while in the latter , the stellar distribution of the galaxy responds to the formation of the sidm core by increasing its own stellar core relative to the cdm case . this regime is therefore more promising to derive constraints for the nature of dm . for the different dm models . the stars trace the evolution of dm and also form a core . the size of the stellar core is closely related to the size of the dm core . this can be seen most prominently for the sidm1 and sidm10 models.,scaledwidth=49.0% ] next we are interested in the time evolution of the core radii . it was already obvious from figure [ fig : histories1000 ] that for the largest cross section cases , the core should already be present early on during the formation history of the galaxy . this is indeed the case as we demonstrate more clearly in figure [ fig : coresize ] , where the evolution of the core sizes are shown as a function of time . as a measure of core radius , we fit burkert profiles @xcite at each time , for each of the models , to extract the core size @xmath42 : @xmath43 we note that we use this two - parameter fit for simplicity to fit all sidm models and give a measure of the core size . as we explored in detail above , the different sidm models are actually better fitted by different radial profiles . however , our purpose here is not to rigorously define a core size but simply to present an evolutionary trend for the different models . this trend is clearly visible in the figure as well as the dependence of the amplitude of the core size on the scattering cross section . figure [ fig : coresize ] shows the core radii determined by these two - parametric burkert fits for all dm models with ( solid lines ) and without ( dashed lines ) the effects baryons . , in the dm - only simulations ( dashed ) with the simulations including baryons ( solid ) . baryons have only a tiny effect on the evolution and size of the cores . the largest effect can be seen for sidm10 , where the shallow dm profile allows sne feedback to expand the core a bit more compared to the dm - only case.,scaledwidth=49.0% ] figure [ fig : coresize ] also demonstrates that the actual impact of baryons on the dm distribution relative to the dm - only case is minor , as we discussed already above ( see lower right panel of figure [ fig : halo_profiles ] ) . in the case of cdm this is not surprising since : ( i ) our star formation model is less bursty compared to models where the cusp - core transformation is efficient and ( ii ) for the mass scale we are considering , halo mass @xmath3 for halo da , the energy released by sne is not expected to be sufficient to create sizeable dm cores @xcite , although see @xcite . figure [ fig : coresize ] demonstrates that our star formation and feedback model creates only a slightly larger core for the sidm10 model . this is because expelling gas in this case is easier due to the reduced potential well caused by dm collisions . we stress again that these results are sensitive to the model used for sne - driven energy injection into the dm particles ( both efficiency and time scales ) . larger efficiencies of energy injection into shorter timescales would result in a larger removal of dm mass from the inner halo . according to figure [ fig : coresize ] a sizeable core is already present very early on . by @xmath44 all the models already have cores more than half of their present day size . furthermore , figure [ fig : coresize ] also demonstrates , that none of our sidm models lead to the gravothermal catastrophe where the core collapses following the outward flux of energy caused by collisions . this is consistent with the findings in vzl , where only one subhalo , with similar total dark matter mass as halo da , of the analogous sidm10 mw - size simulation was found to enter that regime towards @xmath2 . . the vdsidm models have a weaker impact.,scaledwidth=49.0% ] as a consequence of the dm core settling early on in the formation history of the galaxy , the star formation rate within the central @xmath4 is reduced significantly at late times in the cases with constant cross section . this results in a stellar population that is in average older than in the case of cdm . this is clearly shown in figure [ fig : metals ] , where we plot the time evolution of the ratio of the metallicity averaged within the central @xmath4 , relative to the cdm case . the difference today is @xmath45 . interestingly , in the vdsidm cases , there is an excess in star formation within @xmath4 in the last stages of the evolution resulting in a younger stellar population since the last @xmath46 ( see also figure [ fig : coresize ] ) . we will investigate this issue , and in general the properties of the central @xmath47 region , in a follow - up paper using simulations with increased resolution ( zavala & vogelsberger , in prep ) . as a function of total stellar mass . bottom panel : dm density slope at @xmath48 as a function of total stellar mass . the different dm models lead to significantly different slopes and masses at and within @xmath48 . at this radius even the vdsidm models clearly deviate from the cdm case . both the mass and the slope clearly scale with the cross section and allow to disentangle the different dm models . observational estimates from a combined sample of dwarf galaxies @xcite and from the things survey @xcite are also shown in the top and bottom panels , respectively.,title="fig:",scaledwidth=49.0% ] as a function of total stellar mass . bottom panel : dm density slope at @xmath48 as a function of total stellar mass . the different dm models lead to significantly different slopes and masses at and within @xmath48 . at this radius even the vdsidm models clearly deviate from the cdm case . both the mass and the slope clearly scale with the cross section and allow to disentangle the different dm models . observational estimates from a combined sample of dwarf galaxies @xcite and from the things survey @xcite are also shown in the top and bottom panels , respectively.,title="fig:",scaledwidth=49.0% ] in figure [ fig : encmass500 ] we focus on a region even closer to the halo centre and show the total mass within @xmath48 ( top ) and the slope of the density profile measured at this radius ( bottom ) . we compare both to observational estimates using samples of dwarf galaxies compiled in @xcite ( top ) and from the things survey ( bottom , * ? ? ? * ) . at these small radii , the change in the enclosed mass is still more dramatic for the constant cross section sidm models having a deficit in mass by a factor @xmath49 relative to the cdm case , while the vdsidm cases , although close to cdm , still deviate visibly . the logarithmic slope of the density profile at this radius varies between @xmath50 ( sidm10 ) and @xmath51 ( cdm ) . figure [ fig : encmass500 ] shows that given the large dispersion in the data , all dm models are essentially consistent with observations . there is however some tension with the cdm simulation of halo da having a slightly too large total mass , and a slightly too steep dm density slope at @xmath52 . on the other hand , the sidm10 case might be to cored for the stellar mass of halo db ( @xmath53 ) . taking both haloes into account , and looking at the two relations of figure [ fig : encmass500 ] only , it seems that sidm1 agrees best with these observations . we stress however , that our dwarf sample is far to small to draw any conclusions based on this result and these observations are in any case , too uncertain to use them as constraints . self - interacting dark matter ( sidm ) is one the most viable alternatives to the prevailing cold dark matter ( cdm ) paradigm . current limits on the elastic scattering cross section between dm particles are set at @xmath54 @xcite . at this level , the dm phase space distribution is altered significantly relative to cdm in the centre of dm haloes . the impact of dm self - interactions on the baryonic component of galaxies that form and evolve in sidm haloes has not been explored so far . recently , @xcite analytically estimated the dm equilibrium configuration that results from a stellar distribution added to the centre of a halo in the case of sidm . these authors studied the regime where the stellar component dominates the gravitational potential and concluded that the dm core sizes ( densities ) are smaller ( higher ) than observed in dm - only sidm simulations . this might have important consequences on current constraints of sidm models since they have been derived precisely in the baryon - dominated regime . in this paper we explore the opposite regime , that of dwarf galaxies where dm dominates the gravitational potential even in the innermost regions . our analysis is based on the first hydrodynamical simulations performed in a sidm cosmology . we focus most of the analysis on a single dwarf with a halo mass @xmath55 . we study two cases with a constant cross section : sidm1 and sidm10 , @xmath56 , respectively , and two cases with a velocity - dependent cross section : vdsidma - b , that were also studied in detail in vzl and @xcite . except for sidm10 , all these models are consistent with astrophysical constraints , solve the `` too big to fail '' problem and create @xmath57(1 kpc ) cores in dwarf - scale haloes . our simulations include baryonic physics using the implementation described in @xcite employing the moving mesh code arepo @xcite . we use the same model that was set up to reproduce the properties of galaxies at slightly larger mass - scales . our intention in this first analysis is not to match the properties of dwarf galaxies precisely , but rather to compare sidm and cdm with a single prescription for the baryonic physics , which has been thoroughly tested on larger scales . our most important findings are : * impact of sidm on global baryonic properties of dwarf galaxies : * the stellar and gas content of our simulated dwarfs agree reasonably well with various observations including the stellar mass as a function of halo mass , the luminosity metallicity relation , the neutral hydrogen content , and the cumulative star formation histories . the latter are similar to those of local isolated group dwarf galaxies with similar stellar masses . we find that the stellar mass , the gas content , the stellar metallicities and star formation rates are only minimally affected by dm collisions in allowed sidm models . the allowed elastic cross sections are too small to have a significant global impact on these quantities , and the relative differences between the different dm models are typically less than @xmath58 . in most cases these changes are not systematic as a function of the employed dm model . the modifications in the global baryonic component of the galaxies can therefore not be used to constrain sidm models since the effects are too small and not systematic . * impact of sidm on the inner halo region : * within @xmath31 , we find substantial differences driven by the collisional nature of sidm . besides the well - known effect of sidm on the dm density profiles , we also find that at these scales the distribution of baryons is significantly affected by dm self - interactions . both stars and gas show relative differences up to @xmath59 in the density , the velocity dispersion , and the gas temperature . most of the effects increase with the size of the cross section in the central region . the strongest correlation with the cross section can be found for the stellar profiles , where the central stellar density profile clearly correlates with the central cross section leading to lower central densities for dm models with larger central cross sections . * impact of baryons on the inner halo region : * we find that the impact of baryons on the dm density profile is small for the dm - dominated dwarf ( @xmath60 ) studied here . however , this result is also connected to our smooth star formation model , which is not as bursty as models where a significant core formation is observed due to baryonic feedback . the size of the dm core and the central density are therefore essentially the same as in our simulations that have no baryons , although the core size is slightly larger in the former than in the latter . * disentangling different sidm models : * for the cases where the scattering cross section is constant , the combination of two key processes : ( i ) an early dm core formation such that by @xmath44 , the dm cores already have half of their size today ; and ( ii ) a star formation history dominated by the period after the formation of the dm core , result in the following characteristics of the stellar distribution of sidm galaxies : ( a ) the development of a central stellar core with a size that correlates with the amplitude of the scattering cross section . for instance , for the sidm1 case with @xmath61 , the density within the stellar core is a factor of @xmath41 smaller than for the cdm case . ( b ) a reduced stellar mass in the sub - kpc region ( @xmath62 ) as a byproduct of the reduced dm gravitational potential due to self - scattering . ( c ) a reduced central stellar metallicity ; by @xmath45 at @xmath2 compared to the cdm case . around @xmath63 the metallicity can be reduced by up to @xmath64 . for the cases where the scattering cross section is velocity - dependent , even though a sizeable dm core can still be created ( @xmath65 ) , the effect in the stellar distribution at all scales is minimal relative to cdm . this is likely because the amplitude of the cross section within the inner region of the dwarf is not large enough to produce a dm core that is larger than the stellar core that forms in the cdm case . whether the latter could be the result of numerical resolution is something we will investigate in a forthcoming paper . any changes that we found in the vdsidm cases seem to be only related to the stochastic nature of the simulated star formation and galactic wind processes . these conclusions are key predictions of sidm that can in principle be tested to either constrain currently allowed models , particularly constant cross section models , or to find signatures of dm collisions in the properties of the central stellar distributions of dwarf galaxies . in future works we will explore these possibilities in more detail . we thank daniel weisz for providing cumulative star formation histories of local group dwarfs to us , and michael boylan - kolchin for help with the initial conditions . we further thank volker springel for useful comments and giving us access to the arepo code . the dark cosmology centre is funded by the dnrf . jz is supported by the eu under a marie curie international incoming fellowship , contract piif - ga-2013 - 627723 . the initial conditions were made using the dirac data centric system at durham university , operated by the institute for computational cosmology on behalf of the stfc dirac hpc facility ( www.dirac.ac.uk ) . the dirac system is funded by bis national e - infrastructure capital grant st / k00042x/1 , stfc capital grant st / h008519/1 , stfc dirac operations grant st / k003267/1 , and durham university . dirac is part of the re : green card uk national e - infrastructure .

we present the first cosmological simulations of dwarf galaxies , which include dark matter self - interactions and baryons . we study two dwarf galaxies within cold dark matter , and four different elastic self - interacting scenarios with constant and velocity - dependent cross sections , motivated by a new force in the hidden dark matter sector . our highest resolution simulation has a baryonic mass resolution of @xmath0 and a gravitational softening length of @xmath1 at @xmath2 . in this first study we focus on the regime of mostly isolated dwarf galaxies with halo masses @xmath3 where dark matter dynamically dominates even at sub - kpc scales . we find that while the global properties of galaxies of this scale are minimally affected by allowed self - interactions , their internal structures change significantly if the cross section is large enough within the inner sub - kpc region . in these dark - matter - dominated systems , self - scattering ties the shape of the stellar distribution to that of the dark matter distribution . in particular , we find that the stellar core radius is closely related to the dark matter core radius generated by self - interactions . dark matter collisions lead to dwarf galaxies with larger stellar cores and smaller stellar central densities compared to the cold dark matter case . the central metallicity within @xmath4 is also larger by up to @xmath5 in the former case . we conclude that the mass distribution , and characteristics of the central stars in dwarf galaxies can potentially be used to probe the self - interacting nature of dark matter . [ firstpage ] cosmology : dark matter galaxies : halos methods : numerical

the parent compounds of cuprate superconductors are identified as mott insulators @xcite , in which the lack of conduction arises from the strong electron - electron repulsion . superconductivity then is obtained by adding charge carriers to insulating parent compounds with the superconducting ( sc ) transition temperature @xmath0 has a dome - shaped doping dependence @xcite . since the discovery of superconductivity in cuprate superconductors , the search for other families of superconductors that might supplement what is known about the sc mechanism of doped mott insulators has been of great interest . fortunately , it has been found @xcite that there is a class of cobaltate superconductors na@xmath3coo@xmath4h@xmath5o , which displays most of the structural and electronic features thought to be important for superconductivity in cuprate superconductors : strong two - dimensional character , proximity to a magnetically ordered nonmetallic state , and electron spin 1/2 . in particular , @xmath0 in cobaltate superconductors has the same unusual dome - shaped dependence on charge - carrier doping @xcite . however , there is one interesting difference : in the cuprate superconductors @xcite , cu ions in a square array are ordered antiferromagnetically , and then spin fluctuations are thought to play a crucial role in the charge - carrier pairing , while in the cobaltate superconductors @xcite , co ions in a triangular array are magnetically frustrated , and therefore this geometric frustration may suppress @xmath0 to low temperatures . it has been argued that the triangular - lattice cobaltate superconductors are probably the only system other than the square - lattice cuprate superconductors where a doped mott insulator becomes a superconductor . the heat - capacity measurement of the specific - heat can probe the bulk properties of a superconductor , which has been proven as a powerful tool to investigate the low - energy quasiparticle excitations , and therefore gives information about the charge - carrier pairing symmetry , specifically , the existence of gap nodes at the fermi surface @xcite . in conventional superconductors @xcite , the absence of the low - energy quasiparticle excitations is reflected in the thermodynamic properties , where the specific - heat of conventional superconductors is experimentally found to be exponential at low temperatures , since conventional superconductors are fully gaped at the fermi surface . however , the situation in the triangular - lattice cobaltate superconductors is rather complicated , since the experimental results obtained from different measurement techniques show a strong sample dependence @xcite . thus it is rather difficult to obtain conclusive results . the early specific - heat measurements @xcite showed that the specific - heat in the triangular - lattice cobaltate superconductors reveals a sharp peak at @xmath0 , and can be explained phenomenologically within the bardeen - cooper - schrieffer ( bcs ) formalism under an unconventional sc symmetry with line nodes . however , by contrast , the latest heat - capacity measurements @xcite indicated that among a large number of the gap symmetries that have been suggested @xcite , the sc - state with d - wave ( @xmath6 pairing ) symmetry without gap nodes at the fermi surface is consistent with the observed specific - heat data . furthermore , by virtue of the magnetization measurement technique , the value of the upper critical field and its temperature dependence have been observed for all the temperatures @xmath7 @xcite , where the temperature dependence of the upper critical field follows qualitatively the bcs type temperature dependence . on the theoretical hand , there is a general consensus that superconductivity in the triangular - lattice cobaltate superconductors is caused by the strong electron correlation @xcite . using the resonating - valence - bond mean - field approach , it has been suggested that the spin fluctuation enhanced by the dopant dynamics leads to a d - wave sc - state @xcite . based on the mean - field variational approach with gutzwiller approximation , a d - wave sc - state is realized in the parameter region close to the triangular - lattice cobaltate superconductors @xcite . within the framework of the kinetic - energy - driven sc mechanism @xcite , it has been demonstrated that charge carriers are held together in d - wave pairs at low temperatures by the attractive interaction that originates directly from the kinetic energy by the exchange of spin excitations @xcite . moreover , superconductivity with the d - wave symmetry has been explored by a large - scale dynamical cluster quantum monte carlo simulation on the triangular - lattice hubbard model @xcite . in particular , using the diagram technique in the atomic representation , the sc phase with the d - wave symmetry in an ensemble of the hubbard fermions on a triangular lattice has been discussed @xcite , where the domelike shape of the doping dependence of @xmath0 is obtained . however , to the best of our knowledge , the thermodynamic properties of the triangular - lattice cobaltate superconductors have not been treated starting from a microscopic sc theory , and no explicit calculations of the doping dependence of the upper critical field have been made so far . in this case , a challenging issue for theory is to explain the thermodynamic properties of the triangular - lattice cobaltate superconductors . in our recent study @xcite , the electromagnetic response in the triangular - lattice cobaltate superconductors is studied based on the kinetic - energy - driven sc mechanism @xcite , where we show that the magnetic - field - penetration depth exhibits an exponential temperature dependence due to the absence of the d - wave gap nodes at the charge - carrier fermi surface . moreover , in analogy to the dome - shaped doping dependence of @xmath2 , the superfluid density increases with increasing doping in the lower doped regime , and reaches a maximum around the critical doping , then decreases in the higher doped regime . in this paper , we start from the theoretical framework of the kinetic - energy - driven superconductivity , and then provide a natural explanation to the thermodynamic properties in the triangular - lattice cobaltate superconductors . we evaluate explicitly the internal energy , and then qualitatively reproduced some main features of the heat - capacity and magnetization measurements on the triangular - lattice cobaltate superconductors @xcite . in particular , we show that a sharp peak in the specific - heat of the triangular - lattice cobaltate superconductors appears at @xmath0 , and then the specific - heat varies exponentially as a function of temperature for the temperatures @xmath1 due to the absence of the d - wave gap nodes at the charge - carrier fermi surface , which is much different from that in the square - lattice cuprate superconductors @xcite , where the characteristic feature is the existence of the gap nodes on the charge - carrier fermi surface , and then the specific - heat in the square - lattice cuprate superconductors decreases with decreasing temperatures as some power of the temperature in the temperature range @xmath1 . moreover , the upper critical field follows qualitatively the bcs type temperature dependence , and has the same dome - shaped doping dependence as @xmath2 . the rest of this paper is organized as follows . we present the basic formalism in section [ framework ] , and then the quantitative characteristics of the thermodynamic properties in the triangular - lattice cobaltate superconductors are discussed in section [ thermodynamic ] , where we show that although the pairing mechanism is driven by the kinetic energy by the exchange of spin excitations @xcite , the sharp peak of the specific - heat in the triangular - lattice cobaltate superconductors at @xmath0 can be described qualitatively by the kinetic - energy - driven d - wave bcs - like formalism . finally , we give a summary in section [ conclusions ] . in the triangular - lattice cobaltate superconductors , the characteristic feature is the presence of the two - dimensional coo@xmath5 plane @xcite . in this case , a useful microscopic model that has been widely used to describe the low - energy physics of the doped coo@xmath5 plane is the @xmath8-@xmath9 model on a triangular lattice @xcite . this @xmath8-@xmath9 model is defined through only two competing parameters : the nearest - neighbor ( nn ) hopping integral @xmath8 in the kinetic - energy term , which measures the electron delocalization through the lattice , and the nn spin - spin antiferromagnetic ( af ) exchange coupling @xmath9 in the magnetic - energy part , which describes af coupling between localized spins . in particular , the nn hopping integral @xmath8 is much larger than the af exchange coupling constant @xmath9 in the heisenberg term , and therefore the spin configuration is strongly rearranged due to the effect of the charge - carrier hopping @xmath8 on the spins , which leads to a strong coupling between the charge and spin degrees of freedom of the electron . since the triangular - lattice cobaltate superconductors are viewed as an electron - doped mott insulators @xcite , this @xmath8-@xmath9 model is subject to an important local constraint @xmath10 to avoid zero occupancy , where @xmath11 ( @xmath12 ) is the electron creation ( annihilation ) operator . in the hole - doped side , the local constraint of no double electron occupancy has been treated properly within the fermion - spin approach @xcite . however , for an application of the fermion - spin theory to the electron - doped case , we @xcite should make a particle - hole transformation @xmath13 , where @xmath14 ( @xmath15 ) is the hole creation ( annihilation ) operator , and then the local constraint @xmath10 without zero occupancy in the electron - doped case is replaced by the local constraint of no double occupancy @xmath16 in the hole representation . this local constraint of no double occupancy now can be dealt by the fermion - spin theory @xcite , where the hole operators @xmath17 and @xmath18 are decoupled as @xmath19 and @xmath20 , respectively , with the charge degree of freedom of the hole together with some effects of spin configuration rearrangements due to the presence of the doped charge carrier itself that are represented by the spinful fermion operator @xmath21 , while the spin degree of freedom of the hole is represented by the spin operator @xmath22 . the advantage of this fermion - spin approach is that the local constraint of no double occupancy is always satisfied in actual calculations . based on the @xmath8-@xmath9 model in the fermion - spin representation , the kinetic - energy - driven sc mechanism has been developed for the square - lattice cuprate superconductors in the doped regime without an af long - range order ( aflro ) @xcite , where the attractive interaction between charge carriers originates directly from the interaction between charge carriers and spins in the kinetic energy of the @xmath8-@xmath9 model by the exchange of spin excitations in the higher powers of the doping concentration . this attractive interaction leads to the formation of the charge - carrier pairs with the d - wave symmetry , while the electron cooper pairs originated from the charge - carrier d - wave pairing state are due to the charge - spin recombination @xcite , and they condense into the d - wave sc - state . furthermore , within the framework of the kinetic - energy - driven superconductivity , the doping dependence of the thermodynamic properties in the square - lattice cuprate superconductors has been studied @xcite , and then the striking behavior of the specific - heat in the square - lattice cuprate superconductors are well reproduced . the triangular - lattice cobaltate superconductors on the other hand are the second known example of superconductivity arising from doping a mott insulator after the square - lattice cuprate superconductors . although @xmath0 in the triangular - lattice cobaltate superconductors is much less than that in the square - lattice cuprate superconductors , the strong electron correlation is common for both these materials , which suggest that these two oxide systems may have the same underlying sc mechanism . in this case , the kinetic - energy - driven superconductivity developed for the square - lattice cuprate superconductors has been generalized to the case for the triangular - lattice cobaltate superconductors @xcite . the present work of the discussions of the thermodynamic properties in the triangular - lattice cobaltate superconductors builds on the kinetic - energy - driven sc mechanism developed in refs . @xcite and @xcite , and only a short summary of the formalism is therefore given in the following discussions . in our previous discussions in the doped regime without aflro , the full charge - carrier diagonal and off - diagonal green s functions of the @xmath8-@xmath9 model on a triangular lattice in the charge - carrier pairing state have been obtained explicitly as @xcite , [ bcsgf ] @xmath23 where the charge - carrier quasiparticle coherent weight @xmath24 , the charge - carrier quasiparticle coherence factors @xmath25 and @xmath26 , the charge - carrier quasiparticle energy spectrum @xmath27 , and the charge - carrier excitation spectrum @xmath28 . in the early days of superconductivity in the triangular - lattice cobaltate superconductors , some nmr and nqr data are consistent with the case of the existence of a pair gap over the fermi surface @xcite , while other experimental nmr and nqr results suggest the existence of the gap nodes @xcite . in particular , it has been argued that only involving the pairings of charge carriers located at the next nn sites can give rise to the nodal points of the complex gap appearing inside the brillouin zone @xcite . moreover , the nodal points of the complex gap has been obtained theoretically by considering the interaction between the hubbard fermions @xcite . however , although the recent experimental results @xcite obtained from the specific - heat measurements do not give unambiguous evidence for either the presence or absence of the nodes in the energy gap , the experimental data of the specific - heat @xcite are consistent with these fitted results obtained from phenomenological bcs formalism with the d - wave symmetry without gap nodes . furthermore , some theoretical calculations based on the numerical simulations indicate that the d - wave state without gap nodes is the lowest state around the electron - doped regime where superconductivity appears in triangular - lattice cobaltate superconductors @xcite . in particular , the recent theoretical studies based on a large - scale dynamical cluster quantum monte carlo simulation @xcite and a combined cluster calculation and renormalization group approach @xcite show that the d - wave state naturally explains some sc - state properties as indicated by experiments . in this case , we only consider the case with the d - wave pairing symmetry as our previous discussions @xcite , and then the d - wave charge - carrier pair gap @xmath29 in eq . ( [ bcsgf ] ) has been given in ref . @xcite . since the spin part in the @xmath8-@xmath9 model in the fermion - spin representation is anisotropic away from half - filling @xcite , two spin green s functions @xmath30 and @xmath31 have been defined to describe properly the spin part , and can be obtained explicitly as , [ sgf ] @xmath32 where the function @xmath33 and spin excitation spectrum @xmath34 in the spin green s function ( [ mfsgf ] ) have been given in ref . @xcite , while the function @xmath35 , and the spin excitation spectrum @xmath36 in the spin green s function ( [ mfsgfz ] ) is obtained as , @xmath37(\gamma_{\bf k } -1),\end{aligned}\ ] ] where @xmath38/3 $ ] , the parameters @xmath39 , @xmath40 , @xmath41 , the decoupling parameter @xmath42 , and the spin correlation function @xmath43 have been also given in ref . @xcite . in particular , the charge - carrier quasiparticle coherent weight @xmath24 , the charge - carrier pair gap parameter @xmath44 , all the other order parameters , and the decoupling parameter @xmath42 have been determined by the self - consistently calculation @xcite . in spite of the pairing mechanism driven by the kinetic energy by the exchange of spin excitations , the results in eq . ( [ bcsgf ] ) are the standard bcs expressions for a d - wave charge - carrier pair state . now we turn to evaluate the the internal energy of the triangular - lattice cobaltate superconductors . the internal energy in the charge - spin separation fermion - spin representation can be expressed as @xcite @xmath45 , where @xmath46 and @xmath47 are the corresponding contributions from charge carriers and spins , respectively , and can be obtained in terms of the charge - carrier spectral function @xmath48 , and the spin spectral functions @xmath49 and @xmath50 . following the previous work for the case in the square - lattice cuprate superconductors @xcite , it is straightforward to find the internal energy of the triangular - lattice cobaltate superconductors in the sc - state as , @xmath51\nonumber\\ & + & { z_{\rm af}\over n}\sum_{\bf k}\bar{\xi}_{\bf k } + 6j_{\rm eff}(\chi+\chi^{\rm z}),\end{aligned}\ ] ] where @xmath52 with the doping concentration @xmath53 , while the spin correlation function @xmath54 has been given in ref . @xcite . in the normal - state , the charge carrier pair gap @xmath55 , and in this case , the sc - state internal energy ( [ es ] ) can be reduced to the normal - state case as , @xmath56\nonumber\\ & + & { z_{\rm af}\over n}\sum_{\bf k}\bar{\xi}_{\bf k } + 6j_{\rm eff}(\chi+\chi^{\rm z}).\end{aligned}\ ] ] we are now ready to discuss the thermodynamic properties in the triangular - lattice cobaltate superconductors . the charge - carrier pair gap parameter @xmath44 is one of the characteristic parameters in the triangular - lattice cobaltate superconductors , which incorporates both the pairing force and charge - carrier pair order parameter , and therefore measures the strength of the binding of two charge carriers into a charge - carrier pair . in particular , the charge - carrier pair order parameter and the charge - carrier pair macroscopic wave functions in the triangular - lattice cobaltate superconductors are the same within the framework of the kinetic - energy - driven sc mechanism @xcite , i.e. , the charge - carrier pair order parameter is a _ magnified _ version of the charge - carrier pair macroscopic wave functions . for the convenience in the following discussions , we plot the charge - carrier pair gap parameter @xmath44 as a function of temperature at the doping concentration @xmath57 for parameter @xmath58 in fig . [ pair - gap - parameter - temp ] . it is shown clearly that the charge - carrier pair gap parameter follows qualitatively a bcs - type temperature dependence , i.e. , it decreases with increasing temperatures , and eventually vanishes at @xmath0 . for @xmath58 . [ pair - gap - parameter - temp ] ] for @xmath58 and @xmath59mev . the dashed line is obtained from a numerical fit @xmath60 $ ] , with @xmath61 and @xmath62 . inset : the corresponding experimental data of na@xmath3coo@xmath5@xmath63h@xmath5o taken from ref [ specific - heat ] ] one of the characteristics quantites in the thermodynamic properties is the specific - heat , which can be obtained by evaluating the temperature - derivative of the internal energy as , [ heat ] @xmath64 in the sc - state and normal - state , respectively , where @xmath65 and @xmath66 are the temperature dependence of the specific - heat coefficients in the sc - state and normal - state , respectively . in fig . [ specific - heat ] , we plot the specific - heat @xmath67 ( solid line ) as a function of temperature at @xmath68 for @xmath58 and @xmath59mev . for comparison , the corresponding experimental result @xcite of na@xmath3coo@xmath5@xmath63h@xmath5o is also shown in fig . [ specific - heat ] ( inset ) . apparently , the main feature of the specific - heat observed experimentally on the triangular - lattice cobaltate superconductors @xcite is qualitatively reproduced . as can be seen from fig . [ specific - heat ] , the specific - heat anomaly ( a jump ) at @xmath0 appears . the sc transition is reflected by a sharp peak in the specific - heat at @xmath0 , however , the magnitude of the specific - heat decreases dramatically with decreasing temperatures for the temperatures @xmath1 . moreover , the calculated result of the specific - heat difference @xmath69/c^{(\rm n)}_{\rm v}(t_{\rm c})=4.7 $ ] for the discontinuity in the specific - heat at @xmath0 , which is roughly consistent with the experimental data @xcite @xmath70 observed on na@xmath3coo@xmath5@xmath63h@xmath5o . for a better understanding of the physical properties of the specific - heat in the triangular - lattice superconductors , we have fitted our present theoretical result of the specific - heat for the temperatures @xmath1 , and the fitted result is also plotted in fig . [ specific - heat ] ( dashed line ) , where we found that @xmath71 varies exponentially as a function of temperature ( @xmath60 $ ] with @xmath61 and @xmath62 ) , which is an expected result in the case without the d - wave gap nodes at the charge - carrier fermi surface , and is in qualitative agreement with experimental data @xcite . however , this result in the triangular - lattice superconductors is much different from that in the square - lattice cuprate superconductors , where the characteristic feature is the existence of the gap nodes on the charge - carrier fermi surface , and then the specific - heat of the square - lattice cuprate superconductors decreases with decreasing temperatures as some power of the temperature for the temperatures @xmath1 . for @xmath58 and @xmath59mev . [ econd ] ] in the framework of the kinetic - energy - driven sc mechanism @xcite , the exchanged bosons are spin excitations that act like a bosonic glue to hold the charge - carrier pairs together , and then these charge - carrier pairs ( then electron pairs ) condense into the sc - state . as a consequence , the charge - carrier pairs in the triangular - lattice cobaltate superconductors are always related to lower the total free energy . the condensation energy @xmath72 on the other hand is defined as the energy difference between the normal - state free energy , extrapolated to zero temperature , and the sc - state free energy , @xmath73\nonumber\\ & -&[u^{(\rm s)}(t)-ts^{(\rm s)}(t)],\end{aligned}\ ] ] where the related entropy of the system is evaluated from the specific - heat coefficient in eq . ( [ heat ] ) as , @xmath74 where @xmath75 , @xmath76 referring to the sc - state and normal - state , respectively . we have made a calculation for the condensation energy ( [ condensation - energy ] ) , and the result of @xmath72 as a function of temperature at @xmath77 for @xmath58 and @xmath59mev is plotted in fig . [ econd ] . in comparison with the result of the temperature dependence of the charge - carrier pair gap parameter shown in fig . [ pair - gap - parameter - temp ] , we therefore find that in spite of the pairing mechanism driven by the kinetic energy by the exchange of spin excitations , the condensation energy of the triangular - lattice cobaltate superconductors follows qualitatively a bcs type temperature dependence . for @xmath58 and @xmath59mev . [ bc - doping ] ] a quantity which is directly related to the condensation energy @xmath72 in eq . ( [ condensation - energy ] ) is the upper critical field @xmath78 , @xmath79 this upper critical field @xmath78 is a fundamental parameter whose variation as a function of doping and temperature provides important information crucial to understanding the details of the sc - state . in fig . [ bc - doping ] , we plot the upper critical field @xmath78 as a function of doping with @xmath80 for @xmath58 and @xmath59mev . it is shown clearly that the upper critical field takes a dome - shaped doping dependence with the underdoped and overdoped regimes on each side of the optimal doping , where @xmath78 reaches its maximum . moreover , the calculated upper critical field at the optimal doping is @xmath81 , which is not too far from the range @xmath82 estimated experimentally for different samples of na@xmath3coo@xmath4h@xmath5o @xcite . for a superconductor , the upper critical field is defined as the critical magnetic field that destroys the sc - state at zero temperature , which therefore means that the upper critical field also measures the strength of the binding of charge carriers into the charge - carrier pairs . in this case , the domelike shape of the doping dependence of @xmath78 is a natural consequence of the domelike shape of the doping dependence of @xmath44 and @xmath0 as shown in ref . @xcite . to further understand the intrinsic property of the upper critical field @xmath78 in the triangular - lattice cobaltate superconductors , we have also performed a calculation for @xmath78 at different temperatures , and the result of @xmath78 as a function of temperature at @xmath77 for @xmath58 and @xmath59mev is plotted in fig . [ bc - temp ] in comparison with the corresponding experimental result @xcite of na@xmath3coo@xmath4h@xmath5o ( inset ) . it is thus shown that @xmath78 varies moderately with initial slope . in particular , as in the case of the temperature dependence of the condensation energy shown in fig . [ econd ] , the upper critical field @xmath78 also follows qualitatively the bcs type temperature dependence , i.e. , it decreases with increasing temperature , and vanishes at @xmath0 , which is also qualitatively consistent with the experimental results @xcite . for @xmath58 and @xmath59mev . insets : the corresponding experimental data of na@xmath3coo@xmath5@xmath63h@xmath5o taken from ref . [ bc - temp ] ] the coherence length @xmath83 also is one of the basic sc parameters of the triangular - lattice cobaltate superconductors , and is directly associated with the upper critical field as @xmath84 $ ] , where @xmath85 is the magnetic flux quantum . in fig . [ coherence ] , we plot the coherence length @xmath83 as a function of doping with @xmath80 for @xmath58 and @xmath59mev . since the coherence length @xmath83 is inversely proportional to the upper critical field @xmath78 , the coherence length @xmath83 in the triangular - lattice cobaltate superconductors reaches a minimum around the optimal doping , then grows in both the underdoped and overdoped regimes . in particular , at the optimal doping , the anticipated coherence length @xmath86 nm approximately matches the coherence length @xmath87 nm observed in the optimally doped na@xmath3coo@xmath5@xmath63h@xmath5o @xcite . this coherence length @xmath86 nm at the optimal doping estimated from the upper critical field using the ginzburg - landau expression also is qualitatively consistent with that obtained based on the microscopic calculation @xmath88 nm , where @xmath89 is the charge carrier velocity at the fermi surface . this relatively short coherence length is surprising for a superconductor with such a low @xmath0 , but is consistent with the narrow bandwidth in the triangular - lattice superconductors @xcite , since the charge - carrier quasiparticle spectrum @xmath90 in the full charge - carrier diagonal green s function ( [ bcsdgf ] ) and off - diagonal green s function ( [ bcsodgf ] ) has a narrow bandwidth @xmath91 . for @xmath58 and @xmath59mev . [ coherence ] ] within the framework of the kinetic - energy - driven sc mechanism , we have discussed the doping dependence of the thermodynamic properties in the triangular - lattice cobaltate superconductors . we show that the specific - heat anomaly ( a jump ) appears at @xmath0 , and then the specific - heat varies exponentially as a function of temperature for the temperatures @xmath1 due to the absence of the d - wave gap nodes at the charge - carrier fermi surface , which is much different from that in the square - lattice cuprate superconductors @xcite , where the characteristic feature is the existence of the gap nodes on the charge - carrier fermi surface , and then the specific - heat of the square - lattice cuprate superconductors decreases with decreasing temperatures as some power of the temperature in the temperature range @xmath1 . on the other hand , both the condensation energy and the upper critical field in the triangular - lattice cobaltate superconductors follow qualitatively the bcs type temperature dependence . in particular , in analogy to the dome - shaped doping dependence of @xmath0 , the maximal upper critical field occurs around the optimal doping , and then decreases in both underdoped and overdoped regimes . incorporating the present result @xcite with that obtained in the square - lattice cuprate superconductors , it is thus shown that the dome - shaped doping dependence of the upper critical field is a universal feature in a doped mott insulator , and it does not depend on the details of the geometrical spin frustration . since the knowledge of the thermodynamic properties in the triangular - lattice cobaltate superconductors is of considerable importance as a test for theories of superconductivity , the qualitative agreement between the present theoretical results and experimental data also provides an important confirmation of the nature of the sc phase of the triangular - lattice cobaltate superconductors as a conventional bcs - like with the d - wave symmetry , although the pair mechanism is driven by the kinetic energy by the exchange of spin excitations . yoshihiko ihara , kenji ishida , hideo takeya , chishiro michioka , masaki kato , yutaka itoh , kazuyoshi yoshimura , kazunori takada , takayoshi sasaki , hiroya sakurai , and eiji takayama - muromachi , j. phys . . jpn . * 75 * , 013708 ( 2006 ) .

the study of superconductivity arising from doping a mott insulator has become a central issue in the area of superconductivity . within the framework of the kinetic - energy - driven superconducting mechanism , we discuss the thermodynamic properties in triangular - lattice superconductors . it is shown that a sharp peak in the specific - heat appears at the superconducting transition temperature @xmath0 , and then the specific - heat varies exponentially as a function of temperature for the temperatures @xmath1 due to the absence of the d - wave gap nodes at the charge - carrier fermi surface . in particular , the upper critical field follows qualitatively the bardeen - cooper - schrieffer type temperature dependence , and has the same dome - shaped doping dependence as @xmath2 .

many samples of uv ceti type stars posses the stellar spot activity known as by dra syndrome . the by dra syndrome among the uv ceti stars was first found by @xcite . he reported the existence of sinusoidal - like variations at out - of - eclipses of the eclipsing binary star yy gem . @xcite explained this sinusoidal - like variation at out - of - eclipse as a heterogeneous temperature on star surface , which was called by dra syndrome by @xcite . this interpretation of by dra syndrome in terms of dark regions of the surface of rotating stars was confirmed , based on more rigorous arguments , by later works of @xcite . since the most of solar flares occur over solar spot regions , in the stellar case it is also expected to find a correlation between the frequency of flares and the effects caused by spots in the light curve . in order to determine a similar relation among the stars , lots of studies have been made using uv ceti type stars showing stellar spot activity such as by dra . one of these studies was reported by @xcite on yy gem . @xcite did not find a clear correlation between the location and extent in longitude of flares and spots on yy gem . moreover , he notes that the longitudinal extent he derived for a flare - producing region is in good agreement with the longitudinal extents of starspots previously calculated for by dra and cc eri by @xcite . in another study , @xcite compared longitudes of the stellar spots obtained from two years observations of yz cmi and ev lac with longitudes of flare events and distributions of flare energy and frequencies along the longitude obtained in the same work . direct comparisons and statistical tests are not able to reveal positive relationships between flare frequency or flare energy and the position of the spotted region . in another extended work , @xcite looked for whether there is any relation between stellar spots and flares observed from 1967 to 1977 in the observations of ev lac . the authors were able to find a relation in the year 1970 . they could not find any relation in other observing seasons because of the higher threshold of the system used for flare detection . in the last years , @xcite found some flares occurring in the same active area with other activity patterns with using simultaneous observations . since no correlation is found between stellar spots and flares , a hypothesis about fast and slow flares was put forward . the hypothesis is based on the work named as fast electron hypothesis . according to this hypothesis , the shape of a flare light variation depends on the location of the event on the star surface in respect to direction of observer . if the flaring area is on the front side of the star according to the observer , the light variation shape looks as a fast flare . if the flaring area is on the opposite side of the star according to the observer , the light variation shape looks as a slow flare @xcite . in addition , @xcite described two types of flares to model flare light curves . @xcite indicated that thermal processes are dominant in the processes of slow flares , which are 95@xmath0 of all flares observed in uv ceti type stars . non - thermal processes are dominant in the processes of fast flares , which are classified as `` other '' flares . according to @xcite , there is a large energy difference between these two types of flares . moreover , @xcite developed a rule to the classifying of fast and slow flares . when the ratios of flare decay times to flare rise times are computed for two types of flares , the ratios never exceed 3.5 for all slow flares . on the other hand , the ratios are always above 3.5 for fast flares . it means that if the decay time of a flare is 3.5 times longer than its rise time at least , the flare is a fast flare . if not , the flare is a slow flare . in this paper , the results obtained from johnson ubvr observations of ad leo , ev lac and v1005 ori will be discussed . @xcite and @xcite reported that v1005 ori is a flare star that exhibits rotational modulation due to stellar spots . the authors found an amplitude variation of @xmath1 with a period of @xmath2 in v band . besides , @xcite examined photometric data in the time series analyses and found 4 periods for rotational modulation . the period of @xmath3 is suggested as the most probable period among them . on contrary , in b band observations of 1981 , a @xmath4 day period variation with an amplitude of @xmath5 was found @xcite . no important light curve changes are seen in the years 1996 and 1997 , while the minimum phases of rotation modulation is varied from @xmath6 to @xmath7 . the amplitude of the curves is @xmath8 in the year 1996 , but in the year 1997 it gets larger than the previous ones @xcite . in the case of ad leo , it is a debate issue whether ad leo has any stellar spot activity , or not . @xcite and @xcite show that ad leo does not exhibit any rotational modulation caused by stellar spots . besides , @xcite found no variations at the @xmath9 magnitude level during the period 1978 , may 10 to 17 . however , @xcite reveals that ad leo demonstrates by dra syndrome with a period of @xmath10 days . in addition , @xcite confirmed this period of ad leo for by dra variation . on the other hand , ev lac is a well known active star with both high level flare and stellar spot activities . @xcite indicated that the star has no variation caused by rotational modulation in the observations in b band from 1972 to 1976 . however , @xcite showed a rotational modulation with a period of @xmath11 and an amplitude of @xmath12 . @xcite , based on continuous observations from 1979 to 1981 , renewed the ephemerides of the variation as a period of @xmath13 and an amplitude of @xmath1 . this indicates that the light curves of ev lac were almost constant for 2.5 years because the spot groups on the star are stable during these 2.5 years . using the renewed ephemerides , @xcite found the amplitude of light curve enlarging from @xmath1 to @xmath5 in the year 1986 . on the other hand , no variation was seen in the light curve of the year 1987 . @xcite showed that the spotted area is located in the same semi - sphere on ev lac for 10 years . comparing the phases of the light curve minima caused by rotational modulation with the flare frequencies and the distribution of the flare equivalent durations for yz cmi and ev lac , @xcite showed that there is no relation between the flare activity and stellar spot activity on these stars . eq peg is classified as a metal - rich star and it is a member of the young disk population in the galaxy @xcite . eq peg is a visual binary @xcite . both components are flare stars @xcite . angular distance between components is given as a value between 3@xmath14.5 and 5@xmath14.2 @xcite . one of the components is 10.4 mag and the other is 12.6 mag in v band @xcite . observations show that flares on eq peg generally come from the fainter component @xcite . @xcite proved that 65@xmath0 of the flares come from faint component and about 35@xmath0 from the brighter component . the fourth star in this study is v1054 oph , whose flare activity was discovered by @xcite . @xcite demonstrated that eq peg has a variability with the period of @xmath15 . v1054 oph (= wolf 630abab , gliese 644abab ) is a member of wolf star group @xcite . wolf 630abab , wolf 629ab (= gliese 643ab ) and vb8 (= gliese 644c ) , are the members of the main triplet system , whose scheme is shown in fig.1 given by @xcite . the masses were derived for each components of wolf 630abab by @xcite . the author showed that the masses are 0.41 @xmath16 for wolf 629a , 0.336 @xmath16 for wolf 630ba and 0.304 @xmath16 for wolf 630bb . in addition , @xcite demonstrated that the age of the system is about 5 gyr . in this study , for each program stars , we analyse the variations at out - of - flare for each light curves obtained in johnson ubvr observations , or not . although all of them show high flare activity , ev lac , v1005 ori and eq peg exhibit stellar spot activity . on the other hand , the spot activity is not obvious for ad leo . it is discussed whether ad leo has any stellar spot activity , or not . finally , this work do not demonstrate any variation from rotational modulations . to perform this kind of studies we would require a long term observing program . as a part of this study , the phase distributions for both fast and slow flares are examined in terms of the minimum phases of rotational modulation . thus , hypothesis developed by @xcite is tested . the observations were acquired with a high - speed three channel photometer attached to the 48 cm cassegrain type telescope at ege university observatory . observations were grouped in two schedules . using a tracking star in second channel of the photometer , flare observations were only continued in standard johnson u band with exposure times between 2 and 10 seconds . the same comparison stars were used for all observations . the second observation schedule was used for determining whether there was any variation out - of - flare . pausing flare patrol of program stars , we observed them once or twice a night , when they were close to the celestial meridian . using a tracking star in second channel of the photometer , the observations in this schedule were made with the exposure time of 10 seconds in each band of standard johnson ubvr system , respectively . there were any delay between the exposure in different filters due to the high - speed three channel photometer . although the program and comparison stars are so close on the sky , differential atmospheric extinction corrections were applied . the atmospheric extinction coefficients were obtained from the observations of the comparison stars on each night . moreover , the comparison stars were observed with the standard stars in their vicinity and the reduced differential magnitudes , in the sense variable minus comparison , were transformed to the standard system using procedures outlined by @xcite . the standard stars are listed in the catalogues of @xcite and @xcite . and also , the de - reddened colour of the systems were computed . heliocentric corrections were also applied to the times of observations . the mean averages of the standard deviations are @xmath17 , @xmath18 , @xmath19 and @xmath19 for the observations acquired in standard johnson ubvr bands , respectively . to compute the standard deviations of observations , we use the standard deviations of the reduced differential magnitudes in the sense comparisons ( c1 ) minus check ( c2 ) stars for each night . there is no variation in the standard brightness comparison stars . [ cols="<,^,^,^",options="header " , ] ( 130mm,60mm)figure12.ps all the flares of ad leo detected in three seasons were again combined for this analysis . the same histograms were derived for both the fast and slow flares of ad leo . they are shown in figure 12 . as it is seen from the analyses of the histogram in the figure , the phase of mfor is @xmath20 , while it is @xmath21 for the slow flares . there is a difference of @xmath22 between two types . according to @xcite , it is expected that there should be a difference of @xmath23 between them . ( 120mm,60mm)figure13.ps in the case of ev lac , it was seen that the phase distribution of the fast flares is not enough to compare it with slow flares for the season 2005 . this is must be because the frequency of the fast flares is not as high as that of the slow flares , as mentioned by @xcite . we only compared them for the season 2004 and 2006 . all histograms of ev lac are shown in figure 13 . the phase of mfor for the fast flares is @xmath24 , while it is @xmath25 for the slow flares in the season 2004 . the difference between the phases of mfor is @xmath26 for two types in this season . the phase of mfor is @xmath27 for the fast flares , while it is @xmath28 for the slow flares in the season 2006 . the difference between the phases of mfor is @xmath29 for both types in the season 2006 , as expected . ( 130mm,60mm)figure14.ps in the case of v1005 ori , comparison could be done for the season 2005/2006 . the histograms of v1005 ori are shown in figure 14 . as it is seen from the analyses of the histogram in the figure , the phase of mfor for the fast flares is @xmath30 , while it is @xmath31 for the slow flares . there is a difference of @xmath32 between two types . ( 130mm,60mm)figure15.ps the same comparison was done for the seasons 2004 and 2005 for eq peq . the histograms of eq peg are shown in figure 15 . as it is seen from the analyses of the histogram in the figure , the phase of mfor for the fast flares is about @xmath33 , while it is @xmath7 for the slow flares . there is a difference of @xmath6 between two types . this value is an acceptable value and close to the expected value according to the hypothesis discussed by @xcite . most of the uv ceti type stars are full convective red dwarfs with sudden - high energy emitting . as it can be seen in the literature , by dra syndrome at out - of - flares is seen in a few stars among 463 flare stars catalogued by @xcite . ev lac and v1005 ori can be given as two examples because the studies in the literature and this study indicate that both stars show the variation due to rotational modulation at out - of - flares . in the case of ev lac , the time series analyses show that the period of rotational modulation found for each data set is range from @xmath34 to @xmath11 . the periods found are similar to those found by @xcite and @xcite . although the periods found for each season are a little bit different , this difference is relatively small . when the amplitudes of the light curves are examined for ev lac , the amplitude of this variation was dramatically decreasing from the year 2004 to 2005 , while the amplitude was clearly larger than ever in this study . however , the mean average of brightness in the light curves was slowly decreasing from the year 2004 to 2006 . the minima phases of the light curves for the three seasons were computed and , it was found as @xmath35 for the season 2004 , @xmath36 for 2005 and @xmath37 for 2006 . in the case of v1005 ori , the periods of the rotational modulation for each season are range from @xmath38 to @xmath39 . in the literature , @xcite found four possible periods varied from @xmath40 to @xmath41 . on the other hand , @xcite found a period of @xmath42 . as it is seen , the periods found in this study are close to the period found by @xcite . when the amplitudes of the light curves were examined , the amplitude observed in the season of 2004/2005 was so smaller than the ones observed in the previous and later seasons that there was no minimum in the light curve . although the mean average of brightness in the light curves was not changing , the minimum phases of the light curves were varying . the minimum phase of the curve for the season 2004/2005 was about @xmath43 and about @xmath44 for the season 2006/2007 . it is hard to say that the minimum phase of the light curves for the season 2005/2006 was about @xmath45 . the case of ad leo is different from the other two stars . the time series analyses do not show any regular variation over the @xmath46 level in one season . on the other hand , the mean brightness levels were increased a value of @xmath47 from the first season to the second and a value of @xmath48 from the second to the last season . this can be because of the stellar polar spots . if the literature is considered , the stellar spots can be carried to polar regions in the case of rapid rotation in the young stars @xcite . according to @xcite , ad leo is at the age of 200 myr . the range of equatorial rotational velocity ( @xmath49 ) given in the literature is between @xmath50 - @xmath51 and @xmath52 @xmath53 for ad leo @xcite . besides , considering these values of @xmath49 , the real rotational velocities must be larger than these values . if both the age and equatorial rotational velocity value parameters in these papers are considered , according to @xcite and @xcite , some spots might be located on the polars for ad leo . in fact , @xcite indicate that by dra had spotted area near polar region , which was stable for 14 years and ev lac has a similar area for 10 years . if the studies made by @xcite and @xcite are considered , ad leo might sometimes show rotational modulation due to the spotted area occurring near the equatorial regions . on the other hand , there is another probability . if the colour index of v - r is considered , it is seen that the star gets bluer from a season to next one , when the star get brighter . besides , no amplitude is seen in the light curves . these can be some indicators that all the surface of the star is covered by cool spots and the efficiency of the spots gets weaker from one season to next one . the colour curves of both ev lac and v1005 ori sometimes exhibit a clear colour excess around the minimum phases of the light curves for some observing seasons . this can be an indicator of some bright areas such as faculae on the surface of these young stars . the effects of the bright areas such as faculae can be seen in the variations of b - v and sometimes v - r colour , while these effects are not seen in the variations in the light curves of bvr bands due to cool spots . the cool spots are more efficient in the light curves of b , especially v and r bands . the same effect is seen in the variations caused by the flare activity . although there is some small effects or no effect of the flare activity in v and r bands , but there is some clear variations in u band light and u - b colour curves . in this study , we observed the stars in u band to investigate the variations out - of - flares . in a sense , u band observations is used to control whether there is any flare activity in the observing durations . if there is some variations in u band light and u - b colour , we did not used the observation to investigate the variation out - of - flares . @xcite showed that eq peg has a variability with the period of @xmath15 . in this study , the time series analyses supported this period . according to our analyses , eq peg exhibits short - term variability with the period of @xmath54 . however , it is seen that there is not any variability in the colour indexes . analysing the light curve of eq peg , it was found as @xmath55 for the minimum phase of the rotational modulation . there are many studies about whether the flares of uv ceti type stars showing by dra syndrome are occurring at the same longitudes of stellar spots , or not . having the same longitudes of flare and spots is an expected case for these stars , because solar flares are mostly occurring in the active regions , where spots are located on the sun @xcite . in the respect of stellar - solar connection , a result of the @xmath56 @xmath57 @xmath58 project of mount wilson observatory @xcite , if the areas of flares and spots are related on the sun , the same case might be expected for the stars . in fact , @xcite have found some evidence to demonstrate this relations . besides , @xcite have found a variations of both the rotational modulation and the phase distribution of flare occurence rates in the same way for the observations in the year 1970 . on the other hand , no clear relation between stellar flares and spots has been found by @xcite . however , @xcite did not draw firm conclusions because of being a non - uniqueness problem . in this study , the flare occurence rates , the ratio of flare number to monitoring time , were computed in intervals of 0.10 phase length as the same method used by @xcite with just one difference . the flare maximum times were used to compute the phases due to main energy emitting in this part of the flare light curves . we observed ad leo for 79.61 @xmath59 and detected 119 flares in three seasons . ev lac was observed 109.63 @xmath59 and 93 flares were detected in three seasons . v1005 ori was observed for 44.75 @xmath59 and 44 flares were detected in two seasons . eq peg was observed for 100.26 @xmath59 and 73 u band flare were detected . since no rotational modulation was found to compare for ad leo , all the flares detected in three season were combined in order to just find whether there is any phase , in which the flare occurence rate gets a peak . on the other hand , we examined flare phase distributions for each season for both ev lac and v1005 ori . in the case of these stars , if the distribution of flares did not cover almost all phases in an observing season of a star , the season is neglected for the comparison of flare and spot activity . consequently , for both ev lac and v1005 ori , we chose the seasons , in which the best flare distributions were obtained . thus , we only used the seasons , in which there is enough data to get reliable conclusions about flare occurrence distributions . in addition , to determine the phases of mfor , all the distributions were modelled with the polynomial function . resolving these models , maximum flare occurrence rates and their phase were found for all program stars . in the case of ev lac , no relation is seen between the minimum phase of the rotational modulation and the phase , in which flare activity reaches the mfor . the minimum phase of the rotational modulation observed in the season 2004 is @xmath35 , while the phase of mfor is @xmath60 . the minimum phase of rotational modulation is @xmath36 , while the flare occurrence rate reaches maximum level in about the phase of @xmath24 for the season 2005 . in the last season of ev lac , rotational modulation minimum is seen in @xmath37 , as mfor is in @xmath61 . in the case of v1005 ori , there is enough data in only one season to compare . as it is seen , the minimum phase of rotational modulation is @xmath45 , while phase of mfor is about @xmath27 for the season 2005/2006 . in this study , the time series analyses indicated that ad leo does not have any rotational modulation . therefore , any minimum time could not have been determined from the observations of three seasons for ad leo . because of this , we could not compare the rotational modulation with flare activity in the case of ad leo . on the other hand , using combined data of three seasons , we found that the mfor is seen in @xmath24 . this phases was computed with using the ephemeris given in equation ( 1 ) taken from @xcite . the time series analyses do not show any short - term variation in the light curves of ad leo . because of this , we waited that there is no any phase , in which the flare activity gets higher levels . on the other hand , as it is seen from the histogram and its normal gaussian model for ad leo , there is a phase for mfor . considering the phase of mfor , the active region(s ) in some particular part of the surface can be more active than the others on the surface of the star . considering the light and colour curves of ad leo , almost all surface of the star may be covered by stellar spots , while it is seen that some region(s ) in the surface of the star can be more active than the remainder of the surface . in the case of eq peg , the minimum phase of the rotational modulation is @xmath55 , while the phase of mfor is @xmath31 . the results acquired from ev lac and v1005 ori demonstrated that flare activity can reach high levels at almost the same longitudes , in which stellar spots occur . on the other hand , there is a considerable difference between the phases of stellar spot and mfor for the observing season 2007 of ev lac . in conclusion , it is seen that there is a longitudinal relation between stellar spot and flare activities in general manner . nevertheless , there are some differences and this makes difficult to do a definite conclusion . moreover , in the case of eq peg , the mfor gets the minimum towards the minimum phase of the rotational modulation . all these cases can be because of a dynamo which is working in the red dwarf stars . in spite of the sun , red dwarf stars are mostly known to have a different dynamo because of full convective outer atmosphere . however , in the last years , some studies showed that flares on the sun do not have to be located upon the spotted areas on the sun @xcite . in addition , it should be kept in mind that most of the studies have been done with using the data obtained from white - light flare observations , but a white - light flare does not have to occur in a flare process . recent studies have shown that non white - light flares may be so common in uv ceti - type stars as they are in the sun @xcite . in this point , it can be mentioned that the analyses of data obtained from only white - light flare observations are not sufficiently qualified . for instance , @xcite found some flares occurring in the same active area with other activity patterns with using simultaneous observations . using the inverse compton event , @xcite developed a hypothesis called fast electron hypothesis , in which red dwarfs generate only fast flares on their surface . on the other hand , according to the flare region on the surface of the star in respect to direction of observer , the shapes of the flare light variations can be seen like a slow flare @xcite . if the scenario in this hypothesis is working , it is expected that the fast and slow flares should collected into two phases in the light curves of uv ceti type stars showing by dra syndrome . it is also expected that these two phases are separated from each other with intervals of @xmath23 in phase . in this study , according to the rule described by @xcite , the flares are classified as fast and slow flares . then the phase distributions of fast flares were compared with the phases of slow flares in order to find out whether there is any separation as expected in this respect . when the phases of both fast and slow flares are examined one by one , it is clear that both of them can occur in any phase . to reach a definite result , the phase distributions of both fast and slow flares are statistically investigated . as it is stated in the previous section , if the distribution of flares did not cover almost all phases in an observing season of a star , the season is neglected for that star . consequently , we chose the seasons , in which there is enough data to get reliable conclusions about flare occurrence distributions for both fast and slow flares . in the case of ad leo and eq peg , we combined all the fast flares of three seasons as we made for the slow flares . for both fast and slow flares , using equation ( 5 ) , the number of flares occurring per an hour in intervals of 0.10 phase length was computed . the obtained occurrence rates for both fast and slow flares are shown by histograms in figures 12 , 13 , 14 and 15 . once again , all the distributions were modelled with the polynomial function . resolving these models , maximum flare occurrence rates and their phase of both slow and fast flares were found for all program stars . in the case of ad leo , the analyses show that both fast and slow flares have a difference of @xmath22 between the phases , in which flare occurrence rates in intervals of 0.10 phase length reach maximum amplitudes . the same difference is @xmath62 for ev lac in the season of 2004 . although these differences are acceptable as low values according to fast electron hypothesis , the difference seen in the season of 2006 is @xmath23 for ev lac . this value is the expected value in respect of fast electron hypothesis . in the case of v1005 ori , slow and fast flares could be compared only for the season of 2005/2006 . the result is that both fast and slow flares have a difference of @xmath63 between the phases of maximum flare occurrence rates . in the case of eq peg , the phase difference between mfors of slow and fast flares is about @xmath6 . the value obtained from eq peg is also the expected value in respect of fast electron hypothesis . it should be noted that in the case of eq peg , it is seen just one clear peak for the distribution of mfor for the fast flares , while there are several peaks for the slow flares . as it is seen from the analyses , both the fast and the slow flares sometimes the same longitudinal distributions and sometimes different . this makes difficult to say that there is a regular longitudinal division between these two types of flares as expected according to @xcite . this means that , when a slow flare is observed , it does not have to be a fast flare occurred on the opposite side of the star in respect to observer direction . the authors acknowledge generous allotments of observing time at the ege university observatory . we thank both dr . hayal boyaciolu , who gave us important suggestions about statistical analyses , and professor m. can akan , who gave us valuable suggestions that improved the language of the paper . we also thank the referee for useful comments that have contributed to the improvement of the paper . we finally thank the ege university research found council for supporting this work through grant no . 2005/fen/051 . 63 amado , p. j. , zboril , m. , butler , c. j. & byrne , p. b. , 2001 , coska , 31 , 13 anderson , c. m. , 1979 , , 91 , 202 baliunas , s.l . , donahue , r.a . , soon , w.h . , horne , j.h . , frazer , j. , woodard - 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in this study , we discuss stellar spots , stellar flares and also the relation between these two magnetic proccess that take place on uv ceti stars . in addition , the hypothesis about slow flares described by @xcite will be discussed . all these discussions are based on the results of three years of observations of the uv ceti type stars ad leo , ev lac , v1005 ori , eq peg and v1054 oph . first of all , the results show that the stellar spot activity occurs on the stellar surface of ev lac , v1005 ori and eq peg , while ad leo does not show any short - term variability and v1054 oph does not exhibits any variability . we report new ephemerides , for ev lac , v1005 ori and eq peg , obtained from the time series analyses . the phases , computed in intervals of 0.10 phase length , where the mean flare occurence rates get maximum amplitude , and the phases of rotational modulation were compared to investigate whether there is any longitudinal relation between stellar flares and spots . although , the results show that flare events are related with spotted areas on the stellar surfaces in some of the observing seasons , we did not find any clear correlation among them . finally , it is tested whether slow flares are the fast flares occurring on the opposite side of the stars according to the direction of the observers as mentioned in the hypothesis developed by @xcite . the flare occurence rates reveal that both slow and fast flares can occur in any rotational phases . the flare occurence rates of both fast and slow flares are varying in the same way along the longitudes for all program stars . these results are not expected based on the case mentioned in the hypothesis .

recently five - dimensional ( 5d ) supergravity ( sugra ) on the orbifold @xmath1 has been studied as an interesting theoretical framework for physics beyond the sm . it has been noted that 5d orbifold sugra with a @xmath0 symmetry gauged by the @xmath2-odd graviphoton can provide the supersymmetric randall - sundrum ( rs ) model @xcite in which the weak to planck scale hierarchy can arise naturally from the geometric localization of 4d graviton @xcite , and/or yukawa hierarchy can be generated by the quasi - localization of the matter zero modes in extra dimension where we generically have an interesting correlation between the flavor structure in the sparticle spectra and the hierarchical yukawa couplings @xcite . in the former case , the bulk cosmological constant and brane tensions which are required to generate the necessary ads@xmath4 geometry appear in the lagrangian as a consequence of the @xmath0 fi term with @xmath2-odd coefficient . in this talk we consider a more generic orbifold sugra which contains a @xmath2-even 5d gauge field @xmath5 participating in the @xmath0 gauging @xcite . if 4d @xmath6 susy is preserved by the compactification , the 4d effective theory of such model will contain a gauged @xmath0 symmetry associated with the zero mode of @xmath5 , which is not the case when the 5d @xmath0 is gauged only through the @xmath2-odd graviphoton . based on the known off - shell formulation @xcite , we formulate a gauged @xmath0 sugra on @xmath1 in which both @xmath5 and the graviphoton take part in the @xmath0 gauging and then analyze the structure of fi terms allowed in such model . as expected , introducing a @xmath2-even @xmath0 gauge field accompanies new bulk and boundary fi terms in addition to the known integrable boundary fi term which could be present in the absence of any gauged @xmath0 symmetry @xcite . as we will see , those new fi terms can have interesting implications to the quasi - localization of the matter zero modes in extra dimension and the susy breaking @xcite and also to the radion stabilization . for a minimal setup , we introduce two vector multiplets and two hypermultiplets in the off - shell formulation of 5d ( conformal ) sugra @xcite : @xmath7 and @xmath8 with the norm function @xmath9 and the hypermultiplet gauging @xmath10 where we adopt the @xmath11 matrix notations omitting @xmath12 index and @xmath13 indices @xmath14 , and the hyperscalars satisfy the reality condition @xmath15 , @xmath16 . the @xmath2-even bosonic ( non - auxiliary ) components are @xmath17 , @xmath18 , @xmath19 , @xmath20 and @xmath21 , and @xmath22 , @xmath23 are the graviphoton vector multiplet and the compensator hypermultiplet respectively . the @xmath2-odd coefficient @xmath24 in the hypermultiplet gauging is consistently introduced by the mechanism proposed in @xcite . the nonzero value of the charge @xmath25 corresponds to the @xmath0 symmetry gauged by @xmath2-even vector field @xmath19 . the bosonic part of the lagrangian is given by @xmath26 @xmath27 \nonumber \\ & & -{\textstyle \frac{1}{2 } } { \rm tr}\big [ { \cal n}_{ij } y^{i\dagger}y^j -4y^{i\dagger } \big ( { \cal a}^\dagger { t}_i { \cal a } -\phi^\dagger { t}_i \phi \big)\big ] , \nonumber \\ % % e_{_{(4)}}^{-1}{\cal l}_{\partial \epsilon } & = & -2\alpha \big ( 3k + { \textstyle \frac{3}{2}}k\ , { \rm tr } \left [ \phi^\dagger \phi \right ] + c\ , { \rm tr } \left [ \phi^\dagger \sigma_3 \phi \sigma_3 \right ] \big ) \nonumber \\ & & \qquad \times \left ( \delta(y)-\delta(y-\pi r ) \right ) , \nonumber \\ e_{_{(4)}}^{-1 } { \cal l}_{n=1 } & = & m_{_{(4)}}^2 \big [ \ , -2r \big(\ , 2y^{x(3)}-e^{-1}e_{_{(4)}}\partial_y\beta \ , \big ) -{\textstyle \frac{1}{2}}r^{(4 ) } \big ] \nonumber \\ & & \qquad \times \left ( \lambda_0 \delta(y ) + \lambda_\pi \delta(y-\pi r ) \right ) , \nonumber\end{aligned}\ ] ] where the matrix notations are employed again , @xmath28 , @xmath29 , @xmath30 \big)^{2/3}$ ] and @xmath31 . here we have included only 4d @xmath6 _ pure _ sugra action at the orbifold fixed points without any khler and superpotentials for simplicity . we remark that after the superconformal gauge fixing , @xmath32/2 } , \nonumber\end{aligned}\ ] ] we find the bulk fi term @xmath33 in @xmath34 and the boundary fi term @xmath35 in @xmath36 for the auxiliary fields @xmath37 in the vector multiplets . we are interested in the 4d poincar invariant background geometry , @xmath38 and the gravitino- , hyperino- and gaugino - killing parameters on this background are given respectively by @xmath39 where @xmath40 , \nonumber\end{aligned}\ ] ] and @xmath41 is the physical gauge scalar field parameterizing the ( very special ) manifold of vector multiplet determined by @xmath42 with the metric @xmath43 . we choose @xmath44 and @xmath45 in the following . the real and diagonal component of the quaternionic hyperscalar field @xmath46 is represented by @xmath47 in the killing parameters , and zero vacuum values are assumed for the other components for simplicity . in terms of these killing parameters , the 4d energy density is found to be @xmath48 and it is obvious that the killing condition @xmath49 determines a stationary point of the 4d scalar potential if the solution exists . now we examine some physical consequences of the 5d gauged @xmath0 supergravity on @xmath1 which can have the bulk and the boundary fi term , for the supersymmetric vacuum configurations , @xmath49 . first we consider the case that we have a charged hypermultiplet @xmath46 with the charge satisfying @xmath50 . for @xmath51 that results in @xmath52 , the vacuum values of the scalar fields are given by @xmath53 for @xmath54 , and @xmath55 for @xmath56 , where @xmath57 and @xmath58 . we find a nontrivial @xmath59-dependent vacuum values for the latter case due to the boundary fi term . notice that the vacuum value of the gauge scalar @xmath60 gives the @xmath59-dependent mass for the charged hypermultiplets which results in nontrivial zero - mode wavefunctions for them . we will show the zero - mode profile in the next more simple but interesting case . next we consider the case there are charged chiral multiplets @xmath61 with the charge @xmath62 at the orbifold fixed points @xmath63 respectively , but no hypermultiplets with the charge @xmath50 in bulk . we introduce minimal khler potential and no superpotential for them at the fixed points . for @xmath51 , the vacuum values of the scalar fields are given by @xmath64 where @xmath65 . we find a linear profile of @xmath41 in the @xmath59-direction due to the bulk fi term , which results in the gaussian form of the zero - mode wavefunction for the charged hypermultiplet , @xmath66 the ratio of the wavefunction values between two fixed points are then shown to be @xmath67 . some numerical plots are shown in fig . [ fig:1 ] for @xmath68 but @xmath69 and in fig . [ fig:2 ] for both @xmath70 . from these figures we find that the nonvanishing @xmath25 ( i.e. , gauging @xmath0 by @xmath2-even vector field ) as well as the bare kink mass @xmath71 affects the zero - mode profiles of the charged hypermultiplets significantly . the nonvanishing charge @xmath72 changes the linear profile of @xmath41 resulting in a more / less severe localization of the charged hypermultiplet zero - mode , depending on the sign of @xmath73 . @xmath60 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 @xmath60 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 + @xmath81 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 @xmath81 + and the matter zero mode @xmath74 for some cases with @xmath69 and @xmath75 . here we choose @xmath76 . for the matter zero mode profile , the solid- , dotted- and dashed - curves represent the case with @xmath77 , @xmath78 and @xmath79 , respectively . all the curves are shown within @xmath80.,title="fig:"]@xmath59 + ( a ) @xmath82 , @xmath69 ( b ) @xmath83 , @xmath69 @xmath60 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 @xmath60 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 + @xmath81 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 @xmath81 + and @xmath74 for @xmath84 , @xmath75 and @xmath76 . again the solid- , dotted- and dashed - curves represent the case @xmath77 , @xmath78 and @xmath79 , respectively . note that @xmath85 in this supersymmetric solution.,title="fig:"]@xmath59 + ( a ) @xmath82 , @xmath86 ( b ) @xmath83 , @xmath87 we have studied a 5d gauged @xmath0 supergravity on @xmath1 in which both a @xmath2-even @xmath3 gauge field and the @xmath2-odd graviphoton take part in the @xmath0 gauging . based on the off - shell 5d supergravity of ref . @xcite , we examined the structure of fayet - iliopoulos ( fi ) terms allowed by such theory . as expected , introducing a @xmath2-even @xmath0 gauging accompanies new bulk and boundary fi terms in addition to the known integrable boundary fi term which could be present in the absence of any gauged @xmath0 symmetry . the new ( non - integrable ) boundary fi terms originate from the @xmath6 boundary supergravity , and thus are free from the bulk supergravity structure in contrast to the integrable boundary fi term which is determined by the bulk structure of 5d supergravity @xcite . we have examined some physical consequences of the @xmath2-even @xmath0 gauging in several simple cases . it is noted that the fi terms of gauged @xmath2-even @xmath0 can lead to an interesting deformation of vacuum structure which can affect the quasi - localization of the matter zero modes in extra dimension and also the susy breaking and radion stabilization . thus the 5d gauged @xmath0 supergravity on orbifold has a rich theoretical structure which may be useful for understanding some problems in particle physics such as the yukawa hierarchy and/or the supersymmetry breaking @xcite . for such phenomenological study and for the analysis of the radion stabilization , the @xmath6 superfield description @xcite will be useful . when one tries to construct a realistic particle physics model within gauged @xmath0 supergravity , one of the most severe constraint will come from the anomaly cancellation condition . in some cases the green - schwarz mechanism might be necessary to cancel the anomaly , which may introduce another type of fi term into the theory @xcite . these issues will be studied in future works . k. choi , d. y. kim , i. w. kim and t. kobayashi , eur . j. c * 35 * , 267 ( 2004 ) [ hep - ph/0305024 ] ; 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h. abe and y. sakamura , jhep * 0410 * , 013 ( 2004 ) [ hep - th/0408224 ] ; h. abe and y. sakamura , phys . rev . d * 71 * , 105010 ( 2005 ) [ hep - th/0501183 ] .

we discuss a gauged @xmath0 supergravity on five - dimensional orbifold ( @xmath1 ) in which a @xmath2-even @xmath3 gauge field takes part in the @xmath0 gauging , and show the structure of fayet - iliopoulos ( fi ) terms allowed in such model . some physical consequences of the fi terms are examined . address = department of physics , kyoto university , kyoto 606 - 8502 , japan

the discovery of iron - based high - temperature superconductors has ignited intensive studies.@xcite the superconducting transition temperature ( @xmath4 ) has risen up to 56 k in @xmath5feaso@xmath6f@xmath7 ( @xmath5=sm , nd , pr , ... ) .@xcite on the other hand , the sibling ni - based compounds only possess relatively low @xmath4 , i.e. , laonip ( @xmath4=3 k),@xcite laonias ( @xmath4=2.75 k),@xcite bani@xmath0p@xmath0 ( @xmath4=2.4 k),@xcite bani@xmath0as@xmath0 ( @xmath4=0.7 k),@xcite and srni@xmath0as@xmath0 ( @xmath4=0.62 k).@xcite . it is intriguing to understand why @xmath4 is low in the ni - based compounds , which may facilitate understanding the high-@xmath4 in iron - based ones . bafe@xmath0as@xmath0 , a typical parent compound of iron - based superconductor with the thcr@xmath0si@xmath0 structure , exhibits a structural transition from tetragonal to orthorhombic , concomitant with a spin - density - wave ( sdw ) transition at 140 k.@xcite the structural transition was suggested to be driven by the magnetic degree of freedom.@xcite bani@xmath0as@xmath0 with the same structure , displays a structural transition around 131 k , however , from a tetragonal phase to a lower symmetry triclinic phase.@xcite no evidence of sdw is reported in bani@xmath0as@xmath0 so far . on the other hand , it was pointed out that the transition in bani@xmath0as@xmath0 is a first - order one , while that is more second - order - like in bafe@xmath0as@xmath0.@xcite moreover , the c - axis resistivity drops by two orders of magnitude and a thermal hysteresis of in - plane resistivity is present at the transition of bani@xmath0as@xmath0.@xcite although there are intriguing resemblance as well as differences between bafe@xmath0as@xmath0 and bani@xmath0as@xmath0 , the electronic structure of the latter is still not exposed . here we report the angle - resolved photoemission spectroscopy ( arpes ) study of the electronic structure of bani@xmath0as@xmath0 . our data are compared to the band structure calculation of bani@xmath0as@xmath0 and the results of iron pnictides reported before , revealing their similarities and differences . particularly , no band folding is found in the electronic structure of bani@xmath0as@xmath0 , confirming that there is no collinear sdw type of magnetic ordering . because of the intimate relation between superconductivity and magnetism,@xcite the absence of magnetic ordering is possibly related to the low-@xmath4 of bani@xmath0as@xmath0 . furthermore , a hysteresis is observed for the band shift , resembling the hysteresis in the resistivity data . the band shift can be accounted for by the significant lattice distortion in bani@xmath0as@xmath0 , in contrast to iron pnictides , where the band shift is largely caused by the magnetic ordering . as@xmath0 in the triclinic phase taken with 100 ev incident electrons . , width=94 ] bani@xmath0as@xmath0 single crystals were synthesized by self - flux method , and a similar synthesis procedure has been described in ref . . its stoichiometry was confirmed by energy dispersive x - ray ( edx ) analysis . arpes measurements were performed ( 1 ) with circularly - polarized synchrotron light and randomly - polarized 8.4 ev photons from a xenon discharge lamp at beamline 9 of hiroshima synchrotron radiation center ( hsrc ) , ( 2 ) with linearly polarized synchrotron light at the surface and interface spectroscopy ( sis ) beamline of swiss light source ( sls ) , and ( 3 ) with randomly - polarized 21.2 ev photons from a helium discharge lamp . scienta r4000 electron analyzers are equipped in all setups . the typical energy resolution is 15 mev , and angular resolution is 0.3@xmath8 . the samples were cleaved _ in situ _ , and measured in ultrahigh vacuum better than 5@xmath9 mbar . the high quality sample surface was confirmed by the clear pattern of low - energy electron diffraction ( leed ) , where no sign of surface reconstruction is observed ( fig . [ leed ] ) . -@xmath3 in the triclinic phase . ( a)-(e ) photoemission intensity plots of cuts 1 - 5 as indicated in panel g , taken in hsrc with circularly polarized 21.2 ev photons at 20 k. ( f ) photoemission intensity plot of cut 6 as indicated in panel g , taken with randomly polarized 21.2 ev photons from a helium lamp at 25 k. ( g ) the mdcs corresponding to panel f. ( h ) cuts 1 - 6 are indicated in the projected 2d brillouin zone . the dashed curves and markers trace the band dispersions.,width=321 ] -@xmath1 measured at 10 k in the triclinic phase with 100 ev linearly polarized light at sls . ( a ) experimental setup for the s and p polarization geometries , and the indication of the @xmath2-@xmath1 cut in the projected 2d brillouin zone . ( b ) and ( c ) photoemission intensity plots measured in the s and p polarization geometries respectively . the image contrast in the rectangular region as enclosed by dash - dotted lines in panel c is adjusted to reveal the bands in this region . ( d ) stack of mdcs in the s and p polarization geometries . each mdc is normalized by its integrated weight . dashed curves and markers trace the band dispersions . labels are explained in the text.,width=321 ] figures [ gm](a ) and [ gm](c ) show the fermi surface maps measured with circularly polarized 22.5 ev photons in the tetragonal and triclinic phases , respectively . there are four patches near the @xmath2 point as indicated by the arrows , and two electron pockets around the @xmath1 point . sixteen cuts from @xmath2 to @xmath1 are presented in fig . [ gm](b ) to illustrate the electronic structure evolution in the tetragonal phase , and the corresponding data in the triclinic phase are shown in fig . [ gm](d ) . in the tetragonal phase , the parabolic - shaped band in cuts 1 - 2 around @xmath2 is referred as @xmath10 . the @xmath10 band appears to be @xmath11-shaped in cuts 3 - 4 . from cut 1 to cut 4 , the parabolic part shrinks continuously , and eventually only the inverted parabolic part is observable in cuts 5 - 7 . the evolution of the electronlike bands around @xmath1 are shown in cuts 8 - 16 , many of which are complex due to the rapid change of dispersions . nonetheless , two bands ( @xmath12 and @xmath13 ) can be resolved as indicated by the dashed curves in cut 16 and will be further elaborated in fig . [ aroundm ] . the data in the triclinic phase [ fig . [ gm](d ) ] are generally similar to those in the tetragonal phase . nonetheless , we note that @xmath10 already shows some bending near the fermi energy ( @xmath14 ) in the tetragonal phase as indicated by the arrows on the data taken along cuts 1 - 2 . on the other hand , @xmath10 just passes through @xmath14 without bending in the triclinic phase . moreover , the band top of @xmath10 is below @xmath14 in cuts 6 - 7 of fig . [ gm](b ) as indicated by the arrows , but barely touches @xmath14 in fig . [ gm](d ) , where cuts 6 - 7 pass through one of the four patches around @xmath2 . therefore , the four spectral weight patches around @xmath2 [ as marked by four arrows in fig . [ gm](a ) ] are due to the residual spectral weight of the @xmath10 band in the tetragonal phase . however , the @xmath10 band shifts up and they evolve into small holelike fermi surfaces in the triclinic phase . this will be further illustrated in fig . [ gx ] . of note , from this complete set of data , we do not observe any sign of band folding or splitting like that in the iron pnictides.@xcite to further illustrate the electronic structure of bani@xmath0as@xmath0 , figs . [ gx](a)-[gx](e ) present photoemission intensities along five cuts parallel to the @xmath2-@xmath3 direction in the triclinic phase . the @xmath11-shaped feature originated from @xmath10 simply moves towards @xmath14 from cut 1 to cut 5 , touching @xmath14 at cuts 4 - 5 , which confirms that the four small fermi surfaces around @xmath2 are holelike . note that the downward part of @xmath10 [ indicated by the arrow in fig . [ gx](e ) ] is clearly resolved here , while it is barely observable in the same momentum region when the cuts are along the @xmath2-@xmath1 direction , as shown in cuts 1 - 2 of figs . [ gm](b ) and [ gm](d ) . it highlights the matrix element effects since the 3@xmath15 orbitals have specific orientations . figure [ gx](f ) shows the photoemission intensity along @xmath2-@xmath3 , taken with randomly polarized 21.2 ev photons from a helium lamp in the triclinic phase . the determined band structure is traced by dashed curves , where the broad spectral weight around @xmath3 are attributed to an electronlike band @xmath16 , which is further shown by markers in the corresponding momentum distribution curves ( mdcs ) [ fig . [ gx](g ) ] . therefore , there is an electron fermi pocket around @xmath3 . similar to iron pnictides , the bands near @xmath14 are quite complicated and mainly contributed to by the ni 3@xmath15 electrons in bani@xmath0as@xmath0 . to resolve the complex bands around @xmath1 , we utilize the linearly polarized light , which could only detect bands with certain symmetry , so that the measured partial electronic structure helps reducing the complexity in analysis.@xcite figure [ aroundm ] presents data along @xmath2-@xmath1 , taken with linearly polarized 100 ev photons in sls in the triclinic phase . two polarization geometries ( s and p ) are illustrated in fig . [ aroundm](a ) . in the s polarization geometry , we resolve two bands , whose dispersions are depicted by dashed curves in the photoemission intensity plot [ fig . [ aroundm](b ) ] . while in the p polarization geometry , one intense parabolic electronlike band around @xmath1 is resolved with the band bottom at about -0.57 ev . the dispersions in both geometries are marked in the corresponding mdcs [ fig . [ aroundm](d ) ] . the asymmetry of the dispersion indicated by triangles may be due to the slight sample misalignment . the image contrast in the dash - dotted region in fig . [ aroundm](c ) is adjusted to highlight the @xmath10 feature . the observed @xmath10 feature is consistent with the data in figs . [ gm ] and [ gx ] . by comparing with the fermi crossings observed in cuts 1 and 16 of fig . [ gm](d ) , we attribute the three bands to @xmath10 , @xmath12 , and @xmath13 as shown in fig . [ aroundm](d ) , where @xmath12 and @xmath13 are two electronlike bands around @xmath1 . moreover , since the experimental setup under the s ( p ) polarization geometry detects states with odd ( even ) symmetry with respect to the mirror plane , the @xmath12 band is of mainly odd symmetry while @xmath13 is of even symmetry . we note that @xmath10 is observed in both geometries [ fig . [ aroundm](b ) and [ aroundm](c ) ] , suggesting that the @xmath10 band has mixed symmetries . = 0 for data shown in panels b and c. data were taken in hsrc with circularly polarized 22.5 ev photons for panels b - d . ( e ) edcs at @xmath17=0 for a cooling - warming - cooling cycle , measured with randomly polarized 8.4 ev photons from a xenon discharge lamp . the short bars indicate the peak positions . ( f ) summary of peak positions obtained from panel e , which exhibits a hysteresis.,width=321 ] orbitals . ( b)-(f ) contributions of the @xmath18 , @xmath19 , @xmath20 , @xmath21 , and @xmath22 orbitals to the calculated band structure of bani@xmath0as@xmath0 respectively . the contribution is represented by both the size of the symbols and the color scale.,width=321 ] to study the first order transition of bani@xmath0as@xmath0 , the temperature dependence is presented in fig . the photoemission intensity plots along @xmath2-@xmath3 are shown in figs . [ loop](b ) and [ loop](c ) for the tetragonal and the triclinic phases respectively . the corresponding energy distribution curves ( edcs ) at @xmath17=0 are stacked in fig . [ loop](d ) . interestingly , the band bottom of the @xmath11-shaped feature is moved from -200 mev in the tetragonal phase to -170 mev in the triclinic phase . in other words , the @xmath11-shaped band moves towards @xmath14 and its electronic energy is raised up . however , another feature at higher binding energies shifts away from @xmath14 . its band top is moved from -350 mev at 145 k to -390 mev at 12 k , which partially saves the electronic energy . since the resistivity shows a hysteresis loop,@xcite it is intriguing to investigate whether a similar hysteresis could be observed for the electronic structure . data in figs . [ loop](e)-[loop](f ) are taken with randomly polarized 8.4 ev photons from a xenon discharge lamp , in a cooling - warming - cooling cycle . the edcs at @xmath17=0 across the transition are stacked in fig . [ loop](e ) , where the peak positions are indicated by short bars . the temperature dependence of peak positions is summarized in fig . [ loop](f ) , showing a clear hysteresis with the band shift as much as 25 mev . such electronic structure demonstration of a hysteresis of 3 k is so far the most obvious . a hysteresis in the electronic structure has been observed in fete , but with a loop width of only 0.5 k.@xcite our observation here is consistent with the bulk transport properties , which indicates that the measured electronic structure reflects the bulk properties . the measured band structure and fermi surface are summarized in figs . [ calc](a ) , [ calc](d ) and [ calc](e ) . for comparison , local density approximation calculations which have been reported before in ref . are reproduced in figs . [ calc](b ) , [ calc](f ) and [ calc](g ) . the notations for bands near @xmath14 are labeled in figs . [ calc](a ) and [ calc](b ) . qualitatively , although not all calculated bands were observed , the main features of the experiments are captured by the calculation , such as the dispersion nature of the bands . the @xmath10 , @xmath12 , and @xmath13 bands of the experimental results in the @xmath2-@xmath1 direction are similar to the numerical results in the @xmath23-@xmath24 direction . as shown in fig . [ calc](c ) , the measured @xmath10 band along @xmath2-@xmath1 matches the calculation well after the calculated bands are renormalized by a factor of 1.66 and shifted down by 0.08 ev . this renormalization factor is consistent with the results of optical measurements.@xcite although not all bands could match , it may suggest that the correlation in bani@xmath0as@xmath0 is weaker than that in iron pnictides.@xcite along @xmath2-@xmath3 , the observed @xmath10 and @xmath16 bands partially resemble the calculated dispersions along both @xmath23-@xmath5 and @xmath25-@xmath26 , as highlighted by the shaded regions [ fig . [ calc](b ) ] , but the energy positions do not match . our data along in this direction might correspond to a @xmath27 between @xmath25 and @xmath23 . as expected from the differences in the experimentally determined and calculated band structures , the fermi surface topologies are quite different in both the experiments and the calculations . in our data [ figs . [ calc](d ) and [ calc](e ) ] , we observe four small fermi pockets around @xmath2 only in the triclinic phase . around @xmath1 , two electronlike fermi pockets are resolved in both phases . around @xmath3 , the observed fermi crossings are from an electronlike pocket . as a comparison , in the calculated fermi surface of high - t tetragonal phase [ figs . [ calc](f ) ] , there are two large warped cylinders of electron pockets around the zone corner , a pocket interconnected from the zone center to a large deformed cylinder around the zone corner , a 3d electron pocket around @xmath23 , and 3d pockets located between @xmath26 and @xmath5 . in the low - t triclinic phase [ figs . [ calc](g ) ] , 3d pockets around @xmath23 and between @xmath26 and @xmath5 are gapped out . the large electron fermi pockets around @xmath1 observed in our data are generally consistent with that in the calculation , which is a direct consequence of two more electrons from ni than fe . note that the @xmath27-dispersion is significant in the calculation . however , we have measured with four different photon energies , including the more bulk - sensitive 8.4 ev photons , and no obvious differences in dispersion have been observed . therefore , the @xmath27-dispersions in bani@xmath0as@xmath0 may be weaker than calculated . for a multiband and multiorbital superconductor , it is crucial to understand the orbital characters of the band structure . because of the symmetry of 3@xmath15 orbitals with respect to the mirror plane , the s polarization geometry in photoemission can only detect the @xmath18 and @xmath19 orbitals while the p polarization geometry can only detect the @xmath20 , @xmath21 , and @xmath22 orbitals [ fig . [ orbital](a)].@xcite the contributions of the five 3@xmath15 orbitals to the calculated band structure are presented in fig . [ orbital](b)-[orbital](f ) , which therefore can be compared to our polarization dependent data . along @xmath23-@xmath24 , the @xmath10 band is consisted of mainly the odd @xmath18 orbital and some contributions of odd @xmath19 and even @xmath22 , thus can be observed in both the s and p polarization geometries ; while the @xmath13 band is consisted of the even @xmath20 and @xmath21 orbitals , thus can only be observed in the p polarization geometry . they are in good agreement with our observation . the @xmath12 band is consisted of the odd @xmath18 , @xmath19 and even @xmath22 orbitals in the calculation , thus should be observed in both the s and p polarization geometries . however , @xmath12 is mainly detected in the s polarization geometry , possibly because in the p polarization geometry it is buried in the intense peak of @xmath13 . the consistency between our data and the calculated orbital characters confirms that our data along @xmath2-@xmath1 match the band structure calculation along @xmath23-@xmath24 . it was observed in the optical data that the phase transition leads to a reduction of conducting carriers , consistent with the removal of small fermi surfaces shown by the calculation.@xcite however , we do not observe such behavior by arpes . on the contrary , instead of the disappearance of small fermi surfaces in the triclinic phase , we observe that bands shift up in energy , leading to additional four fermi surfaces . the inconsistency between the optical data and our photoemission data suggests that the changes in optical data across the phase transition are possibly an integrated effect of band structure reorganization over the entire brillouin zone , instead of the disappearance of certain fermi surface sheets ; but it is also possible that only limited @xmath28-space has been probed in the current photoemission study . as a sibling compound of iron pnictides , bani@xmath0as@xmath0 exhibits quite different properties and electronic structure . the parent compounds of iron pnictides show a second - order - like transition that is the sdw transition concomitant with a structural transition . however , bani@xmath0as@xmath0 shows a strong first - order - like structural transition , without magnetic ordering reported to date . from the aspect of electronic structure , iron pnictides possess several hole pockets around @xmath2 , and several electron pockets around @xmath1 , but have no pockets around @xmath3 , while the band structure of bani@xmath0as@xmath0 is dramatically different from that of the iron pnictides . moreover , no signature of folding could be found in our data , confirming that no collinear magnetic ordering exists in bani@xmath0as@xmath0 . because of the intimate relation between the magnetism and superconductivity,@xcite the absence of magnetic ordering might be related to the low-@xmath4 in bani@xmath0as@xmath0 . across the structural transition in bani@xmath0as@xmath0 , the ni - ni distance changes from 2.93 @xmath29 to 2.8 @xmath29 ( or 3.1 @xmath29 ) , corresponding to a lattice distortion as much as @xmath305% in average.@xcite a rough estimation can be made for the hopping parameter @xmath31 between certain @xmath15-@xmath15 orbitals after the lattice distortion according to ref . , @xmath32 where @xmath16 is the relative lattice distortion ; @xmath33 is the hopping parameter before the distortion ; @xmath34 is the induced hopping parameter change . therefore , the 5% lattice distortion would cause @xmath3017.5% of change to @xmath33 . since the measured bandwidth of @xmath10 is at least 200 mev , the induced band shift would be larger than 35 mev , more than enough to account for the measured band shift of 25 mev . note that the differences between the calculated band structures of tetragonal and triclinic phases in fig . [ calc](f ) are solely induced by considering the different lattice parameters . for instance , the band along @xmath23-@xmath5 near @xmath14 has a shift of 25% of the bandwidth , generally consistent with our observation . therefore , our bani@xmath0as@xmath0 data provide a prototypical experimental showcase of band shift due to significant lattice distortion . as a comparison , the lattice distortion in nafeas is 0.36% , which would induce only 1 mev of band shift , much smaller than the observed 16 mev by arpes.@xcite similar results can be found in other iron pnictides.@xcite the minor lattice distortion can not account for the large band shift observed in iron pnictides , therefore it has been concluded that the only promising explanation left is that the band shift is related to the magnetism.@xcite to summarize , we report the first electronic structure study of bani@xmath0as@xmath0 by arpes . in comparison with the band calculation of bani@xmath0as@xmath0 and reports of iron pnictides , we conclude several points as following : 1 . we observe four small fermi pockets around @xmath2 only in the triclinic phase , an electronlike pocket around @xmath3 and two electronlike pockets around @xmath1 in both tetragonal and triclinic phases . the main features of the measured band structure along @xmath2-@xmath1 is qualitatively captured by the band calculations , however differences exist along @xmath2-@xmath3 . the electronic structure of bani@xmath0as@xmath0 is also distinct from the that of iron pnictides . moreover , the correlation effects in bani@xmath0as@xmath0 seems to be weaker than that in iron pnictides , as the band renormalization factor is smaller for bani@xmath0as@xmath0 . 2 . unlike iron pnictides , we do not observe any sign of band folding in bani@xmath0as@xmath0 , confirming no collinear sdw related magnetic ordering . since the magnetism intimately relates to the superconductivity , possibly this is why the @xmath4 is much lower in bani@xmath0as@xmath0 than in iron pnictides . the sdw / structural transition in iron pnictides is second - order - like , while the structural transition in bani@xmath0as@xmath0 is first - order and a thermal hysteresis is observed for its band shift . the band shift in bani@xmath0as@xmath0 is caused by the significant lattice distortion . on the other hand , the band shifts in the iron pnictides can not be accounted for by the minor lattice distortion there , but are related to the magnetic ordering . part of this work was performed at the surface and interface spectroscopy beamline , swiss light source , paul scherrer institute , villigen , switzerland . we thank c. hess and f. dubi for technical support . this work was supported by the nsfc , moe , most ( national basic research program no . 2006cb921300 and 2006cb601002 ) , stcsm of china . l. x. yang , b. p. xie , y. zhang , c. he , q. q. ge , x. f. wang , x. h. chen , m. arita , j. jiang , k. shimada , m. taniguchi , i. vobornik , g. rossi , j. p. hu , d. h. lu , z. x. shen , z. y. lu , and d. l. feng , phys . b * 82 * , 104519 ( 2010 ) . a. j. drew , ch . niedermayer , p. j. baker , f. l. pratt , s. j. blundell , t. lancaster , r. h. liu , g. wu , x. h. chen , i. watanabe , v. k. malik , a. dubroka , m. rssle , k. w. kim , c. baines , and c. bernhard , nat . mater . * 8 * , 310 ( 2009 ) . a. d. christianson , e. a. goremychkin , r. osborn , s. rosenkranz , m. d. lumsden , c. d. malliakas , i. s. todorov , h. claus , d. y. chung , m. g. kanatzidis , r. i. bewley , and t. guidi , nature * 456 * , 930 ( 2008 ) . y. zhang , j. wei , h. w. ou , j. f. zhao , b. zhou , f. chen , m. xu , c. he , g. wu , h. chen , m. arita , k. shimada , h. namatame , m. taniguchi , x. h. chen , and d. l. feng , phys . lett . * 102 * , 127003 ( 2009 ) . l. x. yang , y. zhang , h. w. ou , j. f. zhao , d. w. shen , b. zhou , j. wei , f. chen , m. xu , c. he , y. chen , z. d. wang , x. f. wang , t. wu , g. wu , x. h. chen , m. arita , k. shimada , m. taniguchi , z. y. lu , t. xiang , and d. l. feng , phys . lett . * 102 * , 107002 ( 2009 ) . bo zhou , yan zhang , le - xian yang , min xu , cheng he , fei chen , jia - feng zhao , hong - wei ou , jia wei , bin - ping xie , tao wu , gang wu , masashi arita , kenya shimada , hirofumi namatame , masaki taniguchi , x. h. chen , and d. l. feng , phys . b * 81 * , 155124 ( 2010 ) . liu , h .- y . liu , l. zhao , w .- t . zhang , x .- w . jia , j .- q . meng , x .- dong , j. zhang , g. f. chen , g .- wang , y. zhou , y. zhu , x .- y . wang , z .- y . xu , c .- t . chen , and x. j. zhou , phys . b * 80 * , 134519 ( 2009 ) . y. zhang , b. zhou , f. chen , j. wei , m. xu , l. x. yang , c. fang , w. f. tsai , g. h. cao , z. a. xu , m. arita , h. hayashi , j. jiang , h. iwasawa , c. h. hong , k. shimada , h. namatame , m. taniguchi , j. p. hu , d. l. feng , arxiv:0904.4022 ( unpublished ) . fei chen , bo zhou , yan zhang , jia wei , hong - wei ou , jia - feng zhao , cheng he , qing - qin ge , masashi arita , kenya shimada , hirofumi namatame , masaki taniguchi , zhong - yi lu , jiangping hu , xiao - yu cui , and d. l. feng , phys . b , * 81 * , 014526 ( 2010 ) .

bani@xmath0as@xmath0 , with a first order phase transition around 131 k , is studied by the angle - resolved photoemission spectroscopy . the measured electronic structure is compared to the local density approximation calculations , revealing similar large electronlike bands around @xmath1 and differences along @xmath2-@xmath3 . we further show that the electronic structure of bani@xmath0as@xmath0 is distinct from that of the sibling iron pnictides . particularly , there is no signature of band folding , indicating no collinear sdw related magnetic ordering . moreover , across the strong first order phase transition , the band shift exhibits a hysteresis , which is directly related to the significant lattice distortion in bani@xmath0as@xmath0 .

transitional disks have a small or no excess from @xmath0@xmath1 m to @xmath0@xmath2 m relative to their full disk cousins , but a significant excess at longer wavelength @xcite , suggesting cleared out inner disk . this interpretation dates back to the era of the _ infrared astronomical satellite _ ( @xcite , @xcite ) , and was later developed with the help of detailed near - infrared ( nir ) to mid - infrared ( mir ) spectra provided by the infrared spectrograph ( irs ) on - board _ spitzer space telescope_. detailed radiative transfer modeling suggests that this kind of sed is consistent with disk models which harbor a central ( partially ) depleted region ( i.e. a cavity or a gap ) , while a `` wall - like '' structure at the outer edge of this region can be responsible for the abrupt rise of the sed at mir @xcite . the disk+cavity model based on sed - only fitting usually contains large uncertainties , because the sed samples the emission from the whole disk ; by tuning the ingredients in the fitting , one could fit the irs sed with different models ( see the example of ux tau a , @xcite and ( * ? ? ? * hereafter a11 ) ) . better constraints on the disk structure can be obtained from resolved images of the transitional disks . using the _ submillimeter array _ ( sma ) interferometer @xcite , resolved images of transitional disks at sub - mm wavelength have provided direct detections of these cavities @xcite , and measurements of their properties . recently , a11 observed a sample of 12 nearby transitional disks ( at a typical distance of @xmath0@xmath3 pc ) . combining both the sma results and the sed , they fit detailed disk+cavity model for each object , and the cavity size ( @xmath0@xmath4 au ) is determined with @xmath0@xmath5 uncertainty . they concluded that large grains ( up to @xmath0mm - sized ) inside the cavity are depleted by at least a factor of 10 to 100 ( the `` depletion '' in this work is relative to a `` background '' value extrapolated from the outer disk ) . under the assumption that the surface density of the disk is described by their model , the infrared spectral fitting demands that the small grains ( micron - sized and smaller ) inside the cavity to be heavily depleted by a factor of @xmath0@xmath6 . recently , most objects in this sample have been observed by the subaru high - contrast coronographic imager for adaptive optics ( hiciao ) at nir bands , as part of the the strategic explorations of exoplanets and disks with subaru project , seeds , @xcite . seeds is capable of producing polarized intensity ( pi ) images of disks , which greatly enhances our ability to probe disk structure ( especially at the inner part ) by utilizing the fact that the central source is usually not polarized , so that the stellar residual in pi images is much smaller than in full intensity ( fi ) images @xcite . the seeds results turned out to be a big surprise in many cases the polarized nir images do not show an inner cavity , despite the fact that the inner working angle of the images ( the saturation radius or the coronagraph mask size , @xmath7 , or @xmath0@xmath8 au at the distance to taurus @xmath0@xmath3 pc , see section [ sec : psf ] ) is significantly smaller than the cavity sizes inferred from sub - mm observations . high contrast features such as surface brightness excesses or deficits exist in some systems , but they are localized and do not appear to be central cavities . instead , the image is smooth on large scales , and the azimuthally averaged surface brightness radial profile ( or the profile along the major axis ) increases inward smoothly until @xmath9 , without any abrupt break or jump at the cavity edge ( the slope may change with radius in some systems ) . examples include rox 44 ( m. kuzuhara et al . 2012 , in prep . ) , sr 21 ( k. follette et al . 2012 , in prep . ) , gm aur ( j. hashimoto et al . 2012a , in prep . ) , and sao 206462 @xcite ; see also the sample statistics ( j. hashimoto et al . 2012b , in prep . ) . some objects such as ux tau a also do not show a cavity ( r. tanii et al . 2012 , in prep . ) , however the inner working angle of their seeds images is too close to the cavity size , so the status of the cavity is less certain . we note that lkca 15 also does not exhibit a clear cavity in its pi imagery ( j. wisniewski et al . 2012 , in prep . ) , but does exhibit evidence of the wall of a cavity in its fi imagery @xcite . this apparent inconsistency between observations at different wavelengths reveals something fundamental in the transitional disk structure , as these datasets probe different components of protoplanetary disks . at short wavelengths ( i.e. nir ) where the disk is optically thick , the flux is dominated by the small dust ( micron - sized or so ) at the surface of the disk ( where the stellar photons get absorbed or scattered ) , and is sensitive to the shape of the surface ; at long wavelength ( i.e. sub - mm ) , disks are generally optically thin , so the flux essentially probes the disk surface density in big grains ( mm - sized or so ) , due to their large opacity at these wavelengths @xcite . combining all the three pieces of the puzzle together ( sed , sub - mm observation , and nir imaging ) , we propose a disk model that explains the signatures in all three observations simultaneously : the key point is that the spatial distributions of small and big dust are decoupled inside the cavity . in this model , a well defined cavity ( several tens of au in radius ) with a sharp edge exists only in spatial distribution of the big dust and reproduces the central void in the sub - mm images , while no discontinuity is found for the spatial distribution of the small dust at the cavity edge . inside the cavity , the surface density of the small dust does not increase inwardly as steeply as it does in the outer disk ; instead it is roughly constant or declines closer to the star ( while maintaining an overall smooth profile ) . in this way , the inner region ( sub - au to a few au ) is heavily depleted in small dust , so that the model reproduces the nir flux deficit in the sed ( but still enough small dust surface density to efficiently scatter near - ir radiation ) . modeling results show that the scattered light images for this continuous spatial distribution of the small dust appear smooth as well , with surface brightness steadily increasing inwardly , as seen in many of the seeds observations . the structure of this paper is as follows . in section [ sec : modeling ] we introduce the method that we use for the radiative transfer modeling . in section [ sec : image ] we give the main results on the scattered light images : first a general interpretation of the big picture through a theoretical perspective , followed by the modeling results of various disk+cavity models . we investigate the sub - mm properties of these models in section [ sec : submm ] , and explore the degeneracy in the disk parameter space on their model sed in section [ sec : sed ] . we summarize the direct constraints put by the three observations on this transitional disk sample in section [ sec : discussion ] , as well as the implications of our disk models . our generic solution , which qualitatively explains the signatures in all the three observations , is summarized in section [ sec : summary ] . in this section , we introduce the model setup in our radiative transfer calculations , and the post processing of the raw nir polarized scattered light images which we perform in order to mimic the observations . the purpose of this modeling exercise is to `` translate '' various physical disk models to their corresponding nir polarized scattered light images , sub - mm emission images , and sed , for comparison with observations . we use a modified version of the monte carlo radiative transfer code developed by @xcite , @xcite , and b. whitney et al . 2012 , in prep . ; for the disk structure , we use a11 and @xcite for references . the nir images ( this section ) and sed ( section [ sec : sed ] ) are produced from simulations with @xmath10 photon packets , and for the sub - mm images ( section [ sec : submm ] ) we use @xmath11 photon packets . by varying the random seeds in the monte carlo simulations , we find the noise levels in both the radial profile of the convolved images ( section [ sec : psf ] ) and the sed to be @xmath12 in the range of interest . in our models , we construct an axisymmetric disk ( assumed to be at @xmath0140 pc ) 200 au in radius on a @xmath13 grid in spherical coordinates ( @xmath14 ) , where @xmath15 is in the radial direction and @xmath16 is in the poloidal direction ( @xmath17 is the disk mid - plane ) . we include accretion energy in the disk using the shakura & sunyaev @xmath18 disk prescription @xcite . disk accretion under the accretion rate assumed in our models below ( several @xmath19 yr@xmath20 ) does not have a significant effect on the sed or the images ( for simplicity , accretion energy from the inner gas disk is assumed to be emitted with the stellar spectrum , but see also the treatment in @xcite ) . we model the entire disk with two components : a thick disk with small grains ( @xmath0@xmath21m - sized and smaller , more of less pristine ) , and a thin disk with large ( grown and settled ) grains ( up to @xmath0mm - sized ) . figure [ fig : sigma ] shows the schematic surface density profile for both dust population . the parametrized vertical density profiles for both dust populations are taken to be gaussian ( i.e. @xmath22 , isothermal in the vertical direction @xmath23 ) , with scale heights @xmath24 and @xmath25 being simple power laws @xmath26 ( we use subscripts `` s '' and `` b '' to indicate the small and big dust throughout the paper , while quantities without subscripts `` s '' and `` b '' are for both dust populations ) . following a11 , to qualitatively account for the possibility of settling of big grains , we fix @xmath27 in most cases to simplify the models , unless indicated otherwise . radially the disk is divided into two regions : an outer full disk from a cavity edge @xmath28 to 200 au , and an inner cavity from the dust sublimation radius @xmath29 to @xmath28 ( @xmath29 is determined self - consistently as where the temperature reaches the sublimation temperature @xmath301600 k , @xcite , usually around 0.1@xmath310.2 au ) . at places in the disk where a large surface area of material is directly exposed to starlight , a thin layer of material is superheated , and the local disk `` puffs '' up vertically @xcite . to study this effect at the inner rim ( @xmath29 ) or at the cavity wall , we adopt a treatment similar to a11 . in some models below we manually raise the scale height @xmath32 at @xmath29 or @xmath28 by a certain factor from its `` original '' value , and let the puffed up @xmath32 fall back to the underlying power law profile of @xmath32 within @xmath00.1 au as @xmath33 . we note that these puffed up walls are vertical , which may not be realistic @xcite . for the surface density profile in the outer disk , we assume @xmath34 where @xmath35 is the surface density at the cavity edge ( normalized by the total disk mass ) , @xmath36 is a characteristic scaling length , and the gas - to - dust ratio is fixed at 100 . following a11 , we take @xmath37 of the dust mass to be in large grains at @xmath38 . for the inner disk ( i.e. @xmath39 ) three surface density profiles have been explored : & & _ i(r)=(_cav_cav)e^(r_cav - r)/r_c ( rising _ i(r ) ) , [ eq : sigmai - andrews ] + & & _ i(r)=_cav_cav ( flat _ i(r ) ) , and [ eq : sigmai - flat ] + & & _ i(r)=(_cav_cav ) ( declining _ i(r ) ) , [ eq : sigmai - linearneg ] and their names are based on their behavior when moving inward inside the cavity . we note that equation and together form a single @xmath40 scaling relation for the entire disk ( with different normalization for the inner and outer parts ) , as in a11 . we define the depletion factor of the total dust inside the cavity as @xmath41 where @xmath42 is found by extrapolating @xmath43 from the outer disk , i.e. evaluating equation at @xmath39 ( or equation with @xmath44 ) . in addition , we define @xmath45 and @xmath46 as the cavity depletion factors for the small and big dust respectively as @xmath47 where 0.15 and 0.85 are the mass fractions of the small and big dust in the outer disk . we note that unlike previous models such as a11 , our cavity depletion factors are radius dependent ( a constant @xmath45 or @xmath46 means a uniform depletion at all radii inside the cavity ) . specifically , we define the depletion factor right inside the cavity edge as @xmath48 with the same @xmath49 , different models with different @xmath50 profiles ( equations - ) have similar @xmath50 ( and @xmath51 ) in the outer part of the cavity , but very different @xmath50 ( and @xmath51 ) at the innermost part . lastly , the mass averaged cavity depletion factor @xmath52 is defined as @xmath53 sma observations have placed strong constraints on the spatial distribution of the big dust , while the constraints on the small grains from the sed are less certain , especially beyond @xmath54 au . based on this , we adopt the spatial distribution of big grains in a11 ( i.e. equation , and no big grains inside the cavity ) , and focus on the effect of the distribution of small grains inside the cavity . therefore , the sub - mm properties of our models are similar to those of the models in a11 ( section [ sec : submm ] ) , since large grains dominate the sub - mm emission . we tested models with non - zero depletion for the big grains , and found that they make no significant difference as long as their surface density is below the sma upper limit . from now on we drop the explicit radius dependence indicator @xmath55 from various quantities in most cases for simplicity . for the small grains we try two models : the standard interstellar medium ( ism ) grains ( @xcite , @xmath0micron - sized and smaller ) , and the model that @xcite employed to reproduce the hh 30 nir scattered light images , which are somewhat larger than the ism grains ( maxim size @xmath020 @xmath21 m ) . these grains contain silicate , graphite , and amorphous carbon , and their properties are plotted in figure [ fig : dust ] . the two grain models are similar to each other , and both are similar to the small grains model which a11 used in the outer disk and the cavity grains which a11 used inside the cavity and on the cavity wall . we note that for detailed modeling which aims at fitting specific objects , the model for the small grains needs to be turned for each individual object . for example , the strength and shape of the silicate features indicate different conditions for the small grains in the inner disk ( @xcite , and @xcite , who also pointed out that the silicate features in transitional disks typically show that the grains in the inner disk are dominated by small amorphous silicate grains similar to ism grains ) . however , since we do not aim at fitting specific objects , we avoid tuning the small dust properties and assume ism grains @xcite for the models shown below , to keep our models generalized and simple . for the large grains we try three different models , namely models 1 , 2 , and 3 from @xcite . the properties of these models are plotted in figure [ fig : dust ] . they adopt a power - law size distribution ( i.e. as in @xcite ) with an exponential cutoff at large size , and the maxim size is @xmath01 mm . these grains are made of amorphous carbon and astronomical silicates , with solar abundances of carbon and silicon . these models cover a large parameter space , however we find that they hardly make any difference in the scattered light image and the irs sed , due to their small scale height and their absence inside the cavity . for this reason we fix our big grains as described by model 2 in @xcite ( which is similar to the model of the big grains in a11 ) . we note that small grains have much larger opacity than big grains at nir , and it is the other way around at sub - mm ( figure [ fig : dust ] ) . to obtain realistic images which can be directly compared to seeds observations , the raw nir images of the entire disk+star system from the radiative transfer simulations need to be convolved with the point spread function ( psf ) of the instrument . seeds can obtain both the fi and the pi images for any object , either with or without a coronagraph mask . the observation could be conducted in several different observational modes , including angular differential imaging ( adi , @xcite ) , polarization differential imaging ( pdi , @xcite ) , and spectral differential imaging ( sdi , @xcite ) . for a description of the instrument see @xcite and @xcite . in this work , we produce both the narrow band 880 @xmath21 m images and @xmath56 band nir images . while at 880 @xmath21 m we produce the full intensity images , for the nir scattered light images we focus on the pi images ( produced in the pdi mode , both with and without a coronagraph mask ) . this is because ( 1 ) pdi is the dominate mode for this sample in seeds , and ( 2 ) it is more difficult to interpret fi ( adi ) images since its reduction process partially or completely subtracts azimuthally symmetric structure . other authors had to synthesize and reduce model data in order to test for the existence of features like cavities @xcite or spatially extended emission @xcite . for examples of pdi data reduction and analysis , see @xcite . when observing with a mask , @xmath9 in the pi images is the mask size ( typically @xmath57 in radius ) , and when observing without a mask , @xmath9 is determined by the saturation radius , which typically is @xmath0@xmath58 . to produce an image corresponding to observations made without a mask , we convolve the raw pi image of the entire system with an observed unsaturated hiciao @xmath56-band psf . the resolution of the psf is @xmath0@xmath59 ( @xmath01.2@xmath60 for an 8-m telescope ) and the strehl ratio is @xmath040% @xcite . the integrated flux within a circle of radius @xmath61 is @xmath080% of the total flux ( @xmath090% for a circle of radius @xmath62 ) . we then carve out a circle at the center with @xmath58 in radius to mimic the effect of saturation . we call this product the convolved unmasked pi image . to produce an image corresponding to observations with a mask , we first convolve the part of the raw pi image which is not blocked by the mask with the above psf . we then convolve the central source by an observed pi coronagraph stellar residual map ( the psf under the coronagraph ) , and add this stellar residual to the disk images ( the flux from the inner part of the disk which is blocked by the mask , @xmath020 au at @xmath0140 pc , is added to the star ) . lastly , we carve out a circle @xmath57 in radius from the center from the combined image to indicate the mask . we call this product the convolved masked pi image . we note that the stellar residual is needed to fully reproduce the observations , but in our sample the surface brightness of the stellar residual is generally well below the surface brightness of the disk at the radius of interest , so it does nt affect the properties of the images much . in this study , the disk is assumed to be face - on in order to minimize the effect of the phase function in the scattering , so that we can focus on the effect of the disk structure . this is a good approximation since most objects in this irs / sma / subaru sample have inclinations around @xmath025@xmath63 ( i.e. minor to major axis ratio @xmath00.9 . an observational bias towards face - on objects may exist , since they are better at revealing the cavity ) . additional information about the scattering properties of the dust could be gained from analyzing the detailed azimuthal profile of the scattered light in each individual system , which we defer to the future studies . to calculate the azimuthally averaged surface brightness profiles , we bin the convolved images into a series of annuli @xmath59 in width ( the typical spatial resolution ) , and measure the mean flux within each annulus . with the tools described above , we investigate what kinds of disk structure could simultaneously reproduce the gross properties of all three kinds of observations described in section [ sec : introduction ] . in this section , we first investigate the properties of the scattered light images from a semi - analytical theoretical point of view ( section [ sec : imagetheory ] ) , then we present the model results from the monte carlo simulations ( section [ sec : imageresult ] ) . in a single nir band , when the ( inner ) disk is optically thick ( i.e. not heavily depleted of the small grains ) , the scattering of the starlight can be approximated as happening on a scattering surface @xmath64 where the optical depth between the star and surface is unity ( the single scattering approximation ) . this surface is determined by both the disk scale height ( particularly @xmath65 in the simple vertically - isothermal models ) , and the radial profile of the surface density of the grains . the surface brightness of the scattered light @xmath66 scales with radius as @xcite @xmath67 where @xmath68 is the stellar luminosity at this wavelength , @xmath69 is a geometrical scattering factor , @xmath70 is the polarization coefficient for pi ( @xmath71 for fi ) , and @xmath72 is the grazing angle ( the angle between the impinging stellar radiation and the tangent of the scattering surface ) . we note that both @xmath69 and @xmath70 depend on azimuthal angle , inclination of the disk , and the scattering properties of the specific dust population responsible for scattering at the particular wavelength . however , if the disk is relatively face - on and not too flared , they are nearly position independent , because the scattering angle is nearly a constant throughout the disk and the dust properties of the specific dust population do not change much with radius . the grazing angle is determined by the curvature of the scattering surface . in axisymmetric disks , assuming the surface density and scale height of the small dust to be smooth functions of radius , the grazing angle is also smooth with radius , and two extreme conditions can be constrained as follows : 1 . for disks whose scattering surface is defined by a constant poloidal angle @xmath16 ( such as a constant opening angle disk ) , @xmath73 ( @xcite , where @xmath74 is the radius of the star ) , so the brightness of the scattered light scales with @xmath15 as @xmath75 2 . for flared disks ( but not too flared , @xmath76 ) , the grazing angle can be well approximated as @xmath77 in this case , @xcite explicitly calculated the position of the scattering surface @xmath64 at various radii ( see also @xcite ) , and found @xmath78 where the coefficient @xmath79 a few and is nearly a constant . combined with the fact that @xmath80 ( @xmath81 as typical values in irradiated disks , @xcite ) , we have the intensity of the scattered light scales with radius as @xmath82 although the above calculations are under two extreme conditions ( for complete flat or flared disks ) , and they are based on certain assumptions and the observed images have been smeared out by the instrument psf , the radial profiles ( azimuthally averaged , or along the major axis in inclined systems ) of seeds scattered light images for many objects in this sample lie between equations ( [ eq : ir-3 ] ) and ( [ eq : ir-2 ] ) in the radius range of interest . in order to guide the eye and ease the comparison between modeling results and observations , we use the scaling relation @xmath83 to represent typical observational results , and plot it on top of the radiative transfer results , which will be presented in section [ sec : imageresult ] ( with arbitrary normalization ) . on the other hand , an abrupt jump in the surface density or scale height profile of the small dust in the disk produces a jump in @xmath72 at the corresponding position . the effect of this jump will be explored in section [ sec : imageresult ] . first , we present the simulated @xmath56 band pi images for a face - on transitional disk with a uniformly heavily depleted cavity with @xmath84 as equation ( figure [ fig : image - andrews ] ) , and the associated surface brightness radial profiles ( the thick solid curves in figure [ fig : rp - andrews ] ) . in each figure , the three panels show the raw image , the convolved unmasked image , and the convolved masked image , respectively . this model is motivated by the disk+cavity models in a11 ; we therefore use parameters typical of those models . experiments show that the peculiarities in each individual a11 disk model hardly affect the qualitative properties of the images and their radial profiles , as long as the cavity is large enough ( @xmath85 ) and the disk is relatively face - on . this disk harbors a giant cavity at its center with @xmath86 au ( @xmath0@xmath87 at 140 pc ) . the disk has the same inwardly rising @xmath88 scaling both inside and outside the cavity as equations ( [ eq : sigmao ] ) and ( [ eq : sigmai - andrews ] ) . outside the cavity both dust populations exist , with the big / small ratio as @xmath89 . inside the cavity there is no big dust ( @xmath90 ) , and the small dust is uniformly heavily depleted to @xmath91 ( @xmath92 ) . the surface density profiles for both dust populations can be found in figure [ fig : sigma ] . the disk has total mass of @xmath93 ( gas - to - dust mass ratio 100 ) , @xmath94 at 100 au with @xmath95 , @xmath96 au , and accretion rate @xmath97 yr@xmath20 . the central source is a 3 @xmath98 , @xmath99 , 5750 k g3 pre - main sequence star . we puff up the inner rim and the cavity wall by 100% and 200% , respectively . the most prominent features of this model in both the unmasked and masked pi images are the bright ring at @xmath28 , and the surface brightness deficit inside the ring ( i.e. the cavity ) . correspondingly , the surface brightness profile increases inwardly in the outer disk , peaks around @xmath28 , and then decreases sharply . this is very different from many seeds results ( such as the examples mentioned in section [ sec : introduction ] ) , in which both the bright ring and the inner deficit are absent , and the surface brightness radial profile keeps increasing smoothly all the way from the outer disk to the inner working angle , as illustrated by the scaling relation ( [ eq : seeds ] ) in figure [ fig : rp - andrews ] . this striking difference between models and observations suggests that the small dust can not have such a large depletion at the cavity edge . figure [ fig : rp - andrews ] shows the effect of uniformly filling the cavity with small dust on the radial profile for unmasked images ( left ) and masked images ( right ) , leaving the other model parameters fixed . as @xmath100 gradually increases from @xmath0@xmath101 to 1 , the deviation in the general shape between models and observations decreases . the @xmath102 model corresponds to no depletion for small grains inside the cavity ( i.e. a full small dust disk ) ; this model agrees much better with the scattered light observations , despite a bump around the cavity edge produced by its puffed up wall ( section [ sec : image - puffingup ] ) , although this model fails to reproduce the transitional - disk - like sed ( section [ sec : sed ] ) . we note that this inconsistency between uniformly heavily depleted cavity models and observations is intrinsic and probably can not be solved simply by assigning a high polarization to the dust inside the cavity . the polarization fraction ( pi / fi ) in the convolved disk images of our models ranges from @xmath00.3 to @xmath00.5 , comparable to observations ( for example @xcite ) . even if we artificially increase the polarization fraction inside the cavity by a factor of 10 , by scaling up the cavity surface brightness in the raw pi images ( which results in a ratio pi / fi greater than unity ) while maintaining the outer disk unchanged , the convolved pi images produced by these uniformly heavily depleted cavity models still have a prominent cavity at their centers . we also note that the contrast of the cavity ( the flux deficit inside @xmath28 ) and the strength of its edge ( the brightness of the ring at @xmath28 ) are partially reduced in the convolved images compared with the raw images . this is due both to the convolution of the disk image with the telescope psf , which naturally smooths out any sharp features in the raw images , and to the superimposed seeing halo from the bright innermost disk ( especially for the unmasked images ) . in the rest of section [ sec : image ] we focus on the small grains inside the cavity while keeping the big grains absent , and study the effect of the parameters @xmath84 , @xmath103 , and the puffing up of the inner rim and cavity wall on the scattered light images . the convolved images for three representative models are shown in figure [ fig : image - variation ] , and the radial profiles for all models are shown in figure [ fig : rp - variation ] . each model below is varied from one standard model , which is shown as the top panel in figure [ fig : image - variation ] and represented by the thick solid curve in all panels in figure [ fig : rp - variation ] . this fiducial model has a flat @xmath84 with no discontinuity at the cavity edge for the small dust ( i.e. @xmath104 ) , and no puffed up rim or wall , but otherwise identical parameters to the models above . first we study the effect of three surface density profiles inside the cavity , namely rising ( equation ) , flat ( equation ) , and declining @xmath84 ( equation ) . the surface densities for these models are illustrated in figure [ fig : sigma ] . panel ( a ) in figure [ fig : rp - variation ] shows the effect of varying the surface density on the image radial profiles . except for shifting the entire curve up and down , different @xmath84 produce qualitatively very similar images and radial profiles , and all contain the gross features in many seeds observations ( illustrated by the scaling relation ( [ eq : seeds ] ) ) . this is due to both the fact that smooth surface density and scale height profiles yield a smooth scattering surface , and the effect of the psf . we note that there is some coronagraph edge effect at the inner working angle in the masked images , which is caused by that the part of the disk just outside ( but not inside ) the mask is convolved with the psf . this results in a narrow ring of flux deficit just outside the mask . in general , the flux is trustable beyond about one fwhm of the psf from @xmath9 ( @xmath105 ) ( * ? ? ? * some instrumental effects in observations may also affect the image quality within one fwhm from the mask edge as well ) . we investigate the effect of different @xmath103 with flat @xmath84 ( [ eq : sigmai - flat ] ) . panel ( b ) in figures [ fig : rp - variation ] shows the effect on the radial profile of the convolved images . when deviating from the fiducial model with @xmath104 ( i.e. a continuous small dust disk at @xmath28 ) , a bump around the cavity edge and a surface brightness deficit inside the cavity gradually emerge . we quantify this effect by measuring the relative flux deficit at @xmath106 ( @xmath107 ) as a function of @xmath103 , the small - dust discontinuity at the cavity edge ( subpanel in each plot ) . for each model we calculate the ratio of the flux at @xmath106 , @xmath108 , to @xmath109 , the flux at @xmath110 ( @xmath111 ) , normalized by @xmath112 in the fiducial , undepleted model . for the model with a @xmath113 discontinuity in @xmath114 at @xmath28 ( @xmath115 ) , @xmath112 is @xmath015% lower than in the fiducial model in the convolved unmasked images , and @xmath020% lower in the convolved masked images . the latter is larger because the mask suppresses the halo of the innermost disk , thus the relative flux deficit in the masked image is closer to its intrinsic value , i.e. the deficit in the raw images . in this sense the masked images are better at constraining the discontinuity of the small dust than the unmasked images . the middle row in figure [ fig : image - variation ] shows the model images for @xmath116 ( illustrated by the thin solid curve in figure [ fig : sigma ] ) . the edge of the cavity is quite prominent in the raw image , while it is somewhat smeared out but still visible in the convolved images . for many objects in this irs / sma / subaru cross sample , the seeds images are grossly consistent with a continuous small dust disk at the @xmath28 , while in some cases a small discontinuity may be tolerated . for the purpose of comparison with the observations , we now discuss the detectability of a finite surface density discontinuity at the cavity edge , given the sensitivity and noise level of the seeds data ( the numerical noise level in our simulations is well below the noise level in the observations , see section [ sec : setup ] ) . in typical seeds observations with an integration time of several hundred seconds , the intrinsic poisson noise of the surface brightness radial profile due to finite photon counts is usually a few tenth of one percent at radius of interests , smaller than the error introduced by the instrument and the data reduction process . @xcite estimated the local noise level of the surface brightness to be @xmath010% at @xmath117 for the seeds sao 206462 pi images , which should be an upper limit for the noise level in the azimuthally averaged surface brightness ( more pixels ) and in the inner region of the images ( brighter ) . if this is the typical value for the instrument , then our modeling results indicate that seeds surface brightness measurements should be able to put relatively tight constraints on the surface density discontinuity for the small dust at @xmath28 . for example , seeds should be able to distinguish a disk of small dust continuous at @xmath28 from a disk of small dust with a @xmath113 density drop at @xmath28 . thus , a lower limit on the small dust depletion factor at the cavity edge can be deduced from detailed modeling for each individual object . this lower limit is likely to be higher than the upper limit from sma on the depletion factor of the big dust ( @xmath118 ) , for objects with relatively smooth radial profiles . if this is confirmed for some objects in which the two limits are both well determined , it means that the density distribution of the small dust needs to somehow _ decouple _ from the big dust at the cavity edge . we will come back to this point in section [ sec : discussion ] . lastly , we explore the effects of the puffed up inner rim and the cavity wall on the images , by comparing the fiducial model ( no puffing up anywhere ) to a model with the inner rim puffed up by 100% , and another model with the cavity wall puffed up by 200% . panel ( c ) in figure [ fig : rp - variation ] shows the effects . puffing up the inner rim has little effect on the image , while the puffed up wall produces a bump at the cavity edge , similar to the effect of a gap edge . images for the model with the puffed up wall are shown in figure [ fig : image - variation ] ( bottom row ) , where the wall is prominent in the raw image while been somewhat smeared out but still visible in the convolved images . scattered light images should be able to constrain the wall for individual objects . without digging deeply into this issue , we simply note here that the seeds images for many objects in this sample are consistent with no or only a small puffed up wall . for all models shown in section [ sec : image ] , the disk is optically thin in the vertical direction at 880 @xmath21 m ( @xmath119 , where @xmath120 is the opacity per gram at @xmath121 ) . in this case , the intensity @xmath122 at the surface of the disk at a given radius may be expressed as : @xmath123 where @xmath124 ghz at 880 @xmath21 m , @xmath125 is the planck function at @xmath126 ( the temperature at @xmath23 ) , and @xmath127 is the vertical density distribution ( @xmath128 and @xmath129 depend on @xmath15 as well ) . outside the cavity , the big dust dominates the 880 @xmath21 m emission , since the big dust is much more efficient at emitting at @xmath0880 @xmath21 m than the small dust ( @xmath130 at these wavelengths ) . inside the cavity there is _ no _ big dust by our assumption , so the sub - mm emission comes only from the small dust . the thick curves in figure [ fig : intensity880 ] show the 880 @xmath21 m intensity as a function of radius for the models in section [ sec : image - sigmai ] ( i.e. continuous small dust disks with rising , flat , or declining @xmath84 ) . in other words , this is an 880 @xmath21 m version of the surface brightness radial profile for the `` raw '' image , without being processed by a synthesized beam dimension ( the sma version of the `` psf '' ) . the bottom thin dashed curve is for the uniformly heavily depleted cavity model as in figure [ fig : image - andrews ] ( @xmath91 , rising @xmath84 ) and no big dust inside the cavity , which represents the sub - mm behavior of the models for most systems in a11 . since the disk is roughly isothermal in the vertical direction near the mid - plane where most dust lies @xcite , @xmath126 may be approximated by the mid - plane temperature @xmath131 . the planck function can be approximated as @xmath132 at this wavelength due to @xmath133 ( 880 @xmath21 m @xmath016 k , marginally true in the very outer part of the disk ) . in addition , since the two dust populations have roughly the same mid - plane temperature , but very different opacities ( @xmath134 ) , equation can be simplified to @xmath135 for @xmath38 and @xmath136 for @xmath39 . for our continuous small disk models with a complete cavity for the big dust , the intensity at 880 @xmath21 m drops by @xmath02.5 orders of magnitude when moving from outside ( @xmath137 ) to inside ( @xmath138 ) the cavity edge , due to both the higher opacity of the big dust and the fact that big dust dominates the mass at @xmath38 . inside the cavity , the intensity ( now exclusively from the small dust ) is determined by the factor @xmath139 . for an irradiated disk @xmath131 increases inwardly , typically as @xmath140 @xcite , while @xmath84 in our models could have various radial dependencies ( equations - ) . in the flat @xmath84 models ( equation ) , the intensity inside the cavity roughly scales with @xmath15 as @xmath141 the same as \{@xmath142 } and is 1.5 orders of magnitude lower than @xmath137 in the innermost disk ( around the sublimation radius ) . on the other hand , if there is no depletion of the small dust anywhere inside the cavity ( i.e. the rising @xmath84 with @xmath102 case ) , the intensity at the center ( thick dashed curve ) can exceed @xmath137 by two orders of magnitude . in a11 , for 880 @xmath21 m images of models with no big dust inside the cavity , the residual emission near the disk center roughly traces the quantity @xmath143 . in most cases , a11 found those residuals to be below the noise floor . due to the sensitivity limit , the constraint on @xmath143 inside the cavity is relatively weak ; nevertheless , a11 were able to put an upper limit equivalent to @xmath144 for the mm - sized dust ( with exceptions such as lkca 15 ) . here we use a mock disk model to mimic this constraint . the top thin dashed curve in figure [ fig : intensity880 ] is from a model with uniform depletion factors @xmath145 for _ both _ dust populations inside the cavity ( so the entire disk has the same dust composition everywhere ) . the result shows that , qualitatively , various models with a continuous small dust disk and a complete cavity for the big dust are all formally below this mock sma limit , though a quantitative fitting of the visibility curve is needed to constrain @xmath100 and @xmath103 , in terms of upper limits , on an object by object basis . this may line up with another sed - based constraint on the amount of small dust in the innermost disk , as we will discuss in the section [ sec : sed ] . while the intensity discussion qualitatively demonstrates the sub - mm properties of the disks , figure [ fig:880um ] shows the narrow band images at 880 @xmath21 m for two disk models . the top row is from the model which produces figure [ fig : image - andrews ] ( also the bottom thin dashed curve in figure [ fig : intensity880 ] and the left panel in figure [ fig : sigma ] ) , which is an a11 style model with a uniformly heavily depleted cavity with rising @xmath84 , @xmath91 , and no big dust inside the cavity . the bottom row is from the fiducial model in section [ sec : imageresult ] ( which produces the top row in figure [ fig : image - variation ] , and the thick solid curve in figure [ fig : intensity880 ] and in the left panel in figure [ fig : sigma ] ) , which has a continuous distribution for the small dust with flat @xmath84 and @xmath104 , and no big dust at @xmath39 as well . the panels are the raw images from the radiative transfer simulations ( left ) , images convolved by a gaussian profile with resolution @xmath0@xmath87 ( middle , to mimic the sma observations , a11 ) and @xmath0@xmath58 ( right , to mimic future alma observations ( section [ sec : future ] ) . both models reproduce the characteristic features in the sma images of this transitional disk sample : a bright ring at the cavity edge and a flux deficit inside , agree with the semi - analytical analysis in section [ sec : submm - intensity ] , but _ very different _ nir scattered light images . the intrinsic reason for this apparent inconsistency is , as we discussed above , that big and small dust dominate the sub - mm and nir signals in our models , respectively . thus two disks can have similar images at one of the two wavelengths but very different images at the other , if they share similar spatial distributions for one of the dust populations but not the other . lastly , we comment on the effect of big to small dust ratio , which is fixed in this work as 0.85/0.15 to simplify the model ( see the discussion of depletion of the small dust in the surface layer of protoplanetary disks , @xcite ) . the scattering comes from the disk surface and is determined by the grazing angle , which only weakly depends on the small dust surface density , if it is continuous and smooth ( section [ sec : image - sigmai ] ) . changing the mass fraction of the big dust in the outer disk from 0.85 to 0.95 in our @xmath104 models ( effectively a factor of 3 drop in surface density of the small dust everywhere ) introduces a @xmath020% drop in the surface brightness of the scattered light images , but a factor of 3 drop in the cavity 880 @xmath21 m intensity ( @xmath146 at @xmath39 ) . lastly , we note that since the big - to - small dust sub - mm emission ratio is @xmath147 , the small grains must contain more than 90% of the total dust mass to dominate the sub - mm emission . in this section , we explore the parameter degeneracy in reproducing the transitional - disk - like sed with their distinctive nir - mir dips . sed fitting ( particularly of the irs spectrum ) can only provide constraints on the spatial distribution of the small dust within a few or a few tens au from the center , and it contains strong degeneracy in the parameter space ( a11 ) . below , we show that disk models with different cavity structures can produce roughly the same sed , containing the transitional disk signature , as long as their innermost parts are modestly depleted ( by a factor of @xmath01000 or so ) . except for the specifically mentioned parameters , the other parameters of these models are the same as for the fiducial model in section [ sec : imageresult ] ; in particular there is no big dust inside the cavity . figure [ fig : sed ] shows the sed for four disk models varied based on the fiducial model in section [ sec : imageresult ] . the model for the thick dashed curve has a uniformly heavily depleted cavity with rising @xmath84 , @xmath148 , and no big dust inside the cavity ( illustrated by the thin dashed curve in the left panel of figure [ fig : sigma ] ) . the scale height profile has @xmath149 and @xmath150 at 100 au . the inner rim is puffed up by 100% and the outer wall is puffed up by 200% . the inclination is assumed to be @xmath151 , @xmath152 au , and @xmath153 au . this model is motivated by the a11 disk+cavity structure . the full small dust disk model ( the thin dashed curve ) has otherwise identical properties but @xmath102 ( i.e. completely filled cavity for the small dust ) . the other two smooth small - dust disk models have much more massive inner disks with @xmath104 ( i.e. a continuous small dust disk ) and no puffed up inner rim or cavity wall . the solid curve model has flat @xmath84 ( equation , illustrated by the thick solid curve in the left panel of figure [ fig : sigma ] ) , @xmath154 and @xmath155 at 100 au . the dash - dotted curve model has declining @xmath84 ( equation , illustrated by the dash - dotted curve in the left panel of figure [ fig : sigma ] ) , @xmath156 , and @xmath157 at 100 au . the two smooth small dust disk models with flat or declining @xmath84 produce qualitatively similar sed as the uniformly heavily depleted model ( in particular , roughly diving to the same depth at nir , and coming back to the same level at mir , as the signature of transitional disks ) , despite the fact that they have very different structures inside the cavity . the minor differences in the strength of the silicate feature and the nir flux could be reduced by tuning the small dust model and using a specifically designed scale height profile at the innermost part ( around the sublimation radius or so ) . the main reasons for the similarity are : 1 . the depletion factor ( or the surface density ) at the innermost part ( from @xmath29 to @xmath01 au or so ) . while the two smooth small dust disk models differ by @xmath05 orders of magnitude on the depletion factor ( or the surface density ) at the cavity edge from the uniformly heavily depleted model , the difference is much smaller at the innermost disk , where most of the nir - mir flux is produced . at the innermost disk , the small dust is depleted by @xmath03 orders of magnitude in the flat @xmath84 model , @xmath05 orders of magnitude for the declining @xmath84 model , and @xmath05 orders of magnitude in the uniformly heavily depleted model ( with rising @xmath84 ) . on the other hand , the integrated depletion factor @xmath158 for the small dust is @xmath00.3 for the flat @xmath84 model , @xmath00.2 for the declining @xmath84 , and @xmath010@xmath159 for the uniformly heavily depleted model , more in line with @xmath103 , because most of the mass is at the outer part of the cavity . we note that the total amount of small dust is not as important as its spatial distribution inside the cavity , and the amount of dust in the innermost part , in determining the nir - mir sed . the scale height of small grains @xmath25 at the innermost part . the two smooth small dust disk models are more flared than the uniformly heavily depleted model . while the three have roughly the same scale height outside the cavity , the difference increases inward . at 1 au , @xmath25 for the uniformly heavily depleted model is 1.7@xmath160 that of the flat @xmath84 model and @xmath161 that of the declining @xmath84 model . 3 . the puffed up inner rim . the inner rim scale height is doubled in the heavily depleted model , which increases the nir flux and reduces the mir flux since the puffed up rim receives more stellar radiation and shadows the disk behind it . the puffed up inner rim is removed in the flat or declining @xmath84 models . the surface density ( or the depletion factor ) and the scale height at the innermost part are considerably degenerate in producing the nir to mir flux in the sed ( a11 ) . in general , a disk which has a higher surface density and scale height at the innermost part and a puffed up inner rim intercepts more stellar radiation at small radii , and has more dust exposed at a high temperature , so it produces more nir flux . on the other hand , the shadowing effect cast by the innermost disk on the outer disk causes less mir emission @xcite . in this way , changes in some of these parameters could be largely compensated by the others so that the resulting sed are qualitatively similar . however , in order to reproduce the characteristic transitional disk sed , the value of the depletion factor inside the cavity can not be too high . the increasing surface density at small radii would eventually wipe out the distinctive sed deficit , and the resulting sed evolves to a full - disk - like sed , as illustrated by the full small dust disk model in figure [ fig : sed ] . in our experiments with not too flared @xmath65 ( comparing with the canonical @xmath81 in irradiated disk , @xcite ) , we find an upper limit on the order of @xmath162 for the depletion factor in the innermost part in our smooth disk models . we note that this limit depends on the detailed choices of the disk and cavity geometry , such as @xmath36 and @xmath28 , and the big - to - small - dust ratio in the outer disk . the inner rim is puffed up in a11 to intercept the starlight and to shadow material at larger radii . at a given radius , the scale height @xmath32 of the gas disk scales as @xmath163 where @xmath164 is the orbital frequency and @xmath165 is the isothermal sound speed in the disk @xmath166 where @xmath128 is the ( mid - plane ) temperature and @xmath21 is the average weight of the particles . if the scale heights of the dust and the gas are well coupled ( e.g. for well - mixed - gas - dust models or a constant level of dust settling ) , tripling the rim scale height ( not unusual in a11 ) means an order of magnitude increase in its temperature . due to the sudden change of the radial optical depth from @xmath00 to unity in a narrow transition region directly illuminated by the star , some puffing up may be present , but probably not that significant . in addition , @xcite pointed out that a realistic puffed up rim has a curved edge ( away from the star ) instead of a straight vertical edge due to the dependence of @xmath167 on pressure , which further limits the ability of the rim to shadow the outer disk . based on these reasons , we choose not to have the rim puffed up in our models , though we note that a relatively weak puffing up , as in the uniformly heavily depleted model here , does not make a major difference in the results . similar idea applies to the puffed up wall as well . the typical @xmath65 value assumed in a11 in this sample ( @xmath168 ) is small compared with the canonical values for irradiated disk models ( @xmath81 , @xcite ) . this leads to that the temperature determined by the input scale height ( @xmath169 , equation and ) may increase inwardly too steeply compared with the output mid - plane temperature calculated in the code ( @xmath170 ) . @xmath169 at 100 au in the three models are close to each other due to their similar scale height there , and all agree with @xmath170 ( @xmath030 k ) within @xmath171 . however , at 1 au , while @xmath169 in our smooth small dust disk models is close to @xmath170 ( @xmath0220 k , within @xmath172 ) , the input temperature in the uniformly heavily depleted model appears to be too high by a factor of @xmath03 . first , we review the _ direct _ , model - independent constraints on the disk structure which the three observations the infrared sed , sma sub - mm observations , and seeds nir polarized scattered light imaging put on many transitional disks in this cross sample : 1 . irs reveals a distinctive dip in the spectra around 10 @xmath21 m , which indicates that the small dust ( @xmath0@xmath21m - sized or so ) in the inner part of the disk ( from @xmath29 to several au ) must be moderately depleted . however , due to degeneracy in parameter space , the detailed inner disk structure is model dependent . models with different cavity depletion factors , @xmath84 , and scale heights at the innermost disk could all reproduce the transitional disk signature . the irs spectra are not very sensitive to the distribution of big dust . the sma images show a sub - mm central cavity , which indicates that the big dust ( mm - sized or so , responsible for the sub - mm emission ) is heavily depleted inside the cavity . however , while the observations can effectively constrain the spatial distribution of the big dust outside the cavity , they can place only upper limits on its total amount inside the cavity . sma observations do not place strong constraints on the distribution of the small dust , though a weak upper limit for the amount of small dust inside the cavity may be determined based on the sma noise level . seeds nir polarized scattered light images are smooth on large scales , and have no clear signs of a central cavity . the radial profiles of many images increase inwardly all the way from the outer disk to the inner working angle without sudden jumps or changes of slope , indicating that the scattering surfaces and their shapes are smooth and continuous ( outside @xmath9 ) . on the other hand , scattered light images are not very sensitive to the detailed surface density profiles and the total amount of small dust inside the cavity . the nir images normally do not provide significant constraints on the distribution of the big dust . in this work , we propose a generic disk model which grossly explains all three observations simultaneously . previous models in the literature which assume a full outer disk and a uniformly heavily depleted inner cavity can reproduce ( 1 ) and ( 2 ) , but fail at ( 3 ) , because they also produce a cavity in the scattered light images , which contradicts the new seeds results . through radiative transfer modeling , we find that qualitatively ( 3 ) is consistent with a smooth disk of small dust with little discontinuity in both surface density and scale height profile . table [ tab : models ] summarizes the key points in various models and compares their performances in these three observations . since we focus on generic disk models only which reproduce the gross features in observations , and we do not try to match the details of specific objects , we more or less freeze many nonessential parameters in sections [ sec : image]-[sec : sed ] which do not qualitatively change the big picture for simplicity . the important ingredients include the dust properties ( mostly for the small dust , both the size distribution and the composition ) , @xmath114 , the big - to - small - dust ratio , and @xmath173 ( both the absolute scale and @xmath65 ) . nir scattered light images are able to provide constraints on some of these parameters ( particularly @xmath173 and @xmath65 ) , due to the dependence of the position and the shape of the disk surface on them . these parameters were not well constrained previously using sub - mm observations and sed due to strong degeneracies ( a11 ) . we note that alternative models for explaining the scattered light images exist , but generally they require additional complications . as one example , if the small dust is not depleted in the outer part of the sub - mm cavity , but is heavily depleted inside a radius smaller than @xmath9 ( @xmath174 au ) , then it is possible to fit all the three observations , in which case the _ small dust cavity _ does not reveal itself in the scattered light images due to its small size . however , in this case one needs to explain why different dust populations have different cavity sizes . future scattered light imaging with even smaller @xmath9 may test this hypothesis . as another example , while we achieve a smooth scattering surface by having continuous surface density and scale height profiles for the small dust , it is possible to have the same result with discontinuities in both , but with just the right amount such that the combination of the two yields a scattering surface inside the cavity smoothly joining the outer disk . this may work if the cavity is optically thick ( i.e. not heavily depleted ) , so that a well - defined scattering surface inside the cavity exists . experiments show that for the models in section [ sec : imageresult ] , uniformly depleting the small dust by a factor of @xmath01000 and tripling the scale height inside the cavity would roughly make a smooth scattering surface . however , fine tuning is needed to eliminate the visible edge from a small mismatch in the two profiles . also , the thicker cavity shadows the outer disk , and makes it much dimmer in scattered light ( by about one order of magnitude ) . lastly , without tuning on the scale height and/or surface density in the innermost part , this model produces too much nir - mir flux and too small flux at longer wavelengths in its sed , due to its big scale height at small @xmath15 and the subsequent shadowing effect . there are several important conclusions that can be drawn based on our modeling of the transitional disks at different wavelengths . first , as we discussed in section [ sec : image - deltacav015 ] , for some objects the _ lower _ limit for the depletion of the _ small _ dust at the cavity edge ( as constrained by the scattered light images ) is likely to be above the _ upper _ limit for the _ big _ dust constrained by the sma , based on the modeling results and the noise level in the two instruments . this essentially means that the small dust has to spatially _ decouple _ from the big dust at the cavity edge . this is the first time that this phenomenon has been associated with a uniform sample in a systematic manner . detailed modeling of both images for individual objects is needed , particularly in order to determine how sharp the big dust cavity edge is from the sub - mm observations , to pin down the two limits and check if they are really not overlapping . while we defer this to future work , we note that having the big dust the same surface density as the small dust inside the cavity probably can not reproduce the sub - mm images . experiments show that even with our declining @xmath84 ( [ eq : sigmai - linearneg ] ) , a fixed big / small dust ratio and a continuous surface density for both throughout the disk ( i.e. @xmath175 ) produce a sub - mm cavity with a substantially extended edge , and the central flux deficit disappears in the smeared out image . if this is confirmed , it further leads to two possibilities : ( a ) whatever mechanism responsible for clearing the cavity have different efficiency for the small and big dust , or ( b ) there are other additional mechanisms which differentiate the small and big dust after the cavity clearing process . at the moment it is not clear which one of the two possibilities is more likely , and both need more thorough investigations . second , as we argued in section [ sec : sed ] , in order to reproduce the distinctive nir deficit in the transitional disk sed , an effective `` upper limit '' of @xmath100 at the innermost region is required , which in experiments with our disk parameters is on the order of @xmath162 . this is far from the _ lower limit _ of @xmath100 at the cavity edge ( close to 1 ) , constrained by the scattered light images . together , the two limits indicate that the spatial distribution of the small grains is very different inside and outside the cavity specifically , @xmath114 tends to be flat or even decrease inwardly inside the cavity . in addition , this implies that the gas - to - dust ratio needs to increase inwardly , given that most of these objects have non - trivial accretion rates ( @xmath176yr , a11 ) . for a steady shakura & sunyaev disk , the accretion rate @xmath177 is related to the gas surface density @xmath178 as : @xmath179 where @xmath18 is the shakura - sunyaev viscosity parameter , @xmath164 is the angular velocity of the disk rotation , and @xmath165 is given by equation . at @xmath00.1 au , equation predicts @xmath180@xmath181 g @xmath182 , assuming a temperature @xmath183 k , @xmath184yr , @xmath185 , and @xmath186 as typical t tauri values . this is very different from our upper limit of @xmath1801 g @xmath182 in the innermost disk , obtained assuming a fixed gas - to - dust ratio of 100 ( the flat @xmath84 models in figure [ fig : sigma ] ) . to simultaneously have large @xmath178 but small @xmath187 in the innermost disk , the gas - to - dust ratio needs to increase substantially from the nominal value of 100 ( by a factor of @xmath188 in our models , echos with @xcite ) . this could put constraints on the cavity depletion mechanism or dust growth and settling theory . at the moment , the mechanism(s ) which are responsible for clearing these giant cavities inside transitional disks are not clear ( see summary of the current situation in a11 ) . regarding the applications of our model on this subject , we note two points here . the flat / declining surface density of the small grains inside the cavity in our models is consistent with the grain growth and settling argument , that the small grains in the inner disk are consumed at a faster rate due to higher growth rate there @xcite . also , the so - called dust filtration mechanism seems promising for explaining why small dust but not big dust is present inside the cavity @xcite , since it could effectively trap the big dust at a pressure maximum in the disk but filter through the small dust . particularly , combining the two ( dust growth and dust filtration ) , @xcite proposed a transitional disk formation model from a theoretical point of view to explain the observations , and their predicted spatial distribution of both dust populations in the entire disk is well consistent with the ones here . imaging is a very powerful tool for constraining the structure of protoplanetary disks and the spatial distribution of both the small and big dust , and there are many ongoing efforts aiming at improving our ability to resolve the disks . in the direction of optical - nir imaging , updating existing coronagraph and adaptive optics ( ao ) systems , such as the new coronagraphic extreme adaptive optics ( scexao ) system on subaru ( @xcite , which could raise the strehl ratio to @xmath00.9 ) , are expected to achieve better performance and smaller @xmath9 in the near future . to demonstrates the power of the optimal performances of these next generation instruments in the nir imaging , figure [ fig : nextgenerationpsf ] shows the surface brightness radial profile of several masked @xmath56-band disk images convolved from the _ same _ raw image by _ different _ psf . except the dotted curve , all the other curves are from the model corresponding to the middle row in figure [ fig : image - variation ] , which has a @xmath87 radius cavity , flat @xmath84 , and @xmath116 ( a @xmath189 drop in @xmath114 at @xmath28 ) . we use three psfs : the current hiciao psf in h band , which could be roughly approximated by a diffraction limited core of an 8-m telescope ( resolution @xmath0@xmath59 ) with a strehl ratio of @xmath00.4 plus an extended halo ; mock psf i ( to mimic scexao ) , which is composed of a diffraction limited core of an 8-m telescope with a strehl ratio of 0.9 , and an extended halo similar in shape ( but fainter ) as the current hiciao psf ; model psf ii ( to mimic the next generation thirty - to - forty meter class telescopes ) , which is composed of a diffraction limited core of a 30-m telescope ( resolution @xmath0@xmath190 ) with a strehl ratio of 0.7 , and a similar halo as the previous two . compared with the full small dust disk case , all the convolved images of the @xmath116 model shows a bump at @xmath28 and a relative flux deficit at @xmath39 . however , from the current hiciao psf to model psf i and ii , the contrast level of the cavity becomes higher and higher , and closer and closer to the raw image ( which essentially has an infinite spatial resolution ) . with these next generation instruments , the transition of the spatial distribution of the small dust at the cavity edge will be better revealed . on the other hand , in radio astronomy , the atacama large millimeter array ( alma ) is expected to revolutionize the field , with its much better sensitivity level and exceptional spatial resolution ( @xmath0@xmath58 or better ) . as examples , the right panels in figure [ fig:880um ] show images convolved by a gaussian profile with resolution @xmath0@xmath58 , which mimic the ability of alma and show two prominent improvements over the images under the current sma resolution ( @xmath0@xmath87 , middle panels ) . first , the edge of the cavity is much sharper in the mock alma images . this will make the constraint on the transition of the big dust distribution at the cavity edge much better . second , while the weak emission signal in the bottom model is overwhelmed by the halo of the outer disk in the mock sma image , resulting in that the bottom model is nearly indistinguishable from the top model ( which essentially produces zero cavity sub - mm emission ) , the mock alma image successfully resolves the signal as an independent component from the outer disk , and separates the two models . this weak emission signal traces the spatial distribution of the dust ( both populations ) inside the cavity , which is the key in understanding the transitional disk structure . at this stage , the total number of objects which have been observed by all three survey - scale projects ( using irs / sma / subaru ) is still small . increasing the number in this multi - instrument cross sample will help clear the picture . in addition , future observations which produce high spatial resolution images at other wavelengths , such as uv , optical , or other nir bands ( for example using hst , @xcite , or the future jwst ) , or using interferometer ( such as the astrometric and phase - referencing astronomy project on keck , @xcite , and amber system on very large telescope interferometer , @xcite ) should also be able to provide useful constraints on the disk properties . we summarize this paper by coming back to the question which we raised at the beginning : what kind of disk structure is consistent with and is able to reproduce the characteristic signatures in all three observations of transitional protoplanetary disks : a high contrast cavity in sub - mm images by sma , a nir deficit in sed by spitzer irs , and a smooth radial profile in nir polarized scattered light images by subaru hiciao . we propose one generic solution for this problem , which is feasible but by no means unique . the key points are : 1 . a cavity with a sharp edge in the density distribution of big grains ( up to @xmath0mm - sized ) and with a depletion factor of at least 0.1 - 0.01 inside is needed to reproduce the sma sub - mm images , as pointed out by a11 . 2 . right inside the cavity edge ( @xmath015 - 70 au ) , the surface density for the small dust ( @xmath0micron - sized and smaller ) does not have a big sudden ( downward ) jump ( a small discontinuity may exist ) . the seeds nir scattered light images , which typically detect the disk at @xmath19115 au ( the inner working angle in seeds ) , generally require continuous / smooth profiles for the surface density and scale height of the small dust . the small dust in the innermost region ( i.e. within a few au , on a scale smaller than measured by seeds ) has to be moderately depleted in order to produce the transitional - disk - like sed , assuming the disk is not too flared , but the exact depletion factor is uncertain and model dependent . as we discussed in section [ sec : ourfeature ] , combining all the above points , our model suggests that the spatial distributions of the big and small dust are _ decoupled _ inside the cavity ( particularly at the cavity edge ) . also , our model argues that the surface density of the small dust inside the cavity is flat or decreases with radius , consistent with the predictions in dust growth models . combined with the accretion rate measurement of these objects , it further implies that the gas - to - dust ratio increases inwardly inside the cavity of transitional disks . r.d . thanks sean andrews , nuria calvet , eugene chiang , bruce draine , catherine espaillat , elise furlan , and jim stone for useful conversations and help . this work is partially supported by nsf grant ast 0908269 ( r. d. , z. z. , and r. r. ) , ast 1008440 ( c. g. ) , ast 1009314 ( j. w. ) , ast 1009203 ( j. c. ) , nasa grant nnx22sk53 g ( l. h. ) , and sloan fellowship ( r. r. ) . we thank pascale garaud and doug lin for organizing the international summer institute for modeling in astrophysics ( isima ) at kavli institute for astronomy and astrophysics , beijing , which facilitated the discussion of radiative transfer modeling among r. d. , l. h. , t. m. , and z. z .. we would also like to thank the anonymous referee for suggestions that improved the quality of the draft .

transitional circumstellar disks around young stellar objects have a distinctive infrared deficit around 10 microns in their spectral energy distributions ( sed ) , recently measured by the _ spitzer infrared spectrograph _ ( irs ) , suggesting dust depletion in the inner regions . these disks have been confirmed to have giant central cavities by imaging of the submillimeter ( sub - mm ) continuum emission using the _ submillimeter array _ ( sma ) . however , the polarized near - infrared scattered light images for most objects in a systematic irs / sma cross sample , obtained by hiciao on the subaru telescope , show no evidence for the cavity , in clear contrast with sma and spitzer observations . radiative transfer modeling indicates that many of these scattered light images are consistent with a smooth spatial distribution for micron - sized grains , with little discontinuity in the surface density of the micron - sized grains at the cavity edge . here we present a generic disk model that can simultaneously account for the general features in irs , sma , and subaru observations . particularly , the scattered light images for this model are computed , which agree with the general trend seen in subaru data . decoupling between the spatial distributions of the micron - sized dust and mm - sized dust inside the cavity is suggested by the model , which , if confirmed , necessitates a mechanism , such as dust filtration , for differentiating the small and big dust in the cavity clearing process . our model also suggests an inwardly increasing gas - to - dust - ratio in the inner disk , and different spatial distributions for the small dust inside and outside the cavity , echoing the predictions in grain coagulation and growth models .

at specific molecular sites . the vertical position @xmath0 controls the tip - molecule coupling @xmath1 , while the molecule - substrate coupling is fixed . ( b ) planar approach : by embedding a bottom - up synthesized magnetic molecule into a solid - state device , one can control its energy levels through a gate - voltage @xmath2 . this _ scanning in energy space _ grants access to both regimes of `` real '' ( redox ) charging and `` virtual '' charging ( scattering ) and their nontrivial crossover . ( c ) conductance map showing a range of features in the regime ( center ) , regimes ( far left and right ) as well as the crossover regime . these are analyzed in detail in fig . [ fig:10 ] and fig . [ fig:15 ] . ] both the fundamental and applied studies on transport phenomena in electronic devices of molecular dimensions have bloomed over the past decade@xcite . an interesting aspect of this development is that it has increasingly hybridized the diverse fields of chemistry , nanofabrication and physics with the primary ambition of accessing properties like high spin and large exchange couplings , vibrational modes , large charging energies and long electronic / nuclear spin coherence times , subtle electronic orbital interplay , self - organisation @xcite and chirality @xcite . this is rendered possible by the higher energy scales of the molecular systems a direct consequence of their size and their complex , chemically tailorable , inner structures which have proven to be effective in addressing , for instance , the spin - phonon@xcite , shiba@xcite and kondo physics@xcite , quantum interference effects@xcite and nuclear spin manipulation@xcite . in most of the works , in particular those concerning molecular _ spin _ systems , two complementary approaches have contributed to explore these effects . on the one hand stands transport spectroscopy , which is the major tool of choice in the scanning - tunneling microscopy ( stm ) approach to nanoscale spin systems @xcite , depicted in fig . [ fig:1](a ) . is also dominant in the field of mechanically - controlled break junctions ( mcbj)@xcite to study vibrations @xcite and , less often , spin effects@xcite . on the other hand , transport spectroscopy , originating in the multi - terminal fabrication of quantum dots ( qds , fig . [ fig:1](b))@xcite , is a well - developed tool applied to a broad range of excitations in nanostructures , @xcite including spin.@xcite l l l l * regime * ' '' '' & * section * & * qd community * & * stm / mcbj community * + ' '' '' & sec . [ sec : qd ] & single - electron tunneling ( set ) & resonant tunneling + & & sequential / incoherent tunneling & + & sec . [ sec : stm ] & ( in)elastic co - tunneling ( cot ) & ( in)elastic electron tunneling spectroscopy ( eets / iets ) + & & coherent tunneling & + & & schrieffer - wolff ( transformation ) & appelbaum ( hamiltonian ) + & sec . [ sec : cot ] & & pump - probe ( co)tunneling spectroscopy + crossover & sec . [ sec : crossover ] & cotunneling - assisted single - electron & + & & tunneling ( coset , cast ) & + the key difference between and approaches is the former s reliance on energy - level control _ independent _ of the transport bias , i.e. , true _ gating _ of the molecular levels@xcite , which should be distinguished from the capacitive level shift in stm which is _ caused _ by the bias . in terms of physical processes , this difference corresponds to spectroscopy relying on `` real '' charging of the molecule and transport involving only `` virtual '' charging . in this contribution we discuss a comprehensive picture of transport applicable to a large family of nanoscale objects . this is motivated by the experimental spectrum of a molecular junction depicted in fig . [ fig:1](c ) . such a conductance map is so full of detail that it warrants a systematic joint experimental and theoretical study . in particular , we discuss several effects which are often overlooked despite their importance to electron transport spectroscopy and despite existing experimental @xcite and theoretical works @xcite . for instance , it turns out that `` inelastic '' or `` off - resonant '' transport is _ not _ simply equivalent to the statement that `` resonant processes play no role '' . in fact , we show that generally less than @xmath3 of the parameter regime of applied voltages that nominally qualified as `` off - resonant '' is actually described by the widely used inelastic ( co)tunneling ( or iets ) picture . although in many experiments to date this has not been so apparent , our experimental evidence suggests that this needs consideration . in theoretical considerations , and transport regimes are often taken as complementary . our measurements illustrate how this overlooks an important class of relaxation processes . the breakdown of the picture in the regime presents , in fact , new opportunities for studying the relaxation of molecular spin - excitations which are of importance for applications . interestingly , these resonances are qualitative indicators of a device of high quality , e.g. , for applications involving spin - pumping . we illustrate _ experimentally _ the ambiguities that the sole modeling of conductance curves can run into . for instance , we show that this may lead one to infer quantum states that do not correspond to real excitations , but are simply _ mirages _ of lower lying excitations , including their zeeman splittings . although elaborated here for a spin system , our conclusions apply generally , for example to electronic @xcite and vibrational excitations in nano electro - mechanical systems ( nems ) @xcite . the outline of the paper is as follows : in sec . [ sec : theory ] we review the physical picture of electron tunneling spectroscopy and outline how a given spectrum manifests itself in and transport spectra . in sec . [ sec : crossover ] we discuss how these two spectra continuously transform into each other as the energy levels are varied relative to the bias voltage . with this in hand , we put together a physical picture capturing all discussed effects which will be subsequently applied to describe the experiment in sec . [ sec : experiment ] . in sec . [ sec : experiment ] we follow the reverse path of experimental transport spectroscopy : we reconstruct the excitation spectrum of a high - spin molecular junction based on the feature - rich transport spectra as a function of bias voltage , magnetic field , and gate voltage . starting from the analysis , we use the boundary conditions imposed by the spectrum to resolve a number of ambiguities in the state - assignment . with the full model in hand we highlight two informative transport features : ( i ) _ nonequilibrium _ , i.e. , a pump - probe spectroscopy using the electronic analog of raman transitions and ( ii ) _ mirages _ of resonances that occur well inside the regime . we conclude with an outlook in sec . [ sec : discussion ] . since we aim to bring the insights from various communities together , we summarize in table [ tab : compare ] the different but equivalent terminology used . for clarity reasons we set @xmath4 for the rest of this discussion . with different charge @xmath5 and further quantum numbers denoted by @xmath6 , as sketched in the lower panel . due to the capacitive coupling to a gate electrode these energy differences can be tuned to be ( b ) on - resonance and ( c ) off - resonance with the electrode continuum . ( b ) `` real '' charging : absorption of an electron , reduces the molecule for real , @xmath7 , going from the ground state @xmath8 for charge @xmath5 to an excited state @xmath9 with charge @xmath10 . since this is a one - step process , the rate scales with @xmath11 , the strength of the tunnel coupling . ( c ) `` virtual charging '' : the `` scattering '' of an electron `` off '' or `` through '' the molecule proceeds via any virtual intermediate state , for example , starting from the ground state @xmath8 and ending in a final excited state @xmath9 , @xmath12 . the rate of such a two - step process scales as @xmath13 . in this case charging is considered only `` virtual '' , as no redox reaction takes place : although energy and angular momentum are transferred onto the molecule , the electron number remains fixed to @xmath10 . ] the two prevalent conceptual approaches to transport through molecular electronic devices are characterized by the simple physical distinction , sketched in fig . [ fig:2 ] , between `` real '' charging chemical reduction or oxidation and `` virtual '' charging electrons `` scattering '' between contacts `` through '' a molecular `` bridge '' . theoretically , the distinction rests on whether the physical processes appear in the leading or next - to - leading order in the tunnel coupling strength , @xmath11 , relative to the thermal fluctuation energy @xmath14 . experimentally , this translates into distinct applied voltages under which these processes turn on . these conditions are the primary spectroscopic indicators , allowing the distinction between `` real '' and `` virtual '' transport processes , and take precedence over line shape and lifetime broadening . for reviews on theoretical approaches to molecular transport see refs . . real charging forms the starting point of what we will call the picture of transport ( see table [ tab : compare ] for other nomenclature ) . its energy resolution is limited by the heisenberg lifetime set by the tunnel coupling @xmath15 @xmath16 allowing for sharp transport spectroscopy of weakly coupled systems . this relation has a prominent place in the field of qds which covers artificial structures as `` artificial atoms '' and `` artificial molecules '' with redox spectra @xcite very similar to real atoms @xcite and simple molecules @xcite . resonant transport also plays a role in stm although its energy resolution is often limited by the strong coupling typical of the asymmetric probe - substrate configuration . given sufficient weak coupling / energy resolution , much is gained when the energy - level dependence of these transport spectra , can be mapped out as function of _ gate - voltage_. this dependence allows a detailed model to be extracted involving just a few electronic orbitals @xcite , their coulomb interactions @xcite and their interaction with the most relevant degrees of freedom ( e.g. , isotropic @xcite and anisotropic spins @xcite , quantized vibrations @xcite , and nuclear spins @xcite ) . in particular electronic @xcite , spin - orbit @xcite structure as well as electro - mechanical coupling @xcite of cnts have been very accurately modeled this way . in molecular electronics transport spectroscopy takes a prominent role since imaging of the device is challenging . by moving to molecular - scale gated structures one often compromises real - space imaging . in this paper we highlight the advantages that such structures offer . nevertheless , electrical gates that work simultaneously with a scanning tip @xcite or a mcbj @xcite have been realized , but with rather low gate coupling . notably , mechanical gating @xcite by lifting a single molecule from the substrate has been demonstrated , resulting in stability diagrams where the role of @xmath17 taken over by the tip - height @xmath0 in fig . [ fig:1 ] . a scanning quantum - dot @xcite has also been realized using a single - molecule @xcite . in the transport regime one considers processes of the leading order in the tunnel coupling @xmath11 , cf . ( [ eq : lifetime - set ] ) . although most of this is in principle well - known , we review this approach @xcite since some of its basic consequences for the _ regime _ discussed below are often overlooked . typically , analysis of spectra requires a model hamiltonian @xmath18 that involves at most tens of states in the most complex situations @xcite . its energies @xmath19 are labeled by the charge number @xmath5 and a further quantum numbers ( orbital , spin , vibrational ) collected into an index @xmath6 . crucial for the following discussion is the voltage - dependence of this energy spectrum . we assume it is uniform , i.e. , @xmath20 , independent of further quantum numbers @xmath6 . this can be derived from a capacitive description of the coulomb interactions between system and electrodes referred to as the _ constant interaction model _ @xcite . in this case , @xmath21 where @xmath19 are constants and @xmath22 ( @xmath23 ) is the potential applied at source ( drain ) electrode . here , @xmath24 for @xmath25 , @xmath26,@xmath27 are capacitive parameters of which only two are independent since @xmath28 . in sec . [ sec : break ] we discuss corrections to this often good assumption@xcite . unless stated otherwise , we will set for simplicity @xmath29 , i.e. , the negative shift of the energy levels equals the gate voltage . the bias is applied to the electron source , @xmath30 , and the drain is grounded , @xmath31 , giving @xmath32 and @xmath33 with constant @xmath34 . unless stated otherwise , schematics are drawn assuming @xmath35 , corresponding to symmetric and dominant source - drain capacitances @xmath36 . the hamiltonian for the complete transport situation takes the generic form @xmath37 where @xmath38 is a sum of tunneling hamiltonians that each transfers a single electron across one of the junctions to either metal electrodes . the electrodes , labeled by @xmath39l(left ) , r(right ) , are described by @xmath40 essentially through their densities of states and by their electrochemical potentials @xmath41 and temperature @xmath14 . for the present purposes this level of detail suffices , e.g. , see ref . for details . for a tunneling process involving such a transfer of precisely one electron , one of the electrochemical potentials has to fulfill @xmath42 in order for the electron to be injected into an @xmath5-electron state @xmath6 , resulting in the final @xmath43-electron state @xmath44 . below this threshold the state @xmath45 is `` unstable '' , and decays back to @xmath46 by expelling the electron back into the electrode . the rate for the injection process , @xmath47 , is given by familiar `` golden rule '' expressions and depends on the difference of both sides of eq . ( [ eq : set - res ] ) relative to temperature @xmath14 . when the process `` turns on '' by changing @xmath48 , it gives rise to a peak in the differential conductance , , corresponding to a sharp step in current , of width @xmath14 and height @xmath49 ( in units of @xmath50 ) since we are assuming weak coupling and high temperature . if the total system conserves both the spin and its projection along some axis ( e.g. , the @xmath51-field axis ) , the rate involves a selection - rule - governed prefactor . this prefactor is zero unless the change of the molecular spin and its projection satisfy @xmath52 these conditions reflect the fact that only a single electron is available for transferring spin to the molecule . incidentally , we note that this picture is very useful even beyond the weak couplings and high temperatures assumed here . close to the resonance defined by condition ( [ eq : set - res ] ) the transport still shows a peak which is , however , modified by higher - order corrections . the width of the current step becomes broadened @xmath53 , giving a conductance peak @xmath54 in units of @xmath50 . its energy position may shift on the order of @xmath11 . ) ( shaded ) for two charge states @xmath5 and @xmath43 . the boundary lines ( bold ) , where @xmath55 for @xmath56 , have slopes @xmath57 and @xmath58 , respectively , allowing the capacitive parameters to be determined . the green lines , offset horizontally by @xmath59 , indicate the window of accessibility of the excited state @xmath60 and are defined by @xmath61 . ( b ) similar to figure ( a ) , for three charge states @xmath5 , @xmath43 and @xmath62 . this adds a copy of the bias window of ( a ) that is horizontally offset by the energy @xmath63 [ eq . ( [ eq : u ] ) ] with boundaries @xmath64 for @xmath65 . the excitation lines on the right ( green ) are mirrored horizontally , @xmath66 for @xmath56 , since electron processes relative to @xmath43 have become hole processes . ] it is now clear in which regime of applied voltages the above picture applies . in fig . [ fig : set ] this is sketched in the plane of applied bias ( @xmath48 ) and gate voltage ( @xmath17 ) . here , we call such a ( schematic ) intensity plot also known as `` stability diagram '' or `` coulomb - diamond'' a _ transport spectrum_. the indicated vertical line cuts through this diagram correspond to traces measured in stm or mcbj experiments . applied to the ground states of subsequent charge states labeled by @xmath8 , eq . ( [ eq : set - res ] ) gives the two inequalities @xmath67 these define the shaded bias window in fig . [ fig : set](a ) , delimited by the `` cross '' . here , a single electron entering from the left can exit to the right , resulting in a net directed current . it is now tempting to naively define the _ regime _ as the complement of the grey regime in fig . [ fig : set](a ) , i.e. , by moving across its boundaries by more than @xmath14 or @xmath11 . a key point of our paper is that this simple rationale is not correct already for a small finite bias matching some excitation at energy @xmath68 , indicated by green lines in . only in the linear - response regime@xcite around @xmath69 the regime can be defined as the complement of the resonant regime : @xmath70 in subsequent charge states analogous considerations apply : transitions between charge states @xmath43 and @xmath62 give rise to a shifted `` copy '' of the bias window as shown in fig . [ fig : set](b ) . the shift experimentally directly accessible is denoted by : @xmath71 this includes the charging energy of the molecule , but also the magnitude of orbital energy differences and the magnetic field . and magnetic field @xmath51 one finds @xmath72 due to the opposite spin - filling enforced by the pauli principle . ] the above rules are substantiated by a simple master equation for the stationary - state occupations @xmath73 of the states with energy @xmath19 that can be derived from the outlined model , see , e.g. , ref . . this approach is used in sec . [ sec : qd - exp ] to model part of our experiment . for the @xmath74 resonance regime the stationary - state equation reads ( for notational simplicity we here set @xmath75 ) @xmath76 here , @xmath77 is the matrix of transition rates @xmath78 between states @xmath79 and @xmath80 , and analogously for @xmath81 . for example , one of the equations , @xmath82 describes the balance between the gain in occupation probability due to all transitions @xmath83 , and the leakage @xmath84 from the state @xmath80 . the entries of the diagonal matrices @xmath85 and @xmath86 have negative values @xmath87 and @xmath88 , respectively , such that probability normalization @xmath89 is preserved in eq . ( [ eq : set - master ] ) . in the leading order in @xmath11 , the rate matrix has separate contributions from the left ( @xmath90 ) and right ( @xmath91 ) electrode : @xmath92 . these allow the stationary current to be computed by counting the electrons transferred by tunnel processes through the @xmath93-th junction , @xmath94_{f , i}^{n_f , n_i } p^{n_i}_{i } , \label{eq : set - current}\end{aligned}\ ] ] where stationarity guarantees @xmath95 . we note that , because we are considering only single - electron tunneling processes ( first order in @xmath11 ) , the primed sum is constrained to @xmath96 by charge conservation . we now take the opposite point of view and consider transport entirely due to `` virtual charging '' or `` scattering through '' the molecule . the resulting transport spectroscopy , alternatively called cotunneling ( ) or iets spectroscopy , dates back to lambe and jacklevic @xcite . the discussion of the precise conditions under which the picture applies is postponed to sec . [ sec : crossover ] . throughout we will denote by the label unless stated otherwise _ inelastic _ cotunneling . the attractive feature of relative to spectroscopy is the higher energy resolution as we explain below [ eq . ( [ eq : lifetime - cot ] ) ff . ] . exploiting this in combination with the stm s imaging capability has allowed chemical identification @xcite . this in turn has enabled atomistic modeling of the junction using _ ab - initio _ calculations @xcite , also including strong interaction effects @xcite , giving a detailed picture of transport on the atomic scale @xcite . in recent years , spectroscopy has been also intensively applied to spin systems @xcite in more symmetric @xcite stm configurations . however , it is sometimes not realized that the same spectroscopy also applies to gated molecular junction , and more generally to qds @xcite . in fact , motivated by the enhanced energy resolution , spectroscopy of discrete spin - states was introduced in gate - controlled semiconductor qds @xcite before it was introduced in stm as `` spin - flip '' spectroscopy @xcite , see also @xcite . spectroscopy is also used to study molecular properties other than spin , e.g. , vibrational states @xcite . ( b ) , now indicating the `` equilibrium '' resonance ( green horizontal line ) in which the excitation @xmath97 is reached from the ground state @xmath98 by . this horizontal line always connects to the resonances for @xmath99 and @xmath100 ( green tilted lines ) . ( b ) `` nonequilibrium '' resonances corresponding to the transition @xmath101 . case ( i ) and ( ii ) are discussed in the text . note that there is never a corresponding resonance ( crossed - out red dashed line ) connecting to a non - equilibrium excitation as is the case for `` equilibrium '' lines in ( a ) . ] in the picture , one considers transport due to next - to - leading order processes , i.e. of order @xmath13 in the tunnel rates . this involves elastic ( inelastic ) processes involving two electrons from the electrodes and a zero ( net ) energy transfer of energy . when the maximal energy supplied by the electrons one electron coming in from , say , @xmath90 at high energy @xmath102 , and the other outgoing to @xmath91 at low energy @xmath34 exceeds a discrete energy difference of the molecule , @xmath103 transport may be altered with @xmath48 . importantly , on the right hand side , all @xmath48 and @xmath17 dependences of the energies cancel out [ cf . [ sec : qd - gate ] ] since we assumed that the applied voltages _ uniformly _ shift the excitation spectrum for fixed charge.@xcite the occurrence of such a process depends on whether the initial state @xmath6 is occupied or not by another already active process . it thus depends on whether we are in the `` equilibrium '' or `` nonequilibrium '' regime , both of which are accessible in our experiment in sec . [ sec : experiment ] . the spectroscopy rules require the following separate discussion . [ [ equilibrium - inelastic ] ] `` equilibrium '' inelastic + + + + + + + + + + + + + + + + + + + + + + + + + + + already in the linear transport regime , @xmath104 and @xmath11 ) , ] there is scattering through the molecule in a fixed stable charge state in the form of _ elastic _ @xcite , see table [ tab : compare ] for the varied nomenclature . this gives rise to a small current scaling @xmath105 . with increasing bias @xmath48 , this mechanism yields a nonlinear background current which is , however , featureless . when the voltage provides enough energy to reach the lowest excitation @xmath9 of the @xmath43-electron ground state @xmath8 , the transition @xmath106 is enabled , cf . [ fig:2](c ) . this occurs when the _ gate - voltage independent _ criterion set by eq . ( [ eq : cot - res ] ) with @xmath107 and @xmath108 is satisfied : @xmath109 the above energy condition is the tell - tale sign of an process : as sketched in fig . [ fig : cot](a ) , this allows for a clear - cut distinction from processes with a gate dependent energy condition ( [ eq : set - res ] ) . importantly , such a feature always connects to the gate - dependent resonance corresponding to excitation @xmath110 . as in the regime , we stress that criterion ( [ eq : cot - res2 ] ) uses the peak position in the ( @xmath111 plane as a primary indicator . the line shape along a vertical cut in the figure , as measured in stm , may be less clear . in theoretical modeling the line shape is also not a unique indicator . the line shape is a good secondary indicator of the nature of a process . [ [ nonequilibrium - inelastic - electronic - pump - probe - spectroscopy . ] ] `` nonequilibrium '' inelastic : electronic pump - probe spectroscopy. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the above `` equilibrium '' picture of transport has been successfully applied in many instances . however , as the first excited state @xmath97 is accessed , the rules of the game change . if the relaxation induced by sources other than transport is weak enough @xcite , the occupation of the excited states can become non - negligible . in such a case , as illustrated in fig . [ fig : cot](b ) , a secondary inelastic process from the excited state @xmath9 to an even higher excited state @xmath112 should be considered . such secondary processes , with the generic condition : @xmath113 indicate a device with an intrinsic relaxation rate small compared to rates @xmath114 . as discussed in fig . [ fig : cot](b ) , such excitations _ never _ connect to a corresponding excitation in the transport spectrum . at this point , two cases have to be considered , both of which are relevant to the our experiment in sec . [ sec : pump - probe ] . \(i ) if @xmath115 i.e . , the gaps in the energy spectrum grow with energy an extra _ `` nonequilibrium '' inelastic _ resonance at bias @xmath116 appears , as illustrated in panel ( i ) of fig . [ fig : cot](b ) . this extra resonance is very useful since it provides a further consistency check on the excitations @xmath117 and @xmath118 observed independently in the . is not allowed by a selection rule , the secondary resonance may be the only evidence of this state . ) ] clearly , the intensity of such secondary `` nonequilibrium '' resonances is generally expected to be lower than the primary ones that start from the ground state . in sec . [ sec : pump - probe ] we will experimentally control this sequential `` electronic pump - probe '' excitations by tuning a magnetic field . \(ii ) in the opposite case , @xmath119 , no _ extra _ excitation related to @xmath112 appears : there is no change in the current at the lower voltage @xmath120 because the initial state @xmath97 only becomes occupied at the _ higher _ voltage @xmath117 . this is illustrated in panel ( ii ) of fig . [ fig : cot](b ) . examples of both these cases occur in the spectra of molecular magnets due to the interesting interplay of their easy - axis and transverse anisotropy , see the supplement of ref . . similar to the case , the conditions ( [ eq : cot - res])-([eq : cot - res - neq ] ) are incorporated in a simple stationary master equation for transport whose derivation we discuss further below . in particular , the occupation probabilities @xmath121 in the stationary transport state are determined by ( as previously , we put @xmath75 ) @xmath122 here , @xmath86 is a matrix of rates @xmath123 for transitions between states @xmath124 . since in the regime charging is only `` virtual '' , these transitions now occur for a fixed charge state . the matrix takes the form @xmath125 , including rate matrices @xmath126 for back - scattering from the molecule ( to the same electrode , @xmath127 ) and scattering through it ( between electrodes @xmath128 ) . the current is obtained by counting the net number of electrons transferred from one electrode to the other : @xmath129 the inclusion into this picture of the above discussed `` non - equilibrium '' effects depends whether one solves the master equation ( [ eq : cot ] ) or not . to obtain the simpler description of `` equilibrium '' inelastic one can insert _ by hand _ equilibrium populations @xmath130 directly into eq . ( [ eq : stm - current ] ) . solving , instead , eq . ( [ eq : cot ] ) without further assumptions gives the `` nonequilibrium '' inelastic case @xcite discussed above [ case ( ii ) ] . in practice , these two extreme limits both computable without explicit consideration of intrinsic relaxation are always useful to compare since any more detailed modeling of the intrinsic relaxation will lie somewhere in between. the electron tunneling rates in eq . ( [ eq : cot ] ) are made up entirely of contributions of order @xmath13 . there are two common ways of computing these rates , and we now present the underlying physics relevant for the discussion in sec . [ sec : crossover ] . [ [ appelbaum - schrieffer - wolff - hamiltonian . ] ] appelbaum - schrieffer - wolff hamiltonian. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + a conceptual connection between the `` virtual '' charging picture and the picture of `` real '' charging in sec . [ sec : qd ] emerges naturally when applying the unitary transformation @xcite due to appelbaum @xcite , schrieffer and wolff @xcite ( ) to the transport _ hamiltonian _ @xmath131 [ cf . [ sec : qd ] ] . the effective model obtained in this way allows one to easily see the key features of the spectroscopy . in this approach , the one - electron tunneling processes described by the hamiltonian @xmath15 , are transformed away and the charge state is fixed by hand to a definite integer . with that , also all the gate - voltage dependence of resonance _ positions _ [ eq . ( [ eq : cot - res2 ] ) ff . ] drops out . this new model is obtained by applying a specially chosen unitary transformation @xmath132 to the original hamiltonian such that : @xmath133 the term @xmath134 is effectively replaced by @xmath135 , which involves only @xmath13 processes and represents exclusively scattering of electrons `` off '' and `` through '' the molecule . in many cases of interest @xcite this coupling @xmath135 contains terms describing the potential ( scalar ) and exchange ( spin - spin ) scattering of electrons and holes with amplitudes @xmath136 and @xmath137 , respectively : @xmath138 in the equation above , the operators @xmath139 ( @xmath140 ) describe spin-(in)dependent intra- [ @xmath127 ] and inter - electrode [ @xmath128 ] scattering of electrons , see ref . for details . this scattering is coupled to the molecule through its charge and spin ( @xmath143 ) . _ selection rules . _ the coupling @xmath144 has selection rules that differ from the original single - electron tunnel coupling @xmath38 : @xmath145 these reflect that the two electrons involved in the scattering process have integer spin 0 or 1 available for exchange with the molecule . we will apply this in sec . [ sec : spectro ] . this is illustrated by the example model ( [ eq : ha ] ) where the spin - operator @xmath143 has matrix elements that obey eq . ( [ eq : rules - cot ] ) . _ lifetime . _ after transforming to this new effective picture , scattering becomes the leading order transport mechanism . the `` golden rule '' approach can be then applied analogously to the case of of the regime , but now with respect to the scattering @xmath144 . in this way eq . ( [ eq : cot ] ) is obtained together with an expression for the corresponding rate matrix @xmath86 . the given by eq . ( [ eq : stm - current ] ) shows gate - voltage - independent _ steps _ at energies set by eq . ( [ eq : cot - res ] ) . although at high temperatures these steps get thermally broadened @xcite , at low enough @xmath14 their broadening is smaller than that of the peaks . while calculation of this lineshape requires higher - order contributions to @xmath86 , the relevant energy scale ( inverse lifetime ) is given by the magnitude of the `` golden rule '' rates _ for the effective coupling @xmath144 _ scaling as @xmath146 this results in a much larger lifetime compared to the one from due to the role of the interactions on the molecule suppressing charge fluctuations . the smaller intrinsic broadening is a key advantage of vs. spectroscopy @xcite . _ line shape . _ due to nonequilibrium effects i.e . , the voltage - dependence of the occupations obtained by solving eq . ( [ eq : cot]) a small peak can develop on top of the step @xcite . moreover , processes beyond the leading - order in @xmath144 , which is all the approach accounts for , can have a similar effect . these turn the tunneling step into a peak and are in use for more precise modeling of experiments @xcite . spin - polarization @xcite and spin - orbit effects @xcite , however , also affect the peak shape and asymmetry . at low temperatures and sufficiently strong coupling a nonequilibrium kondo effect develops which has been studied in great detail @xcite . these works show that the peak amplitude is then enhanced nonperturbatively in the tunnel coupling , in particular for low lying excitations . this requires nonequilibrium renormalization group methods beyond the present scope and we refer to various reviews @xcite . in particular , it requires an account of the competition between the kondo effect and the current - induced decoherence @xcite in the ( generalized ) quantum master equation for the nonequilibrium density operator @xcite . from the present point of view of spectroscopy , the kondo effect can be considered as a limit of an inelastic feature at @xmath147 as @xmath148 , see fig . [ fig : cot](a ) . its position is simply @xmath149 at gate voltages sufficiently far between adjacent resonances by criterion ( [ eq : naive ] ) . in particular , for transport spectroscopy of atomic and molecular spin systems the kondo effect and its splitting into features @xcite is very important especially in combination with strong magnetic anisotropy @xcite . we refer to reviews on stm @xcite and qd @xcite studies . [ [ golden-rule-mathcalt-matrix-rates.sectmatrix ] ] `` golden rule '' @xmath150-matrix rates.[sec : tmatrix ] + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + a second way of arriving at the master equation ( [ eq : cot ] ) and the rates in @xmath86 is the so - called @xmath150-matrix approach . in essence , here is regarded as a scattering process : in the `` golden rule '' the next - to - leading order @xmath150-matrix @xcite , @xmath151 is used instead of the coupling @xmath15 , where @xmath152 is the scattering energy . the main shortcoming of this approach is that the @xmath150-matrix rates so obtained are infinite . the precise origin of the divergences was identified in ref . to the neglect of contributions that formally appear in _ first - order in @xmath11 _ but which effectively contribute only in second order to the stationary state @xcite . by taking these contributions consistently into account @xcite , finite effective rates@xcite for the master equation for the probabilities are obtained . in both the and approach these contributions are ignored and , instead , finite expressions for the rates are obtained only after _ ad - hoc _ infinite subtractions @xcite . this regularization `` by hand '' can and in practice does lead to rates _ different _ from the consistently - computed finite rates , see ref . for explicit comparisons . these problems have also been related @xcite to the fact that calculation of stationary transport using a density matrix ( occupations ) is _ not _ a scattering problem although it can be connected to it @xcite in the following sense : the coupling to the electrodes is never adiabatically turned off at large times ( i.e. , there is no _ free _ `` outgoing state '' ) . as we discuss next , such a consistent first _ plus _ second order approach is not only technically crucial but this also leads to additional physical effects that we measure in sec . [ sec : experiment ] . ( a ) , now indicating the regimes where the excited state @xmath97 relaxes by ( blue ) or by ( red ) . ( b ) relaxation mechanisms after excitation by : _ left panel _ : relaxation in two steps via `` real '' occupation of charge state @xmath5 . _ right panel _ : when the process `` set 1 '' is energetically not allowed excitation and relaxation proceeds by using charge state @xmath5 only `` virtually '' . ] having reviewed the two prominent , complementary pictures of transport due to `` real '' and `` virtual '' charging , we now turn to the crossover regime where these two pictures coexist . this has received relatively little attention , but our experiment in sec . [ sec : experiment ] highlights its importance . as we have seen , despite the fact that charging is only `` virtual '' , an energy exchange between molecule and scattering electrons can occur . depending on the energy - level positions , this `` virtual '' tunneling can `` heat '' the molecule so as to switch on `` real '' charging processes even well _ outside _ the regime . however , in contrast to real heating , which leads to smearing of transport features , this nonequilibrium effect actually results in sharp features in the transport as a function of bias voltage . it thus becomes a new tool for _ spectroscopy_. we first consider the simple case of a single excited state at energy @xmath153 for @xmath43 electrons . in fig . [ fig : crossover](a ) we see that the resulting resonance at @xmath147 ( red ) connects to the excited - state resonance @xmath154 ( blue ) , see also fig . [ fig : cot](a ) . the other resonance condition for the excited state , @xmath155 defines the green line dividing the inelastic regime @xmath156 into two regions shaded red and blue . in the one shaded blue , at the point marked with a circle , the excited state created by a process is stable , that is , it can not decay by a single - electron process since @xmath157 . as shown in the right panel of fig . [ fig : crossover](b ) , the relaxation of this stable state can then only proceed by another process via `` virtual '' charging and it is thus slow ( @xmath105 ) . essentially , this means that the molecule is not `` hot '' enough to lift the coulomb blockade of the _ excited _ state . in contrast , in the red shaded area , at the point marked with a star , this stability is lost as @xmath158 . now the relaxation proceeds much faster through a single - electron process ( order @xmath11 ) as sketched in the left panel of fig . [ fig : crossover](b ) . the molecule gets charged for `` real '' ( either @xmath5 or @xmath62 ) and quickly absorbs / emits an electron returning to the stable @xmath43 electron _ ground _ state , where the system idles waiting for the next excitation . notably , this quenching of the excited state takes place far away from the resonant transport regime in terms of the resonance width , i.e. , violating the linear - response criterion ( [ eq : naive ] ) for being `` off - resonance '' . the enhanced relaxation induced by first - order tunneling , occurring when moving from the circle to the star in fig . [ fig : crossover](a ) , leads to a change in current if no other processes ( e.g. , phonons , hyperfine coupling , etc . ) dominate this relaxation channel ( @xmath53 ) . as a results , the presence of such a resonance signals a `` good '' molecular device , i.e. , one in which the intrinsic relaxation is small compared to the `` transport - coupling '' @xmath11 . we refer to this resonance , first pointed out in refs . and studied further @xcite , as cotunneling - assisted or . the resonance has both and character . on the one hand , the geometric construction in fig . [ fig : crossover](a ) and fig . [ fig : mirage ] shows that it stems from _ the same _ excitation as the step at @xmath147 . however , its position @xmath159 has the same strongly gate - voltage dependence as a resonance , in contrast to the original resonance at @xmath147 . yet , the peak requires to appear and its amplitude is relatively weak , whereas the peak is strong and does not require . for this reason , the peak can be seen as a _ `` mirage '' _ of the excitation and a `` mirror image '' of the @xmath99 peak , as constructed in fig . [ fig : mirage](a ) . the resulting mirrored energy conditions can easily be checked in an experiment cf . [ fig:13] and impose constraints on spectroscopic analysis : if shows a resonance as a function of bias _ outside _ the regime , a resonance at the mirrored position _ inside _ the regime should be present . the vertical cut on the right shows a resonance at @xmath147 and its _ mirage _ at some bias @xmath160 . to identify the latter as such , a corresponding resonance must be present at the mirrored gate voltage @xmath161 , as in the vertical cut shown on the left . note that the indicated construction works for nonsymmetric capacitive coupling . for symmetric coupling , one can literally mirror the gate - voltage position relative to @xmath68 on the horizontal axis . ] besides the appearance of mirages , the crossover regime provides further important pieces of spectroscopic information by constraining how and spectra continuously connect as the gate voltage is varied . this is discussed in sec . [ sec : theory - qd ] , [ sec : stm ] and later on in sec . [ sec : qd - exp ] , but we summarize the rules here . first , only `` equilibrium '' transitions the only ones connecting to an resonance as we explained in fig . [ fig : crossover] can exhibit a mirage . second , excited - excited transitions ( i.e. , for the same charge state @xmath43 ) _ never _ connect to a corresponding feature , as we illustrated in panel ( i ) of fig . [ fig : cot](b ) . finally , transitions between excited states with different charge visible in the regime _ never _ connect to a feature as will be illustrated in fig . [ fig:11 ] . these are strict consistency requirements when analyzing the transport spectra in the -crossover regime . we are now in the position to determine the region in which the physical picture of scattering through the molecule of sec . [ sec : stm ] applies . this is illustrated in fig . [ fig : validity ] . the key necessary assumption of the approach often not stated precisely is that all excited states @xmath97 that are _ accessible _ from the ground state @xmath98 must be `` stable '' with respect to _ first - order _ relaxation processes : @xmath162 this is the case if the condition ( [ eq : set - res ] ) additionally holds for the _ excited _ states , i.e. , for @xmath163 in eq . ( [ eq : set - res ] ) : @xmath164 for both @xmath65 . we note that in theoretical considerations , it is easy to lose sight of condition ( [ eq : stableex ] ) when `` writing down '' an hamiltonian model ( or only @xmath150-matrix rates for ) [ sec . [ sec : theory - stm ] ] and assuming the couplings to be fitting _ parameters _ of the theory . ) that `` accessible '' means also accessible via nonequilibrium cascades of transitions ( `` nonequilibrium ' ' ) , but we will not discuss this further complication . ) ] in fig . [ fig : crossover](a ) we already shaded in light blue the region bounded by the first condition ( [ eq : stableex ] ) where the picture applies . in fig . [ fig : validity](a ) we now show that the full restrictions imposed by both `` virtual '' charge states @xmath5 and @xmath62 in ( [ eq : stableex ] ) strongly restrict the validity regime of the approach for states with `` real '' occupations and charge @xmath43 . in fig . [ fig : validity](b ) and its caption we explain that for any individual excitation @xmath165 the picture _ always _ breaks down in the sense that it works only for _ elastic _ , i.e. , for @xmath166 . this amounts to @xmath3 of the _ nominal _ regime . when accounting for several excited states below the threshold @xmath167 , a sizeable fraction of this region must be further excluded . in fig . [ fig : validity](c ) we construct the regime of validity ( blue ) for some example situations . the shape and size of this validity regime ( light blue ) depends on the details of the excitation spectrum . the center panel illustrates that for a harmonic spectrum the picture in fact applies in only @xmath168 of the nominal regime ( i.e. , obtained by taking the complement of the regime ) . the left and right panel in fig . [ fig : validity](c ) show how this changes for anharmonic spectra characteristic of quantum spins with positive and negative magnetic anisotropy , respectively . ( a ) for an excitation @xmath169 . regions where the approach is valid ( fails ) are colored blue ( red ) , as in fig . [ fig : crossover ] . in the light blue region @xmath166 there is only elastic ( dashed black construction lines are not resonances ) , but for @xmath170 inelastic does excite the molecule . however , the approach only applies in the darker blue triangle where both excitation _ and relaxation _ proceed by `` virtual '' charging . this regime is restricted from both sides [ eq . ( [ eq : stableex ] ) ] and shrinks as @xmath68 increases . ( b ) for @xmath171 the picture works only for _ elastic _ , i.e. , for @xmath166 , which amounts to @xmath172 of the _ nominal _ regime . the inset explains the threshold @xmath169 : for fixed gate voltage at the center ( @xmath173 ) the best - case scenario for the picture to work there is no relaxation by as long as the bias satisfies @xmath174 . requiring this to hold at the onset of excitation by , @xmath147 , gives the threshold . ( c ) several excitations from a superharmonic ( left ) , harmonic ( center ) and subharmonic ( right ) spectrum for charge @xmath43 . in the limit of vanishing harmonic energy spacing , the blue region where the picture works approaches @xmath175 of the nominal off - resonant regime . ] in summary , processes _ always _ dominate the relaxation of excitations at energy @xmath176 populated by excitation because they are `` too hot '' : for such excitations there is no `` deep '' or `` far off - resonant '' regime where considerations based on the picture alone are valid . for lower - energy excitations , @xmath177 , there is a triangular - shaped region in which one is still truly `` far off resonance '' and excitations are not quenched . the size of that region varies according to ( [ eq : stableex ] ) and is much smaller than naively expected by extending the linear - response criterion ( [ eq : naive ] ) . although theoretical @xcite and experimental @xcite studies on exist , this point seems to have been often overlooked and is worth emphasizing . experimentally , to be sure that the picture applies to unidentified excitation one must at least have an estimate of the gap @xmath63 and of the level position or , preferably , a map of the dependence of transport on the level position independent of the bias as in gated experiment discussed in sec . [ sec : experiment ] or stm situations allowing for mechanical gating @xcite . due to their hybrid character , mirages do not emerge in a picture of either `` real '' or `` virtual '' charging alone . in particular , processes are omitted when deriving the rates by means of the transformation [ sec . [ sec : stm ] ] , and , for this reason , that picture can not account for these phenomena . instead , a way to capture these effects is to extend eqs . ( [ eq : set - master ] ) and ( [ eq : cot ] ) to a master equation which simultaneously includes transition rates of leading ( @xmath11 ) and next - to - leading order ( @xmath13 ) . this has been done using the @xmath150-matrix approach @xcite , requiring the _ ad - hoc _ regularization by hand mentioned in sec . [ sec : tmatrix ] . a systematic expansion which avoids these problems is , however , well - known @xcite . and we refer to refs . for calculation of the rates . relevant to our experiment in sec . [ sec : experiment ] is that with the computed rates in hand , a stationary master equation needs to be solved to obtain the occupation of the states and from these the current . we stress that _ even when far off - resonance _ where naively speaking `` the charge is fixed '' to , say , @xmath43 a description of the transport requires a model which also includes _ both the @xmath5 and @xmath62 charge states _ , together with their relative excitations . this is essential to correctly account for the relaxation mechanisms that visit these states `` for real '' and not `` virtual . '' and @xmath137 in eq . ( [ eq : ha ] ) , ] the mirages are missed since @xmath144 only accounts for scattering processes . the minimal master equation required for transport thus takes then the form : @xmath178 where as before @xmath75 for simplicity . here the rates for the various processes change whenever one of the energetic conditions ( [ eq : set - res ] ) and ( [ eq : cot - res ] ) is satisfied . examination of the various contributions in the expression of the rate matrices @xcite reveals that the following effects are included : * @xmath77 is a matrix of rates that change when condition ( [ eq : set - res ] ) is met . it also includes @xmath13-corrections that _ shift and broaden _ the resonance . * @xmath86 is matrix of both and rates . the latter change when condition ( [ eq : cot - res ] ) is met . * @xmath179 and @xmath180 are matrices of _ pair - tunneling _ rates , e.g. , @xmath181 for transitions between states differing by _ two _ electrons , @xmath182 . these lead to special resonances discussed in sec . [ sec : break ] . the solution of the full stationary master equation ( [ eq : full ] ) requires some care @xcite due to the fact that it contains both small rates and large rates whose interplay produces the mirages . even though the _ ( first - order ) rates _ are large , they have a small albeit non - negligible effect since , in the stationary situation , the initial states for these transitions may have only small occupations . these occupations , in turn , depend on the competition between _ all _ processes / rates in the stationary limit . this is the principal reason why one can not avoid solving the master equation ( [ eq : full ] ) with both first and second order processes included . to conclude , eq . ( [ eq : full ] ) captures the delicate interplay of ( ) and ( ) processes leading to mirages ( ) . the appearance of such mirages indicates that _ intrinsic _ relaxation rates are smaller than transport rates ( @xmath53 ) . `` nonequilibrium '' is also included in this approach and the appearance of its additional features in our experiment signals a molecular device with even lower intrinsic relaxation rates , i.e. , smaller than the relaxation rates ( @xmath105 ) . -dependence . upon crossing the red line , the ground state changes from singlet to triplet , see main text . ( b - c ) electron - pair tunneling resonance ( red ) for ( b ) repulsive electron interaction @xmath183 and ( c ) effectively attractive interaction @xmath184 . ( d ) transport feature ( red ) due to coherent spin - dynamics on a single , interacting orbital coupled to nearly antiparallel ferromagnets . although it looks like a resonance with anomalous gate and bias dependence it does not correspond to any state on the system . it is instead a sharp amplitude modulation caused by the _ orientation _ of the accumulated nonequilibrium spin relative to the electrode polarization vectors . ] the above account of the basic rules of transport spectroscopy , although extensive , is by no means exhaustive . the key conditions are eq . ( [ eq : set - res ] ) and ( [ eq : cot - res ] ) , determining the _ resonance positions _ as a function of applied voltages . readers interested mostly in the application of these rules to a high - resolution transport experiment can skip the remainder of this section and proceed directly to sec . [ sec : experiment ] . here , we give an overview of a variety of additional effects that bend or break these rules , found in experimental and theoretical studies . in fig . [ fig : break ] we sketch a number of transport spectra that can not be understood from what we have learned in the previous discussion . [ [ nonuniform - level - shifts - due - to - voltages . ] ] nonuniform level shifts due to voltages. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the assumption made so far [ sec . [ sec : qd - gate ] ] that all energy levels are uniformly shifted by applied gate and bias voltages may not be valid in case of local electric field gradients . in fact , this was already seen in the first experiment on gated spectroscopy of a single - triplet semiconductor dot @xcite due to the change of the confining potential with gate voltage . in molecular junctions this has also been observed . figure [ fig : break](a ) schematizes how the transport spectrum in refs . displays such effects . in this case , the resonances can still be identified as weakly gate - dependent resonances , which is not a trivial issue as the experiments in ref . show . however , a qualitatively new and strongly gate - dependent resonance@xcite ( red line ) appears upon ground state change . piecing together all the evidence , it was shown that this effect originates from a change in amplitude of the background , without requiring the introduction of any additional states into the model . these effects are included in eq . ( [ eq : full ] ) , which was shown @xcite to reproduce the experimental data of ref . in detail . [ [ pair - tunneling . ] ] pair tunneling. + + + + + + + + + + + + + + + + in all the schematics so far we left out resonances that are caused by electron _ pair tunneling_. these are described @xcite by the rates @xmath179 and @xmath180 included in the master equation ( [ eq : full ] ) . in fig . [ fig : break](b ) we sketch where these pair - tunneling resonances ( red lines ) are expected to appear : their positions are obtained by taking the _ bias - averaged positions _ of the two _ subsequent resonances . _ this condition follows by requiring the maximal energy of an electron pair in the electrode @xmath93 to match a corresponding molecular energy change . for example , for a single orbital at energy @xmath185 one obtains @xmath186 where @xmath187 is the charging energy . this gives a bias window in which pair tunneling @xmath188 can contribute to transport , @xmath189 provided that the @xmath5 and / or the @xmath62 state is occupied . the effective _ charging energy _ for each electron is _ halved _ since the energy @xmath187 is available for both electrons together in a single process . although small ( comparable with ) its distinct resonance position and shape clearly distinguish the pair - tunneling current from current ref . that dominates in the regime where it occurs . [ [ electron - attraction . ] ] electron attraction. + + + + + + + + + + + + + + + + + + + + + clearly , pair tunneling effects are expected to become important if the effective interaction energy @xmath187 is attractive @xcite . such attraction in fact appears in various systems . in molecular systems this is known as electrochemical `` potential - inversion '' @xcite . in artificial qds a negative @xmath187 have been observed experimentally @xcite in transport spectra of the type sketched in fig . [ fig : break](c ) , see also ref . . interestingly , in this case the ground state has either @xmath5 or @xmath62 electrons and never @xmath43 since starting from @xmath190 the single - electron transition energies @xmath191 and @xmath192 are higher than electron - pair transition energy _ per electron _ @xmath193 . this is also included in the approach ( [ eq : full ] ) , see ref . . [ [ coherence - effects . ] ] `` coherence '' effects. + + + + + + + + + + + + + + + + + + + + + + + finally , we turn to the assumption used in sec . [ sec : theory - qd ] that the molecular state is described by `` classical '' occupation probabilities of the quantum states ( statistical mixture ) . for instance , each degenerate spin multiplet is treated as an `` incoherent '' mixture of different spin projections ( no quantum superpositions of spin - states states ) . equivalently , the spin has no average polarization in the direction transverse to the quantization axis . however , when in contact with , e.g. , spin - polarized electrodes , such polarization does arise already in order @xmath11 . in that case one must generalize eq . ( [ eq : full ] ) to include off - diagonal density - matrix in the energy eigenbasis . , see discussion in sec . [ sec : tmatrix ] . ) ] in physical terms , this means that one must account for the coupled dynamics of charge , spin - vector and higher - rank spin tensors @xcite . in the regime , such effects can lead to a nearly 100% modulation of the transport current @xcite due to quantum interference . this emphasises that @xcite the first order approximation in @xmath11 is not `` incoherent '' or `` classical '' as some of the nomenclature in table [ tab : compare ] seems to imply . similar coherence effects can arise from orbital polarization in qds @xcite and stm configurations @xcite , from an interplay between spin and orbital coherence @xcite , or from charge superpositions of electron pairs . finally , for high - spin systems coherence effects of tensorial character can arise . this leads to the striking effect that in contact with ferromagnets ( vector polarization ) they can produce a magnetic anisotropy ( tensor ) @xcite , see also related work @xcite . an extension of the approach ( [ eq : full ] ) also describes these effects @xcite . the perhaps most striking effect of spin - coherence is depicted in fig . [ fig : break](d ) : resonances can _ split _ for no apparent reason @xcite and wander off deep into the regime @xcite ( red line ) . depending on the junction asymmetry , this feature of coherent nonequilibrium spin dynamics can appear as a pronounced gate - voltage dependent current peak or as a feature close to the linear response regime , mimicking a kondo resonance , see also ref . . in the second part of this paper we present feature - rich experimental transport spectra as a function of gate - voltage and magnetic field . their analysis requires all the spectroscopic rules that we outlined in the first part of the paper . we show how the underlying hamiltonian model can be reconstructed from the transport data , revealing an interesting high - spin quantum system with low intrinsic relaxation . the molecule used to form the junction is a @xmath194 single - molecule magnet ( smm ) with formula [ fe@xmath195(l)2(dpm)6 ] @xmath196 et2o where hdpm is 2,2,6,6- tetramethyl - heptan-3,5-dione . here , h3l is the tripodal ligand 2-hydroxymethyl-2-phenylpropane-1,3-diol , carrying a phenyl ring @xcite . after molecular quantum - dot formation , the device showed interesting isotropic high - spin behavior and the clearest signatures of to date @xcite for any quantum - dot structure . before turning to the measurements and their analysis , we first discuss specific challenges one faces probing spin - systems using either or spectroscopy . isotropic , _ high - spin _ molecules have molecular states labeled by the spin length @xmath197 and spin - projection @xmath198 . to detect them two types of selection rules are frequently used in stm and qd studies . using these we construct the possible spectroscopic and fingerprints that we can expect to measure . spectroscopy using conductance as a function of magnetic field @xmath51 ( `` spin - flip spectroscopy''@xcite ) has been a key tool in both stm and break - junction studies . this approach assumes that `` virtual '' charging processes dominate . these processes involve two electrons for which the selection rules ( [ eq : rules - cot ] ) apply . however , for high - spin molecules considered here , there can be multiple spin - spectrum assignments that fit the same transport spectrum . an indication for this is that in the present experiment some of the spectra are very similar to those of entirely different nanostructures @xcite . to see how this comes about we construct in fig . [ fig : spectro](a)-(c ) the three possible different fingerprints that two spin - multiplets can leave in the transport spectrum based on selection rules ( [ eq : rules - cot ] ) alone . for simplicity , we assume that all processes start from the ground state @xmath199 , i.e. , in the `` equilibrium '' situation discussed in sec . [ sec : cot ] . this figure shows that one can determine only whether the spin value changes by 1 or remains the same upon excitation , but not on the _ absolute _ values of the spin lengths ( unless the ground state has spin zero ) . ) vs. a magnetic field @xmath51 . the right panels show the corresponding transport spectra , i.e. , the resonant bias positions in matching an energy difference ( @xmath200 ) . ( a ) if the spin increases upon excitation , @xmath201 , there is a _ three - fold _ splitting of the transport - spectrum ( blue ) starting at @xmath147 for @xmath202 due to the transitions to the excited multiplet . the ground multiplet gives a line ( green ) starting at @xmath149 and increasing with @xmath51 if @xmath203 . only for @xmath204 this green line is _ missing_. ( b ) if the spin length does not change upon excitation , @xmath205 , the excited multiplet appears in the transport spectrum through a _ double _ line starting at @xmath147 . the ground multiplet gives a line ( green ) starting at @xmath149 and increasing with @xmath51 if @xmath203 . clearly , for @xmath206 the @xmath51-dependent lines are _ missing_. ( c ) if the spin length decreases upon excitation , @xmath207 , the excited multiplet appears in the transport spectrum through a single line ( blue ) starting at @xmath147 , increasing with @xmath51 . since in this case the ground spin @xmath208 is always nonzero , there is an intra - multiplet line ( green ) starting at @xmath149 . ] a second key tool in the study of spin effects is the transport in the regime @xcite . this provides additional constraints that reduce the nonuniqueness in the spin - assignment . in the regime , the linear - transport part is governed by the transition between the two ground - state multiplets with different charge , @xmath190 and @xmath98 , for which selection rules ( [ eq : rules - set ] ) hold . as sketched in fig . [ fig : spin_blockade ] , if linear transport is observed , then the ground - state spin values are necessarily linked by @xmath209 this constraint , used in refs . , restricts the set of level assignments inferred through spectroscopy on each of the _ two _ subsequent charge states , by fixing the relative ground state spins @xmath210 and @xmath211 . their absolute values remain , however , undetermined , unless one of two happens to be zero . arguments based on the presence of the additional spin - multiplets can then be used to motivate a definite assignment of spin values . molecules for which eq . ( [ eq : nospinblock ] ) fails can be identified by a clear experimental signature : the transport is blocked up to a finite bias as explained in fig . [ fig : spin_blockade ] . such _ spin - blockade _ has been well - studied experimentally @xcite and theoretically @xcite and finds application in spin - qubits ( `` pauli - spin blockade '' ) . it has been reported also for a molecular junction @xcite . clearly , when several excited spin multiplets / charge states are involved , both the and spin - spectroscopy become more complex . however , selection rules similar to eq . ( [ eq : nospinblock ] ) also apply to _ excited _ states and thus `` lock '' the two spin spectra together . in addition , the nonequilibrium occupations of the states contribute to further restricts@xcite the set of possible spin - values as we will now illustrate in our experimental spectroscopic analysis . , then transport is suppressed ( red dashed cross ) . transport sets in only when a finite bias makes the lowest _ spin - compatible _ excitation energetically accessible . this can be either and @xmath43 state with @xmath212 ( shown ) or an @xmath5 electron state with @xmath213 ( not shown ) . ] molecular junctions are produced starting from a three - terminal solid - state device @xcite consisting of an oxide - coated metallic local gate electrode with a thin gold nanowire deposited on top . on such a device , a low - concentration solution of molecules ( @xmath214 mm ) is drop - casted . the nanowire is then electromigrated at room temperature and allowed to self - break @xcite so that a clean nanogap is formed , with a width of @xmath215 nm . the solution is evaporated and the electromigrated junctions are cooled down in a dilution fridge ( @xmath216 mk ) equipped with a vector magnet and low - noise electronics . all the measurements are performed in a two - probe scheme either by applying a dc bias @xmath48 and recording the current @xmath217 or by measuring @xmath218 with a standard lock - in ac modulation of the bias . a molecular junction as sketched in fig . [ fig:1](b ) is formed when a molecule physisorbs@xcite on the gold leads , and thus establishes a tunneling - mediated electrical contact . the presence of the molecule in the junction is signaled by large transport gaps @xmath63 exceeding @xmath219 mev and low - bias inelastic fingerprints . numerous molecular systems have been investigated in this configuration . as a side remark , the fact that we do not observe pronounced magnetic anisotropy effects is not unexpected : the formation of a molecular junction may involve surface interactions . in several cases previously studied clear spectroscopic signatures of the `` bare '' molecular structure ( before junction formation ) , such as the magnetic anisotropy @xcite , were observed also in junctions . however , depending on the mechanical and electrical robustness of the molecule , this and other spin - related parameters may undergo quantitative @xcite or qualitative changes @xcite and sometimes offer interesting opportunities for molecular spin control @xcite . image - charge stabilization effects , for example , can lead to entirely new spin structure such as a singlet - triplet pair @xcite on opposite sides of a molecular bridge . we now turn to the analysis of the feature - rich transport spectrum anticipated in fig . [ fig:1](c ) and reproduced in fig . [ fig:10 ] . it consists of two regimes on the left and right with fixed charge states provisionally labeled @xmath5 and @xmath43 and a regime in the center surrounded by a significant crossover regime . we first separately identify the electronic spectrum for each of the two accessible charge states @xmath5 and @xmath43 using the approach discussed in sec . [ sec : stm ] . in fig . [ fig:10 ] ( a ) we show the color map and the corresponding steps for fixed @xmath220 v as a function of magnetic field , @xmath51 . two steps ( peaks in ) starting from @xmath221 mev and @xmath222 mev at @xmath223 t shift upward in energy and parallel to each other as the magnetic field increases . in the standard picture each step signal to the opening of an inelastic transport channel through the molecule . transport takes place via virtual charging involving a real spin - flip excitation with selection rules on spin - length @xmath224 and magnetization @xmath225 . the charge of the molecule remains fixed , and is labeled @xmath43 . the shift in magnetic field of both steps indicates a nonzero spin ground state multiplet with spin @xmath226 . according to sec . [ sec : spectro ] , the presence of only one other finite - bias excitation shifting in magnetic field relates the spin - values as @xmath227 , but leaves their absolute values undetermined . as we will see later , other spectroscopic information constrains the ground spin to be a triplet @xmath14 , @xmath228 , with a singlet excited state labeled @xmath197 . from the excitation voltage , a ferromagnetic ( fm ) interaction energy @xmath229 mev can be extracted . such type of excitation has been seen in other molecular structures @xcite . spectra of this kind have also been obtained earlier in other quantum - dot heterostructures , such as few - electron single and double quantum dots , albeit typically characterized by smaller and antiferromagnetic couplings @xcite . we now change the gate voltage to more negative values so that the molecule is oxidized @xmath230 , i.e. , we extract exactly one electron from the molecule . this can be inferred from the transport regime that we traverse along the way . in this new charge state we perform an independent spectroscopy . in fig . [ fig:10](b ) we show the for @xmath231 v as a function of the magnetic field @xmath51 with corresponding line cuts . at @xmath223 t two sets of peaks in appear at @xmath232 mev and @xmath233 mev and split each in three peaks at higher magnetic fields . a weak excitation shifting upwards in @xmath51 from @xmath234 v is also present . with the help of fig . [ fig : spectro](a ) the weak excitation and the first set of peaks are associated to @xmath235 , while the second set , corresponding instead to the spectrum depicted in , fixes the spin to @xmath236 . the crucial information provided by the clear absence of spin blockade in the intermediate regime eventually constrains @xmath210 to @xmath237 or @xmath238 according to ( [ eq : nospinblock ] ) . the only two spin configurations compatible with the observations are therefore : a ground doublet @xmath239 , an excited quartet @xmath240 and a second doublet @xmath241 or , alternatively , a ground quartet , an excited sextuplet and a quartet . as we will see in the next section , the latter can be rigorously ruled out by analyzing the spectrum . the presence of the excited quartet state @xmath240 implies that the charge state @xmath5 is a _ three - spin _ system , @xmath242 , as sketched in the top panel of fig . [ fig:10](b ) . the system with one extra electron in fig . [ fig:10](a ) is thus actually a @xmath243 electron system with one closed shell , as sketched in the figure . upon extraction of an electron , the spectrum of the molecular device changes drastically , transforming from a ferromagnetic _ high - low _ spin spectrum for @xmath243 into a nonmonotonic _ low - high - low _ spin excitation sequence for @xmath242 . the spin - excitation energies extracted from the two independent analyses are : @xmath244 and @xmath245 these energy differences provide the starting point of a more atomistic modeling of the magnetic exchanges in the two charge states . we stress that for the transport spectroscopy this is not necessary and it goes beyond the present scope . we only note that while the @xmath243 state requires only one fixed ferromagnetic exchange coupling @xmath246 [ fig . [ fig:10](a ) ] together with the assumption that two other electrons occupy a closed shell ; the @xmath242 spectrum requires , in the most general case , three distinct exchange couplings between the three magnetic centers [ fig . [ fig:10](b ) ] . these relate to the two available energy differences through @xmath247 and @xmath248 to a complicated function @xmath249 . since this involves three unknowns for two splittings , only microscopic symmetry considerations or detailed consideration of the transport current magnitude are needed to uniquely determine the microscopic spin structure . this type of microscopic modeling has proven successful in many instances , see ref . and references therein . however , the underlying assumptions on localized spins and fixed charge occupations can only be made when sufficiently far away from resonance , i.e. , such that does not take place as expressed by conditions ( [ eq : stableex_rates ] ) and ( [ eq : stableex ] ) . using the ability to control the energy levels with the gate , the analysis can be complemented by a spectroscopy in the central part of fig . [ fig:10](c ) . here , `` real '' charging processes dominate . for example , starting from the ground state @xmath239 , addition of a single electron leads to occupation of the @xmath14 ground state . this is evidenced by the clear presence of a regime of transport down to the linear - response limit . inside the regime additional lines parallel to the edges of the cross appear as well . as we explained in fig . [ fig : set ] , these correspond to `` real '' charging processes where excess ( deficit ) energy is used to excite ( relax ) the molecule . these additional lines , schematized for our experiment in fig . [ fig:11](a ) , fall into two categories according to the criteria : 1 . lines terminating at the boundary of the regime correspond to the _ ground _ @xmath5 to excited @xmath250 transitions or _ vice versa_. 2 . lines that never reach the boundary , but terminate inside the regime at a line parallel to this boundary . these correspond to _ excited _ @xmath5 to excited @xmath250 transitions . their earlier termination indicates that that the initial excited state must become first occupied through another process . the line _ at which _ it terminates corresponds to the onset of this `` activating '' process . . transitions between a _ ground and excited _ state ( blue , red ) reach the boundary of the regime at the black circle from where they continue horizontally as a excitation . the inset depicts the chemical potential configuration at such a black circle where and connect . the transitions between _ two excited _ states ( orange , green ) do not connect to any excitation . ( b ) transport spectrum computed using the master equations ( [ eq : set - master])-([eq : set - current ] ) . the energies are extracted independently from the two spectra in fig . [ fig:10 ] and the capacitive parameters @xmath251 , @xmath252 @xmath253 are fixed by the observed slopes of the lines [ cf . [ fig : set](a ) ] , leaving the tunnel rates ( [ eq : rates_t])-([eq : rates_s ] ) as adjustable parameters . the broadening of the peaks in the experiment is due to tunneling , @xmath254 ( fwhm ) , rather than temperature , @xmath255 . ( [ eq : set - master])-([eq : set - current ] ) do not include this @xmath11-broadening and we crudely simulate it by an effective higher temperature @xmath256 . the master equations ( [ eq : set - master])-([eq : set - current ] ) are valid for small effective tunnel coupling @xmath257 , which only sets the overall scale of plotted current and not the relative intensities of interest . the caption to fig . [ fig:15 ] explains that @xmath258 should not be adjusted to match the larger experimental current magnitude . ] in fig . [ fig:10](c ) and fig . [ fig:11](a ) the transitions @xmath259 , @xmath260 and @xmath261 fall into category ( a ) , while the @xmath262 and @xmath263 transitions belong to ( b ) . due to the large difference in spin - length values of the spin - spectra the latter transition , marked in dashed - green , is actually forbidden by the selection rules ( [ eq : rules - set ] ) . following this line , we find that it terminates at a strong negative differential conductance ( ndc ) feature ( white in the stability diagram in fig . [ fig:10 ] ) marking the onset of the transition @xmath259 . to test our earlier level assignment based , we now compute the expected transport spectrum the first - order ( @xmath11 ) master equations ( [ eq : set - master])-([eq : set - current ] ) and by adjusting the result , we extract quantitative information about the tunnel coupling . the model hamiltonian is constructed from the energies ( [ eq : energies_3])-([eq : energies_4 ] ) and their observed spin - degeneracies . assuming that spin is conserved in the tunneling , the rates between magnetic sublevels are fixed by clebsch - gordan spin - coupling coefficients @xcite incorporating both the and selection rules eqs . ( [ eq : rules - set ] ) and ( [ eq : rules - cot ] ) . the tunnel parameters in units of an overall scale @xmath258 are adjusted to fit the relative experimental intensities : @xmath264 and @xmath265 their relative magnitudes provide further input the further microscopic modeling of the 3 - 4 spin system mentioned at the end of sec . [ sec : stm - exp ] . as shown in fig . [ fig:11](b ) , the _ ( ) part _ of the experimental conductance in fig . [ fig:10](c ) , as schematized in fig . [ fig:11](a ) is reproduced in detail . this includes transitions exciting the molecule from its ground states , but also a transition between excited states.@xcite the ndc effect is explained in more detail later on together with the full calculation in fig . [ fig:14 ] . as discussed in fig . [ fig : cot]-[fig : crossover ] and indicated in fig . [ fig:11](a ) the excitations corresponding to the ground @xmath5 to excited @xmath250 transitions connect continuously to the excitations . those corresponding to two excited states , each of a different charge state , has no corresponding excitation to connect to . in this sense , the spectrum effectively ties the two separately - obtained spin spectra and allows a consistency check on their respective level assignments , cf . [ sec : connecting ] . for instance , from the fact that the @xmath260 transition is clearly visible marked red in fig . [ fig:10](c) we conclude that the first excited multiplet of the @xmath5 charge state _ can not _ be a sextuplet ( @xmath266 ) since such transition would be spin - forbidden and thus weak . another example is given by the presence of the @xmath267 transition [ orange in fig . [ fig:10](c ) ] , which implies that the second excited multiplet of the @xmath5 charge state be a quartet . the fact that this transition does not continue into any of the ones is also consistent with its excited - to - excited character . these two exclusions considerations were anticipated in sec . [ sec : stm - exp ] and are crucial for our assignment in the three - electron state and has now allowed us to reverse - engineer the effective many - electron molecular hamiltonian . with this in hand , we turn to the main experimental findings and investigate the `` nonequilibrium '' through the molecule [ sec . [ sec : pump - probe ] ] and the crossover regime where `` real '' and `` virtual '' tunneling nontrivially compete in the relaxation of spin excitations [ sec . [ sec : coset ] ] . , we highlight here the transitions that are involved in the spin pumping process ( green dotted lines ) . ( a ) spectra measured as a function of @xmath51-field at @xmath268 v in the @xmath243 charge state . the @xmath269 nonequilibrium spin - excitation shows up as a weak , field - independent step vanishing at higher field . for @xmath270 the intra - triplet transition ( red arrow ) requires lower energy than the `` nonequilibrium '' @xmath269 transition . for @xmath271 , the intra - triplet is unlocked ( activated ) at an energy higher than the @xmath269 and only one transition of the cascade is visible . ( b ) spectra measured as a function of @xmath51-field at @xmath272 v in the @xmath242 charge state . here the nonequilibrium excitation has a negative slope . for @xmath270 the excited state of the ground - state doublet @xmath239 is populated enough to promote a second , nonequilibrium excitation to the excited doublet @xmath241 ( green dotted line ) . as @xmath271 the @xmath273 transition crosses over , lowering , in consequence , the population of the spin - up state . this results into a quench of the nonequilibrium excitation . due to the proximity to regime as compared to fig . [ fig:10](a ) , a feature ( orange dotted line ) appears as a mirage of a spin - excitation . ] we first investigate how spectrum evolves as we further _ approach _ the regime from either side . [ fig:12](a ) shows the analogous of fig . [ fig:10](a ) but closer to the regime , at @xmath268 v. a horizontal , @xmath51-field independent line appears ( dotted green line in the center - panel schematic ) that terminates at @xmath274 t , precisely upon crossing the intra - triplet excitation ( blue line ) . this indicates that the excited triplet ( spin @xmath275 perpendicular to the field , @xmath276 ) lives long enough for a secondary process to excite the system to the singlet state ( reducing the spin length to @xmath277 ) . strong evidence for this is the termination of this line : once the initial state ( @xmath276 excited triplet ) for this transition is no longer accessible for @xmath271 , the `` nonequilibrium '' cascade of transitions is interrupted . we consistently observe this effect , also when approaching the regime from the side of the other charge state ( @xmath242 ) with different spin . in fig . [ fig:12](b ) we show the magnetic field spectrum taken at @xmath278 v. here the lowest @xmath241 excitation gains strength@xcite relative to fig . [ fig:10](b ) . in this case , the excited @xmath239 state is the starting point of a `` nonequilibrium '' cascade . as for the previous case , it terminates when levels cross at @xmath279 t for similar reasons : once the @xmath240 state gains occupation for @xmath271 ( since the @xmath273 transition becomes energetically more favorable ) the excited @xmath198-substates of the @xmath239 multiplet are depleted causing the line to terminate . in both charge states , the observed current gives an estimate for the spin - relaxation time , @xmath280 s. nonequilibrium transitions can thus give rise to clear excitations at _ lower _ energy than expected from the simple selection - rule plus equilibrium arguments of sec . [ sec : spectro ] . in this type of processes , two event ( @xmath13 ) happen in sequence , so that a total of four electrons are involved.@xcite in this sense , the phenomena can be regarded as a single - molecule _ electronic pump - probe _ experiment , that is , the excess energy left behind by the first process ( pump ) allows the second process to reach states ( probe ) that would be otherwise inaccessible at the considered bias voltage . this has been successfully applied in stm studies @xcite for dynamical spin - control . [ [ mirages . ] ] mirages. + + + + + + + + + we now further reduce the distance to resonance , again coming from either side , and _ enter _ the crossover regime discussed in sec . [ sec : crossover ] . we are , however , still `` well away from resonance '' by the linear - response condition ( [ eq : naive ] ) . in the upper panel of fig . [ fig:13](a ) we show traces taken at various magnetic fields for a constant gate voltage . at high bias voltage the steeply rises due to the onset of the main resonance . below this onset , we note a step - like excitation at @xmath281 mev ( black arrow ) which shifts up in magnetic field with the same @xmath8-factor ( @xmath282 ) as the other lower - lying excitations.@xcite if one adopts the picture this excitation is attributed to the opening of an independent `` channel '' . this attribution proves to be erroneous : keeping @xmath283 t fixed and varying the gate voltage ( fig . [ fig:13](a ) , lower panel ) , we observe that the lower excitations are left unchanged , whereas the higher one under consideration _ shifts linearly _ with @xmath17 , revealing that it is _ not _ a excitation . this attribution to can be further ruled out by looking at the full gate - voltage dependence in the stability diagram shown in the left panel of fig . [ fig:13](b ) . the excitation ( red arrow ) has the same gate dependence as the resonances , even though it is definitely not in the regime by the linear - response criterion ( [ eq : naive ] ) . in fact , it is a mirage of the _ same _ lowest gate - voltage independent excitation as we explained in fig . [ fig : mirage ] . its bias ( energy ) position does not provide information about the excitation energy @xmath68 : depending on the energy level position the mirage s excitation voltage @xmath159 can lie anywhere above the threshold voltage @xmath147 , see sec . [ sec : crossover ] . in the stability diagram in the right panel of fig . [ fig:13](b ) , we connect by dashed lines all the resonances to their corresponding excitations according to the scheme in fig . [ fig : mirage ] . we find that mirages appear for virtually all spin - related excitations of the molecule . the stability diagram in fig . [ fig:13](c ) [ same color coding as in ( b ) ] shows that at high magnetic field @xmath284 t these mirages persist . the clearly visible resonances mark the lines where the relaxation mechanism changes from `` virtual '' ( ) to `` real '' ( ) charging . they indicate that any intrinsic relaxation is comparable or slower than . mirages are thus a signature of slow intramolecular relaxation , in particular they indicate that the intrinsic relaxation time is bounded from below by the magnitude of the observed currents @xmath285 s , consistent with the sharper lower bound we obtained above from nonequilibrium spectroscopy . [ [ spin - relaxation . ] ] spin relaxation. + + + + + + + + + + + + + + + + + to shed light on what the relaxation mechanism by transport entails in our device , we return to the stability diagram for @xmath202 t , which is shown as in the right panel of fig . [ fig:13](d ) . highlighted at negative bias are the two crossover - regime bands within which , rather than , dominates the relaxation . the left panel shows the different relaxation paths for these two bands . focusing on the orange band , we start out on the far left of fig . [ fig:13](d ) moving at fixed bias @xmath286 mev along the onset of inelastic . fig . [ fig:13](e ) depicts the corresponding energies ( left ) and energy differences ( right ) . here , the molecule is in the spin - doublet @xmath239 ground state and is occasionally excited to the high - spin quartet @xmath240 by from where it relaxes via path 1 ( @xmath287 s ) , again by . when reaching the circle ( @xmath173 ) in fig . [ fig:13](d ) the _ relaxation _ mechanism changes : path 1 is overridden by the faster relaxation path 2 ( @xmath288 s ) which becomes energetically allowed [ eq . ( [ eq : stableex ] ) ] . the top panel of fig . [ fig:13](e ) illustrates that although the ground state @xmath239 is off - resonant ( highlighted in red ) , after exciting it by to @xmath240 increasing the spin - length the system has enough _ spin - exchange _ energy ( green ) to expel a single electron in a `` _ _ real _ _ '' tunneling processes leaving a _ charged _ triplet state behind . at the star ( @xmath289 ) in fig . [ fig:13](d ) the _ excitation _ mechanism changes from to , leaving the relaxation path unaltered . now the ground state @xmath239 becomes unstable with respect to `` real '' charging : there is enough energy to expel an electron to the right electrode and sequentially accept another one from the left . we thus have an transport cycle , i.e. , the stationary state is a statistical mixture of the @xmath5 and @xmath43 ground states . ) and its corresponding current formula ( not shown , see refs . ) for the same parameters as in fig . [ fig:11](b ) . ( a ) transport spectrum for @xmath202 corresponding to fig . [ fig:10](c ) . ( b ) corresponding color plots of the occupation probabilities of the five spin multiplets ( probabilities summed of degenerate levels ) . the effective coupling @xmath258 merely an overall scale factor in fig . [ fig:11](b) now controls the magnitude of the and current corrections _ relative _ to the current . although elaborate , these corrections still neglect nonperturbative broadening effects and must kept small for consistency by explicitly setting @xmath290 . more advanced master equation approaches based on renormalization - group@xcite ( rg ) or hierarchical@xcite ( hqme ) methods can deal with both this broadening and the corresponding larger currents . ] the regime is delimited by mirage resonances and situated between the two positions @xmath173 and @xmath289 . failure to identify the difference between this `` band '' and the pure happening on the left of @xmath173 , besides yielding a wrong qualitative spin multiplet structure , leads to an overestimation of the relaxation time : in the regime the spin - excitations created by inelastic are _ quenched_. [ [ quenching - of - spin - excitations . ] ] quenching of spin - excitations. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we now assess this quenching in detail for the experimental situation by a calculation based on the master equation @xcite ( [ eq : full ] ) that includes all @xmath11 and @xmath13 processes using the model determined earlier [ eqs . ( [ eq : energies_3])-([eq : rates_s ] ) ] , simulating the broadening as before by an effective temperature [ fig . [ fig:11 ] ] . the computed conductance for @xmath202 is shown in fig . [ fig:14](a ) . besides the excitations including the ndc effect obtained earlier in fig . [ fig:11](c ) , we capture the main features of the experimental data in fig . [ fig:11](c ) and fig . [ fig:13](d ) : the three horizontal excitations and two prominent lines . we can now explore the nonequilibrium occupations of the five spin - multiplets as the transport spectrum is traversed . these are shown in fig . [ fig:14](b ) . the lowest panels show in the left ( right ) regime the ground multiplet @xmath239 with @xmath5 electrons ( @xmath14 with @xmath43 electrons ) is occupied with probability 1 at low bias voltage ( black regions ) . in contrast , in the regime these two ground states are both partially occupied due to processes . we compare the occupations along three different vertical line cuts in fig . [ fig:14](a ) . \(i ) increasing the bias voltage in the regime , starting from @xmath291 v , one first encounters in fig . [ fig:14](a ) a dip ( ndc , white ) . this is caused by the occupation of the @xmath197 state , as the @xmath197-panel in fig . [ fig:14](b ) shows . this drains so much probability from the @xmath14 multiplet [ with a higher transition rate to the @xmath239 multiplet , eq . ( [ eq : rates_t])-([eq : rates_s ] ) ] that the current goes down . increasing the bias further depopulates the @xmath197 state again , thereby restoring the current through a series of peaks . \(ii ) increasing the bias voltage starting from the right regime the excited @xmath197-state becomes populated by decreasing the average spin - length of the molecule . when crossing the resonance at higher bias this excitation is _ completely quenched _ ( white diagonal band ) well before reaching the regime , _ enhances _ the molecular spin , restoring the triplet . \(iii ) when starting from the left regime , the population of the excited @xmath240-state enhances the average spin - length of the molecule . as before , crossing the resonance at higher bias _ quenches _ this excitation . now this _ reduces _ molecular spin , restoring the doublet . along the way , the @xmath241 state also becomes occupied by and subsequently quenched by . because of its higher energy , the white band in the @xmath241-panel of fig . [ fig:14](b ) is much broader . ) characteristic of molecular qd devices . ( b ) strongly asymmetric couplings , ( @xmath292 ) . this is typical for molecular stm junctions , where the energy levels `` pin '' to one electrode ( substrate ) , leaving the tip electrode to act as a probe . in this case the energy @xmath293 represents the _ level alignment _ with the fermi - energy . ] the results show that the widths of the two bands where the excitations are quenched by are unrelated to the width of the resonances , set by the maximum of @xmath11 and @xmath14 . they are , instead , set by the _ excitation spectrum _ one wishes to probe . in fig . [ fig:15](a ) we quantify how far the energy level has to be detuned from resonance in order avoid this quenching in our molecular qd device structure . when this detuning lies in the window @xmath294 one is sure to run into the band with increasing bias . only for @xmath295 there is a finite window where the excitation is not quenched . for the excitations @xmath14 , @xmath240 , @xmath241 in our experiment , this amounts to @xmath296 , @xmath296 , and @xmath296 times the resonance width . in fig . [ fig:15](b ) we show the corresponding construction for strong capacitive asymmetry typical of stm setups . to avoid quenching for any bias polarity , one now needs to stay further away from resonance @xmath297 . interestingly , for @xmath298 excitation at forward @xmath147 is not quenched , whereas at reverse bias @xmath299 it is . for asymmetric junctions , the mechanism thus leads to a _ strong bias - polarity dependence _ of relaxation of excitations in the nominal regime . for @xmath300 one is sure to run into the band for forward bias . whereas in the present experiment we encountered relatively low - lying spin - excitations ( @xmath301 few mev ) atomic and molecular devices can boast such excitations up to tens of mev . to gauge the impact of mirage resonances , consider an excitation at @xmath302 mev that we wish to populate by , e.g. , for the purpose of spin - pumping @xcite . to avoid the quenching of this excitation @xmath147 the distance to the fermi - energy at @xmath149 ( level - alignment ) needs to exceed _ room temperature _ , even when operating the device at mk temperatures . for vibrational and electronic excitations on the 100 mev scale the implications are more severe . moreover , even for excitations that do satisfy these constraints , cascades of `` nonequilibrium '' excitation may if even higher excitations are available ( e.g. , vibrations) provide a path to excitations that do decay by processes . whereas all these effects can be phrased loosely as `` heating '' in this paper we demonstrated the discrete nature of these processes , their _ in - situ _ tuneability , and the role they play as a spectroscopic tool . we have used electron transport on a single - molecule system to comprehensively characterize the spin degree of freedom and its interaction with the tunneling electrons . three key points applicable to a large class of systems emerged with particular prominence : \(i ) combining and spectroscopy in a single stable device provides new tools for determining spin properties _ within _ and _ across _ molecular redox states . this is crucially relevant for the understanding of the different spin - relaxation mechanisms , even in a _ single _ redox state . \(ii ) nonequilibrium pump - probe electron excitation using two processes ( four electrons ) was demonstrated in our three - terminal molecular device and signals a substantial intrinsic spin relaxation time of about 1 ns , much larger than the transport times . \(iii ) mirages of resonances arise from the nontrivial interplay of and . these resonances signal a sharp increase of the relaxation rate and can occur far away from resonance ( many times the resonance width ) . this limits the regime where spin - pumping works by quenching nonequilibrium populations created by a current . the appearance of a mirage of a certain excitation indicates that the relaxation of the corresponding molecular degree of freedom dominates over all possible unwanted , intrinsic mechanism . thus , `` good devices show mirages '' and `` even better devices '' show nonequilibrium transitions . energy level control turns out to be essential for `` imaging '' in energy space , distinguishing mirages from real excitations . whereas real - space imaging seems to be of little help in this respect , the mechanical gating possible with scanning probes overcomes this problem . however , even when energy - level control is available , spectroscopy of molecular junctions still requires extreme care as we illustrated in sec . [ sec : break ] by several examples that break spectroscopic rules . moreover , our work underlines that level alignment has to be treated on a more similar footing as as coupling ( @xmath11 ) and temperature ( @xmath14 ) broadening in the engineering of molecular spin structures and their spin - relaxation rates @xcite . beyond electron charge transport , recent theoretical work @xcite has pointed out that importance of is amplified when moving to nanoscale transport of _ heat _ @xcite . whereas in charge transport all electrons carry the same charge , in energy transport electrons involved in processes effectively can carry a quite different energy from that acquired in a process only and therefore dominate energy currents @xcite . thus , the sensitivity to spin - relaxation processes is dramatically increased in heat transport , indicating an interesting avenue @xcite for a _ spin - caloritronics _ @xcite on the nanoscale . we thank a. cornia for the synthesis of the molecules , m. leijnse and m. josefsson for assistance with the calculations , and s. lounis , m. dos santos dias and t. esat for discussions . we acknowledge financial support by the dutch organization for fundamental research ( nwo / fom ) and an advanced erc grant ( mols@mols ) . m. m. acknowledges financial support from the polish ministry of science and higher education through a young scientist fellowship ( 0066/e-336/9/2014 ) , and from the polish ministry of science and education as iuventus plus project ( ip2014 030973 ) in years 2015 - 2017 . thanks funds from the eu fp7 program , project 618082 acmol through a nwo - veni fellowship . 265ifxundefined [ 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 , , and , eds . , @noop _ _ , ( , , ) @noop * * , ( ) , ed . , @noop _ _ ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1021/nl2021637 [ * * , ( ) ] link:\doibase 10.1146/annurev - physchem-040214 - 121554 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase doi 10.1126/science.1202204 [ * * , ( ) ] @noop ( ) @noop * * , ( ) link:\doibase 10.1021/acs.nanolett.5b02188 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.77.125306 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , ed . 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( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevb.77.201406 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , http://stacks.iop.org/0957-4484/19/i=6/a=065401 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , oxford graduate texts ( , , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.105.086103 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' @noop * * , ( ) @noop * * , ( ) and , eds . , @noop _ _ , , vol . 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molecular systems can exhibit a complex , chemically tailorable inner structure which allows for targeting of specific mechanical , electronic and optical properties . at the single - molecule level , two major complementary ways to explore these properties are molecular quantum - dot structures and scanning probes . this article outlines comprehensive principles of electron - transport spectroscopy relevant to both these approaches and presents a new , high - resolution experiment on a high - spin single - molecule junction exemplifying these principles . such spectroscopy plays a key role in further advancing our understanding of molecular and atomic systems , in particular the relaxation of their spin . in this joint experimental and theoretical analysis , particular focus is put on the crossover between _ resonant _ regime [ single - electron tunneling ( set ) ] and the _ off - resonant _ regime [ inelastic electron ( co)tunneling ( iets ) ] . we show that the interplay of these two processes leads to unexpected _ mirages _ of resonances not captured by either of the two pictures alone . although this turns out to be important in a large fraction of the possible regimes of level positions and bias voltages , it has been given little attention in molecular transport studies . combined with nonequilibrium iets four - electron pump - probe excitations these mirages provide crucial information on the relaxation of spin excitations . our encompassing physical picture is supported by a master - equation approach that goes beyond weak coupling . the present work encourages the development of a broader connection between the fields of molecular quantum - dot and scanning probe spectroscopy .

our story begins with a theorem of gromov , proved in 1980 . [ theorem : polynomial : growth ] let @xmath0 be any finitely generated group . if @xmath0 has polynomial growth then @xmath0 is virtually nilpotent , i.e. @xmath0 has a finite index nilpotent subgroup . gromov s theorem inspired the more general problem ( see , e.g. @xcite ) of understanding to what extent the asymptotic geometry of a finitely generated solvable group determines its algebraic structure . one way in which to pose this question precisely is via the notion of quasi - isometry . a ( coarse ) _ quasi - isometry _ between metric spaces is a map @xmath1 such that , for some constants @xmath2 : 1 . @xmath3 for all @xmath4 . the @xmath5-neighborhood of @xmath6 is all of @xmath7 . @xmath8 and @xmath7 are _ quasi - isometric _ if there exists a quasi - isometry @xmath9 . note that the quasi - isometry type of a metric space @xmath8 is unchanged upon removal of any bounded subset of @xmath8 ; hence the term `` asymptotic '' . quasi - isometries are the natural maps to study when one is interested in the geometry of a group . in particular : the word metric on any f.g . group is unique up to quasi - isometry . any injective homomorphism with finite index image is a quasi - isometry , as is any surjective homomorphism with finite kernel . the equivalence relation generated by these two types of maps can be described more compactly : two groups @xmath10 are equivalent in this manner if and only if they are _ weakly commensurable _ , which means that there exists a group @xmath11 and homomorphisms @xmath12 , @xmath13 each having finite kernel and finite index image ( proof : show that `` weakly commensurable '' is in fact an equivalence relation ) . this weakens the usual notion of _ commensurability _ , i.e. when @xmath0 and @xmath14 have isomorphic finite index subgroups . weakly commensurable groups are clearly quasi - isometric . any two cocompact , discrete subgroups of a lie group are quasi - isometric . there are cocompact discrete subgroups of the same lie group which are not weakly commensurable , for example arithmetic lattices are not weakly commensurable to non - arithmetic ones . the polynomial growth theorem was an important motivation for gromov when he initiated in @xcite the problem of classifying finitely - generated groups up to quasi - isometry . theorem [ theorem : polynomial : growth ] , together with the fact that nilpotent groups have polynomial growth ( see [ section : nilpotent ] below ) , implies that the property of being nilpotent is actually an asymptotic property of groups . more precisely , the class of nilpotent groups is _ quasi - isometrically rigid _ : any finitely - generated group quasi - isometric to a nilpotent group is weakly commensurable to some nilpotent group . sometimes this is expressed by saying that the property of being nilpotent is a _ geometric property _ , i.e. it is a quasi - isometry invariant ( up to weak commensurability ) . the natural question then becomes : [ question : rigidity ] which subclasses of f.g . groups are quasi - isometrically rigid ? for example , are polycyclic groups quasi - isometrically rigid ? metabelian groups ? nilpotent - by - cyclic groups ? in other words , which of these algebraic properties of a group are actually geometric , and are determined by apparently cruder asymptotic information ? a. dioubina @xcite has recently found examples which show that the class of finitely generated solvable groups is _ not _ quasi - isometrically rigid ( see [ section : dioubina ] ) . on the other hand , at least some subclasses of solvable groups are indeed rigid ( see [ section : abc ] ) . along with question [ question : rigidity ] comes the finer classification problem : [ problem : classification ] classify f.g . solvable ( resp . nilpotent , polycyclic , metabelian , nilpotent - by - cyclic , etc . ) groups up to quasi - isometry . as we shall see , the classification problem is usually much more delicate than the rigidity problem ; indeed the quasi - isometry classification of finitely - generated nilpotent groups remains one of the major open problems in the field . we discuss this in greater detail in [ section : nilpotent ] . the corresponding rigidity and classification problems for irreducible lattices in semisimple lie groups have been completely solved . this is a vast result due to many people and , among other things , it generalizes and strengthens the mostow rigidity theorem . we refer the reader to @xcite for a survey of this work . in contrast , results for finitely generated solvable groups have remained more elusive . there are several reasons for this : finitely generated solvable groups are defined algebraically , and so they do not always come equipped with an obvious or well - studied geometric model ( see , e.g. , item 3 below ) . dioubina s examples show not only that the class of finitely - generated solvable groups is not quasi - isometrically rigid ; they also show ( see [ section : abc ] below ) that the answer to question [ question : rigidity ] for certain subclasses of solvable groups ( e.g. abelian - by - cyclic ) differs in the finitely presented and finitely generated cases . there exists a finitely presented solvable group @xmath15 of derived length 3 with the property that @xmath15 has unsolvable word problem ( see @xcite ) . solving the word problem for a group is equivalent to giving an algorithm to build the cayley graph of that group . in this sense there are finitely presented solvable groups whose geometry can not be understood , at least by a turing machine . solvable groups are much less rigid than irreducible lattices in semisimple lie groups . this phenomenon is exhibited concretely by the fact that many finitely generated solvable groups have infinite - dimensional groups of self quasi - isometries , with the operation of composition , becomes a group @xmath16 once one mods out by the relation @xmath17 if @xmath18 in the @xmath19 norm . ] ( see below ) . [ problem : flexible ] for which infinite , finitely generated solvable groups @xmath15 is @xmath20 infinite dimensional ? in contrast , all irreducible lattices in semisimple lie groups @xmath21 have countable or finite - dimensional quasi - isometry groups . at this point in time , our understanding of the geometry of finitely - generated solvable groups is quite limited . in [ section : nilpotent ] we discuss what is known about the quasi - isometry classification of nilpotent groups ( the rigidity being given by gromov s polynomial growth theorem ) . beyond nilpotent groups , the only detailed knowledge we have is for the finitely - presented , nonpolycyclic abelian - by - cyclic groups . we discuss this in depth in [ section : abc ] , and give a conjectural picture of the polycyclic case in [ section : abc2 ] . one of the interesting discoveries described in these sections is a connection between finitely presented solvable groups and the theory of dynamical systems . this connection is pursued very briefly in a more general context in [ section : final ] , together with some questions about issues beyond the limits of current knowledge . this article is meant only as a brief survey of problems , conjectures , and theorems . it therefore contains neither an exhaustive history nor detailed proofs ; for these the reader may consult the references . it is a pleasure to thank david fisher , pierre de la harpe , ashley reiter , jennifer taback , and the referee for their comments and corrections . recall that the _ wreath product _ of groups @xmath22 and @xmath23 , denoted @xmath24 , is the semidirect product @xmath25 , where @xmath26 is the direct sum of copies of b indexed by elements of a , and a acts via the `` shift '' , i.e. the left action of @xmath22 on the index set @xmath22 via left multiplication . note that if @xmath22 and @xmath23 are finitely - generated then so is @xmath24 . the main result of dioubina @xcite is that , if there is a bijective quasi - isometry between finitely - generated groups @xmath22 and @xmath23 , then for any finitely - generated group @xmath27 the groups @xmath28 and @xmath29 are quasi - isometric . dioubina then applies this theorem to the groups @xmath30 where @xmath31 is a finite nonsolvable group . it is easy to construct a one - to - one quasi - isometry between @xmath22 and @xmath23 . hence @xmath32 and @xmath33 are quasi - isometric . now @xmath0 is torsion - free solvable , in fact @xmath34 is an abelian - by - cyclic group of the form @xmath35$]-by-@xmath36 . on the other hand @xmath14 contains @xmath37 , and so is not virtually solvable , nor even weakly commensurable with a solvable group . hence the class of finitely - generated solvable groups is not quasi - isometrically rigid . dioubina s examples never have any finite presentation . in fact if @xmath24 is finitely presented then either @xmath22 or @xmath23 is finite ( see @xcite ) . this leads to the following question . is the class of finitely presented solvable groups quasi - isometrically rigid ? note that the property of being finitely presented is a quasi - isometry invariant ( see @xcite ) . while the polynomial growth theorem shows that the class of finitely generated nilpotent groups is quasi - isometrically rigid , the following remains an important open problem . classify finitely generated nilpotent groups up to quasi - isometry . the basic quasi - isometry invariants for a finitely - generated nilpotent group @xmath0 are most easily computed in terms of the set @xmath38 of ranks ( over @xmath39 ) of the quotients @xmath40 of the lower central series for @xmath0 , where @xmath41 is defined inductively by @xmath42 and @xmath43 $ ] . one of the first quasi - isometry invariants to be studied was the growth of a group , studied by dixmier , guivarch , milnor , wolf , and others ( see @xcite , chapters vi - vii for a nice discussion of this , and a careful account of the history ) . the _ growth _ of @xmath0 is the function of @xmath44 that counts the number of elements in a ball of radius @xmath44 in @xmath0 . there is an important dichotomy for solvable groups : let @xmath0 be a finitely generated solvable group . then either @xmath0 has polynomial growth and is virtually nilpotent , or @xmath0 has exponential growth and is not virtually nilpotent . when @xmath0 has polynomial growth , the degree @xmath45 of this polynomial is easily seen to be a quasi - isometry invariant . it is given by the following formula , discovered around the same time by guivarch @xcite and by bass @xcite : @xmath46 where @xmath47 is the degree of nilpotency of @xmath0 . another basic invariant is that of virtual cohomological dimension @xmath48 . for groups @xmath0 with finite classifying space ( which is not difficult to check for torsion - free nilpotent groups ) , this number was shown by gersten @xcite and block - weinberger @xcite to be a quasi - isometry invariant . on the other hand it is easy to check that @xmath49 where @xmath47 is the degree of nilpotency , also known as the hirsch length , of @xmath0 . as bridson and gersten have shown ( see @xcite ) , the above two formulas imply that any finitely generated group @xmath15 which is quasi - isometric to @xmath50 must have a finite index @xmath50 subgroup : by the polynomial growth theorem such a @xmath15 has a finite index nilpotent subgroup @xmath51 ; but @xmath52 and so @xmath53 which can only happen if @xmath54 for @xmath55 , in which case @xmath51 is abelian . give an elementary proof ( i.e. without using gromov s polynomial growth theorem ) that any finitely generated group quasi - isometric to @xmath50 has a finite index @xmath50 subgroup . as an exercise , the reader is invited to find nilpotent groups @xmath56 which are not quasi - isometric but which have the same degree of growth and the same @xmath57 . there are many other quasi - isometry invariants for finitely - generated nilpotent groups @xmath15 . all known invariants are special cases of the following theorem of pansu @xcite . to every nilpotent group @xmath15 one can associate a nilpotent lie group @xmath58 , called the _ malcev completion _ of @xmath15 ( see @xcite ) , as well as the associated graded lie group @xmath59 . [ theorem : pansu ] let @xmath60 be two finitely - generated nilpotent groups . if @xmath61 is quasi - isometric to @xmath62 then @xmath63 is isomorphic to @xmath64 . we remark that there are nilpotent groups with non - isomorphic malcev completions where the associated gradeds are isomorphic ; the examples are 7-dimensional and somewhat involved ( see @xcite , p.24 , example 2 ) . it is not known whether or not the malcev completion is a quasi - isometry invariant . theorem [ theorem : pansu ] immediately implies : the numbers @xmath65 are quasi - isometry invariants . in particular we recover ( as special cases ) that growth and cohomological dimension are quasi - isometry invariants of @xmath15 . to understand pansu s proof one must consider _ carnot groups_. these are graded nilpotent lie groups @xmath51 whose lie algebra @xmath66 is generated ( via bracket ) by elements of degree one . chow s theorem @xcite states that such lie groups @xmath51 have the property that the left - invariant distribution obtained from the degree one subspace @xmath67 of @xmath66 is a _ totally nonintegrable _ distribution : any two points @xmath68 can be connected by a piecewise smooth path @xmath69 in @xmath51 for which the vector @xmath70 lies in the distribution . infimizing the length of such paths between two given points gives a metric on @xmath51 , called the _ carnot carethodory _ metric @xmath71 . this metric is non - riemannian if @xmath72 . for example , when @xmath51 is the @xmath73-dimensional heisenberg group then the metric space @xmath74 has hausdorff dimension @xmath75 . one important property of carnot groups is that they come equipped with a @xmath76-parameter family of dilations @xmath77 , which gives a notion of _ ( carnot ) differentiability _ ( see @xcite ) . further , the differential @xmath78 of a map @xmath79 between carnot groups @xmath56 which is ( carnot ) differentiable at the point @xmath80 is actually a _ lie group homomorphism _ @xmath81 . * sketch of pansu s proof of theorem [ theorem : pansu ] . * if @xmath82 is a nilpotent group endowed with a word metric @xmath83 , the sequence of scaled metric spaces @xmath84 has a limit in the sense of gromov - hausdorff convergence : @xmath85 ( see @xcite , and @xcite for an introduction to gromov - hausdorff convergence ) . it was already known , using ultralimits , that some subsequence converges @xcite . pansu s proof not only gives convergence on the nose , but it yields some additional important features of the limit metric space @xmath86 : * ( identifying limit ) it is isometric to the carnot group @xmath87 endowed with the carnot metric @xmath71 . * ( functoriality ) any quasi - isometry @xmath88 bewteen finitely - generated nilpotent groups induces a _ bilipschitz homeomorphism _ note that functoriality follows immediately once we know the limit exists : the point is that if @xmath90 is a @xmath91 quasi - isometry of word metrics , then for each @xmath47 the map @xmath92 is a @xmath93 quasi - isometry , hence the induced map @xmath94 is a @xmath95 quasi - isometry , i.e. is a bilipschitz homeomorphism . given a quasi - isometry @xmath96 , we thus have an induced bilipschitz homeomorphism @xmath97 between carnot groups endowed with carnot - carethodory metrics . pansu then proves a regularity theorem , generalizing the rademacher - stepanov theorem for @xmath98 . this general regularity theorem states that a bilipschitz homeomorphism of carnot groups ( endowed with carnot - carethodory metrics ) is differentiable almost everywhere . since the differential @xmath99 is actually a group homomorphism , we know that for almost every point @xmath100 the differential @xmath101 is an isomorphism . the first progress on question [ question : rigidity ] and problem [ problem : classification ] in the non-(virtually)-nilpotent case was made in @xcite and @xcite . these papers proved classification and rigidity for the simplest class of non - nilpotent solvable groups : the _ solvable baumslag - solitar groups _ @xmath102 these groups are part of the much broader class of abelian - by - cyclic groups . a group @xmath15 is _ abelian - by - cyclic _ if there is an exact sequence @xmath103 where @xmath22 is an abelian group and @xmath104 is an infinite cyclic group . if @xmath15 is finitely generated , then @xmath22 is a finitely generated module over the group ring @xmath105 $ ] , although @xmath22 need not be finitely generated as a group . by a result of bieri and strebel @xcite , the class of finitely presented , torsion - free , abelian - by - cyclic groups may be described in another way . consider an @xmath106 matrix @xmath107 with integral entries and @xmath108 . let @xmath109 be the ascending hnn extension of @xmath50 given by the monomorphism @xmath110 with matrix @xmath107 . then @xmath109 has a finite presentation @xmath111=1 , ta_it^{-1}=\phi_m(a_i ) , i , j=1,\ldots , n\rangle\ ] ] where @xmath112 is the word @xmath113 and the vector @xmath114 is the @xmath115 column of the matrix @xmath107 . such groups @xmath109 are precisely the class of finitely presented , torsion - free , abelian - by - cyclic groups ( see @xcite for a proof involving a precursor of the bieri - neumann - strebel invariant , or @xcite for a proof using trees ) . the group @xmath109 is polycyclic if and only if @xmath116 ( see @xcite ) . the results of @xcite and @xcite are generalized in @xcite , which gives the complete classification of the finitely presented , nonpolycyclic abelian - by - cyclic groups among all f.g . groups , as given by the following two theorems . the first theorem in @xcite gives a classification of all finitely - presented , nonpolycyclic , abelian - by - cyclic groups up to quasi - isometry . it is easy to see that any such group has a torsion - free subgroup of finite index , so is commensurable ( hence quasi - isometric ) to some @xmath109 . the classification of these groups is actually quite delicate the standard quasi - isometry invariants ( ends , growth , isoperimetric inequalities , etc . ) do not distinguish any of these groups from each other , except that the size of the matrix @xmath107 can be detected by large scale cohomological invariants of @xmath109 . given @xmath117 , the _ absolute jordan form _ of @xmath107 is the matrix obtained from the jordan form for @xmath107 over @xmath118 by replacing each diagonal entry with its absolute value , and rearranging the jordan blocks in some canonical order . [ theorem : classification ] let @xmath119 and @xmath120 be integral matrices with @xmath121 for @xmath122 . then @xmath123 is quasi - isometric to @xmath124 if and only if there are positive integers @xmath125 such that @xmath126 and @xmath127 have the same absolute jordan form . theorem [ theorem : classification ] generalizes the main result of @xcite , which is the case when @xmath128 are positive @xmath129 matrices ; in that case the result of @xcite says even more , namely that @xmath123 and @xmath124 are quasi - isometric if and only if they are commensurable . when @xmath130 , however , it s not hard to find @xmath131 matrices @xmath128 such that @xmath132 are quasi - isometric but not commensurable . polycyclic examples are given in @xcite ; similar ideas can be used to produce nonpolycyclic examples . the following theorem shows that the algebraic property of being a finitely presented , nonpolycyclic , abelian - by - cyclic group is in fact a geometric property . [ theorem : rigidity ] let @xmath133 be a finitely presented abelian - by - cyclic group , determined by an @xmath106 integer matrix @xmath107 with @xmath134 . let @xmath0 be any finitely generated group quasi - isometric to @xmath15 . then there is a finite normal subgroup @xmath135 such that @xmath136 is commensurable to @xmath137 , for some @xmath106 integer matrix @xmath51 with @xmath138 . theorem [ theorem : rigidity ] generalizes the main result of @xcite , which covers the case when @xmath107 is a positive @xmath129 matrix . the @xmath129 case is given a new proof in @xcite , which is adapted in @xcite to prove theorem [ theorem : rigidity ] . the `` finitely presented '' hypothesis in theorem [ theorem : rigidity ] can not be weakened to `` finitely generated '' , since diuobina s example ( discussed in [ section : dioubina ] ) is abelian - by - cyclic , namely @xmath35$]-by-@xmath36 . one new discovery in @xcite is that there is a strong connection between the geometry of solvable groups and the theory of dynamical systems . assuming here for simplicity that the matrix @xmath107 lies on a @xmath76-parameter subgroup @xmath139 in @xmath140 , let @xmath141 be the semi - direct product @xmath142 , where @xmath143 acts on @xmath98 by the @xmath76-parameter subgroup @xmath139 . we endow the solvable lie group @xmath141 with a left - invariant metric . the group @xmath141 admits a _ vertical flow _ : @xmath144 there is a natural _ horizontal foliation _ of @xmath141 whose leaves are the level sets @xmath145 of time . a quasi - isometry @xmath146 is _ horizontal respecting _ if it coarsely permutes the leaves of this foliation ; that is , if there is a constant @xmath147 so that @xmath148 where @xmath149 denotes hausdorff distance and @xmath150 is some function , which we think of as a _ time change _ between the flows . a key technical result of @xcite is the phenomenon of _ time rigidity _ : the time change @xmath151 must actually be _ affine _ , so taking a real power of @xmath107 allows one to assume @xmath152 . it is then shown that `` quasi - isometries remember the dynamics '' . that is , @xmath153 coarsely respects several foliations arising from the partially hyperbolic dynamics of the flow @xmath154 , starting with the weak stable , weak unstable , and center - leaf foliations . by keeping track of different exponential and polynomial divergence properties of the action of @xmath154 on tangent vectors , the weak stable and weak unstable foliations are decomposed into flags of foliations . using time rigidity and an inductive argument it is shown that these flags are coarsely respected by @xmath153 as well . relating the flags of foliations to the jordan decomposition then completes the proof of : [ theorem : horizontal ] if there is a horizontal - respecting quasi - isometry @xmath146 then there exist nonzero @xmath155 so that @xmath156 and @xmath157 have the same absolute jordan form . the `` nonpolycyclic '' hypothesis ( i.e. @xmath134 ) in theorem [ theorem : classification ] is used in two ways . first , the group @xmath109 has a model space which is topologically a product of @xmath98 and a regular tree of valence @xmath158 , and when this valence is greater than @xmath159 we can use coarse algebraic topology ( as developed in @xcite , @xcite , and @xcite ) to show that any quasi - isometry @xmath160 induces a quasi - isometry @xmath161 satisfying the hypothesis of theorem [ theorem : horizontal ] . second , we are able to pick off _ integral _ @xmath162 by developing a `` boundary theory '' for @xmath109 ; in case @xmath134 this boundary is a self - similar cantor set whose bilipschitz geometry detects the primitive integral power of @xmath163 by cooper s theorem @xcite , finishing the proof of theorem [ theorem : classification ] . [ problem : extend ] extend theorem [ theorem : classification ] and theorem [ theorem : rigidity ] to the class of finitely - presented nilpotent - by - cyclic groups . of course , as the classification of finitely - generated nilpotent groups is still open , problem [ problem : extend ] is meant in the sense of reducing the nilpotent - by - cyclic case to the nilpotent case , together with another invariant . this second invariant for a nilpotent - by - cyclic group @xmath0 will perhaps be the absolute jordan form of the matrix which is given by the action of the generator of the cyclic quotient of @xmath0 on the nilpotent kernel of @xmath0 . the polycyclic , abelian - by - cyclic groups are those @xmath109 for which @xmath116 , so that @xmath109 is cocompact and discrete in @xmath141 , hence quasi - isometric to @xmath141 . in this case the proof of theorem [ theorem : classification ] outlined above breaks down , but this is so in part because the answer is quite different : the quasi - isometry classes of polycyclic @xmath109 are much coarser than in the nonpolycyclic case , as the former are ( conjecturally ) determined by the absolute jordan form up to _ real _ , as opposed to integral , powers . the key conjecture is : [ conjecture : poly1 ] suppose that @xmath164 , and that @xmath107 and @xmath51 have no eigenvalues on the unit circle . then every quasi - isometry of @xmath165 is horizontal - respecting . the general ( with arbitrary eigenvalues ) case of conjecture [ conjecture : poly1 ] , which is slightly more complicated to state , together with theorem [ theorem : horizontal ] easily implies : suppose that @xmath164 . then @xmath109 is quasi - isometric to @xmath137 if and only if there exist nonzero @xmath155 so that @xmath166 and @xmath167 have the same absolute jordan form . here by @xmath166 we mean @xmath168 , where @xmath169 is a @xmath76-parameter subgroup with @xmath170 ( we are assuming that @xmath107 lies on such a subgroup , which can be assumed after squaring @xmath107 ) . now let us concentrate on the simplest non - nilpotent example , which is also one of the central open problems in the field . the 3-dimensional geometry @xmath171 is the lie group @xmath141 where @xmath172 is any matrix with 2 distinct real eigenvalues ( up to scaling , it does nt matter which such @xmath107 is chosen ) . [ conjecture : solv ] the @xmath73-dimensional lie group @xmath171 is quasi - isometrically rigid : any f.g . group @xmath0 quasi - isometric to @xmath171 is weakly commensurable with a cocompact , discrete subgroup of @xmath171 . there is a natural boundary for @xmath171 which decomposes into two pieces @xmath173 and @xmath174 ; these are the leaf spaces of the weak stable and weak unstable foliations , respectively , of the vertical flow on @xmath175 , and are both homeomorphic to @xmath143 . the isometry group @xmath176 acts on the pair @xmath177 affinely and induces a faithful representation @xmath178 whose image consists of the pairs @xmath179 just as quasi - isometries of hyperbolic space @xmath180 are characterized by their quasiconformal action on @xmath181 ( a fact proved by mostow ) , giving the formula @xmath182 , we conjecture : [ conjecture : qigroup ] @xmath183 where @xmath184 denotes the group of bilipschitz homeomorphisms of @xmath143 , and @xmath185 acts by switching factors . there is evidence for conjecture [ conjecture : qigroup ] : the direction @xmath186 is not hard to check ( see @xcite ) , and the analogous theorem @xmath187 was proved in @xcite . by using convergence groups techniques and a theorem of hinkkanen on uniformly quasisymmetric groups ( see @xcite ) , we have been able to show : conjecture [ conjecture : poly1 ] ( in the @xmath188 case ) @xmath189 conjecture [ conjecture : qigroup ] @xmath189 conjecture [ conjecture : solv ] here is a restatement of conjecture [ conjecture : poly1 ] in the @xmath190 case : every quasi - isometry @xmath191 is horizontal respecting . [ conjecture : solvhorizontal ] here is one way _ not _ to prove conjecture [ conjecture : solvhorizontal ] . one of the major steps of @xcite in studying @xmath192 was to construct a model space @xmath193 for the group @xmath192 , study the collection of isometrically embedded hyperbolic planes in @xmath193 , and prove that for any quasi - isometric embedding of the hyperbolic plane into @xmath193 , the image has finite hausdorff distance from some isometrically embedded hyperbolic plane . however , @xmath171 has quasi - isometrically embedded hyperbolic planes which are _ not _ hausdorff close to isometrically embedded ones . the natural left invariant metric on @xmath171 has the form @xmath194 from which it follows that the @xmath195-planes and @xmath196-planes are the isometrically embedded hyperbolic planes . but none of these planes is hausdorff close to the set @xmath197 which is a quasi - isometrically embedded hyperbolic plane . an even stranger example is shown in figure [ figurewindvane ] . these strange quasi - isometric embeddings from @xmath198 to @xmath171 do share an interesting property with the standard isometric embeddings , which may point the way to understanding quasi - isometric rigidity of @xmath171 . we say that a quasi - isometric embedding @xmath199 is _ @xmath22-quasivertical _ if for each @xmath200 there exists a vertical line @xmath201 such that @xmath202 is contained in the @xmath22-neighborhood of @xmath203 , and @xmath203 is contained in the @xmath22-neighborhood of @xmath204 . in order to study @xmath171 , it therefore becomes important to understand whether every quasi - isometrically embedded hyperbolic plane is quasi - vertical . specifically : show that for all @xmath205 there exists @xmath22 such that each @xmath205-quasi - isometrically embedded hyperbolic plane in @xmath171 is @xmath22-quasivertical . [ problemquasivertical ] arguing by contradiction , if problem [ problemquasivertical ] were impossible , fixing @xmath205 and taking a sequence of examples whose quasi - vertical constant @xmath22 goes to infinity , one can pass to a subsequence and take a renormalized limit to produce a quasi - isometric embedding @xmath206 whose image is entirely contained in the upper half @xmath207 of @xmath171 . but we conjecture that this is impossible : there does not exist a quasi - isometric embedding @xmath208 whose image is entirely contained in the upper half space @xmath209 . while we have already seen that there is a somewhat fine classification of finitely presented , nonpolycyclic abelian - by - cyclic groups up to quasi - isometry , this class of groups is but a very special class of finitely generated solvable groups . we have only exposed the tip of a huge iceberg . an important next layer is : the first step in attacking this problem is to find a workable method of describing the geometry of the natural geometric model of such groups @xmath0 . such a model should fiber over @xmath98 , where @xmath47 is the rank of the maximal abelian quotient of @xmath0 ; inverse images under this projection of ( translates of ) the coordinate axes should be copies of the geometric models of abelian - by - cyclic groups . * polycyclic versus nonpolycyclic . * we ve seen the difference , at least in the abelian - by - cyclic case , between polycyclic and nonpolycyclic groups . geometrically these two classes can be distinguished by the trees on which they act : such trees are lines in the former case and infinite - ended in the latter . it is this branching behavior which should combine with coarse topology to make the nonpolycyclic groups more amenable to attack . note that a ( virtually ) polycyclic group is never quasi - isometric to a ( virtually ) nonpolycyclic solvable group . this follows from the theorem of bieri that polycylic groups are precisely those solvable groups satisfying poincare duality , together with the quasi - isometric invariance of the latter property ( proved by gersten @xcite and block - weinberger @xcite ) . * solvable groups as dynamical systems . * the connection of nilpotent groups with dynamical systems was made evident in @xcite , where gromov s polynomial growth theorem was the final ingredient , combining with earlier work of franks and shub @xcite , in the positive solution of the _ expanding maps conjecture _ : every locally distance expanding map on a closed manifold @xmath107 is topologically conjugate to an expanding algebraic endomorphism on an infranil manifold ( see @xcite ) . in [ section : abc ] and [ section : abc2 ] we saw in another way how invariants from dynamics give quasi - isometry invariants for abelian - by - cyclic groups . this should be no big surprise : after all , a finitely presented abelian - by - cyclic group is describable up to commensurability as an ascending hnn extension @xmath109 over a finitely - generated abelian group @xmath50 . the matrix @xmath107 defines an endomorphism of the @xmath47-dimensional torus . the mapping torus of this endomorphism has fundamental group @xmath109 , and is the phase space of the suspension semiflow of the endomorphism , a semiflow with partially hyperbolic dynamics ( when @xmath107 is an automorphism , and so @xmath109 is polycyclic , the suspension semiflow is actually a flow ) . here we see an example of how the geometric model of a solvable group is actually the phase space of a dynamical system . but bieri - strebel @xcite have shown that _ every _ finitely presented solvable group is , up to commensurability , an ascending hnn extension with base group a finitely generated solvable group . in this way every finitely presented solvable group is the phase space of a dynamical system , probably realizable geometrically as in the abelian - by - cyclic case . e. ghys and p. de la harpe , infinite groups as geometric objects ( after gromov ) , in _ ergodic theory , symbolic dynamics and hyperbolic spaces _ , ed . by t. bedford and m. keane and c. series , oxford univ . press , 1991 .

a survey of problems , conjectures , and theorems about quasi - isometric classification and rigidity for finitely generated solvable groups .

many simply connected , smoothable topological 4manifolds ( with @xmath3 odd ) are known to admit infintely many distinct smooth structures . the existence of such _ exotic _ structures are , however , less clear when the euler characteristic of the 4manifold is small , for example if the manifold is homeomorphic to the ( blow up of the ) complex projective plane . using the rational blow down construction @xcite together with knot surgery @xcite in double node neighborhoods @xcite , many new smooth 4manifolds have been discovered with @xmath4 and @xmath5 @xcite . similar ideas can be applied to get examples of irreducible exotic simply connected 4manifolds with @xmath1 and relatively small @xmath6 . the study of exotic structures on simply connected manifolds with @xmath1 has a rich history . recall that the @xmath7 surface @xmath8 is such an example with @xmath9 . applying appropriate logarithmic transformations on @xmath8 it was shown that @xmath10 admits infinitely many smooth structures @xcite . later works , relying on gompf s symplectic normal connected sum operation , together with donaldson theory showed that the topological manifolds @xmath11 with @xmath12 admit infinitely many smooth structures @xcite . more recent results of park @xcite proved the same statement with @xmath13 . our main result in this paper improves this bound : [ t : m1 ] the simply connected topological 4manifold @xmath0 admits infinitely many distinct smooth structures . by modifying the construction used in the proof of theorem [ t : m1 ] , we can define a set of 4manifolds which still have @xmath1 , but their euler characteristic is smaller than the above examples . in these cases , however , we were unable to show that the manifolds are simply connected . [ t : m2 ] there are infinitely many pairwise nondiffeomorphic smooth , closed 4manifolds with vanishing first homology , @xmath1 , @xmath2 and nontrivial seiberg witten invariants . the proof of the results will involve two steps . first we desribe how to construct the manifolds claimed and then we use seiberg witten theory in proving that they are nondiffeomorphic . in the construction we will apply mapping class group arguments and the theory of lefschetz fibrations . using the knot surgery construction then we can identify configurations of curves in the resulting 4manifolds which can be rationally blown down , leading us to the desired examples . recall that a genus1 lefschetz fibration @xmath14 can be described by the word in the mapping class group @xmath15 of the 2torus @xmath16 corresponding to the monodromy presentation of @xmath17 . more precisely , a point @xmath18 where @xmath19 is not onto ( called a singular point of the fibration ) gives rise to a singular fiber @xmath20 , and the monodromy of the fibration around such a fiber can be given by the composition of dehn twists along the circles corresponding to the vanishing cycles of the singular points of the fiber . by traversing through the singular fibers in a counterclockwise manner relative to a fixed base point @xmath21 , we get the above mentioned word describing the fibration . notice that we do not assume that @xmath17 is injective on the set of its singular points , that is , a singular fiber can contain more than one singular points . the assumption that the map @xmath17 is a lefschetz fibration implies that the vanishing cycles corresponding to the singular points in one fixed singular fiber can be chosen to be disjoint . it is known that the mapping class group @xmath15 can be generated by two elements @xmath22 which are subject to the two relations @xmath23 in fact , @xmath15 can be shown to be isomorphic to @xmath24 by mapping @xmath25 to @xmath26 and @xmath27 to @xmath28 . since the forgetful map from the mapping class group @xmath29 of the 2torus with one marked point to @xmath15 is an isomorphism , we get that any genus1 lefschetz fibration admits a section . genus1 lefschetz fibrations were classified by moishezon @xcite , who showed that after a possible perturbation such a fibration over @xmath30 is equivalent to one of the fibrations given by the words @xmath31 ( @xmath32 ) in @xmath15 . the resulting 4manifold is usually called @xmath33 ( the simply connected elliptic surface with section and of holomorphic euler characteristic @xmath34 ) , and @xmath8 is the famous @xmath7 surface . it can be shown that a section of @xmath35 has self intersection @xmath36 . following @xcite we call a fiber with monodromy conjugate to @xmath37 of _ type @xmath38 _ ( @xmath39 ) . when @xmath40 , the corresponding fiber is also called a _ fishtail _ fiber . it is easy to see that topologically a singular fiber of type @xmath41 ( @xmath42 ) is a plumbing of @xmath43 smooth 2spheres of self intersection @xmath44 plumbed along a circle ( see @xcite ) , while a fishtail fiber is an immersed 2sphere with one positive double point . since in @xmath16 nonisotopic simple closed curves necessarily intersect each other , it is easy to see that a genus1 lefschetz fibration can have only @xmath41fibers as singular fibers . the fibration @xmath17 can be perturbed near a singular fiber @xmath45 of type @xmath41 into a fibration @xmath46 which is the same as @xmath17 outside of @xmath45 but breaks @xmath45 into two singular fibers of types @xmath47 and @xmath48 with @xmath49 . this fact implies that a generic genus1 lefschetz fibration admits only fishtail fibers . in our study however , we will find it most helpful to understand what kind of other singular fibers an elliptic fibration can admit . we start with a proposition showing the existence of a particular genus1 fibration on @xmath8 . [ p : mcg ] there exists an elliptic lefschetz fibration on the @xmath7 surface @xmath8 with a section , a singular fiber @xmath45 of type @xmath51 , three singular fibers @xmath52 of type @xmath53 and two further fishtail fibers . it is not hard to see that the word @xmath54 ( defining a genus1 lefschetz fibration on the @xmath7 surface @xmath8 ) in the mapping class group @xmath15 of the torus is equivalent to @xmath55 ( alternatively , by substituting @xmath25 and @xmath27 with the matrices they correspond to under the map @xmath56 , we can check that the above product is equal to the identity matrix . ) by collecting the powers of @xmath25 in the front using conjugation , we get @xmath57 followed by the product of three squares of some conjugates of @xmath27 and two further conjugates of @xmath27 . since a conjugate of @xmath27 by a word @xmath58 corresponds to the dehn twist along the image under the diffeomorphism @xmath59 of the curve inducing @xmath27 , the proposition follows . as we already mentioned , genus1 lefschetz fibrations always admit sections . perturb first the above fibration near the @xmath53 fibers @xmath60 ( @xmath61 ) in a way that these give rise to fishtail fibers @xmath62 with isotopic vanishing cycles . let us denote the resulting lefschetz fibration by @xmath63 . let @xmath64 be three twist knots as depicted in @xcite . let @xmath65 denote the 4manifold we get after performing three knot surgeries with knots @xmath64 along three regular fibers in the fibration on the @xmath7 surface found above . because of the existence of fishtail fibers in the complement with nonisotopic vanishing cycles , we conclude that @xmath66 . if we perform the surgeries in the double node neighborhoods near the fishtail fibers @xmath67 , and @xmath68 and @xmath69 respectively , then we can find a pseudo section @xmath70 as in @xcite which is an immersed sphere of self intersection @xmath71 with three positive double points , intersecting the further fishtail fibers and the @xmath51 fiber @xmath45 transversally . next smooth the intersections of this pesudo section @xmath70 with two further fishtail fibers . the result is an immersed sphere of self intersection 2 having 5 positive double points . now blow up the 4manifold @xmath65 in the five double points of this sphere , and find an embedded sphere of self intersection @xmath72 in @xmath73 . let @xmath74 denote the tubular neighborhood of the linear plumbing of spheres given by this @xmath75sphere together with 14 of the @xmath71spheres in the @xmath51 fiber @xmath45 . it is not hard to see that @xmath74 is diffeomorphic to @xmath76 in the notation of @xcite , cf . also @xcite . it is then easy to show that @xmath77 can be given as the oriented boundary of the rational ball @xmath78 , see @xcite . define @xmath79 as the rational blow down of @xmath73 along @xmath74 , that is , @xmath80 notice that @xmath65 is simply connected , and the complement of @xmath74 in @xmath81 is simply connected since the @xmath71sphere in @xmath45 intersecting the last @xmath71sphere of the linear chain @xmath74 provides a hemisphere which contracts the generator of the fundamental group of @xmath82 . since @xmath83 surjects onto @xmath84 under the natural embedding , van kampen s theorem implies that @xmath79 is simply connected . now simple signature and euler characteristics computation together with freedman s theorem @xcite verifies the result . for short , let @xmath85 denote the 4manifold @xmath79 if @xmath86 the @xmath34twist knot @xmath87 . the following proposition is true in a wider generality , we restrict our attention to the special case @xmath88 in order to keep our discussion as simple as possible . using the results of @xcite the seiberg witten invariants of @xmath85 can be easily computed . this computation immediately shows by ( * ? ? ? * theorem 1.1 ) the seiberg witten invariant of @xmath93 can be computed to be equal to @xmath94 ( use the facts that the seiberg witten function of the @xmath7 surface is equal to 1 and the alexander polynomial of @xmath87 is @xmath95 ) . this result , together with the blow up formula shows that the five fold blow up @xmath96 has exactly two seiberg witten basic classes @xmath97 which evaluate on the @xmath75sphere of the configuration @xmath74 as @xmath98 . moreover , the value of the seiberg witten function on these basic classes is @xmath99 . now * theorem 8.5 ) implies that @xmath85 has two basic classes , on which the value of the seiberg witten function is equal to @xmath99 , verifying the result . the 4manifolds @xmath85 provide an infinite family of smooth 4manifolds all homeomorphic to @xmath0 according to proposition [ p : hom ] , and by corollary [ c : nondiffeo ] these manifolds are pairwise nondiffeomorphic . therefore the set @xmath100 provides an infinite collection of distinct smooth structures on @xmath0 , hence the proof is complete . using a variation of the above procedure , closed 4manifolds with @xmath1 and @xmath2 can be constructed as follows . consider the fibration @xmath63 found above , containing a singular fiber of type @xmath51 and eight fishtail fibers , out of which three pairs @xmath101 ( @xmath61 ) have isotopic vanishing cycles . proceed as before by doing three knot surgeries along three regular fibers with twist knots @xmath64 and using the double node neighborhoods provided by the fishtail fibers @xmath102 identify the pseudo section , which is again an immersed sphere with homological square @xmath44 and has three positive double points . as before , resolve the two positive intersections of this pseudo section with the remaining two fishtail fibers , and find the immersed sphere with 5 double point and homological square 2 . blow up the 4manifold at the double points of the pesudo section , and consider the resulting sphere of square @xmath72 in @xmath65 . this sphere intersects the @xmath51 fiber @xmath45 transversally in a unique point @xmath103 which is on the @xmath71sphere @xmath104 . now @xmath105 is intersected by two other spheres in @xmath45 , let @xmath106 be one of them and denote the intersection point of @xmath105 and @xmath106 by @xmath107 . apply 17 infinitely close blow ups at @xmath107 . the resulting plumbing manifold @xmath108 can be given by the linear plumbing @xmath109 @xmath110 as before , it is routine to see that the boundary @xmath111 can be given as the boundary of the rational ball @xmath112 , hence we can blow it down , resulting the 4manifold @xmath113 . since the normal circle of the pseudo section ( which circle generates the first homology of @xmath111 ) vanishes in the homology of the complement @xmath114 ( as it is shown by a regular fiber ) , the mayer vietoris sequence implies that @xmath115 vanishes . simple euler characteristic and signature computations now imply finally , by computing seiberg witten invariants of these manifolds in the fashion it was done in theorem [ t : sw ] , we get that the 4manifolds @xmath116 are pairwise nondiffeomorphic , all with nonvanishing seiberg witten invariants , verifying the claim of theorem [ t : m2 ] .

we construct an infinite family of simply connected , pairwise nondiffeomorphic 4manifolds , all homeomorphic to @xmath0 . similar ideas provide examples of 4manifolds with @xmath1 , @xmath2 , vanishing first homology and nontrivial seiberg witten invariants .

oscillation experiments over vastly different baselines and a range of neutrino energies have filled up a vast portion of the mass and mixing jigsaw of the neutrino sector . yet , we still remain in the dark with regard to cp - violation in the lepton sector . neither do we know the mass ordering whether it is normal or inverted . further open issues are the absolute mass scale of neutrinos and whether they are of majorana or dirac nature . while we await experimental guidance for each of the above unknowns , there have been many attempts to build models of lepton mass which capture much of what is known . here we propose a neutrino mass model based on the direct product group @xmath1 . the elements of @xmath15 correspond to the permutations of three objects can be found in appendix [ apps3 ] . ] . needless to say , @xmath15-based models of neutrino mass have been considered earlier @xcite . a popular point of view @xcite has been to note that a permutation symmetry between the three neutrino states is consistent with ( see appendix [ apps3 ] ) . so these models entail fine tuning . ] ( a ) a democratic mass matrix , @xmath16 , all whose elements are equal , and ( b ) a mass matrix proportional to the identity matrix , @xmath17 . a general combination of these two forms , e.g. , @xmath18 , where @xmath19 are complex numbers , provides a natural starting point . one of the eigenstates , namely , an equal weighted combination of the three states , is one column of the popular tribimaximal mixing matrix . many models have been presented @xcite which add perturbations to this structure to accomplish realistic neutrino masses and mixing . variations on this theme @xcite explore mass matrices with such a form in the context of grand unified theories , in models of extra dimensions , and examine renormalisation group effects on such a pattern realised at a high energy . other variants of the @xmath15-based models , for example @xcite , rely on a 3 - 3 - 1 local gauge symmetry , tie it to a @xmath20-extended model , or realise specific forms of mass matrices through soft symmetry breaking , etc . as discussed later , the irreducible representations of @xmath15 are one and two - dimensional . the latter provides a natural mechanism to get maximal mixing in the @xmath21 sector @xcite . the present work , also based on @xmath15 symmetry , breaks new ground in the following directions . firstly , it involves an interplay of type - i and type - ii see - saw contributions . secondly , it presents a general framework encompassing many popular mixing patterns such as tribimaximal mixing . further , the model does not invoke any soft symmetry breaking terms . all the symmetries are broken spontaneously . we briefly outline here the strategy of this work . we use the standard notation for the leptonic mixing matrix the pontecorvo , maki , nakagawa , sakata ( pmns ) matrix @xmath22 . @xmath23 where @xmath24 and @xmath25 . the neutrino masses and mixings arise through a two - stage mechanism . in the first step , from the type - ii see - saw the larger atmospheric mass splitting , @xmath26 , is generated while the solar splitting , @xmath27 , is absent . also , @xmath28 , @xmath29 and the model parameters can be continuously varied to obtain any desired @xmath4 . of course , in reality @xmath30 @xcite , the solar splitting is non - zero , and there are indications that @xmath9 is large but non - maximal . experiments have also set limits on @xmath31 . the type - i see - saw addresses all the above issues and relates the masses and mixings to each other . the starting form incorporates several well - studied mixing patterns such as tribimaximal ( tbm ) , bimaximal ( bm ) , and golden ratio ( gr ) mixings within its fold . these alternatives all have @xmath32 and @xmath2 . they differ only in the value of the third mixing angle @xmath4 as displayed in table [ t1 ] . the fourth option in this table , no solar mixing ( nsm ) , exhibits the attractive feature symmetry @xcite which built on previous work along similar lines @xcite . ] that the mixing angles are either maximal , i.e. , @xmath33 ( @xmath9 ) or vanishing ( @xmath3 and @xmath4 ) . .the solar mixing angle , @xmath34 for this work , for the tbm , bm , and gr mixing patterns . nsm stands for the case where the solar mixing angle is initially vanishing . [ cols="^,^,^,^,^",options="header " , ] the group has two 1-dimensional representations denoted by @xmath35 and @xmath36 , and a @xmath37-dimensional representation . @xmath35 is inert under the group while @xmath38 changes sign under the action of @xmath39 . for the 2-dimensional representation a suitable choice of matrices with the specified properties can be readily obtained . we choose @xmath40 where @xmath41 is a cube root of unity , i.e. , @xmath42 . for this choice of @xmath39 and @xmath43 the remaining matrices of the representation are : @xmath44 the product rules for the different representations are : @xmath45 one can see that each of the @xmath46 matrices @xmath47 in eqs . ( [ s3_21 ] ) and ( [ s3_22 ] ) satisfies : @xmath48 where @xmath49 for @xmath50 and @xmath51 for @xmath52 . if @xmath53 and @xmath54 are two field multiplets transforming under @xmath15 as doublets then using eqs . ( [ s3_21 ] ) and ( [ s3inv ] ) : @xmath55 sometimes we have to deal with hermitian conjugate fields . noting the nature of the complex representation ( see , for example , @xmath43 in eq . ( [ s3_21 ] ) ) the conjugate @xmath15 doublet is @xmath56 . as a result , one has in place of ( [ s3_prod ] ) @xmath57 eqs . ( [ s3_prod ] ) and ( [ s3_prod2 ] ) are essential in writing down the fermion mass matrices in sec . [ model ] . as seen in table [ tab1s ] this model has a rich scalar field content . in this appendix we write down the scalar potential of the model keeping all these fields and derive conditions which must be met by the coefficients of the various terms so that the desired _ vev_s can be achieved . these conditions ensure that the potential is locally minimized by this choice . table [ tab1s ] displays the behaviour of the scalar fields under @xmath1 besides the gauged electroweak @xmath58 . the fields also carry a lepton number . the scalar potential is the most general polynomial in these fields with up to quartic terms . our first step will be to write down the explicit form of this potential . here we do not exclude any term permitted by the symmetries . @xmath59 invariance of the terms as well as the abelian lepton number and @xmath60 conservation are readily verified . it is only the @xmath15 behaviour which merits special attention . there are a variety of scalar fields in this model , e.g. , @xmath61 singlets , doublets , and triplets . therefore , the scalar potential has a large number of terms . for simplicity we choose all couplings in the potential to be real . in this appendix we list the potential in separate parts : ( a ) those belonging to any one @xmath61 sector , and ( b ) inter - sector couplings of scalars . the @xmath61 singlet _ vev_s , which are responsible for the right - handed neutrino mass , are significantly larger than those of other scalars . so , in the second category we retain only those terms which couple the singlet fields to either the doublet or the triplet sectors . the @xmath61 singlet sector comprises of two fields @xmath62 and @xmath63 transforming as @xmath64 and @xmath65 of @xmath66 respectively . they have @xmath67 . the scalar potential arising out of these is : @xmath68 ^ 2 + { \lambda_2^s \over 2}\left[\gamma^\dagger\gamma\right]^2 \nonumber \\ & + & { \lambda_3^s \over 2}(\chi^\dagger\chi)(\gamma^\dagger\gamma ) + \lambda_4^s\left\{(\gamma^\dagger \chi)(\gamma^\dagger\chi)+ h.c.\right \ } \;\ ; , \label{vs_s3}\end{aligned}\ ] ] where the coefficient of the cubic term , @xmath69 , carries the same dimension as mass while the @xmath70 are dimensionless . the @xmath61 doublet sector of the model has two fields @xmath71 that are doublets of @xmath15 , in addition to @xmath72 , @xmath73 , and @xmath74 which are @xmath15 singlets . among them , all fields except @xmath75 and @xmath72 ( @xmath76 @xmath77 ) are invariant under @xmath78 . @xmath79 leaving aside @xmath15 properties for the moment , to which we return below , out of any @xmath80 doublet @xmath81 one can construct two quartic invariants @xmath82 and @xmath83 . needless to say , this can be generalised to the case where several distinct @xmath80 doublets are involved . in order to avoid cluttering , in eq . ( [ vd_s3 ] ) we have displayed only the first combination for all quartic terms . the quartic terms involving @xmath84 to @xmath85 in eq . ( [ vd_s3 ] ) are combinations of two pairs of @xmath15 doublets . each pair can combine in accordance to @xmath86 resulting in three terms . the @xmath15 invariant in the potential arises from a combination of the @xmath87 from one pair with the corresponding term from the other pair . thus , for each such term of four @xmath15 doublets , three possible singlet combinations exist ( recall , eq . ( [ s3prodapp ] ) ) and we have to keep an account of all of them . we elaborate on this using as an example the @xmath88 term which actually stands for a set of terms : @xmath89 ^ 2 + \lambda^d_{1_{1'}}\left[(\phi_1^\dagger\phi_1)-(\phi_2^\dagger\phi_2)\right]^2 + \lambda^d_{1_{2}}\left[(\phi_1^\dagger\phi_2)(\phi_2^\dagger\phi_1 ) + ( \phi_2^\dagger\phi_1)(\phi_1^\dagger\phi_2)\right].\ ] ] substituting @xmath90 , @xmath91 and @xmath92 and defining @xmath93 and @xmath94 we get : @xmath95 + { \lambda^d_{a_2}\over2}(v_1^*v_1)(v_2^*v_2 ) . \label{t6}\ ] ] similarly , @xmath96 + { \lambda^d_{b_2}\over2}(v_3^*v_3)(v_4^*v_4 ) \label{t7}\ ] ] where , @xmath97 and @xmath98 . further , @xmath99 & \rightarrow & \lambda^d_{3_{1}}\left[(\phi_1^\dagger\phi_1+\phi_2^\dagger\phi_2)(\phi_3^\dagger\phi_3+\phi_4^\dagger\phi_4)\right ] + \lambda^d_{3_{1'}}\left[(\phi_1^\dagger\phi_1-\phi_2^\dagger\phi_2)(\phi_3^\dagger\phi_3-\phi_4^\dagger\phi_4)\right ] \nonumber\\ & + & \lambda^d_{3_{2}}\left[(\phi_1^\dagger\phi_2)(\phi_4^\dagger\phi_3 ) + ( \phi_2^\dagger\phi_1)(\phi_3^\dagger\phi_4)\right].\end{aligned}\ ] ] substituting the respective @xmath90 and defining @xmath100 , @xmath101 and @xmath102 we get ; @xmath99 & \longrightarrow & { \lambda^d_{{ab}_1}\over 2 } \left[(v_1^*v_1)(v_3^*v_3)+(v_2^*v_2)(v_4^*v_4)\right ] + { \lambda^d_{{ab}_2}\over 2 } \left[(v_1^*v_1)(v_4^*v_4)+(v_2^*v_2)(v_3^*v_3)\right ] \nonumber\\ & + & \lambda^d_{{ab}_3 } \left[(v_1^*v_2)(v_4^*v_3)+(v_2^*v_1)(v_3^*v_4)\right ] . \label{t8}\end{aligned}\ ] ] in a similar fashion the @xmath103 term when expanded will lead to @xmath104 & \longrightarrow & { \tilde{\lambda}^d_{{ab}_1}\over 2 } \left[(v_1^*v_3)(v_3^*v_1)+(v_2^*v_4)(v_4^*v_2)\right ] + { \tilde{\lambda}^d_{{ab}_2}\over 2 } \left[(v_1^*v_3)(v_4^*v_2)+(v_2^*v_4)(v_3^*v_1)\right ] \nonumber\\ & + & \tilde{\lambda}^d_{{ab}_3 } \left[(v_1^*v_4)(v_4^*v_1)+(v_2^*v_3)(v_3^*v_2)\right ] . \label{t9}\end{aligned}\ ] ] adding eqs.([t8 ] ) and eq.([t9 ] ) we get : @xmath99 + { \lambda_4^d\over 2}\left[(\phi_a^\dagger\phi_b)(\phi_b^\dagger\phi_a)\right ] & = & { \hat{\lambda}^d_{{ab}_1}\over 2 } \left[(v_1^*v_1)(v_3^*v_3)+(v_2^*v_2)(v_4^*v_4)\right ] \nonumber\\ & + & { \hat{\lambda}^d_{{ab}_2}\over 2 } \left[(v_1^*v_1)(v_4^*v_4)+(v_2^*v_2)(v_3^*v_3)\right ] \nonumber\\ & + & \hat{\lambda}^d_{{ab}_3 } \left[(v_1^*v_2)(v_4^*v_3)+(v_2^*v_1)(v_3^*v_4)\right ] ; \label{sum_t8nt9}\end{aligned}\ ] ] where , @xmath105 , @xmath106 and @xmath107 . also , summing up the @xmath108 and @xmath109 terms lead to @xmath110 , where @xmath111 . both the @xmath61 triplets present in our model ( @xmath112 , @xmath113 ) that are responsible for majorana mass generation of the left handed neutrinos happen to be @xmath15 invariants and differ only in their @xmath78 properties i.e. , @xmath114 and @xmath115 . @xmath116 ^ 2 + { \lambda_2^t \over 2}\left[\rho_l^\dagger\rho_l\right]^2 + { \lambda_3^t \over 2}(\delta_l^\dagger\delta_l)(\rho_l^\dagger\rho_l ) \nonumber \\ & + & { \lambda_4^t\over 2}(\delta_l^\dagger\rho_l)(\rho_l^\dagger\delta_l ) + { \lambda_5^t \over 2}(\delta_l\rho_l)(\delta_l\rho_l)^\dagger \;\;. \label{vt_s3}\end{aligned}\ ] ] it is noteworthy that when we write the minimized potential in terms of the vacuum expectation values , the @xmath117 , @xmath118 and @xmath119 terms will be providing the same contribution as far as potential minimization is concerned . thus we can club these couplings together as @xmath120 . so far we have listed those terms in the potential which arise from scalars of any specific @xmath61 behaviour singlets , doublets , or triplets . in addition , there can be terms which couple one of these sectors to another . since the vacuum expectation values of the singlet scalars are the largest we only consider here the couplings of this sector to the others . the @xmath61 triplet sector vev is very small and we drop the doublet - triplet cross - sector couplings . couplings between the @xmath61 singlet and doublet scalars in the potential give rise to the terms : @xmath121 + \lambda_2^{ds}\left[(\phi_b^\dagger\phi_a)_{1}\chi + h.c.\right ] + \lambda_3^{ds}\left[(\alpha^\dagger\eta)\chi + h.c.\right ] \nonumber \\ & + & { \lambda_1^{ds}\over 2}(\phi_a^\dagger\phi_a)(\chi^\dagger\chi ) + { \lambda_2^{ds}\over 2}(\phi_a^\dagger\phi_a)(\gamma^\dagger\gamma ) + { \lambda_3^{ds}\over 2}(\phi_b^\dagger\phi_b)(\chi^\dagger\chi ) + { \lambda_4^{ds}\over 2}(\phi_b^\dagger\phi_b)(\gamma^\dagger\gamma ) \nonumber \\ & + & { \lambda_5^{ds}\over 2}(\alpha^\dagger\alpha)(\chi^\dagger\chi ) + { \lambda_6^{ds}\over 2}(\alpha^\dagger\alpha)(\gamma^\dagger\gamma ) + { \lambda_7^{ds}\over 2}(\eta^\dagger\eta)(\chi^\dagger\chi ) + { \lambda_8^{ds}\over 2}(\eta^\dagger\eta)(\gamma^\dagger\gamma ) \nonumber \\ & + & \lambda_9^{ds}\left[(\phi_a^\dagger\phi_b)\chi^2 + h.c.\right ] + \lambda_{10}^{ds}\left[(\phi_a^\dagger\phi_b)\gamma^2 + h.c.\right ] + \lambda_{11}^{ds}\left[(\eta^\dagger\alpha)\chi^2 + h.c.\right ] + \lambda_{12}^{ds}\left[(\eta^\dagger\alpha)\gamma^2 + h.c.\right ] \nonumber \\ & + & \lambda_{13}^{ds}\left[(\phi_a^\dagger\phi_b)_{1'}(\chi\gamma)+ h.c.\right ] + { \lambda_{14}^{ds}\over 2}(\beta^\dagger\beta)(\chi^\dagger\chi ) + { \lambda_{15}^{ds}\over 2}(\beta^\dagger\beta)(\gamma^\dagger\gamma ) \;\;. \label{vds_s3}\end{aligned}\ ] ] the terms in the potential which arise from couplings between the @xmath61 singlet and triplet scalars are : @xmath122 + { \lambda_1^{ts } \over 2 } ( \delta_l^\dagger\delta_l)(\chi^\dagger\chi ) + { \lambda_2^{ts}\over 2}(\delta_l^\dagger\delta_l)(\gamma^\dagger\gamma ) + { \lambda_3^{ts } \over 2}(\rho_l^\dagger\rho_l)(\chi^\dagger\chi ) \nonumber \\ & + & { \lambda_4^{ts}\over 2}(\rho_l^\dagger\rho_l)(\gamma^\dagger\gamma ) + \lambda_5^{ts } \left \ { ( \delta_l^\dagger\rho_l)\chi^2 + h.c.\right \ } + \lambda_6^{ts } \left \ { ( \delta_l^\dagger\rho_l)\gamma^2 + h.c.\right \ } \;\;. \label{vts_s3}\end{aligned}\ ] ] @xmath61 doublets : @xmath125 , @xmath126 , @xmath127 , @xmath128 and @xmath129 . recall that from the structure of the charged lepton mass matrix eq . ( [ vevrat ] ) requires @xmath130 where the real quantity @xmath131 . we often also need @xmath132 . define @xmath137 . @xmath138 \nonumber\\ & + & v_\alpha\left [ { \lambda_5^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_6^{ds}\over2}(u_\gamma^*u_\gamma ) \right ] + v_\eta\left [ \lambda_{17}^d ( v_1^*v_3 ) b + \lambda_3^{ds}u_\chi + \lambda_{11}^{ds}(u_\chi^*)^2 + \lambda_{12}^{ds}(u_\gamma^*)^2 \right ] = 0.\nonumber\\ \label{mint_d1}\end{aligned}\ ] ] @xmath140 \nonumber\\ & + & v_\eta\left [ { \lambda_{7}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{8}^{ds}\over2}(u_\gamma^*u_\gamma ) \right ] + v_\alpha\left [ \lambda_{17}^d(v_3^*v_1)b + \lambda_3^{ds}u_\chi^*+\lambda_{11}^{ds}(u_\chi)^2 + \lambda_{12}^{ds}(u_\gamma)^2 \right ] \nonumber\\ & = & 0 . \label{mint_d3}\end{aligned}\ ] ] @xmath141 \nonumber\\ & + & v_1\left [ \left\{{\lambda_{5}^{d}\over2}(v_\eta^*v_\eta ) + { \lambda_{6}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{7}^{d}\over2}(v_\beta^*v_\beta ) \right \ } + \left\{{\lambda_{1}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{2}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & v_3\left [ \left\{\lambda_{17}^d(v_\alpha^*v_\eta ) \right \ } + \left\ { \lambda_1^{ds}u_\gamma^*+ \lambda_2^{ds}u_\chi^*+ \lambda_9^{ds}(u_\chi)^2 + \lambda_{10}^{ds}(u_\gamma)^2 + \lambda_{13}^{ds}(u_\chi u_\gamma ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d4}\end{aligned}\ ] ] @xmath142 \nonumber\\ & + & av_1\left [ \left\{{\lambda_{5}^{d}\over2}(v_\eta^*v_\eta ) + { \lambda_{6}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{7}^{d}\over2}(v_\beta^*v_\beta ) \right \ } + \left\{{\lambda_{1}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{2}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & av_3\left [ \left\{\lambda_{17}^d(v_\alpha^*v_\eta ) \right \ } + \left\ { -\lambda_1^{ds}u_\gamma^*+ \lambda_2^{ds}u_\chi^*+ \lambda_9^{ds}(u_\chi)^2 + \lambda_{10}^{ds}(u_\gamma)^2 -\lambda_{13}^{ds}(u_\chi u_\gamma ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d5}\end{aligned}\ ] ] @xmath143 \nonumber\\ & + & v_3\left [ \left\ { { \lambda_{8}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{9}^{d}\over2}(v_\beta^*v_\beta ) + { \lambda_{10}^{d}\over2}(v_\eta^*v_\eta ) \right \ } + \left\{{\lambda_{3}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{4}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & v_1\left [ \left\{\lambda_{17}^d(v_\eta^*v_\alpha ) \right \ } + \left\ { \lambda_1^{ds}u_\gamma+ \lambda_2^{ds}u_\chi+ \lambda_9^{ds}(u_\chi^*)^2 + \lambda_{10}^{ds}(u_\gamma^*)^2 + \lambda_{13}^{ds}(u_\chi^ * u_\gamma^ * ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d6}\end{aligned}\ ] ] @xmath144 \nonumber\\ & + & av_3\left [ \left\ { { \lambda_{8}^{d}\over2}(v_\alpha^*v_\alpha ) + { \lambda_{9}^{d}\over2}(v_\beta^*v_\beta ) + { \lambda_{10}^{d}\over2}(v_\eta^*v_\eta ) \right \ } + \left\{{\lambda_{3}^{ds}\over2}(u_\chi^*u_\chi ) + { \lambda_{4}^{ds}\over2}(u_\gamma^*u_\gamma ) \right \ } \right ] \nonumber\\ & + & av_1\left [ \left\{\lambda_{17}^d(v_\eta^*v_\alpha ) \right \ } + \left\ { -\lambda_1^{ds}u_\gamma+ \lambda_2^{ds}u_\chi+ \lambda_9^{ds}(u_\chi^*)^2 + \lambda_{10}^{ds}(u_\gamma^*)^2 -\lambda_{13}^{ds}(u_\chi^ * u_\gamma^ * ) \right \ } \right ] \nonumber\\ & = & 0 . \label{mint_d7}\end{aligned}\ ] ] 100 see , for example , p. f. harrison and w. g. scott , phys . lett . b * 557 * , 76 ( 2003 ) [ hep - ph/0302025 ] . s. l. chen , m. frigerio and e. ma , phys . d * 70 * , 073008 ( 2004 ) erratum : [ phys . d * 70 * , 079905 ( 2004 ) ] [ hep - ph/0404084 ] ; 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( t2k collaboration ) , arxiv:1502.01550v2 [ hep - ex ] ( see fig .

_ we develop a see - saw model for neutrino masses and mixing with an @xmath0 symmetry . it involves an interplay of type - i and type - ii see - saw contributions of which the former is subdominant . the @xmath1 quantum numbers of the fermion and scalar fields are chosen such that the type - ii see - saw generates a mass matrix which incorporates the atmospheric mass splitting and sets @xmath2 . the solar splitting and @xmath3 are absent , while the third mixing angle can achieve any value , @xmath4 . specific choices of @xmath4 are of interest , e.g. , @xmath5 ( tribimaximal ) , @xmath6 ( bimaximal ) , @xmath7 ( golden ratio ) , and @xmath8 ( no solar mixing ) . the role of the type - i see - saw is to nudge all the above into the range indicated by the data . the model results in novel interrelationships between these quantities due to their common origin , making it readily falsifiable . for example , normal ( inverted ) ordering is associated with @xmath9 in the first ( second ) octant . cp - violation is controlled by phases in the right - handed neutrino majorana mass matrix , @xmath10 . in their absence , only normal ordering is admissible . when @xmath10 is complex the dirac cp - phase , @xmath11 , can be large , i.e. , @xmath12 , and inverted ordering is also allowed . the preliminary results from t2k and nova which favour normal ordering and @xmath13 are indicative , in this model , of a lightest neutrino mass of 0.05 ev or more . _ ` key words : neutrino mixing , \theta_{13 } , solar splitting , s3 , see - saw , leptonic cp - violation ` + soumita pramanick@xmath14 , amitava raychaudhuri@xmath14 +

the vast majority of the free electrons in the ism of the milky way reside in a thick ( @xmath9 900-pc scale height ) diffuse layer known as the reynolds layer or the warm ionized medium ( e.g. reynolds 1993 ) . this phase fills about 20% of the ism volume , with a local midplane density of about 0.1 @xmath10 . such a phase is now known to be a general feature of external star - forming galaxies , both spirals ( e.g. walterbos 1997 ; rand 1996 ) and irregulars ( e.g. hunter & gallagher 1990 ; martin 1997 , hereafter m97 ) , where it is commonly referred to as diffuse ionized gas ( dig ) . however , for edge - on spirals , only in the more actively star - forming galaxies does the gas manifest itself as a smooth , widespread layer of emission detectable _ above _ the hii region layer ( rand 1996 ) . one such galaxy , ngc 891 , is an attractive target for study , not only because of its prominent dig layer ( rand , kulkarni , & hester 1990 ; dettmar 1990 ) , but also its proximity ( @xmath11 mpc will be assumed here ) and nearly fully edge - on aspect ( @xmath12 ; swaters 1994 ) . one of the outstanding problems in the astrophysics of the ism is the ionization of these layers . for the reynolds layer , the local ionization requirement ( @xmath13 s@xmath14 per @xmath15 of galactic disk ; reynolds 1992 ) is comfortably exceeded ( by a factor of 6 or 7 ) only by the ionizing output of massive stars . alternatively , the ionization would require essentially all the power put out by supernovae ( reynolds 1984 ) hence , this energy source could contribute at some level but probably can not explain all of the diffuse emission . photo - ionization models , on the other hand , must explain how the ionizing photons can travel @xmath9 1 kpc or more from their origin in the thin disk of massive stars to maintain this distended layer . crucial information on both the ionization and thermal balance of dig comes from emission line ratios . in the reynolds layer , ratios of [ s@xmath2ii ] @xmath16 and [ n@xmath2ii ] @xmath17 to h@xmath1 are generally enhanced relative to their hii - region values , while [ o@xmath2iii ] @xmath18/h@xmath1 is much weaker . these contrasts are in accordance with models in which photons leak out of hii regions and ionize a larger volume , with the radiation field becoming increasingly diluted with distance from the hii region [ mathis 1986 ; domg@xmath19rgen , & mathis 1994 ; sokolowski 1994 ( hereafter s94 ; see also bland - hawthorn , freeman , & quinn 1997 ) ] . the effect of this dilution , measured by the ionization parameter , @xmath20 , is primarily to allow species such as s and o , which are predominantly doubly ionized in hii regions , to recombine into a singly ionized state . the effect may be less noticeable for n because it is mostly singly ionized in hii regions . the wisconsin h@xmath1 mapper ( wham ) has been used to determine [ o@xmath2i]/h@xmath1 in three low - latitude directions , resulting in values @xmath21 to 0.04 ( haffner & reynolds 1997 ) . such low values imply , since the ionization of o and h are strongly coupled by a charge exchange reaction , that the diffuse gas is nearly completely ionized ( reynolds 1989 ) . although weak , [ o@xmath2iii ] emission has been detected in two directions in the reynolds layer at @xmath22 ( reynolds 1985 ) , with the result [ o@xmath2iii]/h@xmath23 . reynolds postulated that the [ o@xmath2iii ] emission does not arise from diluted stellar ionization but from gas at about 10@xmath24 k , presumably the same gas as seen in c@xmath2 iv @xmath251550 and o@xmath2iii ] @xmath26 emission by martin & bowyer ( 1990 ) . the origin of this rapidly cooling gas is unclear . [ o@xmath2iii ] emission from the dig of ngc 891 and the implications for dig ionization is one of the main subjects of this paper . in external spiral galaxies , smooth increases in [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 vs. distance from hii regions have been observed in both the in - plane and vertical directions in accordance with photo - ionization models [ walterbos & braun 1994 ; dettmar & schulz 1992 ; rand 1997a ( hereafter r97 ) ; golla , dettmar , & domg@xmath27rgen 1996 ; greenawalt , walterbos , & braun 1997 ; wang , heckman , & lehnert 1997 ] . the same trends are seen in irregulars ( hunter & gallagher 1990 ; m97 ) . this behavior has been revealed in ngc 891 through spectra using long slits oriented vertically to the plane . dettmar & schulz ( 1992 ) placed a slit at @xmath28 ne of the nucleus , while r97 took a deeper spectrum at @xmath29 ne . r97 found that [ n@xmath2ii]/h@xmath1 rises to a value of 1.4 , implying a very hard ionizing spectrum . s94 , which pays particular attention to modeling the dig of ngc 891 , can predict such a high value only by assuming a stellar imf extending to 120 m@xmath30 , a reduction in cooling efficiency due to elemental depletions , and hardening of the radiation field by the intervening gas . [ o@xmath2i]/h@xmath1 was not detected by dettmar & schulz ( 1992 ) in the halo of ngc 891 at @xmath28 ne , with an upper limit of 0.05 . dettmar ( 1992 ) also reported an upper limit on [ o@xmath2iii]/h@xmath4 of 0.4 at the same location . a wealth of forbidden - line long - slit data on bright dig and hii regions in irregular galaxies has recently been published by m97 . through the use of line - diagnostic diagrams ( e.g. baldwin , phillips , & terlevich 1981 ; veilleux & osterbrock 1987 ) , she finds that while photo - ionization models can explain the line ratio behavior in many galaxies , the rather shallow fall - off of [ o@xmath2iii]/h@xmath4 with distance from hii regions and the sharp rise in [ o@xmath2i]/h@xmath1 seen in some galaxies imply a second source of ionization . shocks are favored as the most likely second source . the forbidden lines , though bright , are sensitive to metallicity and temperature and thus their interpretation in terms of ionization scenarios is complicated by uncertainties in abundances , degree of depletion , and sources of non - ionization heating . a more direct constraint on the ionizing spectrum has come from the very weak he@xmath2i @xmath255876 line . / h@xmath1 is relatively easy to interpret in terms of the ratio of helium- to hydrogen - ionizing photons , allowing the hardness of the ionizing spectrum , the mean spectral type of the responsible stars , and the upper imf cutoff to be inferred , assuming pure stellar photoionization . the results for the reynolds layer ( reynolds & tufte 1995 ) , for hi worms from equivalent radio recombination lines ( heiles et al . 1996 ) and for ngc 891 ( r97 ) all imply a much softer spectrum than do the forbidden lines . further consequences of this discrepancy are discussed in the above three references . the goal of this paper is to make further progress in understanding the ionization of dig in spirals . the motivations are two - fold . first , the dig halo of ngc 891 features several bright filaments and shells . it is likely that some of these are chimney walls ( norman & ikeuchi 1989 ) surrounding regions of space evacuated by many supernovae . in this case , radiation from any continuing star formation near the base of the chimney will have an unimpeded journey to the walls , and thus the filaments may be directly ionized and show a spectrum more like an hii region than diffuse gas , which receives a significant contribution from relatively soft diffuse re - radiation ( norman 1991 ) . if true , then the filaments should show lower [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 than the surrounding gas . to this end , a spectrum has been taken with a slit oriented parallel to the major axis , but offset into the halo gas , traversing several filaments . the second purpose is to study in more detail the dependence of line ratios on @xmath6 beyond the results reported in r97 for he@xmath2i / h@xmath1 and [ n@xmath2ii]/h@xmath1 . by adding measurements of [ s@xmath2ii ] , [ o@xmath2iii ] , [ o@xmath2i ] , and h@xmath4 , one can form diagnostic diagrams and thus constrain the source(s ) of ionization in the spirit of m97 . the spectra were obtained at the kpno 4-m telescope on 1996 december 1213 . the slit positions are shown in figure 1 ( plate 00 ) overlaid on the h@xmath1 image of rkh and on a version of the image in which a median filter has been applied and the resulting smooth image subtracted to reveal the filaments clearly . note also how the filaments connect onto the brightest hii regions in the disk , with few exceptions . for the first night , the slit was oriented parallel to the major axis and offset from it by 15 along the se side of the minor axis . this slit position will be referred to as the parallel slit . the slit position for the second night was the same as in r97 s observations : oriented perpendicular to the major axis and centered @xmath31 on the nw ( approaching ) side of ngc 891 the perpendicular slit . the slit length is 5 , and the spatial scale is 0.69 " per pixel . the kpc-24 grating was used with the t2 kb 2048x2048 ccd , providing a dispersion of 0.53@xmath32 per pixel , a resolution of 1.3@xmath32 , and a useful coverage of about 800 @xmath32 . for observations of red lines , the grating was tilted to give a central wavelength of about 6600@xmath32 , allowing h@xmath1 , [ n@xmath2ii ] @xmath17 , [ s@xmath2ii ] @xmath16 , and [ o@xmath2i ] @xmath33 to be observed . for the blue lines [ o@xmath2iii ] @xmath34 and h@xmath4 , the central wavelength was set to about 5000@xmath32 . for the parallel slit , seven half - hour spectra were taken , along with separate sky exposures because no part of the slit covered pure sky . for the perpendicular slit , seven half - hour spectra were taken covering the blue lines , and three covering the red lines . sky subtraction for these spectra was achieved using regions of pure sky at both ends of the slit . exact slit center positions were varied to allow removal of chip defects in the stacking process . the reduction was carried out with the iraf package . small - scale variations in response were removed using projector flats . the slit illumination correction was determined with sky flats , and the spectral response function with standard stars . arc lamp exposures were used to calibrate the wavelength scale as a function of location along the slit . the final calibrated , sky - subtracted spectra were spatially aligned and stacked . the noise in continuum - free regions of the stacked perpendicular slit exposures is @xmath35 and @xmath36 erg @xmath37 s@xmath14 @xmath38 arcsec@xmath39 in the blue and red spectra , respectively . for the stacked parallel slit exposures , continuum covers the entire slit length and consequently the pixel - to - pixel variations are higher : @xmath40 erg @xmath37 s@xmath14 @xmath38 arcsec@xmath39 . since the spectral lines appear well represented by gaussians , line properties were determined with gaussian fits . the continuum level was estimated from a linear fit to the continuum on each side of the line . error bars reflect the noise in the spectra and the uncertainty in the fit of the spectral response function . line ratios for the perpendicular slit are consistent with those in r97 , given the calibration and slit positioning uncertainties . however , there is a mistake in the velocity scale of figure 10 of r97 : all velocities are about 35 km s@xmath14 too low . the h@xmath1 , [ n@xmath2ii ] , and [ s@xmath2ii ] lines were detected along the entire useable slit length , which corresponds to about 15 kpc at the assumed distance of ngc 891 . reliable parameters could be measured over a 13 kpc region . [ o@xmath2i ] is detected from about 3 kpc sw of the peak in continuum emission to the ne edge of the slit . however , confusion with a sky line limits measurement of reliable parameters on the ne side to a maximum distance of about 5 kpc from the continuum peak . figure 2 shows the runs of [ s@xmath2ii ] @xmath41/h@xmath1 , [ n@xmath2ii ] @xmath256583/h@xmath1 , [ o@xmath2i ] @xmath42/h@xmath1 , [ s@xmath2ii ] @xmath41/[n@xmath2ii]@xmath256583 , and normalized h@xmath1 surface brightness with position along the slit . the data have been averaged over 10 pixels , or about 300 pc , except in the case of [ o@xmath2i]/h@xmath1 , where the averaging is over 20 pixels . figure 1b shows that the slit traverses four bright filaments . these are apparent as peaks in the h@xmath1 profile ( marked in figure 2a ) . the emission is also obviously much brighter on the ne side than on the sw side , confirming the result from the images of rkh and dettmar ( 1990 ) . the ratio of the two [ s@xmath2ii ] lines is consistent with the low - density limit of 1.5 ( osterbrock 1989 ) , even for the filaments . at first glance , it would seem that the data support the expectation outlined in i : there is a definite reduction in [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 at the positions of the four bright filaments . [ o@xmath2i ] @xmath42/h@xmath1 also tends to be lower at the filament crossings . however , this result reflects a more general trend of these line ratios with h@xmath1 surface brightness , as shown in figure 3 . the ratios at the positions of the three brightest filaments define the correlation at h@xmath1 surface brightnesses @xmath43 erg @xmath37 s@xmath14 arcsec@xmath39 . there is no evidence for a discontinuity or steepening of the correlation at these intensities . in fact , the slope becomes nearly flat here . what is the reason for these very good correlations ? very similar results are found for the perpendicular slit data ( figure 7 ) , suggesting , in a stellar photo - ionization scenario , that they reflect the well - established variation ( e.g. s94 ) of line ratios with ionization parameter , @xmath20 , which measures the diluteness of the radiation field ( see 1 ) . if this is the case , then it is implied that @xmath20 is lower along lines of sight with faint dig , even when comparing at the same @xmath6 . gas along lines of sight with faint h@xmath1 emission may be relatively remote from ionizing stars in the disk , so that large columns of intervening gas and geometric dilution result in a low @xmath20 . alternatively , it is possible that such gas is no more remote from ionizing stars , but that @xmath20 is low because the responsible clusters feature fewer such stars , leading to an intrinsically weak emergent ionizing radiation field . the fact that the lines of sight passing through the filaments feature the brightest dig while the filaments are clearly associated with bright visible hii regions in the disk ( figure 1b ) would tend to suggest that the former explanation is correct . however , the dust lane may be hiding numerous fainter hii regions whose stars may be the primary source of ionization for gas along lines of sight with fainter emission . on the other hand , such hii regions should contain fewer massive stars and have , if the imf varies little , a lower probability of containing the most massive stars , resulting in softer spectra on average . a softer spectrum leads to lower line ratios ( s94 ) and would offset the dilution effect to some degree . this question will probably be resolved from studies of dig in more face - on galaxies where the in - plane variations of h@xmath1 surface brightness and line ratios can be related to the distribution of ionizing stars . regardless of the explanation , since the filaments in ngc 891 follow the overall correlation in figure 3 , it can not be claimed that the line ratios are lower in the filaments because they surround _ evacuated _ regions and are directly ionized by hii regions below . this does not imply that the filaments are not chimney walls , but does point out the potential difficulties in deriving information on isolated structures in edge - on galaxies . it is interesting that [ s@xmath2ii]/[n@xmath2ii ] shows almost no spatial variation or dependence on h@xmath1 surface brightness compared to the other line ratios . this ratio will be discussed further in the next section , where it will become clear that there is little dependence on @xmath6 either . figure 4 shows heliocentric velocity centroids , formed from a weighted average of the h@xmath1 , [ s@xmath2ii ] , and [ n@xmath2ii ] line centroids , along with the emission profile . one of the filaments on the ne side and a broad region centered on a filament on the sw side show velocities further from the systemic velocity than expected from the smooth , nearly linear trend of velocity with position . again , this is not necessarily an indication that the filaments have peculiar velocities , but may simply indicate that they are located in the inner disk . if so , then these velocities are more heavily weighted by inner disk material than are adjacent ones . inner disk gas will show velocities further from the systemic velocity compared to outer disk gas because of the greater projection of the rotation velocity vector along the line of sight . hence , the velocity deviations may simply be due to geometrical effects . the h@xmath1 line , the [ n@xmath2ii ] @xmath44 line and the [ s@xmath2ii ] lines are detected up to about @xmath45 kpc on each side of the plane . [ o@xmath2i ] is detected to about half this height , while [ o@xmath2iii ] and h@xmath4 are detected up to about @xmath46 kpc . shown in figure 6 are the vertical runs of [ s@xmath2ii ] @xmath47/h@xmath1 , [ n@xmath2ii ] @xmath44/h@xmath1 , [ o@xmath2i]/h@xmath1 , [ s@xmath2ii ] @xmath41/[n@xmath2ii ] @xmath44 , and [ o@xmath2iii]/h@xmath4 . the h@xmath1 profile is also plotted the local minimum at @xmath7 kpc is due to the dust lane . the data are averaged over 10 pixels . [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 show a smooth and remarkably similar increase with @xmath6 , from about 0.35 in the midplane to over 1.0 at @xmath48 kpc . [ s@xmath2ii]/[n@xmath2ii ] is nearly constant at about 0.6 at all @xmath6 where the uncertainties are not too large . [ o@xmath2i]/h@xmath1 increases from about 0.03 at @xmath7 kpc to 0.08 at @xmath49 kpc . however , the most surprising result is that [ o@xmath2iii]/h@xmath4 _ rises _ from 0.3 in the midplane to 0.8 at @xmath50 kpc . figure 7 shows the correlation of these ratios with h@xmath1 surface brightness . these are very similar to the correlations in figure 3 . again , the ratio of the two [ s@xmath2ii ] lines is everywhere consistent with the low - density limit of 1.5 . again assuming @xmath51k , [ o@xmath2i]/h@xmath1 values imply h is essentially 100% ionized at @xmath7 kpc , decreasing to about 90% at @xmath8 kpc . if @xmath52k , then these ionization fractions are 90% and 80% . except for the discrepancy mentioned in 2 , the mean velocities of the lines are consistent with the results of r97 and give no additional information . therefore , we will not discuss the kinematics further . we now attempt to understand whether the above emission line properties can be understood by massive - star photo - ionization alone . in doing so , we temporarily ignore the problems posed by the low he@xmath2i / h@xmath1 but will return briefly to the reconciliation of this ratio with the forbidden line ratios in 4 . we use unpublished models from s94 since they are the only models which specifically attempt to reproduce the line ratios in the dig of ngc 891 . in these models an ionizing spectrum of radiation from a population of stars with an imf slope of 2.7 ( intermediate between salpeter and miller - scalo values ) and stellar atmospheres from kurucz ( 1979 ) is considered . this radiation field is allowed to propagate through a slab of gas ( representing a clump of halo gas ) in a one - dimensional calculation , the dilution being measured by the ionization parameter , @xmath20 , at the front of the slab . as discussed in 1 , as lower values of @xmath20 are considered , the predominant ionization state of s and o ( and n to a lesser extent ) changes from doubly to singly ionized . as a consequence , [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 rise while [ o@xmath2iii]/h@xmath4 falls . at the end of the model slab , an increasingly neutral zone appears as lower values of @xmath20 are considered , leading to a slow rise in [ o@xmath2i]/h@xmath1 with @xmath20 . in this one - dimensional calculation of pure photo - ionization , @xmath20 is expected to decline exponentially with @xmath6 . this dependence is probably more complicated in a real galaxy , although @xmath20 should generally fall with increasing height . thus , further free parameters are the value of @xmath20 at @xmath7 kpc , and the run of @xmath20 with @xmath6 . both radiation bounded models and matter bounded models with various terminating total atomic hydrogen columns for the clumps were considered by s94 . we will use only his models with the hardest stellar spectrum considered ( with an upper imf cutoff of 120 m@xmath30 ) , hardening of the radiation field as it propagates through the intervening gas before reaching the slab , and heavy element depletions . only these models are able to yield [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 in the range 11.5 . these models have also been recently published by bland - hawthorn et al . we examine as two extremes the matter - bounded model ( which will be referred to as pm ) with the lowest terminal hydrogen column considered for the individual clumps , 2@xmath53 @xmath37 , and the radiation - bounded model ( pr ) . figures 8 and 9 show these ratios in diagnostic diagrams of [ o@xmath2iii]/h@xmath4 and [ o@xmath2i]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 , and [ s@xmath2ii]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 . along the sequence of points , @xmath6 generally increases from 0 kpc at the left end to 2 kpc for plots of [ o@xmath2iii]/h@xmath4 , 1.3 kpc for [ o@xmath2i]/h@xmath1 , and 3 kpc for [ s@xmath2ii]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 at the right end . models pr and pm are shown in figures 8 and 9 as the small open circles joined by solid lines . values of log@xmath54 are marked as explained in the captions . it is immediately obvious that neither model is a good match to the data . most importantly , the flatness of [ o@xmath2iii]/h@xmath4 below @xmath8 kpc and its rise with [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 ( and @xmath6 ) above @xmath8 kpc is at complete odds with the models . also , the typical value of [ o@xmath2iii]/h@xmath4 is poorly predicted by a model chosen to match the observed [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 . in model pr , while [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 require log@xmath54 in the range 2.3 to 2.7 at @xmath7 kpc , and 3.3 to 3.7 at @xmath50 kpc , the predicted [ o@xmath2iii]/h@xmath4 is @xmath55 times that observed for most of this range of @xmath20 . in model pm , on the other hand , if we use [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 to set @xmath20 at @xmath7 kpc , we require log@xmath54 in the range 3.7 to 4.1 , and an extra source of [ o@xmath2iii ] emission at lower @xmath20 . while such a source can be identified and tested ( see below ) , it is not clear that such a low value of @xmath20 should apply to the midplane , given that the disk contains both hii regions and diffuse gas . however , at low @xmath6 the kinematics indicate that we receive emission preferentially from the outer disk because of the absorbing dust layer ( r97 ) . this outer disk gas may , like the high-@xmath6 gas , see a relatively dilute radiation field because star formation is concentrated in the inner disk ( rand 1997b ) . in that case the appropriate @xmath20 for @xmath7 may be quite low and the gradient of @xmath20 with @xmath6 rather shallower than expected in a galaxy without such a dust lane . both models are more successful at reproducing the runs of [ o@xmath2i]/h@xmath1 vs. [ n@xmath2ii]/h@xmath1 , [ o@xmath2i]/h@xmath1 vs. [ s@xmath2ii]/h@xmath1 and [ s@xmath2ii]/[n@xmath2ii ] . if [ n@xmath2ii]/h@xmath1 were lower by about 0.2 dex at low @xmath6 , pr would fit these data quite well , with log@xmath56 at @xmath7 kpc , @xmath57 at @xmath8 kpc , and @xmath58 at @xmath50 kpc . model pm would fit equally well for log@xmath59 at @xmath7 , @xmath60 at @xmath8 kpc , and @xmath61 at @xmath50 kpc . a similar discrepancy in the observed vs. modeled vertical run of [ s@xmath2ii]/[n@xmath2ii ] was noted by golla et al . ( 1996 ) in ngc 4631 . [ n@xmath2ii]/h@xmath1 is expected to show less disk - halo contrast than [ s@xmath2ii]/h@xmath1 because the change in predominant ionization state between hii regions and diffuse gas is smaller for n due to its higher ionization potential . this trend is reflected in the models of both s94 and domg@xmath19rgen , & mathis ( 1994 ) . in both ngc 891 and ngc 4631 , however , the disk - halo contrast in [ s@xmath2ii]/h@xmath1 is rather similar to that in [ n@xmath2ii]/h@xmath1 . one important factor in explaining the common behavior of [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 might be a _ radial _ abundance gradient in league with the dust absorption effect noted above . rubin , ford , & whitmore ( 1984 ) found that log ( [ s@xmath2ii]/[n@xmath2ii ] ) generally increases in hii regions in spirals by 0.3 from inner to outer hii regions . if such a gradient is present in ngc 891 , then the fact that we preferentially observe outer disk gas at low-@xmath6 means that [ s@xmath2ii]/[n@xmath2ii ] should be higher there for a given @xmath20 . the gradual inclusion of more inner disk gas with increasing @xmath6 will then tend to offset the dependence of [ s@xmath2ii]/[n@xmath2ii ] on @xmath20 in the models . however , the parallel slit data suggests little dependence of the ratio on distance along the major axis , although there is a slight trend in the right direction between 2 and 7 kpc from the center on both sides . hence , inasmuch as the major - axis dependence of [ s@xmath2ii]/[n@xmath2ii ] at @xmath0 pc reflects the radial dependence at @xmath7 pc , it would seem that an abundance gradient does not affect the line ratios significantly . other expected effects of an abundance gradient are also not seen in the parallel slit data . domg@xmath19rgen , & mathis ( 1994 ) find that [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 increase with abundance , while [ o@xmath2i]/h@xmath1 shows little dependence . in the presence of a radial gradient , these trends , if strong enough , might be revealed as a decline in [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 with distance along the major axis . the dominant trend there is with h@xmath1 brightness and thus it seems that variations in @xmath20 are the more important effect . again , though , it must be noted that the averaging along the line of sight in the parallel slit data will diminish the observable effects of an abundance gradient . so far , it has been difficult to find much variation in [ s@xmath2ii]/[n@xmath2ii ] in the dig of edge - on galaxies . this question remains open to further exploration . the most surprising result for the perpendicular slit is the behavior of [ o@xmath2iii]/h@xmath4 with @xmath6 . although such high ( and higher ) values of [ o@xmath2iii]/h@xmath4 are found in hii regions , they are a feature of high - excitation ( high @xmath20 ) conditions . if the level of excitation were increasing with @xmath6 , however , we would also expect to see [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 falling , contrary to what is observed . as discussed above , their rise is qualitatively consistent with a smooth transition from predominantly doubly - ionized to singly - ionized states , as expected in the dilute photo - ionization models . hence , it is very unlikely that the [ o@xmath2iii ] emission arises from the same dig component as the [ n@xmath2ii ] and [ s@xmath2ii ] . we therefore require a second source of diffuse h@xmath1 emission which features bright [ 0@xmath2iii ] . two plausible mechanisms for producing such emission , shocks and turbulent mixing layers are discussed in this and the next subsection . respectively . both can produce bright [ o@xmath2iii ] emission . these mechanisms were also considered for dig in irregular galaxies by m97 . it should be noted , though , that the dig in these irregulars is generally much brighter than that studied here . we first consider whether the line ratio data can be explained if some of the dig emission is produced by shock ionization . we consider only the pm model further because the pr model would require a second source of h@xmath1 emission with highly unusual properties : if the value of @xmath20 at the midplane is to be roughly 2.3 to 2.7 , then the second component must dominate the stellar - ionized component by a factor of 30 or more and have essentially no [ o@xmath2iii ] emission if the runs of [ o@xmath2iii]/h@xmath4 vs. [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 are to be explained . low - speed shocks do have this property , but also produce large amounts of [ o@xmath2i ] emission , further confouding the problem . alternatively , if log@xmath62 at @xmath7 kpc , then the second component must have essentially no [ n@xmath2ii ] or [ s@xmath2ii ] emission and account for about 75% of the h@xmath1 emission at @xmath7 kpc in order to reproduce the values of [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 . in either case , it is also difficult to see how the subsequent rise in all the line ratios with @xmath6 would be achieved without the model being highly contrived . with the pm model , there is some hope of matching the data by adding a strong source of [ o@xmath2iii ] emission as long as @xmath20 is low enough in the midplane and the other ratios can be matched . we employ the shock models of shull & mckee ( 1979 ) , as were also considered by m97 . the line ratios in shock models , especially [ o@xmath2iii]/h@xmath4 , are very sensitive to the shock velocity . other variables include the preshock gas density and ionization state , abundance , and transverse magnetic field strength . the gas is assumed to be initially neutral at @xmath63 @xmath10 and subsequently penetrated by a precursor ionization front . more appropriate to our case would be a lower initial density ( of order 0.1 @xmath10 ) and a high initial ionization fraction . we should also consider depleted abundances to be consistent with the stellar ionization models , but sm calculated only one such model to show the general effect of depletions . we do not perform an exhaustive search of parameter space or carry out a statistical test of the goodness of fit , firstly because an examination of figures such as figure 9 can quickly reveal which composite models are most successful , and secondly because some of the fixed parameters are probably inappropriate for the halo of ngc 891 in any case . in the composite models , we still consider that the stellar radiation field is characterized by a decrease of @xmath20 with @xmath6 , and that some fraction of h@xmath1 emission from shock ionization is added , with this fraction possibly changing with @xmath6 . although no model can reproduce the line ratio behavior to within the errors , we find that some of the main characteristics of the data can be reproduced . one of the most successful models is shown overlaid on the data in figure 9 as the dashed lines joining open circles , which mark values of log@xmath54 . the shock speed is 90 km s@xmath14 . at @xmath7 kpc , 7% of the h@xmath1 emission arises from a 90 km s@xmath14 shock , rising to 30% by @xmath50 kpc . the composite model is most successful if log@xmath59 at @xmath7 kpc , @xmath60 at @xmath8 kpc ( the limit of the [ o@xmath2i]/h@xmath1 data ) , and @xmath61 at @xmath50 kpc ( the limit of the [ o@xmath2iii]/h@xmath4 data ) . note that in the figure , circles indicate log@xmath64 4.0 , 4.3 , 4.7 and 5.0 , corresponding to @xmath65 kpc , in all the panels despite the fact that the [ s@xmath2ii]/[n@xmath2ii ] ratio includes data up to @xmath45 kpc and [ o@xmath2i]/h@xmath1 is only detected up to @xmath8 kpc . also shown in figure 9 are the line ratios for the shock models alone , namely , a 90 km s@xmath14 and a 100 km s@xmath14 shock with standard abundances , and a 100 km s@xmath14 shock with depleted abundances . the major shortcoming of this composite model is in reproducing [ s@xmath2ii]/[n@xmath2ii ] at low @xmath6 . for a given amount of [ s@xmath2ii ] emission , the model overpredicts [ n@xmath2ii ] . at high @xmath6 this ratio is somewhat more successfully modeled , but [ n@xmath2ii]/h@xmath1 simply does not show as much disk - halo contrast as observed , as was the case for the pure photo - ionization models . also , the predicted [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 reach a plateau at 1.0 at the lowest @xmath20 considered by s94 , and thus can not match the observed continuing rise beyond @xmath50 kpc ( figure 9e ) . in this regard radiation - bounded models are more successful ( s94 ) , at least for [ s@xmath2ii]/h@xmath1 . in that model , it is still rising at the lowest considered @xmath20 value , so that its observed continued rise may be explainable . finally , a low value of @xmath20 at @xmath7 pc is still required , but again this may not be unreasonable given the arguments mentioned above . there is some freedom for variation of the shock parameters . a 130 km s@xmath14 shock ( the highest speed considered by sm ) contributing 2% of the h@xmath1 emission at @xmath7 kpc , rising to 7% at @xmath50 kpc , produces almost identical curves in the diagnostic diagrams . since only the [ o@xmath2iii]/h@xmath4 ratio is large ( 7.35 ) for this model , adding such a small amount of this emission affects mainly this ratio . the composite model can be further explored by keeping the shock contribution constant with @xmath6 but varying the shock speed . a model with a shock giving rise to 25% of the h@xmath1 emission , with a speed of 60 km s@xmath14 at @xmath7 kpc , and 90 km s@xmath14 at @xmath50 kpc ( but with log@xmath66 now ) can reproduce the ranges of [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/[n@xmath2ii ] , and [ o@xmath2iii]/h@xmath4 as successfully as the above models , but low - speed shocks produce far too much [ o@xmath2i ] emission . sm find that the effect of introducing depleted abundances for a 100 km s@xmath14 shock is to raise [ s@xmath2ii]/[n@xmath2ii ] , while [ o@xmath2iii]/h@xmath4 drops and [ o@xmath2i]/h@xmath1 rises . however , assuming that the fractional changes also apply for a 90 km s@xmath14 shock , depleted abundances do not solve the [ s@xmath2ii]/[n@xmath2ii ] problem , even if the composite model is started at a different value of log@xmath54 . the effect on [ s@xmath2ii]/[n@xmath2ii ] is insufficient because of the small contribution from shocks needed to fit [ o@xmath2iii]/h@xmath4 . further latitude in the composite models may be gained by considering a larger range of values for some of the other variables . for instance , dopita & sutherland ( 1996 ) calculate properties of lower density , magnetized shocks with speeds from 150 to 300 km s@xmath14 . they consider densities and magnetic field strengths obeying the relation @xmath67 g @xmath68 . these shocks have strong radiative precursors which contribute a significant fraction of the emission . a model with shock speed 150 km s@xmath14 , @xmath69 @xmath10 and no magnetic field produces line ratios similar to the 90 km s@xmath14 model considered above , with the exception that [ o@xmath2i]/h@xmath1 is about 80% higher . the main conclusion from this subsection is that shocks are feasible as a secondary source of energy input into the dig of ngc 891 , but it is difficult to constrain their parameters with much confidence . we have not speculated on the origin of the putative shocks . but given that the observed filaments suggest the presence of supershells and chimneys , it is plausible that the shocks originate in such expanding structures . the shock speeds considered above are reasonable when compared to superbubble calculations ( e.g. maclow , mccray , & norman 1989 ) . in fact , the slit passes close to one of the most prominent filaments , which may still have an associated shock . this possibility highlights the importance of observing with many such slit positions . finally , we point out that m97 also found that if a constant shock speed model is used , then the fraction of the dig emission that must come from shocks increases with @xmath6 . turbulent mixing layers ( tmls ) are expected to occur at the interfaces of hot and cold ( or hot and warm ) gas in the ism of galaxies ( begelman & fabian 1990 ; slavin , shull , & begelman 1993 ) . superbubble walls are a likely location for such layers . shear flows are expected along the interface , leading to kelvin - helmholtz instabilities and subsequent mixing of the gas . the result is a layer of intermediate temperature gas , probably at @xmath70 . this gas may produce some fraction of the dig emission in galaxies . like typical dig , it features enhanced [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 relative to hii regions and low [ o@xmath2i]/h@xmath1 . however , unlike stellar - ionized gas , [ o@xmath2iii]/h@xmath4 should be of order 13 . hence , tmls provide a source of enhanced [ o@xmath2iii]/h@xmath4 and therefore may be relevant to the current problem . slavin , shull , & begelman ( 1993 ) calculate line ratios for tmls as a function of shear velocity , mixing layer temperature , initial cold layer temperature , and abundance ( solar vs. depleted ) . we will consider depleted abundances here . again , we will introduce a contribution to the dig emission from tmls at some @xmath6 or @xmath20 , allowing for an increase in this contribution with @xmath6 as @xmath20 continues to decline , and consider only the pm model for the photo - ionized component . the best match to the data features a shear velocity of 25 km s@xmath14 ( the lowest modeled ) , a mixing layer temperature of log@xmath71 , and an initially warm layer at 10@xmath72 k rather than a cold layer . this model is shown overlaid on the diagnostic diagrams in figure 10 . at @xmath7 kpc , 3% of the h@xmath1 emission arises from tmls , rising to 15% at @xmath50 kpc . the rough relation of log@xmath54 with @xmath6 is similar to the previous composite model : log@xmath73 at @xmath7 kpc , @xmath60 at @xmath8 kpc ( the limit of the [ o@xmath2i]/h@xmath1 data ) , and @xmath61 at @xmath50 kpc ( the limit of the [ o@xmath2iii]/h@xmath4 data ) . again , note that in the figure , circles indicate log@xmath64 4.1 , 4.3 , 4.7 and 5.0 , corresponding to @xmath65 kpc , in all the panels despite the varying maximum @xmath6 of the line ratio determinations . the model is reminiscent of the best shock models and appears to be somewhat more successful , but suffers from the same primary shortcoming : the underprediction of [ s@xmath2ii]/[n@xmath2ii ] at low @xmath6 . other tml models will not alleviate this problem because the model in figure 10 already features maximal [ s@xmath2ii]/[n@xmath2ii ] . as is the case for shocks , there is room for flexibility in the parameters . for instance , [ o@xmath2iii]/h@xmath4 is fairly constant in all the models with depleted abundances , and all models regardless of abundance feature very low [ o@xmath2i]/h@xmath1 . [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 show somewhat more variation with the input parameters , but as most of the [ s@xmath2ii ] and [ n@xmath2ii ] emission arises from the stellar - ionized gas , and the tml contribution is small , there is reasonable latitude for variation of the parameters without altering the resulting values of these ratios . again , the main point is to demonstrate the feasibility of tmls as a secondary source of energy input rather than to find the best fitting parameters , or indeed to show whether tmls or shocks are preferred as the second component . shapiro & benjamin ( 1993 ) consider cooling , falling galactic fountain gas initially raised from the midplane by supernovae . the calculation is followed from an initial temperature of 10@xmath74 k to a final value of 10@xmath72 k. while there are not yet predictions of optical emission line ratios from such gas , one can expect [ o@xmath2iii ] emission as the gas cools . the run of [ o@xmath2iii]/h@xmath4 with @xmath6 in a composite model with fountain gas would depend on the fraction of diffuse h@xmath1 emission produced by the latter ( the authors estimate that it could account for perhaps 40% in the milky way ) and the details of the cooling and dynamics , including the interaction with halo gas from other processes . another idea that has not been deeply explored is heating of the halo by microflares from magnetic reconnection events ( raymond 1992 ) . this process may have a role in producing ultraviolet emission and absorption lines , soft x - ray halo emission , and reynolds layer emission . as the theory stands now , a broad range of [ o@xmath2iii]/h@xmath1 values may result from this process , and there are enough uncertainties as not to warrant a detailed comparison with the data at this point . the relevance of microflares should be revealed with further refinements of the theory and additional observations . the problem of explaining low he@xmath2i / h@xmath1 in combination with high [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 has motivated work on other sources of heating for the dig . since the forbidden lines are highly temperature sensitive , small changes in temperature can be important . minter & balser ( 1997 ) find that the dissipation of turbulent energy in the ism could raise the temperature of the dig by about 2000 k without additional ionization of helium , thus providing a reasonable match to these line ratios in the reynolds layer . however , there is still insignificant [ o@xmath2iii ] emission . photo - electric heating from dust grains ( reynolds & cox 1992 ) should also have little effect on [ o@xmath2iii ] emission . finally , scattered light from hii regions could be a source of [ o@xmath2iii ] emission in the halo , but the rise in [ o@xmath2iii]/h@xmath4 with @xmath6 would not be expected . using a slit oriented parallel to , and offset 700 pc above , the major axis of ngc 891 , a spectrum of the dig has been taken which reveals a clear correlation of ratios of forbidden lines to h@xmath1 with the h@xmath1 surface brightness . the original motivation for this observation was to search for variations in line ratios on and off the filaments of dig as further evidence that they are walls around evacuated chimneys . but although the filaments do show reduced [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 relative to gas on adjacent lines of sight , the contrast merely reflects the overall correlation . the relationship probably indicates that regions of brighter h@xmath1 emission receive a radiation field with a higher ionization parameter . also , although some of the filaments show deviations from the observed smooth trend of mean velocity with position along the major axis , these departures could simply be due to geometric effects : if the filaments are inner galaxy features , they will bias the mean velocity for their line of sight away from the systemic velocity . [ s@xmath2ii]/[n@xmath2ii ] surprisingly shows no significant variation along the slit . finally , the h@xmath1-emitting halo gas at this height is about 8095% ionized , based on the observed range of [ o@xmath2i]/h@xmath1 and assuming @xmath3 k. the correlation of [ o@xmath2i]/h@xmath1 with surface brightness probably reflects a higher degree of ionization where the photon field is more intense . results from this observation emphasize the difficulty in interpreting dig observations of edge - on galaxies . confusion is caused by uncertainties in the location of a parcel of gas along the line of sight , its effective distance from a source of ionization and other unrelated gas in the same direction . it is difficult from such observations to draw conclusions about the environment of the filament , for example . spectra from a slit oriented perpendicular to the plane at @xmath29 along the major axis on the ne side show a rise of [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 with @xmath6 . at the midplane , [ o@xmath2i]/h@xmath1 values indicate that h is essentially 100% ionized , dropping to 90% at @xmath8 kpc , assuming @xmath75k . the @xmath6-dependence of these line ratios is expected if the gas is ionized by massive stars in the disk . however , it is unexpectedly found that [ o@xmath2iii]/h@xmath4 also rises with @xmath6 , whereas it should decline with @xmath6 in photo - ionization models . this result necessitates the consideration of secondary sources of ionization . [ s@xmath2ii]/[n@xmath2ii ] unexpectedly shows essentially no dependence on @xmath6 and h@xmath1 surface brightness . put another way , [ n@xmath2ii]/h@xmath1 shows the same disk - halo contrast as that of [ s@xmath2ii]/h@xmath1 , whereas a smaller contrast is expected . strong [ o@xmath2iii ] emission is expected from several energetic procesess . we considered shocks and turbulent mixing layers as sources of such gas . models in which a small fraction of the h@xmath1 emission comes from one of these mechanisms can be made to fit the data reasonably well , but most noticeably the remarkable constancy of [ s@xmath2ii]/[n@xmath2ii ] with @xmath6 is still difficult to reproduce . in the case of shocks , it is difficult to constrain the shock speed or the contributed fraction of the dig emission at this point . of course , the line of sight may sample shocks with a range of speeds and some mean value . there is also significant latitude in the parameters in the case of tmls . other sources of strong [ o@xmath2iii ] emission may include cooling galactic fountain gas and microflares from magnetic reconnection . these should be explored further in light of the current results . because of these facts and possibilities , the results are meant only to indicate the feasibility of such classes of models and the likelihood that one or more physical processes is producing intermediate temperature gas in the halo of ngc 891 . on the other hand , the finding that the second source of line emission becomes more important as @xmath6 increases may be reasonable . in the case of tmls , for example , shull & slavin ( 1994 ) point out that this process may indeed be more common at large @xmath6 , where superbubbles break out of the thin disk gas layers , producing rayleigh - taylor instabilities and shear flows that lead to the mixing . these authors were attempting to explain the larger scale - height of c@xmath2iv uv absorption line gas relative to n@xmath2v ( sembach & savage 1992 ) as an increasing predominance of tmls over sn bubbles with height off the plane the former producing higher c@xmath2iv / n@xmath2v . if the tml process begins only at the approximate height where breakout occurs , while the stellar radiation field is increasingly diluted with @xmath6 , then an increasing fraction of h@xmath1 emission from tmls may be quite reasonable . the rough fractions found in 3.2.3 are comparable to those expected by slavin et al . ( 1993 ) for the milky way diffuse h@xmath1 emission . if the photo - ionized and secondary components of the dig emission both arise in exponential layers with different scale - heights , then the composite models , although illustrative , can be used to estimate roughly the relative scale - heights . in the two shock models and one tml model considered , the fraction of emission arising from the second component is 35 times higher at @xmath50 kpc than at @xmath7 kpc . assuming exponential layers , the scale - height of the second component must be 34 times that of the photo - ionized component . this conclusion is very tentative , however , given the uncertainties in the modeling and the lack of information on [ o@xmath2iii]/h@xmath4 at higher @xmath6 . it it is tempting to identify the second component with the high-@xmath6 tail of h@xmath1 emission found in the deeper spectrum of rand ( 1997a ) . however , the exponential scale - height of this tail is 57 times that of the main component , and it contributes about 50% of the emission at @xmath50 kpc . while these numbers do not quite match those for the second component proposed here , there may yet prove to be a connection . for the photo - ionized component , we found the most success by using the model from s94 with the lowest terminal hydrogen column considered for individual clouds in the dig layer . using larger columns tends to push the model curves towards those of the radiation - bounded case , which is found to be very difficult to incorporate in a successful composite model . this constraint suggests that the dig consists of quite small clumps ( or filaments or sheets , since s94 s calculation is one - dimensional ) of several pc thickness , for a representative density of @xmath76 @xmath10 . if this conclusion is not borne out by future observations , the composite models presented here will need to be reconsidered . the s94 model used here also features the hardest emergent stellar spectrum considered ( in order to produce high [ s@xmath2ii]/h@xmath1 and [ n@xmath2ii]/h@xmath1 ) , but more modest spectra may be allowable if other sources of non - ionization heating are at work ( e.g. minter & balser 1997 ) . despite complications introduced by the second dig component , it should be noted that the observed properties of the three ratios [ s@xmath2ii]/h@xmath1 , [ n@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 with @xmath6 are still reasonably explained _ to first order _ by photo - ionization models alone . their smooth increase with @xmath6 is as predicted as are their rough values . emission from the second component probably has only a secondary effect on these ratios . the @xmath6-independence of [ s@xmath2ii]/[n@xmath2ii ] is not understood in either a pure photo - ionization or composite model . an undesirable aspect of the composite models considered here is that emission from photo - ionized and ( for example ) shock ionized gas is simply added together with no unified physical picture in mind . it would be desirable eventually to have , say , a calculation of the evolution of a superbubble which included photo - ionization , shocks and tmls in a more self - consistent way . for instance , what is the effect of the radiative precursor of a shock which enters gas already ionized by dilute stellar radiation ? the emission line properties revealed by the perpendicular slit share some similarities with those of the halo of the starburst galaxy m82 . in the fabry - perot data of shopbell & bland - hawthorn ( 1997 ) , [ n@xmath2ii]/h@xmath1 shows a general tendency to rise with @xmath6 on the n side , up to the limit of measurability at about @xmath6=750 pc . on the s side , there is little dependence on z , with perhaps 0.6 typical . [ o@xmath2iii]/h@xmath1=0.03 at @xmath7 pc and 0.08 at @xmath77 pc ( values of [ o@xmath2iii]/h@xmath4 are about 3 times higher assuming little extinction ) . in a long - slit spectrum through the halo of m82 , m97 sees higher values of [ o@xmath2iii]/h@xmath4 , reaching 0.7 at @xmath8 kpc . the sequence of points in her diagnostic diagram of [ s@xmath2ii]/h@xmath1 vs. [ o@xmath2iii]/h@xmath4 has a rising slope , as in ngc 891 , although much steeper ( other irregulars show a falling or nearly flat slope ) . [ o@xmath2i]/h@xmath1 also rises with @xmath6 in m82 , and shows a range of values ( about 0.01 to 0.1 ) similar to those reported here . shopbell & bland - hawthorn ( 1997 ) point out that their ratios become more shock - like with distance from the starburst , as also noted by heckman , armus , & miley ( 1990 ) . this behavior is now seen in a spiral halo as well . other evidence for multi - phase halos is provided by the study of ngc 4631 by martin & kern ( 1998 ) . they detect an extensive halo of [ o@xmath2iii ] emission which spatially coexists with the observed soft x - ray and h@xmath1 emitting halos . within this halo are several bright [ o@xmath2iii ] condensations in which the measured [ o@xmath2iii]/h@xmath4 ratio is @xmath78 . on this basis , they argue that the h@xmath1 and [ o@xmath2iii ] emission is tracing distinct components in a multi - phase halo medium . it should be pointed out that [ o@xmath2iii]/h@xmath1 values of 0.120.21 ( comparable to values in ngc 891 ) are seen in the dig of m31 ( greenawalt et al . there is little correlation of this ratio with the brightness of the dig or its remoteness from visible hii regions . the high [ o@xmath2iii]/h@xmath1 in this case may be due to a radiation field not as dilute as in the halo of ngc 891 ( for instance , [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 are substantially lower than in the halo of ngc 891 at @xmath79 kpc ) . greenawalt et al . ( 1997 ) also find that tmls may contribute some fraction of the h@xmath1 emission in regions of very faint dig , but no more than about 20% and most likely only a few percent , similar to the findings for ngc 891 . more light has been shed on the question of [ o@xmath2iii]/h@xmath4 trends in the dig of face - ons by wang et al . ( 1997 ) . for three of their five galaxies with good [ o@xmath2iii ] detections in their dig spectra , they find that [ o@xmath2iii]/h@xmath4 is in the range @xmath80 to 2 and , for a given slit , is systematically higher than in the hii regions in that slit . they also find that the [ o@xmath2iii ] line widths are usually larger than those of [ n@xmath2ii ] , and thus refer to a quiescent photo - ionized dig component which accounts for the bulk of the h@xmath1 , [ n@xmath2ii ] and [ s@xmath2ii ] emission , and a disturbed component ( shocks and tmls are considered ) contributing a minority ( @xmath81 20% ) of the h@xmath1 emission but responsible for the [ o@xmath2iii ] emission . for vertical hydrostatic equilibrium , the contrast in line - widths indicates that the scale - height of the disturbed component is 1.52 times greater than that of the quiescent dig . in ngc 891 , the line - widths are dominated by galactic rotation and thus such an analysis can not be carried out . the existence of a second source of line emission may be relevant for the issue of the low he@xmath2i / h@xmath1 ratio . the value of 0.027 for the lower halo of ngc 891 implies a much softer spectrum than is required to explain the high [ n@xmath2ii]/h@xmath1 and [ s@xmath2ii]/h@xmath1 ( r97 ) . apart from the possibility that the forbidden line emission is complicated by additional sources of ionization and non - ionizing heating , he@xmath2i / h@xmath1 may also be affected by secondary ionization sources if they contribute sufficient h@xmath1 emission . for instance , the ratio is fairly sensitive to shock conditions . the sm 100 km s@xmath14 model gives a value of 0.027 . the ratio is 0.005 for an 80 km s@xmath14 shock , but this model predicts insignificant [ o@xmath2iii ] emission . the dopita & sutherland ( 1996 ) higher velocity , lower density ( n=1 ) , magnetized models give a much more significant ionized precursor . for the lowest velocity considered , 150 km s@xmath14 , the shock itself gives a low ratio of 0.019 , but the line fluxes for the precursor are not given . for a 200 km s@xmath14 shock , the ratio from combined shock and precursor is 0.046 . slavin , shull , & begelman ( 1993 ) do not predict he@xmath2i emission . regardless , if shocks or tmls can provide , say 25% of the h@xmath1 emission at @xmath50 kpc , then there may be a region of parameter space which can produce low enough he@xmath2i / h@xmath1 so that the composite line ratio is significantly reduced below that of the stellar - ionized gas alone . the contributions of these sources required in 3 may not be sufficiently large , but the effect is worth future consideration . the inferred stellar temperature , mean spectral type and upper imf cutoff would then all be underestimated . finally , it is worth re - emphasizing that all emission line fluxes and ratios presented here are averaged along a line of sight through the dig layer , and that local variations in , for example , the derived ionization fraction of h surely exist . also , although the vertical dependence of the line ratios has been very revealing , only one slit position has been observed , covering the halo above the most active region of star formation in the disk , and close to an h@xmath1 filament . a key question is how these halo properties vary with environment . does [ o@xmath2iii]/h@xmath4 show the same behavior above more quiescent parts of the disk ? is this behavior peculiar to the halo of ngc 891 only , or is it a general feature of dig halos ? these questions will be addressed by further observations . the author has benefited from many useful discussions about dig ionization from r. reynolds , r. walterbos ( whose comments as referee also improved the paper ) , j. slavin , r. benjamin , j. shields , and others . the help of the kpno staff is also greatly appreciated .

two long - slit spectra of the diffuse ionized gas in ngc 891 are presented . the first reveals variations parallel to the major axis in emission line ratios in the halo gas at @xmath0 pc . it is found that filaments of h@xmath1 emission show lower values of [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 and [ o@xmath2i]/h@xmath1 . although this result is expected if the filaments represent the walls of evacuated chimneys , it merely reflects a more general correlation of these ratios with h@xmath1 surface brightness along the slit , and may simply arise from radiation dilution effects . halo regions showing low line ratios are probably relatively close to ionizing sources in the disk below . the results highlight difficulties inherent in observations of edge - on galaxies caused by lack of knowledge of structure in the in - plane directions . the [ s@xmath2ii]/[n@xmath2ii ] ratio shows almost no dependence on distance along the major axis or h@xmath1 surface brightness . values of [ o@xmath2i]/h@xmath1 indicate that h is 8095% ionized ( assuming @xmath3 k ) , with the higher ionization fractions correlating with higher surface brightness . much more interesting information on the nature of this gaseous halo comes from the second observation , which shows the vertical dependence of [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 , [ o@xmath2i]/h@xmath1 , and [ o@xmath2iii]/h@xmath4 through the brightest region of the dig halo . the most surprising result , in complete contradiction to models in which the dig is ionized by massive stars in the disk , is that [ o@xmath2iii]/h@xmath4 rises with height above the plane for @xmath5 kpc ( even as [ n@xmath2ii]/h@xmath1 , [ s@xmath2ii]/h@xmath1 , and [ o@xmath2i]/h@xmath1 are rising , in line with expectations from such models ) . the run of [ s@xmath2ii]/[n@xmath2ii ] is also problematic , showing essentially no contrast with @xmath6 . the [ o@xmath2iii ] emission probably arises from shocks , turbulent mixing layers , or some other secondary source of ionization . composite models in which the line emission comes from a mix of photo - ionized gas and shocks or turbulent mixing layers are considered in diagnostic diagrams , with the result that many aspects of the data can be explained . problems with the run of [ s@xmath2ii]/[n@xmath2ii ] still remain , however . there is a reasonably large parameter space allowed for the second component . for the photo - ionized component , only matter - bounded models succeed , putting a fairly strong restriction on the clumpiness of the halo gas . given the many uncertainties , the composite models can do little more than demonstrate the feasibility of these processes as secondary sources of energy input . a fairly robust result , however , is that the fraction of h@xmath1 emission arising from the second component probably increases with @xmath6 . from values of [ o@xmath2i]/h@xmath1 , h is essentially 100% ionized at @xmath7 kpc and 90% ionized at @xmath8 kpc ( again assuming @xmath3 k ) .

since its discovery in 2003 @xcite , the @xmath4 charmonium is subject of many experimental and theoretical efforts aimed at disclosing its nature for a recent review see @xcite . the most recent data on the mass of the @xmath3 is @xcite m_x=(3871.850.27(stat)0.19(syst ) ) , [ xmass ] with a width of @xmath5 . however , the problem of the quantum numbers for the @xmath3 is not fully resolved yet : while the analysis of the @xmath6 decay mode of the @xmath4 yields either @xmath2 or @xmath7 quantum numbers @xcite , the recent analysis of the @xmath8 mode seems to favour the @xmath7 assignment @xcite , though the @xmath2 option is not excluded . this question is clearly very central , for the most promising explanations for the @xmath3 in the @xmath9-wave @xmath10 molecule model @xcite as well as in the coupled - channel model @xcite require the quantum numbers @xmath2 . in addition , the @xmath3 can not be a naive @xmath11 @xmath7 state , for its large branching fraction for the @xmath12 mode @xcite is not compatible with the quark - model estimates for the @xmath7 charmonium @xcite . so , for the @xmath7 quantum numbers , very exotic explanations for the @xmath3 would have to be invoked . the aim of the present paper is to perform a combined analysis of the data on the @xmath13 and @xmath14 mass distribution in the @xmath6 and @xmath8 mode , respectively . we find that the @xmath9-wave amplitudes from the decay of a @xmath2 state provide a better overall description of the data than the @xmath15-wave ones from the @xmath7 , especially when the parameter range is restricted to realistic values . we conclude then that the existing data favour @xmath2 quantum numbers of the @xmath3 , however , improved data in the @xmath8 mode are necessary to allow for definite conclusions regarding the @xmath3 quantum numbers . recently belle announced @xcite the updated results of the measurements for the reaction @xmath16 : _ 2^+&=&[8.610.82(stat)0.52(syst)]10 ^ -6 , + b_2 ^ 0&=&[4.31.2(stat)0.4(syst)]10 ^ -6 , for the charged @xmath17 and neutral @xmath18 mode , respectively , with @xmath19 being the product branching fraction @xmath20 in the corresponding mode . the number of events in the background - subtracted combined distribution is @xmath21 for the decay @xmath22 babar reports @xcite _ 3^+&=&[0.60.2(stat)0.1(syst)]10 ^ -5 , + b_3 ^ 0&=&[0.60.3(stat)0.1(syst)]10 ^ -5 , for the charged mode @xmath23 and for the neutral mode @xmath24 , respectively . similarly to the two - pion case above , @xmath25 stands for the product branching fraction @xmath26 in the corresponding mode . the number of events in the combined distribution is n^sig+bg_3=34.06.6,n_3^bg=8.91.0 , and we assume a flat background . note , the spectrum reported in @xcite and used below appears not to be efficiency corrected . however , since only the shape of this spectrum plays a role for the analysis ( see number - of - event distributions ( [ n23pi ] ) below ) and we can reproduce the theoretical spectra of @xcite , which have the efficiency of the detector convoluted in via a monte carlo simulation , the invariant mass dependence of the efficiency corrections is expected to be mild and therefore should not affect our analysis significantly . thus the updated ratio of branchings reads @xcite = = 0.80.3 . [ brratio ] in our analysis we use the ratio ( [ brratio ] ) as well as n_2=196,n_3=25.1 , [ bandn ] and he corresponding bin sizes are @xmath27 mev and @xmath28 mev . as in previous analyses , we assume that the two - pion final state is mediated by the @xmath31 in the intermediate state , while the three - pion final state is mediated by the @xmath32 . it was shown in @xcite that the description of the @xmath6 spectrum with the @xmath7 assumption is improved drastically if the isospin - violating @xmath31-@xmath32 mixing is taken into account . theoretical issues of the @xmath31-@xmath32 mixing are discussed , for example , in @xcite . here we include this effect with the help of the prescription used in @xcite , where the transition amplitude is described by the real parameter @xmath33 . thus , the amplitudes for the decays @xmath34 and @xmath35 take the form & & a_2=a_xj / g_a_2 + & & + a_xj / g_g_a_2 , + & & a_3=a_xj / g_a_3 + & & + a_xj / g_g_a_3 , where the vector meson propagators are g_v^-1=m_v^2-m^2-im_v_v(m),v= , , with @xmath36 ( @xmath37 ) being the @xmath29 ( @xmath30 ) invariant mass in the @xmath6 ( @xmath8 ) final state . masses of the vector mesons used below are @xcite @xmath38 the complex mixing amplitude multiplying the @xmath32 propagator used , for example , in @xcite to analyse the two - pion spectrum , in our notation reads as @xmath39 ; in particular we reproduce naturally the phase of 95@xmath40 quoted in @xcite . note , as we shall only study the invariant mass distributions of the two final states , we do not need to keep explicitly the vector nature of the intermediate states . for the `` running '' @xmath31 meson width we use @xmath41 ^ 2,\ ] ] where @xmath42 , @xmath43 , with @xmath44 gev@xmath45 and with the nominal @xmath31 meson width @xmath46 mev . for the @xmath32 meson `` running '' width ( the nominal width being @xmath47 mev ) , the @xmath48 and @xmath49 decay modes are summed , with the branchings ( 3)=89.1%,()=8.28% . in particular , ( m)=_^(0)^3 , while , for the @xmath50 , we resort to the expressions derived in @xcite , with a reduced contact term which provides the correct nominal value of the @xmath51 decay width @xcite . the transition amplitudes for the decays @xmath52 are parameterised in the standard way , namely , a_xj / v = g_xvf_lx(p ) , with the blatt - weisskopf `` barrier factor '' f_0x(p)=1 , f_1x(p)=(1+r^2 p^2)^-1/2 , for the @xmath2 and @xmath7 assignment , respectively . here @xmath53 denotes the @xmath54 momentum in the @xmath3 rest frame . the `` radius '' @xmath55 is not well understood . if one associates it with the size of the @xmath56 vertex , it might be related to the range of force . in the quark model this radius is @xmath57 . this is also in line with the inverse mass of the lightest exchange particle allowed between @xmath54 and @xmath58 , namely , @xmath59 . on the other hand , a larger value @xmath60 gev@xmath45 is used in the experimental analysis of @xcite . therefore , in the analysis presented below we use both values @xmath61 gev@xmath45 as well as @xmath60 gev@xmath45 , keeping in mind that smaller values of @xmath55 are preferred by phenomenology . the theoretical invariant mass distributions for the @xmath29 and @xmath30 final state take the form : = bm__2 p^2l+1f_lx^2(p)|r_xg_+g_g_|^2 , + [ br23pi ] + = bm__3 p^2l+1f_lx^2(p)|g_+r_x g_g_|^2,where @xmath62 and the parameter @xmath63 absorbs the details of the short - ranged dynamics of the @xmath3 production . the theoretical number - of - event distributions read n_2(m)= , + [ n23pi ] + n_3(m)=. the @xmath31-@xmath32 mixing parameter @xmath64 is extracted from the @xmath65 decay width ( @xmath66 ) . the corresponding amplitude reads a_2=g_a_2 , and we find that 3.410 ^ -3 ^2 . [ cols="^,^,^,^,^,^,^,^",options="header " , ] the number - of - event distributions ( [ n23pi ] ) possess 3 free parameters : the `` barrier '' factor @xmath55 , the ratio of couplings @xmath67 and the overall normalisation parameter @xmath63 . as outlined above , we perform the analysis for two values of @xmath55 , namely , the preferred value of 1 gev@xmath45 and a significantly larger value of 5 gev@xmath45 used in earlier analyses . since the normalisation factor @xmath63 drops out from the ratio of the two branchings , we extract the ratio @xmath67 directly from the integrated data , that is from the relation ( dm)/(dm).=b_3/b_2 , where the value of the ratio on the right - hand side is fixed by eq . ( [ brratio ] ) , and in the integration above we have cut off the @xmath29 invariant mass at 400 mev , as in @xcite , and the @xmath30 invariant mass at 740 mev , as in @xcite . therefore the norm @xmath63 is our only fitting parameter which governs the overall strength of the signal in both channels simultaneously , while the shape of the curves is fully determined from other sources . in table [ t1 ] , we list the parameters of the 3 combined fits to the data , found for the 2 values of the blatt - weisskopf parameter @xmath55 . the corresponding line shapes and the result of the integration in bins are shown in fig . [ fig1 ] . one can see from table [ t1 ] and fig . [ fig1 ] that the best overall description of the data for the two channels under consideration is provided by the @xmath9-wave fit . the @xmath15-wave fit is capable to provide the description of the data of a comparable ( however somewhat lower ) quality , only for large values of the blatt - weisskopf parameter @xmath55 , @xmath60 gev@xmath45 . the @xmath15-wave fit becomes poorer when the blatt - weisskopf parameter is decreased , and for values of @xmath55 of order 1 gev@xmath45 , the quality of the @xmath15-wave fit is unsatisfactory , which is the result of a very poor description of the two - pion spectrum see the dashed ( green ) curve in fig . [ fig1 ] . varying the ratio of branchings @xmath68 around its central value within the experimental uncertainty interval [ see eq . ( [ brratio ] ) ] leads only to minor changes in the fits and does not affect the conclusions . since no charged partners of the @xmath4 are observed experimentally , it is supposed to be ( predominantly ) an isoscalar . then the ratio @xmath62 measures the strength of the isospin violation in the @xmath69 decay vertex . as discussed above , this ratio is extracted directly from the data on the ratio of the branchings ( [ brratio ] ) . an isospin - violating observable for a compact charmonium is the ratio of the branching fractions for the @xmath70 decays into @xmath71 and @xmath72 final states as r_(2s)== 0.03 , [ rcharm ] where @xmath73 and @xmath74 are the center - of - mass momenta of @xmath75 and @xmath76 , respectively , and the @xmath70 branching fractions are taken from @xcite . however , since here also the denominator violates a symmetry , namely , su(3 ) , and there might be significant meson - loop contributions @xcite , the estimate ( [ rcharm ] ) is to be regarded as a conservative upper bound for the isospin violation strength for compact charmonia . in contrast to this , in the @xmath9-wave molecular picture for the @xmath3 , isospin violation is enhanced significantly compared to the strength ( [ rcharm ] ) for it proceeds via intermediate @xmath77 states and is therefore driven by the mass difference @xmath78 mev of the neutral and charged @xmath77 threshold see , for example , @xcite . an order - of - magnitude estimate is provided by the expression r_x^mol~|| ~ ~0.13 , [ rmol ] where @xmath79 is the @xmath80 meson mass , while @xmath81 and @xmath82 denote the amplitudes corresponding to loop diagrams with neutral and charged @xmath10 intermediate states , respectively , evaluated at the @xmath3 mass . they are composed of two terms , the strongly channel - dependent analytic continuation of the unitarity cut , proportional to the typical momentum of the meson pair , and the weakly channel - dependent principle value term , whose size is identified with the inverse range of forces of order of 1 gev ( see above ) . this estimate is within a factor of 2 consistent with the value @xmath83 found from our @xmath9-wave fit see table [ t1 ] . on the other hand , if the @xmath3 has the quantum numbers @xmath7 , one should expect the isospin violation in the @xmath3 wave function to be of the natural charmonium size , and thus of the order of @xmath84see discussion below eq . ( [ rcharm ] ) , since the @xmath10 loop effects are suppressed by the additional centrifugal barrier : the estimate analogous to eq . ( [ rmol ] ) now reads @xmath85 . thus , for the state with the quantum numbers @xmath7 , one expects values of at most @xmath86 , that are significantly smaller than those following from the data ( see table [ t1 ] ) . one is led to conclude therefore that for @xmath87 , needed for the quantum numbers @xmath7 to be consistent with the data on the @xmath3 decays , a new , yet unknown , isospin violation mechanism would have to be invoked . we conclude therefore that , although the present quality of the data in the @xmath35 channel is not sufficient to draw a definite conclusion concerning the quantum numbers of the @xmath4 , the combined analysis of the existing two- and three - pion spectra favours the @xmath9-wave fit , related to the @xmath2 assignment for the @xmath4 , over the @xmath15-wave fit , related to the @xmath7 assignment . we notice that an acceptable @xmath15-wave fit calls for a large range parameter in the blatt - weisskopf form factor which meets certain difficulties with its phenomenological interpretation . in addition , while the value @xmath62 can be understood theoretically for the @xmath2 assignment , the value extracted for the @xmath7 assignment is too large to be explained from known mechanisms of the isospin violation . we acknowledge useful discussions with e. braaten , s. eidelman , f .- k . guo , and r. mizuk . the work was supported in parts by funds provided from the helmholtz association ( grant nos vh - ng-222 and vh - vi-231 ) , by the dfg ( grant nos sfb / tr 16 and 436 rus 113/991/0 - 1 ) , by the eu hadronphysics2 project , by the rffi ( grant nos rffi-09 - 02 - 91342-nnioa and rffi-09 - 02 - 00629a ) , and by the state corporation of russian federation `` rosatom . 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we re - analyse the two- and three - pion mass distributions in the decays @xmath0 and @xmath1 and argue that the present data favour the @xmath2 assignment for the quantum numbers of the @xmath3 .

the application of the dirac quantization programme to generally covariant systems triggers a ` conceptual problem ' since the quantization of the hamiltonian constraint yields a seemingly ` timeless ' evolution equation @xmath0 and thereby leads to the issue of ` frozen dynamics ' in the quantum theory . while classically the presence of a hamiltonian constraint does not lead to an impeding obstacle because we can always resort to a time coordinate along the flow of the hamiltonian with respect to which one can order physical relations , such a time coordinate is absent in the physical hilbert space . the standard ` conceptual solution ' to this problem is _ relational dynamics _ @xcite , according to which time evolution arises by relating dynamical quantities to other internal variables which we use as clocks to keep track of internal time ; such relational statements are invariant under the gauge flow of the hamiltonian constraint and could thus , in principle , be promoted into well defined operators on the physical hilbert space . nevertheless , even when adopting this ` conceptual solution ' one encounters a whole plethora of _ technical _ problems in the quantum theory @xcite . here we want to address four such problems : 1 . the _ hilbert space problem_. which hilbert space representation is one to use and , in particular , how is one to construct a positive - definite inner product on the space of solutions ? 2 . the _ multiple choice problem_. which relational clock variable should one employ ? ( different relational clocks may yield _ a priori _ different quantum theories . ) the _ global time problem_. good global clocks which parametrize the gauge orbits such that every trajectory intersects every constant clock time slice once and only once may not exist . the _ observable problem_. it is a notoriously difficult challenge to construct explicit observables in generally covariant theories , especially in the quantum theory . in the sequel we summarise the effective approach to the problem of time @xcite which at least in the semiclassical regime deals with some of these problems and circumvents others . rather than employing special clock choices and attempting to justify these , we follow the basic premise that there are to be no distinguished clocks and that , instead , we ought to treat all clocks on an equal footing . our goal consists in employing local internal times , possibly translating between different clocks and thereby evolving ( relational ) data along complete semiclassical trajectories . the central idea of the effective approach is to sidestep the _ hilbert space problem _ altogether ; instead of employing wave functions or density matrices to describe states in a fixed hilbert space , we regard states as ( _ a priori _ complex ) linear functionals on an algebra of kinematical variables , say polynomials in a canonial pair @xmath1 , and use expectation values @xmath2 and @xmath3 , and moments . ] @xmath4 to parametrize states @xcite . this construction immediately carries over to an arbitrary number of canonical pairs . this space of states can be given a ( quantum ) phase space structure via a poisson bracket defined as follows for any operators @xmath5 polynomial in the canonical variables [ poisson ] \{,}:= . this definition can be extended to the moments by linearity and the leibnitz rule and yields ` classical poisson brackets ' for the expectation values of basic canonical variables , i.e. @xmath6 , vanishing poisson brackets between expectation values and moments @xmath7 and more complicated brackets between the moments @xcite . in this summary we focus on finite dimensional systems with a single constraint @xmath8 playing the role of the hamiltonian constraint of general relativity . the dirac programme requires physical states to satisfy @xmath9 . the spectrum of @xmath8 is essential for the construction of the physical hilbert space : if zero lies in the discrete part of the spectrum , the physical hilbert space turns out to be a subspace of the kinematical hilbert space , while a new hilbert space with a new ( positive definite ) inner product must be constructed for solutions if zero lies in the continuous part of the spectrum . there exist techniques for constructing such physical hilbert spaces in the latter case @xcite , however , finding physical hilbert spaces in practice remains a difficult task . the effective techniques , on the other hand , work for both zero in the discrete or continuous part of the spectrum . at the effective level , physical states must clearly satisfy [ qcon ] ( , , ( qp),)=0 , however , this is not sufficient since the expectation value of @xmath8 may be zero even if @xmath10 . furthermore , when solving one ( first class ) constraint classically we can eliminate an entire canonical pair , while on the quantum phase space after solving ( [ qcon ] ) and factoring out its flow we would be left with an infinite tower of ( unconstrained ) moments of the eliminated canonical pair . clearly , we must impose a further set of constraints to account for this . it turns out that [ pol ] c_pol:= = 0 , where @xmath11 stands for all polynomials in the basic canonical operators , provides a correct independent set of constraint functions on quantum phase space which together with ( [ qcon ] ) is then equivalent to the dirac condition @xcite . as a result , we have infinitely many constraints for infinitely many variables . in order to reduce the system to a tractable finite size , we impose a very general semiclassical approximation : assume @xmath12 and truncate the system at @xmath13 by neglecting all terms of higher order . this is consistent with the quantum poisson bracket structure ( [ poisson ] ) which preserves orders in @xmath13 and generates the flows and dynamics on quantum phase space , once suitable initial data for the relevant expectation values and moments has been chosen . rather than defining a quantum theory , the effective approach in its present form provides an effective tool for evaluating the quantum dynamics of given ( finite dimensional ) systems . let us outline the effective approach to semiclassical relational dynamics with an explicit toy model ( for a more general discussion and details see @xcite ) . the model we are considering is the isotropic 2d harmonic oscillator with prescribed total energy , whose classical solutions are closed orbits in phase space such that no global clock exists @xcite . it is subject to the constraint [ quant - rov ] could construct clock functions are the @xmath14 . in order to obtain a notion of evolution from this _ a priori _ timeless system , we may choose a variable as a local clock , say @xmath15 , and deparametrize locally at the classical level by factorizing the constraint c=(p_1+h(q_1))(p_1-h(q_1 ) ) , h(q_1)=. quantization of the first factor ( in the region where it vanishes ) yields a local schrdinger regime [ schrod ] i(q_1,q_2)=(_2,_2 ; q_1)(q_1,q_2)=(q_1,q_2 ) with a non self adjoint @xmath16 ( defined by spectral decomposition ) due to non unitarity in @xmath15 evolution ( @xmath15 is not globally valid ) . given this construction , we can calculate expectation values and moments in @xmath15 time and compare their evolution to the effective treatment . at the effective level , on the other hand , we retain 14 kinematical degrees of freedom at order @xmath13 , namely four expectation values @xmath17 , four spreads @xmath18 , and six covariances @xmath19 , where the @xmath20 are any of the @xmath14 . at order @xmath13 the constraint ( [ quant - rov ] ) translates via ( [ qcon ] ) and ( [ pol])into the following five first class quantum constraint functions @xcite [ effrov ] c&= & p_1 ^ 2+p_2 ^ 2+q_1 ^ 2+q_2 ^ 2+(p_1)^2+(p_2)^2+(q_1)^2+(q_2)^2-m= 0 + c_q_1&= & 2p_1(q_1p_1)+2p_2(q_1p_2)+2q_1(q_1)^2 + 2q_2(q_1q_2)+ip_1= 0 + c_p_1&= & 2p_1(p_1)^2 + 2p_2(p_1p_2)+2q_1(p_1q_1)+2q_2(p_1q_2)-iq_1= 0 + c_q_2&=&2p_1(p_1q_2)+ 2p_2 ( q_2p_2)+2q_1(q_1q_2)+2q_2(q_2)^2+ip_2 = 0 + c_p_2&=&2p_1(p_1p_2)+ 2p_2(p_2)^2 + 2q_1(q_1p_2)+2q_2(q_2p_2)-iq_2 = 0.these five constraints generate only four independent flows since at order @xmath13 we have to deal with a degenerate poisson structure @xcite . it is convenient to fix three of these independent flows . selecting @xmath15 as a relational clock it should not be represented as an operator ; we therefore choose a gauge which ` projects this clock to a parameter ' by setting its fluctuations to zero ( fixing three @xmath21 flows and leaving us with one ` hamiltonian flow ' ) [ zeitgeist ] ( q_1)^2=(q_1q_2)=(q_1p_2)=0 . indeed , the choice of an internal time variable in the effective framework is best described and interpreted in a corresponding gauge @xcite : we refer to such a choice and gauge ( e.g. ( [ zeitgeist ] ) ) as a _ zeitgeist_. after choosing a local relational clock and corresponding zeitgeist , local relational observables at the effective level are given by correlations of expectation values and moments with the expectation value of the chosen clock ( here @xmath22 ) evaluated in its zeitgeist . we call such state dependent local observables _ fashionables _ because they comprise the complete physical information about the system ( at order @xmath13 ) as long as the zeitgeist is valid , but may fall out of fashion when a zeitgeist necessarily changes at turning points of local clocks . it turns out that the fashionables in @xmath15 time of the effective framework agree perfectly with those computed in the @xmath15 schrdinger regime ( [ schrod ] ) @xcite . in its turning region , @xmath15 becomes complex valued and its zeitgeist incompatible with the semiclassical expansion , such that evolution in @xmath15 breaks down _ before _ the turning point a signature of non unitarity . thus , a new local internal time ( here @xmath23 ) is needed and for a full evolution we must switch between @xmath15 and @xmath23 time . furthermore , since a given internal time is best described in a corresponding choice of gauge , we must switch also between the @xmath15 and @xmath23 zeitgeister . explicit gauge transformations between these zeitgeister can , indeed , be constructed and initial data can be evolved consistently around the entire closed semiclassical orbit through the turning points of various clock variables @xcite . it should be emphasized that in each gauge we evolve a _ different _ set of fashionables which highlights the local nature of the relational concept in the absence of global clock functions . the effective approach summarised here sidesteps the technical problems mentioned in the introduction . in particular , the _ hilbert space problem _ is avoided altogether since no use of any hilbert space representation has been made . at the effective level one can make sense of local time evolution and switching between different local clocks can essentially be achieved by an additional gauge transformation which enables one to handle the _ multiple choice _ and _ global time problem _ and implies the equivalence of different clock choices at semiclassical order . note , however , that if the relational clock is non global , it assumes complex values in its turning region @xcite . state dependent fashionables arise naturally in this framework which simplifies the _ observable problem _ due to the classical treatment of the effective system . in the example outlined here and other models , the effective evolution agrees with local deparametrizations at a hilbert space level @xcite . finally , the notion of relational evolution disappears in highly quantum states of systems without global clocks @xcite . the application of this effective framework to the more interesting closed frw model with a massive scalar field will appear elsewhere @xcite . * acknowledgments * the author would like to thank martin bojowald and artur tsobanjan for collaboration on this subject . 9 slides at http://loops11.iem.csic.es/loops11/index.php?option=com_content&view=article&id=101 c. rovelli , _ quantum gravity _ ( cup , cambridge , 2004 ) ; c. rovelli , phys . rev . * d 42 * 2638 ( 1990 ) , phys . rev . * d 43 * 442 ( 1991 ) ; b. dittrich , gen . grav . * 39 * 1891 ( 2007 ) , class . grav . * 23 * 6155 ( 2006 ) k. v. kucha , in _ proc . 4th canadian conference on general relativity and relativistic astrophysics _ , edited by g. kunstatter , d. vincent and j. williams ( world scientific , singapore , 1992 ) ; c. j. isham , in _ integrable systems , quantum groups , and quantum field theories _ ( kluwer academic publishers , london , 1993 ) ; e. anderson ( _ preprint _ arxiv:1009.2157 [ gr - qc ] ) ; j. tambornino ( _ preprint _ arxiv:1109.0740 [ gr - qc ] ) m. bojowald , p. a. hhn and a. tsobanjan , class . quantum grav . * 28 * 035006 ( 2011 ) m. bojowald , p. a. hhn and a. tsobanjan , phys . * d 83 * 125023 ( 2011 ) m. bojowald and a. skirzewski , rev . phys . * 18 * 713745 ( 2006 )

we provide a synopsis of an effective approach to the problem of time in the semiclassical regime . the essential features of this new approach to evaluating relational quantum dynamics in constrained systems are illustrated by means of a simple toy model .

the large values of the solar ( @xmath11 ) and atmospheric ( @xmath12 ) mixing angles may be telling us about some new symmetries of leptons not presented in the quark sector and may provide a clue to the nature of the quark - lepton physics beyond the standard model . if there exists such a flavor symmetry in nature , the tribimaximal ( tbm ) @xcite pattern for the neutrino mixing will be a good zeroth order approximation to reality : @xmath13 for example , in a well - motivated extension of the standard model through the inclusion of @xmath14 discrete symmetry , the tbm pattern comes out in a natural way in the work of @xcite . although such a flavor symmetry is realized in nature leading to exact tbm , in general there may be some deviations from tbm . recent data of the t2k @xcite and minos @xcite collaborations and the analysis based on global fits @xcite of neutrino oscillations enter into a new phase of precise measurements of the neutrino mixing angles and mass - squared differences , indicating that the tbm mixing for three flavors of leptons should be modified . in the weak eigenstate basis , the yukawa interactions in both neutrino and charged lepton sectors and the charged gauge interaction can be written as @xmath15 when diagonalizing the neutrino and charged lepton mass matrices @xmath16 , one can rotate the neutrino and charged lepton fields from the weak eigenstates to the mass eigenstates @xmath17 . then we obtain the leptonic @xmath18 unitary mixing matrix @xmath19 from the charged current term in eq . ( [ lagrangiana ] ) . in the standard parametrization of the leptonic mixing matrix @xmath20 , it is expressed in terms of three mixing angles and three _ cp_-odd phases ( one for the dirac neutrino and two for the majorana neutrino ) @xcite @xmath21 where @xmath22 and @xmath23 , and @xmath24 is a diagonal phase matrix which contains two _ cp_-violating majorana phases , one ( or a combination ) of which can be in principle explored through the neutrinoless double beta ( @xmath25 ) decay @xcite . for the global fits of the available data from neutrino oscillation experiments , we quote two recent analyses : one by gonzalez - garcia _ et al . _ @xcite @xmath26 in @xmath27 ( @xmath7 ) ranges , or equivalently @xmath28 and the other given by fogli _ et al . _ with new reactor neutrino fluxes @xcite : @xmath29 corresponding to @xmath30 the analysis by fogli _ et al . _ includes the t2k @xcite and minos @xcite results . the t2k collaboration @xcite has announced that the value of @xmath8 is non - zero at @xmath31 c.l . with the ranges @xmath32 or @xmath33 for @xmath34 , @xmath35 and the normal ( inverted ) neutrino mass hierarchy . the minos collaboration found @xmath36 with a best fit of @xmath37 for @xmath34 , @xmath35 and the normal ( inverted ) neutrino mass hierarchy . the experimental result of non - zero @xmath38 implies that the tbm pattern should be modified . however , properties related to the leptonic _ violation remain completely unknown yet . the trimaximal neutrino mixing was first proposed by cabibbo @xcite ] if one considers @xmath14 discrete symmetry , it will have two subgroups , namely , @xmath39 and @xmath40 . the trimaximal matrix given in eq . ( [ cabibbo ] ) is obtained under @xmath40 . ] ( see also @xcite ) @xmath41 with @xmath42 being a complex cube - root of unity . this mixing matrix has maximal _ cp _ violation with the jarlskog invariant @xmath43 . however , this trimaximal mixing pattern has been ruled out by current experimental data on neutrino oscillations . in their original work , harrison , perkins and scott ( hps ) @xcite proposed to consider the simple mass matrices @xmath44 that can lead to the tribimaximal mixing , where @xmath45 and @xmath46 are real parameters , in eq . ( [ mass1 ] ) can be in general introduced as complex : e.g. , @xmath47 and @xmath48 . this case has been considered by xing @xcite who pointed out that the off - diagonal terms in @xmath49 will acquire a phase from the complex @xmath50 . it has the interesting implication that a nonzero @xmath51 will result from the phase of @xmath50 . however , the corresponding jarlskog invariant is exactly zero and the absence of intrinsic _ cp _ violation makes this possibility less interesting . ] @xmath52 and @xmath53 . the mass matrices are diagonalized by the trimaximal matrix @xmath54 for charged lepton fields and the bimaximal matrix @xmath49 defined below for neutrino fields , that is , @xmath55 and @xmath56 . the combination of trimaximal and bimaximal matrices leads to the so - called tbm mixing matrix : @xmath57 it is clear by now that the tribimaximal mixing is not consistent with the recent experimental data on the reactor mixing angle @xmath8 because of the vanishing matrix element @xmath58 in @xmath0 . in this work we consider an extension of the tribimaximal mixing by considering small perturbations to the mass matrices @xmath59 and @xmath60 which we will call @xmath61 and @xmath62 , respectively ( see eq . ( [ mass2 ] ) below ) so that @xmath63 is no longer in the bimaximal form and @xmath64 deviates from the trimaximal structure , where @xmath2 is the unitary matrix needed to diagonalize the matrix @xmath65 . as a consequence , @xmath66 small perturbations . hence , the corrections to the tbm pattern arise from both charged lepton and neutrino sectors . inspired by the t2k and minos measurements of a sizable reactor angle @xmath8 , there exist in the literature intensive studies of possible deviations from the exact tbm pattern . however , most of these investigations were focused on the modification of tbm arising from either the neutrino sector @xcite or the charged lepton part @xcite , but not both simultaneously . the paper is organized as follows . in sec . ii , we set up the model by making a general extension to the charged lepton and neutrino mass matrices . then in sec . iii we study the phenomenological implications by considering two different scenarios for the charged lepton mixing matrix . our conclusions are summarized in sec . in order to discuss the deviation from the tbm mixing , let us consider a simple and general extension of the original proposal by hps given in eq . ( [ mass1 ] ) , by taking into account perturbative effects on the mass matrices @xmath67 and @xmath68 . the generalized mass matrices @xmath69 and @xmath70 can be introduced as , dihedral groups , @xmath71 , etc . by considering higher order and radiative effects , the matrices in eq . ( [ mass2 ] ) can be realized . for example , we have shown in ref . @xcite that these matrices can be obtained by introducing dimension-5 operators to the lagrangians . ] @xmath72 where @xmath69 and @xmath70 are defined as the hermitian square of the mass matrices @xmath73 and @xmath74 , respectively , with the subscript @xmath75 denoting charged fermion fields ( charged leptons or quarks ) . due to the hermiticity of @xmath69 and @xmath70 , the parameters @xmath76 are real , while @xmath77 and @xmath78 are complex . the parameters @xmath79 , @xmath78 and @xmath80 represent small perturbations . note that the ( 11 ) , ( 13 ) , ( 22 ) elements ( _ i.e. , _ @xmath81 , @xmath82 and @xmath83 ) in @xmath70 are assumed to contain any perturbative effects on the elements @xmath84 , @xmath50 , and @xmath46 in @xmath68 , respectively . for simplicity , it is assumed that @xmath85 is real just as the other elements in @xmath70 and the vanishing off - diagonal elements in @xmath68 remain zeros in @xmath70 . the parameters @xmath86 and @xmath77 are encoded in @xcite as @xmath87 where the subscript @xmath88 indicates a generation of charged fermion field , and @xmath89 represents a bare mass of @xmath88 , for example , @xmath90 for charged lepton fields . we first discuss the hermitian square of the neutrino mass matrix , @xmath70 , in eq . ( [ mass2 ] ) . it can be diagonalized by @xmath91 with @xmath92 and @xmath93 where the diagonal phase matrix @xmath94 contains two additional phases , which can be absorbed into the neutrino mass eigenstate fields . for a small perturbation @xmath95 , the mixing parameter @xmath96 can be expressed in terms of @xmath97 @xmath3 is then reduced to @xmath98 the neutrino mass eigenvalues are obtained as @xmath99 and their differences are given by @xmath100 from which we have a relation @xmath101 . it is well known that the sign of @xmath102 is positive due to the requirement of the mikheyev - smirnov - wolfenstein resonance for solar neutrinos . the sign of @xmath103 depends on that of @xmath104 : @xmath105 for the normal mass spectrum and @xmath106 for the inverted one . the quantities @xmath107 ( or @xmath108 ) are determined by the four parameters @xmath109 , while the majorana phases in eq . ( [ unu ] ) are hidden in the squared mass eigenvalues . we next turn to the hermitian square of the mass matrix for charged fermions in eq . ( [ mass2 ] ) . this modified charged fermion mass matrix is no longer diagonalized by @xmath54 @xmath110 where @xmath111 corresponding to @xmath112 , respectively , and @xmath113 is composed of the combinations of @xmath79 and @xmath78 . to diagonalize @xmath114 , we need an additional matrix @xmath115 which can be , in general , parametrized in terms of three mixing angles and six phases : @xmath116 where @xmath117 , @xmath118 and a diagonal phase matrix @xmath119 which can be rotated away by the phase redefinition of left - charged fermion fields . the charged fermion mixing matrix now reads @xmath120 . finally , we arrive at the general expression for the leptonic mixing matrix @xmath121 a simple and general extension of the mass matrices given in eq . ( [ mass2 ] ) thus leads to two possible sources of corrections to the tribimaximal mixing : @xmath2 measures the deviation of the charged lepton mixing matrix from the trimaximal form and @xmath3 characterizes the departure of the neutrino mixing from the bimaximal one . the charged lepton mass matrix in eq . ( [ mass2 ] ) or ( [ aa ] ) has 12 free parameters . three of them are replaced by the phases @xmath122 in eq . ( [ vl ] ) which can be eliminated by a redefinition of the physical charged lepton fields . the remaining 9 parameters can be expressed in terms of @xmath123 . > from eqs . ( [ aa ] ) and ( [ vl ] ) the mixing angles and phases can be expressed as @xmath124+\arg(\eta_{12 } ) \ , \end{aligned}\ ] ] with the condition @xmath125 . in the charged fermion sector , there is a qualitative feature that distinguishes the neutrino sector from the charged fermion one . the mass spectrum of the charged leptons exhibits a similar hierarchical pattern to that of the down - type quarks , unlike that of the up - type quarks which show a much stronger hierarchical pattern . for example , in terms of the cabbibo angle @xmath126 , the fermion masses scale as @xmath127 , @xmath128 and @xmath129 . this may lead to two implications : ( i ) the cabibbo - kobayashi - maskawa ( ckm ) matrix @xcite is mainly governed by the down - type quark mixing matrix , and ( ii ) the charged lepton mixing matrix is similar to that of the down - type quark one . therefore , we shall assume that ( i ) @xmath130 and @xmath131 , where @xmath132 is associated with the diagonalization of the down - type ( up - type ) quark mass matrix and @xmath133 is a @xmath18 unit matrix , and ( ii ) the charged lepton mixing matrix @xmath134 has the same structure as the ckm matrix , that is , @xmath135 or @xmath136 . recently , we have proposed a simple _ ansatz _ for the charged lepton mixing matrix @xmath2 , namely , it has the qin - ma - like parametrization in which the _ cp_-odd phase is approximately maximal @xcite . armed with this _ ansatz _ , we notice that the 6 parameters @xmath137 in @xmath2 are reduced to four independent ones @xmath138 . it has the advantage that the tbm predictions of @xmath139 and especially @xmath140 will not be spoiled and that a sizable reactor mixing angle @xmath8 and a large dirac _ cp_-odd phase are obtained in the mixing @xmath141 . the qin - ma ( qm ) parametrization of the quark ckm matrix is a wolfenstein - like parametrization and can be expanded in terms of the small parameter @xmath142 @xcite . however , unlike the original wolfenstein parametrization @xcite , the qm one has the advantage that its _ cp_-odd phase @xmath143 is manifested in the parametrization and is near maximal , _ i.e. , _ @xmath144 . this is crucial for a viable neutrino phenomenology . it should be stressed that one can also use any parametrization for the ckm matrix as a starting point . as shown in @xcite , one can adjust the phase differences in the diagonal phase matrix @xmath145 in eq . ( [ vl ] ) in such a way that the prediction of @xmath146 will not be considerably affected . for @xmath147 , the qm parametrization @xcite can be obtained from eq . ( [ vl ] ) by the replacements @xmath148 : @xmath149 on the other hand , for @xmath150 the qm parametrization is obtained by the replacements @xmath151 : @xmath152 where the superscript @xmath75 denotes @xmath153 ( down - type quarks ) or @xmath154 ( charged leptons ) . from the global fits to the quark mixing matrix given by @xcite we obtain @xmath155 because of the freedom of the phase redefinition for the quark fields , we have shown in @xcite that the qm parametrization is indeed equivalent to the wolfenstein one in the quark sector . finally , the leptonic mixing parameters ( @xmath156 ) except majorana phases can be expressed in terms of five parameters @xmath96 ( or @xmath108 ) , @xmath157 , the last four being the qm parameters in the lepton sector . if we further assume that all the qm parameters except @xmath143 have the same values in both the ckm and pmns matrices , then only two free parameters left in the lepton mixing matrix are @xmath108 and @xmath143 . if @xmath143 is fixed to be the same as the ckm one , then there will be only one free parameter @xmath108 in our calculation . in the next section , we shall study the dependence of the mixing angles @xmath158 and the jarlskog invariant @xmath159 on @xmath143 and @xmath108 . to make our point clearer , let us summarize the reduction of the number of independent parameters in this work . in the leptonic sector , we start with 16 free parameters ( 12 from the charged lepton mass matrix @xmath160 and 4 from the neutrino mass matrix @xmath161 ) as shown in eq . ( [ mass2 ] ) . among the 12 parameters from @xmath160 , three phases can be rotated away by the redefinition of the charged lepton fields . the remaining 9 parameters correspond to three charged lepton masses ( @xmath162 ) and six angles in the charged lepton mixing matrix @xmath163 as shown in eq . ( [ vl ] ) , while the 4 parameters from @xmath161 correspond to three neutrino masses ( @xmath164 ) plus one angle ( @xmath96 or @xmath108 ) in the neutrino mixing matrix @xmath165 as shown in eq . ( [ unu ] ) or ( [ epsilon ] ) . with our _ ansatz _ for @xmath163 discussed before , the 6 angles in @xmath163 are reduced to four qm parameters ( @xmath166 ) . thus , the number of parameters finally becomes five ( @xmath166 plus @xmath96 ( or @xmath108 ) ) , except for the six lepton masses . under the further assumption of the qm parameters @xmath167 having the same values in both the ckm and pmns matrices , these five parameters are reduced to only two ones @xmath143 and @xmath108 . we now proceed to discuss the low energy neutrino phenomenology with the neutrino mixing matrix @xmath168 ( see eq . ( [ unu ] ) ) characterized by the mixing angle @xmath96 or the small parameter @xmath108 and the charged lepton mixing matrix @xmath169 in which @xmath134 is assumed to have the similar expression as the qm parametrization @xcite given by @xmath170 or @xmath4 ( see eq . ( [ vl1 ] ) and eq . ( [ vl2 ] ) , respectively ) . the lepton mixing matrix thus has the form @xmath171 therefore , the corrections to the tbm matrix within our framework arise from the charged lepton mixing matrix @xmath2 characterized by the parameters @xmath138 and the matrix @xmath3 specified by the parameter @xmath108 whose size is strongly constrained by the recent t2k data . indeed , the parameters @xmath172 and @xmath143 in the lepton sector are _ a priori _ not necessarily the same as that in the quark sector . hereafter , we shall use the central values in eq . ( [ eq : qmfh ] ) of the parameters @xmath173 for our numerical calculations . in the following we consider both cases : 0.4 cm * ( i ) * @xmath174 0.5 cm with the help of eqs . ( [ hps ] ) and ( [ vl1 ] ) , the leptonic mixing matrix corrected by the replacements @xmath175 and @xmath176 , can be written , up to order of @xmath177 and @xmath178 , as @xmath179 note that @xmath180 here contains five independent parameters ( @xmath181 and @xmath108 ) . . ] by rephasing the lepton and neutrino fields @xmath182 , @xmath183 , @xmath184 and @xmath185 , the pmns matrix is recast to @xmath186 where @xmath187 is an element of the pmns matrix with @xmath188 corresponding to the lepton flavors and @xmath189 to the light neutrino mass eigenstates . in eq . ( [ pmns2 ] ) the phases defined as @xmath190 , @xmath191 , @xmath192 , @xmath193 and @xmath194 have the expressions : @xmath195 > from eq . ( [ pmns2 ] ) , the neutrino mixing parameters can be displayed as @xmath196 it follows from eqs . ( [ leptona ] ) and ( [ mixing1 ] ) that the solar neutrino mixing angle @xmath11 can be approximated , up to order @xmath197 and @xmath178 , as @xmath198 this indicates that the deviation from @xmath199 becomes small when @xmath200 approaches to zero and the magnitude of @xmath108 is less than @xmath142 . since it is the first column of @xmath2 that makes the major contribution to @xmath201 , this explains why we need a phase of order @xmath202 for the element @xmath203 : when @xmath204 , the present data of the solar mixing angle can be accommodated even for a large @xmath205 ( but less than @xmath142 ) . the behavior of @xmath201 as a function of @xmath143 is plotted in fig . [ fig1 ] where the horizontal dashed lines denote the upper and lower bounds of the experimental data in @xmath7 ranges . the allowed regions for @xmath143 ( in radian ) lie in the ranges of @xmath206 and @xmath207 , recalling that the qm phase is @xmath208 . likewise , the atmospheric neutrino mixing angle @xmath12 comes out as @xmath209\right ) \ . \label{atm } \end{aligned}\ ] ] fig . [ fig1 ] shows a small deviation from the tbm atmospheric mixing angle with @xmath210 for @xmath211 . owing to the absence of corrections to the first order of @xmath142 or @xmath108 in eq . ( [ atm ] ) , the deviation from the maximal mixing of @xmath12 comes mainly from the terms associated with @xmath212 or @xmath213 . especially , for @xmath214 we have the approximation @xmath215 , which implies @xmath216 for @xmath211 . we see from fig . [ fig1 ] that @xmath217 lies in the ranges @xmath218 for @xmath219 . the reactor mixing angle @xmath8 now reads @xmath220 evidently , @xmath51 depends considerably on the parameters @xmath142 and @xmath108 . thus , we have a non - vanishing @xmath8 with a central value of @xmath221 or @xmath222 for @xmath223 @xcite . note that the size of the unknown parameter @xmath108 is constrained by the plot of @xmath51 versus @xmath143 in fig . [ fig1 ] where the horizontal dot - dashed lines represent the present t2k data for the normal neutrino mass hierarchy . for a negative value of @xmath108 , the plot for @xmath51 versus @xmath143 is flipped upside - down . assuming @xmath224 , we see from eq . ( [ masssquare ] ) that a positive ( negative ) value of @xmath108 leads to a normal ( inverted ) neutrino mass spectrum . for example , we find @xmath225 ( @xmath226 ) for @xmath227 and @xmath228 ( @xmath229 ) . leptonic _ cp _ violation can be detected through the neutrino oscillations which are sensitive to the dirac _ cp_-phase @xmath230 , but insensitive to the majorana phases in @xmath20 @xcite . it follows from eqs . ( [ mixing elements2 ] ) and ( [ mixing1 ] ) that the dirac phase @xmath231 has the expression @xmath232 where terms of order @xmath233 have been neglected in both numerator and denominator . assuming @xmath224 , we show in table [ diraccp11 ] the predictions for @xmath234 and @xmath8 as a function of @xmath108 , where we have used the central values of eq . ( [ eq : qmfh ] ) . .[diraccp11 ] predictions of @xmath230 and @xmath8 as a function of @xmath108 in the case of @xmath174 . [ cols="^,^,^",options="header " , ] the strength of @xmath235 violation @xmath159 can be expressed in a similar way to eq . ( [ jcp1 ] ) @xmath236 which can be approximated as @xmath237 . when @xmath214 , it is further reduced to @xmath238 for @xmath239 . assuming @xmath224 , we see from fig . [ fig2 ] that @xmath240 ( @xmath241 ) for @xmath242 ( @xmath243 ) and @xmath227 . in their original work , harrison , perkins and scott proposed simple charged lepton and neutrino mass matrices that lead to the tribimaximal mixing @xmath0 . in this paper we considered a general extension of the mass matrices so that the lepton mixing matrix becomes @xmath1 . hence , corrections to the tribimaximal mixing arise from both charged lepton and neutrino sectors : the charged lepton mixing matrix @xmath2 measures the deviation of from the trimaximal form and the @xmath3 matrix characterizes the departure of the neutrino mixing from the bimaximal one . following our previous work to assume a qin - ma - like parametrization @xmath4 for @xmath2 in which the _ cp_-odd phase is approximately maximal , we study the phenomenological implications in two different scenarios : @xmath5 and @xmath6 . we found that both scenarios are consistent with the data within @xmath7 ranges . especially , the predicted central value of the reactor neutrino mixing angle @xmath222 is in good agreement with the recent t2k data . however , the data of @xmath146 can be easily accommodated in the second scenario but only marginally in the first one . hence , the precise measurements of the solar mixing angle in future experiments will test which scenario is more preferable . the leptonic _ violation characterized by the jarlskog invariant @xmath9 is generally of order @xmath10 . 99 # 1#2#3phys . * b#1 * , ( # 3 ) # 2 # 1#2#3nucl . * b#1 * , ( # 3 ) # 2 # 1#2#3phys . * d#1 * , ( # 3 ) # 2 # 1#2#3phys . # 1 * , ( # 3 ) # 2 # 1#2#3mod . * a#1 * , ( # 3 ) # 2 # 1#2#3phys . rep . * # 1 * , ( # 3 ) # 2 # 1#2#3science * # 1 * , ( # 3 ) # 2 # 1#2#3astrophys . j. * # 1 * , ( # 3 ) # 2 # 1#2#3eur . j. * c#1 * , ( # 3 ) # 2 # 1#2#3jhep * # 1 * , ( # 3 ) # 2 # 1#2#3j . * g#1 * , ( # 3 ) # 2 # 1#2#3int . j. mod . # 1 * , ( # 3 ) # 2 # 1#2#3prog . * # 1 * , ( # 3 ) # 2 p. f. harrison , d. h. perkins and w. g. scott , phys . lett . b * 530 * , 167 ( 2002 ) [ arxiv : hep - ph/0202074 ] ; p. f. harrison and w. g. scott , phys . b * 535 * , 163 ( 2002 ) [ arxiv : hep - ph/0203209 ] . x. g. he , y. y. keum and r. r. volkas , jhep * 0604 * , 039 ( 2006 ) [ arxiv : hep - ph/0601001 ] . m. c. gonzalez - garcia , m. maltoni and j. salvado , jhep * 1004 * , 056 ( 2010 ) [ arxiv:1001.4524v3 [ hep - ph ] ] . g. l. fogli , e. lisi , a. marrone , a. palazzo and a. m. rotunno , phys . d * 84 * , 053007 ( 2011 ) [ arxiv:1106.6028 [ hep - ph ] ] . k. nakamura _ et al_. ( particle data group ) , j. phys . g * 37 * , 075021 ( 2010 ) . y. shimizu , m. tanimoto and a. watanabe , prog . phys . * 126 * , 81 ( 2011 ) [ arxiv:1105.2929 [ hep - ph ] ] ; n. qin and b. q. ma , phys . b * 702 * , 143 ( 2011 ) [ arxiv:1106.3284 [ hep - ph ] ] ; y. j. zheng and b. q. ma , arxiv:1106.4040 [ hep - ph ] ; e. ma and d. wegman , phys . lett . * 107 * , 061803 ( 2011 ) [ arxiv:1106.4269 [ hep - ph ] ] ; x. g. he and a. zee , phys . d * 84 * , 053004 ( 2011 ) [ arxiv:1106.4359 [ hep - ph ] ] ; t. araki , phys . d * 84 * , 037301 ( 2011 ) [ arxiv:1106.5211 [ hep - ph ] ] ; s. morisi , k. m. patel and e. peinado , phys . d * 84 * , 053002 ( 2011 ) [ arxiv:1107.0696 [ hep - ph ] ] ; w. chao and y. j. zheng , arxiv:1107.0738 [ hep - ph ] ; s. dev , s. gupta and r. r. gautam , phys . b * 704 * , 527 ( 2011 ) [ arxiv:1107.1125 [ hep - ph ] ] ; r. d. a. toorop , f. feruglio and c. hagedorn , phys . b * 703 * , 447 ( 2011 ) [ arxiv:1107.3486 [ hep - ph ] ] . s. dev , s. gupta and r. r. gautam , phys . b * 704 * , 527 ( 2011 ) [ arxiv:1107.1125 [ hep - ph ] ] ; p. s. bhupal dev , r. n. mohapatra and m. severson , phys . d * 84 * , 053005 ( 2011 ) [ arxiv:1107.2378 [ hep - ph ] ] . n. qin and b. q. ma , phys . d * 83 * , 033006 ( 2011 ) [ arxiv:1101.4729 [ hep - ph ] ] . l. wolfenstein , phys . rev . lett . * 51 * , 1945 ( 1983 ) . y. koide and h. nishiura , phys . d * 79 * , 093005 ( 2009 ) [ arxiv:0811.2839 [ hep - ph ] ] .

harrison , perkins and scott have proposed simple charged lepton and neutrino mass matrices that lead to the tribimaximal mixing @xmath0 . we consider in this work an extension of the mass matrices so that the leptonic mixing matrix becomes @xmath1 , where @xmath2 is a unitary matrix needed to diagonalize the charged lepton mass matrix and @xmath3 measures the deviation of the neutrino mixing matrix from the bimaximal form . hence , corrections to @xmath0 arise from both charged lepton and neutrino sectors . following our previous work to assume a qin - ma - like parametrization @xmath4 for the charged lepton mixing matrix @xmath2 in which the _ cp_-odd phase is approximately maximal , we study the phenomenological implications in two different scenarios : @xmath5 and @xmath6 . we find that the latter is more preferable , though both scenarios are consistent with the data within @xmath7 ranges . the predicted reactor neutrino mixing angle @xmath8 in both scenarios is consistent with the recent t2k and minos data . the leptonic _ cp _ violation characterized by the jarlskog invariant @xmath9 is generally of order @xmath10 .

in highly anisotropic layered superconductors , tilting a magnetic field at a direction oblique to the cuo@xmath1 planes leads to novel states of vortex matter . bitter decoration@xcite , hall probe microscopy and lorentz microscopy have confirmed that the attraction between josephson vortex ( jv ) stacks and pancake vortices ( pv ) in the vortex solid state of bi@xmath1sr@xmath1cacu@xmath1o@xmath2 leads to the so - called crossing - lattice@xcite . this state is only one of many constituting a particularly rich phase diagram . the occurence of other structural phases of vortex matter , _ e.g. _ , the lattice of tilted pv stacks , the combined perpendicular and tilted pv stacks@xcite , or the pv - soliton lattice , depends on the interplay between the magnetic and josephson coupling contributions to the pv lattice tilt modulus @xcite . the josephson plasma resonance ( jpr ) frequency @xmath4 is sensitive to the superconducting phase difference , @xmath5 , between cuo@xmath1 layers @xmath6 , through @xmath7 where @xmath8 stands for thermal and disorder average . since @xmath5 intimately depends on the alignment of pv stacks , jpr can in principle be used to detect and identify different vortex phases . the bi@xmath1sr@xmath1cacu@xmath1o@xmath2 single crystals ( @xmath3 k ) were cut from a larger underdoped bi@xmath1sr@xmath1cacu@xmath1o@xmath2 crystal , grown by the travelling solvent floating zone technique . the jpr was measured using the cavity perturbation technique in the tm@xmath9 modes , with the microwave electrical field aligned along the sample @xmath0-axis . two orthogonal coils were used to apply field components parallel ( @xmath10 ) and perpendicular ( @xmath11 ) to the cuo@xmath1 layers . varying the mode @xmath12 , allowed us to change the jpr frequency and to probe both the vortex solid and the liquid state . magnetic - field dependance of the microwave absorption ( arbitrary units ) of a bi@xmath1sr@xmath1cacu@xmath1o@xmath2 single crystal obtained at different frequencies . ] figure [ fig : dissipation ] shows the microwave absorption obtained by sweeping @xmath11 at constant @xmath10 and temperature . when @xmath13 , the jpr is identified as the maximum of the microwave absorption , for the field @xmath14 ( arrows in figure [ fig : dissipation ] ) . following the evolution of the absorption , the lineshape is modified by @xmath10 in two ways . first , the intensity of the microwave response decreases for all frequencies . it has been pointed out that the presence of a josephson vortex lattice ( jvl ) strongly modulates the _ c_-axis critical current@xcite . thus , the jpr can not develop in the stacks of jvs . however , for low values of the in - plane field ( @xmath15 , where @xmath16 is the anisotropy ratio and @xmath17 the interlayer distance ) , the distance between two stacks of jvs is sufficiently high so that the phase remains unaffected far from the cores . the remaining microwave absorption mainly arises from jv - free regions , the extent of which decreases linearly with @xmath10 . second , @xmath10 changes the field @xmath14 at which the maximum absorption occurs : the decrease of @xmath14 observed in the vortex solid at 31.2 and 39.4 ghz implies the decrease of @xmath18 and is consistent with the addition of a jvl . however , for 19.2 and 22.9 ghz , @xmath14 , measured in the vortex liquid , _ increases_. the @xmath14-loci for different temperatures are shown in fig . [ fig : result ] together with the pv vortex lattice melting line in @xmath13 . the increase of @xmath14 is only observed in the vortex liquid state and for sufficiently low pv densities . in the solid phase , it has been shown@xcite that the addition of pvs on a dense lattice of jvs ( @xmath19 ) increases the @xmath0-axis critical current : the pvs adjust their positions to the jvl so as to maximize @xmath18@xcite . the same effect could be present in the vortex liquid phase , giving rise to an effective attraction between pvs and jvs and to a _ correlated vortex liquid _ in which the density of pvs is smaller in jv - free regions . since the microwave absorption due to the jpr mainly comes from those regions , increasing @xmath10 also increases @xmath14 . however , even though the presence of well - defined josephson vortices in the presence of a pv liquid is still controversial , we believe that it can be the case for low densities of pvs . the correlated vortex liquid is therefore stable close to the pv melting line , for low values of the perpendicular magnetic field .

by measuring the josephson plasma resonance , we have probed the influence of an in - plane magnetic field on the pancake vortex correlations along the @xmath0-axis in heavily underdoped bi@xmath1sr@xmath1cacu@xmath1o@xmath2 ( @xmath3 k ) single crystals both in the vortex liquid and in the vortex solid phase . whereas the in - plane field enhances the interlayer phase coherence in the liquid state close to the melting line , it slightly depresses it in the solid state . this is interpreted as the result of an attractive force between pancake vortices and josephson vortices , apparently also present in the vortex liquid state . the results unveil a boundary between a correlated vortex liquid in which pancakes adapt to josephson vortices , and the usual homogeneous liquid . josephson plasma resonance , bi:2212 , josephson vortex 74.25.qt , 74.50.+r

low - ionization nuclear emission - line regions ( liners ; heckman 1980 ) are found in many nearby bright galaxies ( e.g. , ho , filippenko , & sargent 1997a ) . extensive studies at various wavelengths have shown that type 1 liners ( liner 1s , i.e. , those galaxies having broad h@xmath3 and possibly other broad balmer lines in their nuclear optical spectra ) are powered by a low - luminosity agn ( llagn ) with a bolometric luminosity less than @xmath14 ergs s@xmath2 ( ho et al . 2001 ; terashima , ho , & ptak 2000a ; ho et al . 1997b ) . on the other hand , the energy source of liner 2s is likely to be heterogeneous . some liner 2s show clear signatures of the presence of an agn , while others are most probably powered by stellar processes , and the luminosity ratio @xmath6/@xmath15 can be used to discriminate between these different power sources ( e.g. , prez - olea & colina 1996 ; maoz et al . 1998 ; terashima et al . it is interesting to note that currently there are only a few liner 2s known to host an obscured agn ( e.g. , turner et al . this paucity of obscured agn in liners may indicate that liner 2s are not simply a low - luminosity extension of luminous seyfert 2s , which generally show heavy obscuration with a column density averaging @xmath10 @xmath16 @xmath11 @xmath12 ( e.g. , turner et al . 1997 ) . alternatively , biases against finding heavily obscured llagns may be important . for example , objects selected through optical emission lines or x - ray fluxes are probably biased in favor of less absorbed ones , even if one uses the x - ray band above 2 kev . in contrast , radio observations , particularly at high frequency , are much less affected by absorption . although an optical spectroscopic survey must first be done to find the emission lines characteristic of a llagn , follow up radio observations can clarify the nature of the activity . for example , vlbi observations of some llagns have revealed a compact nuclear radio source with @xmath17 k , which is an unambiguous indicator of the presence of an active nucleus and can not be produced by starburst activity ( e.g. , falcke et al . 2000 ; ulvestad & ho 2001 ) . a number of surveys of seyfert galaxies at sub arcsecond resolution have been made with the vla ( ulvestad & wilson 1989 and references therein ; kukula et al . 1995 ; nagar et al . 1999 ; thean et al . 2000 ; schmitt et al . 2001 ; ho & ulvestad 2001 ) and other interferometers ( roy et al . 1994 ; morganti et al . 1999 ) , but much less work has been done on the nuclear radio emission of liners . nagar et al . ( 2002 ) have reported a vla 2 cm radio survey of all 96 llagns within a distance of 19 mpc . these llagns come from the palomar spectroscopic survey of bright galaxies ( ho et al . 1997a ) . as a pilot study of the x - ray properties of llagns , we report here a _ chandra _ survey of a subset , comprising 15 galaxies , of nagar et al s ( 2002 ) sample . fourteen of these galaxies have a compact nuclear radio core with a flat or inverted radio spectrum ( nagar et al . 2000 ) . we have detected 13 of the galactic nuclei with _ chandra_. we also examine the `` radio loudness '' of our sample and compare it with other classes of agn . a new measure of `` radio loudness '' is developed , in which the 5 ghz radio luminosity is compared with the 210 kev x - ray luminosity ( @xmath9=@xmath18 ( 5 ghz)/@xmath19(210 kev ) ) rather than with the b - band optical luminosity ( @xmath20(5 ghz)/@xmath8(b ) ) , as is usually done . @xmath9 has the advantage that it can be measured for highly absorbed nuclei ( @xmath10 up to several times @xmath11 @xmath12 ) which would be totally obscured ( @xmath21 up to a few hundred mag for the galactic gas to dust ratio ) at optical wavelengths , and that the compact , hard x - ray source in a llagn is less likely to be confused with emission from stellar - powered processes than is an optical nucleus . this paper is organized as follows . the sample , observations , and data reduction are described in section 2 . imaging results and x - ray source detections are given in section 3 . section 4 presents spectral results . the power source , obscuration in llagns , and radio loudness of llagns are discussed in section 5 . section 6 summarizes the findings . we use a hubble constant of @xmath22 km s@xmath2 mpc@xmath2 and a deceleration parameter of @xmath23 throughout this paper . our sample is based on the vla observations by nagar et al . their sample of 48 objects consists of 22 liners , 18 transition objects , which show optical spectra intermediate between liners and nuclei , and eight low - luminosity seyferts selected from the optical spectroscopic survey of ho et al . ( 1997a ) . the sample is the first half of a distance - limited sample of llagns ( nagar et al . 2002 ) , as described in section 1 . we selected 14 objects showing a flat to inverted spectrum radio core ( @xmath24 , @xmath25 ) according to nagar et al.s ( 2000 ) comparison with longer wavelength radio data published in the literature . one object ( the liner 2 ngc 4550 ) has a flat spectrum radio source at a position significantly offset from the optical nucleus . this object was added as an example of a liner without a detected radio core . the target list and some basic data for the final sample are summarized in table 1 . the distances are taken from tully ( 1988 ) in which @xmath22 km s@xmath2 mpc@xmath2 is assumed . the sample consists of seven liner 1s , four liner 2s , two seyfert 1s , one seyfert 2 , and one transition 2 object . 12 out of these 15 objects have been observed with the vlba and high brightness temperature ( @xmath26 k ) radio cores were detected in all of them ( falcke et al . 2000 ; ulvestad & ho 2001 ; nagar et al . therefore , these objects are strong candidates for agns . results of archival / scheduled _ chandra _ observations of more liners with a compact flat / inverted spectrum radio core found by nagar et al . ( 2002 ) will be presented in a future paper . a log of _ chandra _ observations is shown in table 1 . the exposure time was typically two ksec each . all the objects were observed at or near the aim point of the acis - s3 back - illuminated ccd chip . eight objects were observed in our guaranteed time observation program and the rest of the objects were taken from the _ chandra _ archives . the eight objects were observed in 1/8 sub - frame mode ( frame time 0.4 s ) to minimize effects of pileup ( e.g. , davis 2001 ) . 1/2 sub - frame modes were used for three objects . ciao 2.2.1 and caldb 2.7 were used to reduce the data . in the following analysis , only events with _ asca _ grades 0 , 2 , 3 , 4 , 6 ( `` good grades '' ) were used . for spectral fitting , xspec version 11.2.0 was employed . an x - ray nucleus is seen in all the galaxies except for ngc 4550 and ngc 5866 . some off - nuclear sources are also seen in some fields . the source detection algorithm `` wavdetect '' in the ciao package was applied to detect these nuclear and off - nuclear sources , where a detection threshold of @xmath27 and wavelet scales of 1 , @xmath28 , 2 , @xmath29 , 4 , @xmath30 , 8 , @xmath31 , and 16 pixels were used . source detections were performed in the three energy bands 0.58 kev ( full band ) , 0.52 kev ( soft band ) , and 28 kev ( hard band ) . the resulting source lists and raw images were examined by eye to exclude spurious detections . the source and background counts were also determined by manual photometry and compared with the results of wavdetect . in the few cases that the two methods gave discrepant results , we decided to use the results of the manual photometry after inspection of the raw images . some sources were detected in only one or two energy bands . in such cases , we calculated the upper limits on the source counts in the undetected band(s ) at the 95% confidence level by interpolating the values in table 2 of kraft , burrows , & nousek ( 1991 ) . table 2 shows the positions , detected counts , band ratios ( hard / soft counts ) , fluxes and luminosities in the 210 kev band of the nuclear sources . the same parameters for off - nuclear sources with signal - to - noise ratios greater than three are summarized in table 3 . for bright objects ( @xmath32 40 counts ) , the fluxes were measured by spectral fits presented in the next section . for faint objects , fluxes were determined by assuming the galactic absorption column density and power law spectra , with photon indices determined from the band ratios . when only lower or upper limits on the band ratio were available , a photon index of 2 was assumed if the limit is consistent with @xmath33 . when the band ratio is inconsistent with @xmath33 , the upper or lower limit is used to determine the photon index . luminosities were calculated only if the source is spatially inside the optical host galaxy as indicated by comparing the position with the optical image of the digitized sky survey . possible identifications for off - nuclear sources are also given in the last column of table 3 . the positions of the x - ray nuclei coincide with the radio core positions to within the positional accuracy of _ chandra_. the nominal separations between the x - ray and radio nuclei are in the range 0.05 0.95@xmath34 . inspection of the images shows that the nucleus in most objects appears to be unresolved , while some objects show faint extended emission . the soft and hard band images of the nuclear regions of ngc 3169 and ngc 4278 are shown in fig 1 as examples of extended emission . in the soft band image of ngc 3169 , emission in the nuclear region extending @xmath35 arcsec in diameter is clearly visible , while the nucleus itself is not detected ( table 2 ) . about 25 counts were detected within a 10 pixel ( 4.9 arcsec ) radius in the @xmath36 kev band . this extended emission is not seen in the hard band . the soft band image of ngc 4278 consists of a bright nucleus and a faint elongated feature with a length of @xmath37 and a position angle of @xmath38 . the hard band image is unresolved . the nuclear regions ( 10 arcsec scale ) of the other galaxies look compact to within the current photon statistics . more extended diffuse emission at larger scales ( @xmath39 ) is seen in a few objects . the nuclei of ngc 2787 and ngc 4203 are embedded in soft diffuse emission with diameters of @xmath40 and @xmath41 , respectively . ngc 4579 shows soft diffuse emission with a similar morphology to the circumnuclear star forming ring , in addition to a very bright nucleus ( see also eracleous et al . ngc 4565 shows extended emission along the galactic plane . ngc 5866 has soft extended emission @xmath42 ( 2 kpc ) in diameter and no x - ray nucleus is detected ( table 2 ) . any diffuse emission associated with the other galaxies is much fainter . spectral fits were performed for the relatively bright objects those with @xmath32 40 detected counts in the 0.58 kev band . the spectrum of one fainter object ( ngc 4548 ) showing a large ( = hard ) hardness ratio was also fitted . some objects are so bright that pileup effects are significant . column 7 of table 1 gives the count rates per ccd read - out frame time and can be used to estimate the significance of pileup . in the two objects ngc 3147 and ngc 4278 , the pileup is mild and we corrected for the effect by applying the pileup model implemented in xspec , where the grade morphing parameter @xmath3 was fixed at 0.5 ( after initially treating it as a free parameter [ davis 2001 ] since @xmath3 is not well constrained ) . the pileup effect for the three objects with the largest count rate per frame ( ngc 4203 , ngc 4579 , and ngc 5033 ) is serious and we did not attempt detailed spectral fits . instead , we use the spectra and fluxes measured with _ asca _ for these three objects ( terashima et al . 2002b and references therein ) in the following discussions . we confirmed that the nuclear x - ray source dominates the hard x - ray emission within the beam size of _ asca _ ( see appendix ) . the other objects in the sample are faint enough to ignore the effects of pileup . x - ray spectra were extracted from a circular region with a radius between 4 pixels ( 2.0@xmath34 ; for faint sources ) and 10 pixels ( 4.9@xmath34 ; for bright sources ) depending on source brightness . background was estimated using an annular region centered on the target . a maximum - likelihood method using the c - statistic ( cash 1979 ) was employed in the spectral fits . in the fit with the c - statistic , background can not be subtracted , so we added a background model ( measured from the background region ) with fixed parameters to the spectral models , after normalizing by the ratio of the geometrical areas of the source and background regions . the errors quoted represent the 90% confidence level for one parameter of interest ( @xmath43=2.7 ) . a power - law model modified by absorption was applied and acceptable fits were obtained in all cases ( fig . the best - fit parameters for the nuclear sources are given in table 4 . the observed fluxes and luminosities ( the latter corrected for absorption ) in the 210 kev band are shown in table 2 . the results for a few bright off - nuclear sources are presented in the appendix . the photon indices of the nuclear sources are generally consistent with the typical values observed in llagns ( photon index @xmath44 , e.g. , terashima et al . 2002a , 2002b ) , although errors are quite large due to the limited photon statistics . the spectral slope of ngc 6500 ( @xmath45 ) is somewhat steeper than is typical of llagns . this may indicate that there is soft x - ray emission from a source other than the agn and/or the intrinsic slope of the agn is steep . one galaxy ( ngc 3169 , a liner 2 ) has a large absorption column ( @xmath10=@xmath46 @xmath12 ) , while two galaxies ( ngc 4548 , a liner 2 , and ngc 3226 , a liner 1.9 ) show substantial absorption ( @xmath47 @xmath12 , and @xmath48 @xmath12 , respectively ) . others have small column densities which are consistent with ` type 1 ' agns . no meaningful limit on the equivalent width of an fe k@xmath3 line was obtained for any of the objects because of limited photon statistics in the hard x - ray band . one object ngc 2787 has only 8 detected photons in the 0.58 kev band and is too faint to obtain spectral information . a photon index of 2.0 and the galactic absorption of @xmath49 @xmath12 were assumed to calculate the flux and luminosity which are shown in table 2 . an x - ray nucleus is detected in all the objects except for ngc 4550 and ngc 5866 . we test whether the detected x - ray sources are the high energy extension of the continuum source which powers the optical emission lines by examining the luminosity ratio @xmath6/@xmath15 . the h@xmath3 luminosities ( @xmath15 ) were taken from ho et al . ( 1997a ) and the reddening was estimated from the balmer decrement for narrow lines and corrected using the reddening curve of cardelli , clayton , & mathis ( 1989 ) , assuming the intrinsic h@xmath3/h@xmath50 flux ratio = 3.1 . the x - ray luminosities ( corrected for absorption ) in the @xmath51 kev band are used . the h@xmath3 luminosities and logarithm of the luminosity ratios @xmath6/@xmath15 are shown in table 5 . the @xmath6/@xmath15 ratios of most objects are in the range of agns ( @xmath52 @xmath6/@xmath15 @xmath16 12 ) and in good agreement with the strong correlation between @xmath6 and @xmath15 for llagns , luminous seyferts , and quasars presented in terashima et al . ( 2000a ) and ho et al . this indicates that their optical emission lines are predominantly powered by a llagn . note that this correlation is not an artifact of distance effects , as shown in terashima et al . ( 2000a ) . the four objects ngc 2787 , ngc 4550 , ngc 5866 , and ngc 6500 , however , have much lower @xmath6/@xmath15 ratios ( @xmath52 @xmath6/@xmath15 .5ex0 ) than expected from the correlation ( @xmath52 @xmath6/@xmath15 @xmath53 ) , and their x - ray luminosities are insufficient to power the h@xmath3 emission ( terashima et al . this x - ray faintness could indicate one or more of several possibilities such as ( 1 ) an agn is the power source , but is heavily absorbed at energies above 2 kev , ( 2 ) an agn is the power source , but is currently switched - off or in a faint state , and ( 3 ) the optical narrow emission lines are powered by some source(s ) other than an agn . we briefly discuss these three possibilities in turn . if an agn is present in these x - ray faint objects and absorbed in the hard energy band above 2 kev , only scattered and/or highly absorbed x - rays would be observed , and then the intrinsic luminosity would be much higher than that observed . this can account for the low @xmath6/@xmath15 ratios and high radio to x - ray luminosity ratios ( @xmath54(5 ghz)/@xmath6 ; table 5 and section 5.3 ) . if the intrinsic x - ray luminosities are about one or two orders of magnitude higher than those observed , as is often inferred for seyfert 2 galaxies ( turner et al . 1997 , awaki et al . 2000 ) , @xmath6/@xmath15 and @xmath54(5 ghz)/@xmath6 become typical of llagns . alternatively , the agn might be turned off or in a faint state , with a higher activity in the past being inferred from the optical emission lines , whose emitting region is far from the nucleus ( e.g. , eracleous et al . also , the radio observations were made a few years before the _ chandra _ ones . this scenario might thus explain their relatively low @xmath6/@xmath15 ratios and their relatively high @xmath55/@xmath6 ratios . if this is the case , the size of the radio core can be used to constrain the era of the active phase in the recent past . the upper limits on the size of the core estimated from the beam size ( @xmath56 2.5 mas ) are 0.16 , 0.19 , and 0.48 pc for ngc 2787 , ngc 5866 , and ngc 6500 , respectively ( falcke et al . therefore , the agn must have been active until @xmath570.52 , @xmath570.60 , and @xmath571.6 years , respectively , before the vlba observations ( made in 1997 june ) and inactive at the epochs ( 2000 jan 2002 jan , see table 1 ) of the x - ray observations . this is an ad hoc proposal and such abrupt declines of activity are quite unusual , but it can not be completely excluded . it may also be possible that the ionized gas inferred from the optical emission lines is ionized by some sources other than an agn , such as hot stars . if the observed x - rays reflect the intrinsic luminosities of the agn , a problem with the agn scenario for the three objects ngc 2787 , ngc 5866 , and ngc 6500 is that these galaxies have very large @xmath18(5 ghz)/@xmath6 ratios , and would thus be among the radio loudest llagns . the presence of hot stars in the nuclear region of ngc 6500 is suggested by uv spectroscopy ( maoz et al . maoz et al . ( 1998 ) studied the energy budget for ngc 6500 by using the h@xmath3 and uv luminosity at 1300 a and showed that the observed uv luminosity is insufficient to power the h@xmath3 luminosity even if a stellar population with the salpeter initial mass function and a high mass cutoff of 120@xmath58 are assumed . this result indicates that a power source in addition to hot stars must contribute significantly , and supports the obscured agn interpretation discussed above . the first possibility , i.e. , an obscured low - luminosity agn as the source of the x - ray emission , seems preferable for ngc 2787 , ngc 5866 and ngc 6500 , although some other source(s ) may contribute to the optical emission lines . additional lines of evidence which support the presence of an agn include the fact that all three of these galaxies ( ngc 2787 , ngc 5866 , and ngc 6500 ) have vlbi - detected , sub - pc scale , nuclear radio core sources ( falcke et al . 2000 ) , a broad h@xmath3 component ( in ngc 2787 , and an ambiguous detection in ngc 5866 ; ho et al . 1997b ) , a variable radio core in ngc 2787 , and a jet - like linear structure in a high - resolution radio map of ngc 6500 with the vlba ( falcke et al . only an upper limit to the x - ray flux is obtained for ngc 5866 . if an x - ray nucleus is present in this galaxy and its luminosity is only slightly below the upper limit , this source could be an agn obscured by a column density @xmath10@xmath59 @xmath12 or larger . if the apparent x - ray luminosity of the nucleus of ngc 5866 is _ much _ lower than the observed upper limit , and the intrinsic x - ray luminosity conforms to the typical @xmath6/@xmath15 ratio for llagn ( @xmath52 @xmath6/@xmath15 @xmath60 ) , then the x - ray source must be almost completely obscured . the optical classification ( transition object ) suggests the presence of an ionizing source other than an agn , so the low observed @xmath6/@xmath15 ratio could alternatively be a result of enhanced h@xmath3 emission powered by this other ionizing source . the x - ray results presented above show that the presence of a flat ( or inverted ) spectrum compact radio core is a very good indicator of the presence of an agn even if its luminosity is very low . on the other hand , ngc 4550 , which does not possess a radio core , shows no evidence for the presence of an agn and all the three possibilities discussed above are viable . if the _ rosat _ detection is real ( halderson et al . 2001 ) , the time variability between the _ rosat _ and _ chandra _ fluxes may indicate the presence of an agn ( see appendix ) . it is notable that type 2 liners without a flat spectrum compact radio core may be heterogeneous in nature . for instance , some liner 2s without a compact radio core ( e.g. , ngc 404 and transition 2 object ngc 4569 ) are most probably driven by stellar processes ( maoz et al . 1998 ; terashima et al . 2000b ; eracleous et al . 2002 ) . in our sample , we found at least three highly absorbed llagns ( ngc 3169 , ngc 3226 , and ngc 4548 ) . in addition , if the x - ray faint objects discussed in section 5.1 are indeed agns , they are most probably highly absorbed with @xmath10@xmath61 @xmath12 . among these absorbed objects , ngc 2787 is classified as a liner 1.9 , ngc 3169 , ngc 4548 , and ngc 6500 as liner 2s , and ngc 5866 as a transition 2 object . thus , heavily absorbed liner 2s , of which few are known , are found in the present observations demonstrating that radio selection is a valuable technique for finding obscured agns . along with heavily obscured llagns known in low - luminosity seyfert 2s ( e.g. , ngc 2273 , ngc 2655 , ngc 3079 , ngc 4941 , and ngc 5194 ; terashima et al . 2002a ) , our observations show that at least some type 2 llagns are simply low - luminosity counterparts of luminous seyferts in which heavy absorption is often observed ( e.g. , risaliti , maiolino , & salvati 1999 ) . however , some liner 2s ( e.g. , ngc 4594 , terashima et al . 2002a ; ngc 4374 , finoguenov & jones 2001 ; ngc 4486 , wilson & yang 2002 ) and low - luminosity seyfert 2s ( ngc 3147 ; section 4 and appendix ) show no strong absorption . therefore , the orientation - dependent unified scheme ( e.g. , antonucci 1993 ) does not always apply to agns in the low - luminosity regime , as suggested by terashima et al . . combination of x - ray and radio observations is valuable for investigating a number of areas of agn physics , including the `` radio loudness '' , the origin of jets , and the structure of accretion disks . low - luminosity agns ( liners and low - luminosity seyfert galaxies ) are thought to be radiating at very low eddington ratios ( @xmath62/@xmath63 ) and may possess an advection - dominated accretion flow ( adaf ; see e.g. , quataert 2002 for a recent review ) . a study of radio loudness in llagns can constrain the jet production efficiency by an adaf - type disk . earlier studies have suggested that llagns tend to be radio loud compared to more luminous seyferts based on the spectral energy distributions of seven llagns ( ho 1999 ) and , for a larger sample , on the conventional definition of radio loudness @xmath64(5 ghz)/@xmath8(b ) ( the subscript `` o '' , which stands for optical , is usually omitted but we use it here to distinguish from @xmath9 see below ) , with @xmath65 being radio loud ( kellermann et al . 1989 , 1994 ; visnovsky et al . 1992 ; stocke et al . 1992 ; ho & peng 2001 ) . ho & peng ( 2001 ) measured the luminosities of the nuclei by spatial analysis of optical images obtained with _ hst _ to reduce the contribution from stellar light . a caveat in the use of optical measurements for the definition of radio loudness is extinction , which will lead to an overestimate of @xmath66 if not properly allowed for . although ho & peng ( 2001 ) used only type 11.9 objects , some objects of these types show high absorption columns in their x - ray spectra . in this subsection , we study radio loudness by comparing radio and hard x - ray luminosities . since the unabsorbed luminosity for objects with @xmath10 .5ex@xmath11 @xmath12 can be reliably measured in the 210 kev band , which is accessible to _ asca _ , _ xmm - newton _ , and _ chandra _ , and such columns correspond to @xmath21 .5ex50 mag , it is clear that replacement of optical by hard x - ray luminosity potentially yields considerable advantages . in addition , the high spatial resolutions of _ xmm - newton _ and especially _ chandra _ usually allow the nuclear x - ray emission to be identified unambiguously , while the optical emission of llagn can be confused by surrounding starlight . in the following analysis , radio data at 5 ghz taken from the literature are used since fluxes at this frequency are widely available for various classes of objects . we used primarily radio luminosities obtained with the vla at .5ex@xmath67 resolution for the present sample . high resolution vla data at 5 ghz are not available for several objects . for four such cases , vlba observations at 5 ghz with 150 mas resolution are published in the literature ( falcke et al . 2000 ) and are used here . for two objects , we estimated 5 ghz fluxes from 15 ghz data by assuming a spectral slope of @xmath68 ( cf . nagar et al . the radio luminosities used in the following analysis are summarized in table 5 . since our sample is selected based on the presence of a compact radio core , the sample could be biased to more radio loud objects . therefore , we constructed a larger sample by adding objects taken from the literature for which 5 ghz radio , 210 kev x - ray , and @xmath69 measurements are available . first , we introduce the ratio @xmath70(5 ghz)/@xmath6 as a measure of radio loudness and compare the ratio with the conventional @xmath69 parameter . the x - ray luminosity @xmath6 in the 210 kev band ( source rest frame ) , corrected for absorption , is used . , which utilizes monochromatic b - band luminosities . this alternative provides completely identical results if the x - ray spectral shape is known and the range of spectral slopes is not large . for example , the conversion factor @xmath8(2 kev)/@xmath6 is 0.31 , 0.26 , and 0.22 kev@xmath2 for photon indices of 2 , 1.8 , and 1.6 , respectively , and no absorption . ] we examine the behavior of @xmath9 using samples of agn over a wide range of luminosity , including llagn , the seyfert sample of ho & peng ( 2001 ) and pg quasars which are also used in their analysis . @xmath69 parameters and radio luminosities were taken from ho & peng ( 2001 ) for the seyferts and kellermann et al . ( 1989 ) for the pg sample . the values of @xmath69 in kellermann et al . ( 1989 ) have been recalculated by using only the core component of the radio luminosities . the optical and radio luminosities of the pg quasars were calculated assuming @xmath71 and @xmath72 ( @xmath73 ) . the x - ray luminosities ( mostly measured with _ asca _ ) were compiled from terashima et al . ( 2002b ) , weaver , gelbord , & yaqoob ( 2001 ) , george et al . ( 2000 ) , reeves & turner ( 2000 ) , iwasawa et al . ( 1997 , 2000 ) , sambruna , eracleous , & mushotzky ( 1999 ) , nandra et al . ( 1997 ) , smith & done ( 1996 ) , and cappi et al . note that only a few objects ( ngc 4565 , ngc 4579 , and ngc 5033 ) in our radio selected sample have reliable measurements of nuclear @xmath8(b ) . 3 . compares the parameters @xmath69 and @xmath9 for the seyferts and pg sample . these two parameters correlate well for most seyferts . some seyferts have higher @xmath69 values than indicated by most seyferts . this could be a result of extinction in the optical band . seyferts showing x - ray spectra absorbed by a column greater than @xmath74 @xmath12 ( ngc 2639 , 4151 , 4258 , 4388 , 4395 , 5252 , and 5674 ) are shown as open circles in fig . at least four of them have a value of @xmath69 larger than indicated by the correlation . the correlation between @xmath75 and @xmath76 for the less absorbed seyferts can be described as @xmath75 = 0.88 @xmath77 + 5.0 . according to this relation , the boundary between radio loud and radio quiet object ( @xmath75 = 1 ) corresponds to @xmath78 . the values of @xmath69 and @xmath9 for a few obscured seyferts are consistent with the correlation , indicating that optical extinction is not perfectly correlated with the absorption column density inferred from x - ray spectra . the pg quasars show systematically lower @xmath69 values than those of seyferts at a given @xmath76 . for the former objects , @xmath79 corresponds to @xmath80 . this apparently reflects a luminosity dependence of the shape of the sed : luminous objects have steeper optical - x - ray slopes @xmath81 ( @xmath82 ; e.g. , elvis et al . 1994 , brandt , laor , & wills 2000 ) , where @xmath83 is often measured as the spectral index between 2200 a and 2 kev , while less luminous agns have @xmath84 ( ho 1999 ) . this is related to the fact that luminous objects show a more prominent `` big blue bump '' in their spectra . fig . 8 of ho ( 1999 ) demonstrates that low - luminosity objects are typically 11.5 orders of magnitude fainter in the optical band than luminous quasars for an given x - ray luminosity . note that none of the pg quasars used here shows a high absorption column in its x - ray spectrum below 10 kev . the definition of radio loudness using the hard x - ray flux ( @xmath85 ) appears to be more robust than that using the optical flux because x - rays are less affected by both extinction at optical wavelengths and the detailed shape of the blue bump , as noted above . further , measurements of nuclear x - ray fluxes of seyferts and llagns with _ chandra _ are easier than measurements of nuclear optical fluxes , since in the latter case the nuclear light must be separated from the surrounding starlight , a difficult process for llagns . 4 shows the x - ray luminosity dependence of @xmath9 . in this plot , the llagn sample discussed in the present paper is shown in addition to the seyfert and pg samples used above . this is an `` x - ray version '' of the @xmath75-@xmath86 plot ( fig . 4 in ho & peng 2001 ) . radio galaxies taken from sambruna et al . ( 1999 ) are also added and we use radio luminosities from the core only . our plot shows that a large fraction ( @xmath87% ) of llagns ( @xmath6@xmath88 ergs s@xmath2 ) are `` radio loud '' . this is a confirmation of ho & peng s ( 2001 ) finding . note , however , that our sample is not complete in any sense , and this radio - loud fraction should be measured using a more complete sample . since radio emission in llagns is likely to be dominated by emission from jets ( nagar et al . 2001 ; ulvestad & ho 2001 ) , these results suggest that , in llagn , the fraction of the accretion energy that powers a jet , as opposed to electromagnetic radiation , is larger than in more luminous seyfert galaxies and quasars . since llagns are thought to have an adaf - type accretion flow , such might indicate that an adaf can produce jets more efficiently than the geometrically thin disk believed present in more luminous seyferts . the three llagns with the largest @xmath9 in fig . 4 are the three x - ray faint objects discussed in section 5.1 ( ngc 2787 , ngc 5866 , and ngc 6500 ) and which are most probably obscured agns . if their intrinsic x - ray luminosities are 12 orders of magnitude higher than those observed , their values of @xmath9 become smaller by this factor and are then in the range of other llagns . even if we exclude these three llagns , the radio loudness of llagns is distributed over a wide range : the radio - loudest llagns have @xmath9 values similar to radio galaxies and radio - loud quasars , while some llagns are as radio quiet as radio - quiet quasars . a comparison with blazars is of interest to compare our sample with objects for which the nuclear emission is known to be dominated by a relativistic jet and thus strongly beamed . the average @xmath89 for high - energy peaked bl lac objects ( hbls ) , low - energy peaked bl lac objects ( lbls ) , and flat spectrum radio quasars ( fsrqs ) are 3.10 , 1.27 , and 0.95 , respectively , where we used the average radio and x - ray luminosities for a large sample of blazars given in table 3 of donato et al . the average @xmath76 for hbls is similar to that for llagns in our sample , while the latter two classes are about two orders of magnitude more radio loud than llagns . although llagns and hbls have similar values of @xmath89 , the spectral slope in the x - ray band is different : llagns have a photon index in the range 1.72.0 ( see also terashima et al . 2002 ) , while hbls usually show steeper spectra ( photon index @xmath32 2 , e.g. , fig . 1 in donato et al . 2001 ) , and the x - ray emission is believed to be dominated by synchrotron radiation . furthermore , blazars with a lower bolometric luminosity tend to have a synchrotron peak at a higher frequency and a steeper x - ray spectral slope than higher bolometric luminosity blazars ( donato et al . 2001 ) . we also constructed an @xmath9-@xmath6 plot ( fig . 5 ) using the _ total _ radio luminosities of the radio source ( i.e. including the core , jets , lobes , and hot spots , if present ) . the radio data were compiled from vron - cetty & vron ( 2001 ) , kellermann et al . ( 1989 ) , and sambruna et al . the pg sample and other quasars are shown with different symbols . this plot appears similar to fig . 4 for llagns , seyferts , and radio - quiet quasars since these objects do not possess powerful jets or lobes and off - nuclear radio emission associated with the agn is generally of low luminosity ( ulvestad & wilson 1989 , nagar et al . 2001 , ho & ulvestad 2001 , kellermann et al . 1989 ) . on the other hand , radio galaxies have powerful extended radio emission and consequently the @xmath9 values calculated using the total radio luminosities become higher than if only nuclear luminosities are used . we used the same x - ray luminosities as in fig . 4 , because jets , lobes , and hot spots are almost always much weaker than the nucleus in x - rays . in fact , in our observations of llagns , we found no extended emission directly related to the agn . thus , the differences between fig . 4 and fig . 5 result from the extended radio emission . fourteen galaxies with a nuclear radio source having a flat or inverted spectrum have been observed with _ chandra _ with a typical exposure time of 2 ksec . an x - ray nucleus is detected in all but one object ( ngc 5866 ) . 11 galaxies have x - ray and h@xmath3 luminosities in good accord with the correlation known for agns over a wide range of luminosity , which indicates that these objects are agns and that the agn is the dominant power source of their optical emission lines . their x - ray luminosities are between @xmath0 and @xmath1 ergs s@xmath2 . the three objects ngc 2787 , ngc 5866 , and ngc 6500 have significantly lower x - ray luminosities than expected from the @xmath6-@xmath15 correlation . various observations suggest that these objects are most likely to be heavily obscured agns . these observational results show that radio and hard x - ray observations provide an efficient way to find llagn in nearby galaxies , even if the nuclei are heavily obscured . one object ( the liner 2 ngc 4550 ) , which does not show a radio core , was also observed for comparison . no x - ray nucleus is detected . if the x - ray source detected in this galaxy with _ rosat _ is indeed the nucleus , the nucleus must be variable in x - rays , which would indicate the presence of an agn . we have used the ratio @xmath90(5 ghz)/@xmath6 as a measure of radio loudness and found that a large fraction of llagns are radio loud . this confirms earlier results based on nuclear luminosities in the optical band , but our results based on hard x - ray measurements are much less affected by obscuration and the detailed shape of the `` big blue bump '' . we speculate that the increase in @xmath9 as @xmath6 decreases below @xmath91 ergs s@xmath2 may result from the presence of an advection - dominated accretion flow in the inner part of the accretion flow in low - luminosity objects . however , the steep x - ray spectra in our sample of llagns rule out high temperature thermal bremsstrahlung as the x - ray emission mechanism . is supported by the japan society for the promotion of science postdoctoral fellowship for young scientists . this research was supported by nasa through grants nag81027 and nag81755 to the university of maryland . in this appendix , we compare our results with previously published results particularly in the hard x - ray band obtained with _ asca _ and _ chandra_. the optical spectroscopic classification is given in parentheses after the object name . _ ngc 3147 ( s2)_. this object was observed with _ asca _ in 1993 september and the observed flux was @xmath92 ergs s@xmath2@xmath12 in the 210 kev band ( ptak et al . 1996 , 1999 ; terashima et al . our _ chandra _ image is dominated by the nucleus and shows that the off - nuclear source contribution within the _ asca _ beam is negligible . therefore , a comparison between the observed _ chandra _ flux ( @xmath93 ergs s@xmath2@xmath12 ) , which is 2.3 times larger than that of _ asca _ , implies time variability providing additional evidence for the presence of an agn . in the _ asca _ spectrum , a strong fe - k emission line is detected at @xmath94 kev ( source rest frame ) with an equivalent width of @xmath95 ev . one interpretation of this relatively large equivalent width is that the nucleus is obscured by a large column density and the observed x - rays are scattered emission ( ptak et al . 1996 ) . however , the luminosity ratios @xmath6/@xmath15 and @xmath6/@xmath96\lambda 5007}$ ] suggest small obscuration ( terashima et al . 2002b ) . the observed variability supports the interpretation that the x - ray emission is not scattered emission from a heavily obscured nucleus . this galaxy is an example of a seyfert 2 with only little absorption in the x - ray band . _ ngc 3226 ( l1.9)_. this galaxy was observed with the _ chandra _ hetg in 1999 december ( george et al . they obtained an intrinsic luminosity of @xmath97 ( @xmath98 ergs s@xmath2 , 68% confidence limit ) in the 210 kev band after conversion to a distance of 23.4 mpc . this luminosity is consistent with our value of @xmath99 ( @xmath100 ergs s@xmath2 , 90% confidence range ) after correction for absorption . _ ngc 4203 ( l1.9)_. a result on the same data set is presented in ho et al . the nucleus of this object has a large x - ray flux and pileup is severe in this observation . a bright source is seen 2@xmath101 se of the nucleus which was also separated from the nucleus with _ asca _ sis observations ( iyomoto et al . 1998 ; terashima et al . the _ chandra _ observations show that there is no source confusing the _ asca _ observation of the nucleus . therefore , we used an _ asca _ flux in the discussions . we analyzed archival _ data observed on 1998 may 24 . the effective exposure times after standard data screening were 19.6 ksec for each sis and 23.9 ksec for each gis . spectrum is well fitted with a power law with a photon index 1.85 ( 1.77@xmath1021.94 ) . the best - fit absorption column is @xmath10=0 , with an upper limit of @xmath103 @xmath12 . the observed flux in the 2@xmath10210 kev band is @xmath104 ergs s@xmath2@xmath12 . our _ chandra _ image in the hard energy band is dominated by the nucleus and no bright source is seen in the field . therefore , the hard x - ray measurement with _ asca _ seems reliable . the _ chandra _ flux in the 210 kev band ( @xmath105 ergs s@xmath2@xmath12 ) is about one - third of the _ asca _ flux indicating variability . _ ngc 4550 ( l2)_. this source is not detected with the present _ chandra _ observation . a detection with the _ rosat _ pspc is reported by halderson et al . the observed _ rosat _ flux in the 0.12.4 kev band is @xmath106 ergs s@xmath2 @xmath12 . rosat _ source is offset from the optical nucleus by 10@xmath34 . if this source is indeed the nucleus , our non detection by _ ( @xmath107 ergs s@xmath2 @xmath12 ) indicates time variability . _ ngc 4565 ( s1.9)_. the nuclear region is dominated by two sources : the nucleus and an off - nuclear source which is brighter than the nucleus . the observed _ chandra _ fluxes of these two source in the 210 kev band ( @xmath108 and @xmath109 ergs s@xmath2@xmath12 ) are slightly lower than those obtained with _ asca _ ( @xmath110 and @xmath111 ergs s@xmath2@xmath12 ; mizuno et al . 1999 ; terashima et al . 2002b ) , respectively . these differences appear not to be significant given the statistical , calibration , and spectral - modeling uncertainties . ( the uncertainties on the _ chandra _ fluxes are dominated by the statistical errors , which are @xmath112 % for the off - nuclear source and @xmath1650% for the nucleus , while the error in the _ asca _ fluxes is dominated by calibration uncertainties of @xmath35 % . ) the _ chandra _ spectrum of the off - nuclear source can be fitted by an absorbed power law model with a photon index of @xmath113 and @xmath10 = @xmath114 @xmath12 . a multicolor disk blackbody model also provides a good fit with best - fit parameters @xmath115 kev and @xmath10 = @xmath116 @xmath12 . _ ngc 4579 ( l1.9/s1.9)_. a result on the same data set is presented by ho et al . the nucleus is significantly piled up in the _ chandra _ observation . hard band image is dominated by the nucleus and no bright source is seen in the field . therefore , we used _ fluxes observed in 1995 and 1998 . detailed _ asca _ results are published in terashima et al . ( 1998 , 2000c ) . a long ( 33.9 ksec exposure ) _ chandra _ observation performed in 2000 may is presented in eracleous et al . the 210 kev flux reported is @xmath117 ergs s@xmath2@xmath12 which is similar to that of the second _ asca _ observation in 1998 ( @xmath118 ergs s@xmath2@xmath12 ) . _ ngc 5033 ( s1.5)_. a result on the same data set is presented in ho et al . the nucleus is significantly piled up in the _ chandra _ observation . the five off - nuclear sources shown in table 3 are located within the _ asca _ beam . the sum of the counts from these sources is less than 44 counts in the 28 kev band , while 380 counts are detected from the nucleus before correction for pileup . therefore , the _ asca _ flux ( @xmath119 ergs s@xmath2@xmath12 ; terashima et al . 1999 , 2002b ) is probably larger than the true nuclear flux by @xmath120% or less , unless the off - nuclear sources show drastic time variability . we used the _ asca _ flux without any correction for the off - nuclear source contribution . the 10% uncertainty does not affect any of the conclusions . we performed a spectral fit to the brightest off - nuclear source ( cxou j131329.7 + 363523 ) . an absorbed power law model was applied , and @xmath10 = 0.40 ( @xmath571.2 ) @xmath121 @xmath12 and a photon index @xmath122 were obtained . _ ngc 5866 ( t2)_. extended emission of diameter @xmath56 30@xmath34 ( @xmath1232 kpc ) is seen . the spectrum of this emission may be represented by a mekal plasma model with @xmath124 1 kev and abundance of 0.15 solar . this component could be identified with a gaseous halo of this s0 galaxy . falcke , h. , lehr , j. , barvainis , r. , nagar , n. m. , & wilson , a. s. 2001 , probing the physics of active galactic nuclei by multiwavelength monitoring , eds . b. m. peterson , r. w. pogge , & r. s. polidan , asp conf . series 224 , p.265 , ( asp : san francisco ) ptak , a. , yaqoob , t. , serlemitsos , p.j . , kunieda , h. , & terashima , y. 1996 , , 459 , 542 quataert , e. 2002 , `` probing the physics of active galactic nuclei '' , eds . b. m. peterson , r. w. pogge , and r. s. polidan , ( san francisco : astronomical society of the pacific ) , asp conference proceedings , vol . 224 , p.71 terashima , y. , ho , l .c . , & ptak , a. f. 2000a , , 539 , 161 terashima , y. , ho , l .c . , ptak , a. f. , mushotzky , r. f. , serlemitsos , p. j. , yaqoob , t. , & kunieda , h. 2000b , , 533 , 729 terashima , y. , ho , l .c . , ptak , a. f. , yaqoob , t. , kunieda , h. , misaki , k. , & serlemitsos , p. j. 2000c , , 535 , l79 terashima , y. , iyomoto , n. , ho , l. c. , & ptak , a. f. 2002b , , 139 , 1 cccccccc name & @xmath125 & class & date & exposure & & notes + & ( mpc ) & & & ( s ) & ( s@xmath2 ) & ( frame@xmath2 ) & + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) + ngc 266 & 62.4 & l1.9 & 2001 jun 1 & 2033 & 0.020 & 0.0080 & a + ngc 2787 & 13.3 & l1.9 & 2000 jan 7 & 1050 & 0.0075 & 0.024 & c + ngc 3147 & 40.9 & s2 & 2001 sep 19 & 2202 & 0.54 & 0.21 & a + ngc 3169 & 19.7 & l2 & 2001 may 2 & 1953 & 0.081 & 0.033 & a + ngc 3226 & 23.4 & l1.9 & 2001 mar 23 & 2228 & 0.094 & 0.038 & a + ngc 4143 & 17.0 & l1.9 & 2001 mar 26 & 2514 & 0.063 & 0.025 & a + ngc 4203 & 9.7 & l1.9 & 1999 nov 4 & 1754 & 0.17 & 0.54 & c + ngc 4278 & 9.7 & l1.9 & 2000 apr 20 & 1396 & 0.18 & 0.33 & b + ngc 4548 & 16.8 & l2 & 2001 mar 24 & 2746 & 0.0097 & 0.0039 & a + ngc 4550 & 16.8 & l2 & 2001 mar 24 & 1885 & ... & ... & a + ngc 4565 & 9.7 & s1.9 & 2000 jun 30 & 2828 & 0.045 & 0.081 & b + ngc 4579 & 16.8 & l1.9/s1.9 & 2000 feb 23 & 2672 & 1.1 & 0.47 & a + ngc 5033 & 18.7 & s1.5 & 2000 apr 28 & 2904 & 0.33 & 0.59 & b + ngc 5866 & 15.3 & t2 & 2002 jan 10 & 2247 & ... & ... & a + ngc 6500 & 39.7 & l2 & 2000 aug 1 & 2104 & 0.020 & 0.064 & c + cccccccccl name & ra & dec . & & counts & & hard / soft & flux & luminosity & notes + & ( j2000 ) & ( j2000 ) & ( 0.58 kev ) & ( 0.5 - 2 kev ) & ( 2 - 8 kev ) & band ratio & ( 210 kev ) & ( 210 kev ) + ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) & ( 6 ) & ( 7 ) & ( 8) & ( 9 ) & ( 10 ) + ngc 266 & 0 49 47.81 & 32 16 40.0 & 40.7@xmath1266.4 & 29.8@xmath1265.5 & 10.9@xmath1263.3 & 0.37@xmath1260.13 & 1.6 & 7.5 & a + ngc 2787 & 9 19 18.70 & 69 12 11.3 & 7.9@xmath1262.8 & @xmath5712.0 & @xmath576.4 & ... & 0.25 & 0.053 & b + ngc 3147 & 10 16 53.75 & 73 24 02.8 & 1180.1@xmath12634.4 & 843.4@xmath12629.1 & 333.5@xmath12618.3 & 0.40@xmath1260.03 & 37 & 76 & a , c + ngc 3169 & 10 14 15.05 & 03 27 57.9 & 159.0@xmath12612.6 & @xmath5711.8 & 151.0@xmath12612.3 & @xmath3212.9 & 24 & 26 & a + & & & & & & & 26 & 22 & a , d + ngc 3226 & 10 23 27.01 & 19 53 55.0 & 209.3@xmath12614.5 & 125.2@xmath12611.2 & 80.5@xmath1269.0 & 0.64@xmath1260.09 & 7.6 & 5.5 & a + ngc 4143 & 12 9 36.07 & 42 32 03.0 & 157.4@xmath12612.6 & 121.3@xmath12611.1 & 32.6@xmath1265.7 & 0.27@xmath1260.05 & 3.1 & 1.1 & a + ngc 4203 & 12 15 05.02 & 33 11 49.9 & 294.3@xmath12617.2 & 198.7@xmath12614.1 & 91.7@xmath1269.6 & 0.46@xmath1260.06 & ... & ... & e + ngc 4278 & 12 20 06.80 & 29 16 51.6 & 255.6@xmath12616.4 & 209.6@xmath12614.9 & 52.0@xmath1267.4 & 0.25@xmath1260.04 & 8.1 & 0.91 & a , c + ngc 4548 & 12 35 26.46 & 14 29 46.7 & 26.6@xmath1265.2 & 8.7@xmath1263.0 & 17.7@xmath1264.2 & 2.02@xmath1260.85 & 1.6 & 0.61 & a + ngc 4550 & ... & ... & @xmath127 & @xmath128 & @xmath128 & ... & @xmath129 & @xmath130 & + ngc 4565 & 12 36 20.78 & 25 59 15.7 & 127.3@xmath12611.3 & 92.5@xmath1269.6 & 34.9@xmath1265.9 & 0.38@xmath1260.08 & 3.2 & 0.36 & a + ngc 4579 & 12 37 43.52 & 11 49 05.4 & 3067.9@xmath12655.6 & 2240.5@xmath12647.7 & 812.3@xmath12628.5 & 0.36@xmath1260.01 & ... & ... & e + ngc 5033 & 13 13 27.47 & 36 35 38.1 & 946.5@xmath12630.9 & 562.1@xmath12623.8 & 380.2@xmath12619.5 & 0.68@xmath1260.05 & ... & ... & e + ngc 5866 & ... & ... & @xmath127 & @xmath131 & @xmath132 & ... & @xmath133 & @xmath134 & b + ngc 6500 & 17 55 59.78 & 18 20 18.0 & 42.4@xmath1266.6 & 41.5@xmath1266.5 & @xmath577.6 & @xmath570.18 & 0.28 & 0.55 & a + & & & & & & & 0.69 & 1.3 & a , d + ccccccccccl ngc 2787 & 9 19 23.05 & 69 14 24.4 & j091923.1 + 691424 & 21.0@xmath1264.6 & 17.9@xmath1264.2 & @xmath578.0 & @xmath570.44 & 0.68 & ... & a , b + ngc 3147 & 10 16 51.50 & 73 24 08.9 & j101651.5 + 732409 & 6.7@xmath1262.6 & @xmath576.8 & @xmath579.2 & ... & 0.10 & 0.20 & b + ngc 3169 & 10 14 14.35 & 03 28 10.8 & j101414.3 + 032811 & 6.9@xmath1262.6 & 6.9@xmath1262.6 & @xmath573.0 & @xmath570.43 & 0.11 & 0.051 & b + & 10 14 17.90 & 03 28 55.2 & j101417.9 + 032855 & 10.9@xmath1263.3 & 8.0@xmath1262.8 & @xmath579.4 & @xmath571.18 & 0.18 & 0.084 & b + ngc 3226 & 10 23 26.69 & 19 54 06.8 & j102326.7 + 195407 & 8.9@xmath1263.0 & 7.9@xmath1262.8 & @xmath574.7 & @xmath570.59 & 0.13 & 0.085 & b + ngc 4203 & 12 15 09.20 & 33 09 54.7 & j121509.2 + 330955 & 240.1@xmath12615.5 & 196.4@xmath12614.0 & 40.8@xmath1266.4 & 0.21@xmath1260.04 & ... & ... & a , c , ton 1480 + & 12 15 14.33 & 33 11 04.7 & j121514.3 + 331105 & 11.9@xmath1263.5 & @xmath576.3 & 9.9@xmath1264.3 & @xmath321.57 & 2.0 & ... & a , d j121514.3 + 331105 , g + & 12 15 15.34 & 33 13 54.0 & j121515.3 + 331354 & 6.0@xmath1262.4 & 6.0@xmath1262.4 & @xmath573.0 & @xmath570.50 & 0.10 & ... & a , b + & 12 15 15.64 & 33 10 12.3 & j121515.6 + 331012 & 16.9@xmath1265.2 & 14.9@xmath1265.0 & @xmath576.3 & @xmath570.42 & 0.30 & ... & a , b , star + & 12 15 19.84 & 33 10 12.2 & j121519.8 + 331012 & 15.9@xmath1264.0 & 10.9@xmath1263.3 & @xmath574.7 & @xmath570.43 & 0.29 & ... & a , b + ngc 4550 & 12 35 21.30 & 12 14 04.5 & j123521.3 + 121405 & 6.0@xmath1262.4 & 5.9@xmath1262.4 & @xmath573.0 & @xmath570.50 & 0.10 & ... & a , b + & 12 35 27.76 & 12 13 38.9 & j123527.8 + 121339 & 35.8@xmath1266.0 & 26.6@xmath1265.2 & 8.0@xmath1262.8 & 0.30@xmath1350.12 & 1.1&11000 & a , d , qso 1232 + 125 , h + ngc 4565 & 12 36 14.65 & 26 00 52.5 & j123614.7 + 260052 & 14.9@xmath1265.0 & 8.9@xmath1264.1 & 6.0@xmath1263.6 & 0.67@xmath1260.51 & 0.76 & 0.086 & d , a30 , i + & 12 36 17.40 & 25 58 55.5 & j123617.4 + 255856 & 269.5@xmath12616.4 & 209.7@xmath12614.5 & 59.9@xmath1267.7 & 0.29@xmath1260.04 & 5.8 & 0.67 & e , f , a32 , i + & 12 36 18.64 & 25 59 34.6 & j123618.6 + 255935 & 8.8@xmath1263.0 & @xmath5713.5 & @xmath576.2 & ... & 0.098 & 0.011 & b + & 12 36 19.02 & 25 59 31.5 & j123619.0 + 255932 & 6.9@xmath1262.6 & @xmath5710.7 & @xmath576.2 & ... & 0.077 & 0.009 & b + & 12 36 19.03 & 26 00 27.0 & j123619.0 + 260027 & 19.9@xmath1264.5 & 18.0@xmath1264.2 & @xmath576.4 & @xmath570.36 & 0.22 & 0.025 & b , a33 , i + & 12 36 20.92 & 25 59 26.7 & j123620.9 + 255927 & 5.9@xmath1262.4 & @xmath5710.6 & @xmath574.7 & ... & 0.065 & 0.007 & b + & 12 36 27.39 & 25 57 32.7 & j123627.4 + 255733 & 15.9@xmath1264.0 & 14.9@xmath1263.9 & @xmath574.8 & @xmath570.32 & 0.18 & 0.020 & b , a37 ? , i + & 12 36 28.12 & 26 00 00.9 & j123628.1 + 260001 & 12.9@xmath1263.6 & 11.0@xmath1263.3 & @xmath576.4 & @xmath570.58 & 0.14 & ... & a , b + & 12 36 31.28 & 25 59 36.9 & j123631.3 + 255937 & 12.0@xmath1264.6 & @xmath5718.9 & @xmath574.8 & ... & 0.13 & ... & a , b , a43 ? , i + ngc 5033 & 13 13 24.78 & 36 35 03.7 & j131324.8 + 363504 & 13.9@xmath1263.7 & 10.0@xmath1263.2 & @xmath579.4 & @xmath570.94 & 0.15 & 0.063 & b + & 13 13 28.88 & 36 35 41.0 & j131328.9 + 363541 & 6.7@xmath1262.6 & @xmath5710.2 & @xmath576.3 & ... & 0.072 & 0.030 & b + & 13 13 29.46 & 36 35 17.3 & j131329.5 + 363517 & 34.6@xmath1265.9 & 31.7@xmath1265.7 & @xmath577.9 & @xmath570.25 & 0.37 & 0.16 & b + & 13 13 29.66 & 36 35 23.1 & j131329.7 + 363523 & 47.7@xmath1266.9 & 31.8@xmath1265.7 & 15.9@xmath1265.1 & 0.50@xmath1260.18 & 2.0 & 0.89 & f + & 13 13 35.56 & 36 34 04.4 & j131335.6 + 363404 & 7.0@xmath1262.6 & 6.0@xmath1262.4 & @xmath574.8 & @xmath570.80 & 0.075 & 0.031 & b + ngc 6500 & 17 56 01.59 & 18 20 22.6 & j175601.6 + 182023 & 7.0@xmath1262.6 & @xmath5710.4 & @xmath576.2 & ... & 0.12 & 0.23 & b + ngc 266 & @xmath138 & @xmath139 & 10.6 ( 6 ) + ngc 3147 & @xmath140 & @xmath141 & 45.4 ( 41 ) & a + ngc 3169 & @xmath142 & @xmath143 & 21.3 ( 26 ) + ngc 3226 & @xmath144 & @xmath145 & 8.1 ( 12 ) + ngc 4143 & @xmath146 & @xmath147 & 12.2 ( 11 ) + ngc 4278 & @xmath148 & @xmath149 & 16.9 ( 18 ) & a + ngc 4548 & @xmath150 & @xmath151 & 3.7 ( 5 ) + ngc 4565 & @xmath152 & @xmath153 & 1.6 ( 4 ) + ngc 6500 & @xmath154 & @xmath155 & 8.5 ( 8) + cccccc name & @xmath52@xmath15 & @xmath52@xmath6/@xmath15 & @xmath156(5 ghz ) & @xmath156(5 ghz)/@xmath6 & notes + & ( erg s@xmath2 ) & & ( erg s@xmath2 ) & & + & ( 1 ) & ( 2 ) & ( 3 ) & ( 4 ) & ( 5 ) + ngc 266 & 39.36 & 1.52 & 37.87 & @xmath1023.00 & a + ngc 2787 & 38.56 & 0.16 & 37.22 & @xmath1021.50 & b + ngc 3147 & 40.02 & 1.86 & 38.01 & @xmath1023.87 & c + ngc 3169 & 39.52 & 1.82 & 37.19 & @xmath1024.16 & a + ngc 3226 & 38.93 & 1.81 & 37.20 & @xmath1023.54 & a + ngc 4143 & 38.69 & 1.34 & 37.16 & @xmath1022.87 & b + ngc 4203 & 38.35 & 2.02 & 36.79 & @xmath1023.59 & b , e + ngc 4278 & 39.20 & 0.76 & 37.91 & @xmath1022.05 & b + ngc 4548 & 38.48 & 1.31 & 36.31 & @xmath1023.48 & d + ngc 4550 & 38.50 & @xmath1570.09 & 36.07 & @xmath1582.34 & d + ngc 4565 & 38.46 & 1.10 & 36.15 & @xmath1023.41 & b + ngc 4579 & 39.48 & 1.82 & 37.65 & @xmath1023.59 & b , e + ngc 5033 & 39.70 & 1.67 & 36.79 & @xmath1024.57 & c , e + ngc 5866 & 38.82 & @xmath1570.56 & 36.89 & @xmath1581.18 & b + ngc 6500 & 40.48 & @xmath1020.37 & 38.90 & @xmath1021.21 & a +

the results of _ chandra _ snapshot observations of 11 liners ( low - ionization nuclear emission - line regions ) , three low - luminosity seyfert galaxies , and one -liner transition object are presented . our sample consists of all the objects with a flat or inverted spectrum compact radio core in the vla survey of 48 low - luminosity agns ( llagns ) by nagar et al . ( 2000 ) . an x - ray nucleus is detected in all galaxies except one and their x - ray luminosities are in the range @xmath0 to @xmath1 ergs s@xmath2 . the x - ray spectra are generally steeper than expected from thermal bremsstrahlung emission from an advection - dominated accretion flow ( adaf ) . the x - ray to h@xmath3 luminosity ratios for 11 out of 14 objects are in good agreement with the value characteristic of llagns and more luminous agns , and indicate that their optical emission lines are predominantly powered by a llagn . for three objects , this ratio is less than expected . comparing with properties in other wavelengths , we find that these three galaxies are most likely to be heavily obscured agn . we use the ratio @xmath4(5 ghz)/@xmath5 , where @xmath6 is the luminosity in the 210 kev band , as a measure of radio loudness . in contrast to the usual definition of radio loudness ( @xmath7(5 ghz)/@xmath8(b ) ) , @xmath9 can be used for heavily obscured ( @xmath10 .5ex@xmath11 @xmath12 , @xmath13 mag ) nuclei . further , with the high spatial resolution of _ chandra _ , the nuclear x - ray emission of llagns is often easier to measure than the nuclear optical emission . we investigate the values of @xmath9 for llagns , luminous seyfert galaxies , quasars and radio galaxies and confirm the suggestion that a large fraction of llagns are radio loud .

many biological systems exhibit complex oscillatory dynamics that evolve over multiple time - scales , such as the spiking and bursting activity of neurons , sinus rhythms in the beating of the heart , and intracellular calcium signalling . such rhythms are often described by singularly perturbed systems of ordinary differential equations @xmath2 where @xmath3 is the ratio of slow and fast time - scales , @xmath4 is fast , @xmath5 is slow , and @xmath6 and @xmath7 are smooth functions . a relatively new type of oscillatory dynamic feature discovered in slow / fast systems with @xmath8 is the so - called torus canard @xcite . torus canards are solutions of that closely follow a family of attracting limit cycles of the fast subsystem of , and then closely follow a family of repelling limit cycles of the fast subsystem of for substantial times before being repelled . this unusual behaviour in the phase space typically manifests in the time course evolution as amplitude modulation of the rapid spiking waveform , as shown in figure [ fig : amspiking ] . and @xmath8 fast variables , one of which is @xmath9 . ( a ) the time evolution of the torus canard in this case is an amplitude - modulated spiking rhythm , which consists of rapid spiking ( blue ) wherein the envelope of the waveform ( red ) also oscillates . ( b ) the projection of the torus canard into the slow / fast phase plane shows that the torus canard arises in the neighbourhood of where an attracting family of limit cycles of the fast subsystem ( green , solid ) and a repelling family of limit cycles of the fast subsystem ( green , dashed ) meet ( inset ) . the trajectory alternately spends long times following both the attracting and repelling branches of limit cycles.,title="fig:",width=480 ] ( -364,142)(a ) ( -178,142)(b ) first discovered in a model for the neuronal activity in cerebellar purkinje cells @xcite , torus canards were observed as quasi - periodic solutions that would appear during the transition between bursting and rapid spiking states of the system . further insight into the dynamics of the torus canards in this cell model was presented in @xcite , where a 2-fast/1-slow rotated van der pol - type equation with symmetry breaking was studied . since then , torus canards have been encountered in several other neural models @xcite , such as hindmarsh - rose ( subhopf / fold cycle bursting ) , morris - lecar - terman ( circle / fold cycle bursting ) , and wilson - cowan - izhikevich ( fold / fold cycle bursting ) , where they again appeared in the transition between spiking and bursting states . additional studies have identified torus canards in chemical oscillators @xcite , and have shown that torus canards are capable of interacting with other dynamic features to create even more complicated oscillatory rhythms @xcite . three common threads link all of the examples mentioned above . first and foremost , the torus canards occur in the neighbourhood of a fold bifurcation of limit cycles , also known as a saddle - node of periodics ( snpos ) , of the fast subsystem ( see figure [ fig : amspiking](b ) for instance ) . that is , the torus canards arise in the regions of phase space where an attracting set of limit cycles meets a repelling set of limit cycles . second , the torus canards occur for parameter sets which are @xmath10 close to a torus bifurcation of the full system . third , in these examples , there is only one slow variable , and the torus canards are restricted to exponentially thin parameter sets . in other words , the torus canards in these examples are degenerate . torus canards in @xmath0 require a one - parameter family of 2-fast/1-slow systems in order to be observed , and they undergo a very rapid transition from rapid spiking to bursting ( i.e. , torus canard explosion ) in an exponentially thin parameter window @xmath10 close to a torus bifurcation of the full system . in principle , the addition of a second slow variable unfolds the torus canard phenomenon , making the torus canards generic and robust . this is analogous to the unfolding of planar canard cycles via the addition of a second slow variable . that is , canard solutions in @xmath0 are generic and robust , and their properties are encoded in folded singularities of the reduced flow @xcite . so far , to our knowledge , the only case study of torus canards in systems with more than one slow variable is in a model for respiratory rhythm generation in the pre - btzinger complex @xcite , which is a 6-fast/2-slow system . there , the torus canards were studied numerically by averaging the slow motions over limit cycles of the fast subsystem and examining the averaged slow drift along the manifold of periodics . in particular , folded singularities of the averaged slow flow were numerically identified and the properties of the torus canards were inferred based on canard theory . from their observations , the authors in @xcite conjectured that the average of a torus canard is a folded singularity canard . this leads to the generic torus canard problem , which can be stated simply as follows . there is currently no analytic way to identify , classify , and analyze torus canards in the same way that canards in @xmath0 can be classified and analyzed based on their associated folded singularity . it has been suggested that averaging methods should be used @xcite to reduce the torus canard problem to a folded singularity problem in a related averaged system . however , this approach is not rigorously justified since the averaging method breaks down in a neighbourhood of a fold of limit cycles , which is precisely where the torus canards are located . our main goal then is to extend the averaging method to the torus canard regime and hence solve the generic torus canard problem in @xmath1 . there are three types of results in this article : theoretical , numerical , and phenomenological . the theoretical contribution is that we extend the averaging method to folded manifolds of limit cycles and hence to the torus canard regime . in so doing , we inherit fenichel theory @xcite for persistent manifolds of limit cycles and in particular , we are able to make use of the powerful theoretical framework of canard theory @xcite . we provide analytic criteria for the identification and characterization of torus canards based on an underlying class of novel singularities for differential equations , which we call _ toral folded singularities_. we illustrate our assertions by studying a spatially homogeneous model for intracellular calcium dynamics @xcite . in applying our results to this model , we discover a novel type of bursting rhythm , which we call _ amplitude - modulated bursting _ ( see figure [ fig : ambursting ] for an example ) . we show that these amplitude - modulated bursting solutions can be well - understood using our torus canard theory . in the process , we provide the first numerical computations of intersecting invariant manifolds of limit cycles . the new phenomenological result that stems from our analysis is that we construct the torus canard analogue of a canard - induced mixed - mode oscillation @xcite . ) . the amplitude - modulated bursts alternate between active phases where the trajectory ( blue ) rapidly oscillates , and silent phases where the trajectory remains quiescent . during the active phase , the envelope ( red ) of the rapidly oscillating waveform exhibits small - amplitude oscillations which extend the burst duration . these amplitude - modulated bursts are torus canard - induced mixed - mode oscillations ( section [ sec : tcimmo]).,width=480 ] earlier reports of torus canards have been seen in the literature , even though that terminology was not used . in @xcite , it was remarked that bifurcation delay may result when a trajectory crosses from a set of attracting states to a set of repelling states where the states may be either fixed points or limit cycles . in @xcite , a canonical form for subcritical elliptic bursting near a bautin bifurcation of the fast subsystem was studied . the canonical model consists of two fast ( polar ) variables @xmath11 and a single slow variable @xmath12 . in these polar coordinates , the oscillatory states of the fast subsystem may be identified as stationary radii . within this framework , torus canards occur as canard cycles of the planar @xmath13 subsystem , and in parameter space they arise in the rapid and continuous transition between the spiking and bursting regimes of the canonical model . the outline of the paper is as follows . in sections [ sec : averaging ] and [ sec : classification ] , we give the main theoretical results of the article . namely , we state the generic torus canard problem in @xmath1 in the case of two fast variables and two slow variables , and then combine techniques from floquet theory @xcite , averaging theory @xcite , and geometric singular perturbation theory @xcite to show that the average of a torus canard is a folded singularity canard . in so doing , we devise analytic criteria for the identification and topological classification of torus canards based on their underlying toral folded singularity . we examine the main topological types of toral folded singularities and show that they encode properties of the torus canards , such as the number of torus canards that persist for @xmath14 . we then discuss bifurcations of torus canards and make the connection between torus canards and the torus bifurcation that is often observed in the full system . we apply our results to the politi - hfer model for intracellular calcium dynamics @xcite in sections [ sec : ph ] , [ sec : phtc ] and [ sec : tcimmo ] . we examine the bifurcation structure of the model and identify characteristic features that signal the presence of torus canards . using our torus canard theory , we explain the dynamics underlying the novel class of amplitude - modulated bursting rhythms . we show that the amplitude modulation is organised locally in the phase space by twisted , intersecting invariant manifolds of limit cycles . these sections serve the dual purpose of illustrating the predictive power of our analysis , and also giving a representative example of how to implement those results in practice . in section [ sec : explosion ] , we make the connection between our current work on torus canards and prior work on torus canards in @xmath0 explicit . we show that the theoretical framework developed in sections [ sec : averaging ] and [ sec : classification ] can be used to compute the spiking / bursting boundary in the parameter spaces of 2-fast/1-slow systems by simply tracking the toral folded singularity . we illustrate these results in the morris - lecar - terman , hindmarsh - rose , and wilson - cowan - izhikevich models for neural bursting . in section [ sec : arbitrarydimensions ] , we extend our averaging method for folded manifolds of limit cycles to slow / fast systems with two fast variables and @xmath15 slow variables , where @xmath15 is any positive integer . moreover , we provide asymptotic error estimates for the averaging method on folded manifolds of limit cycles . we then conclude in section [ sec : discussion ] , where we summarize the main results of the article , discuss their implications , and highlight several interesting open problems . in this section , we study generic torus canards in @xmath1 in the case of two fast variables and two slow variables . in section [ subsec : assumptions ] , we state the assumptions of the generic torus canard problem in @xmath1 . within this framework , we develop an averaging method for folded manifolds of limit cycles in section [ subsec : theoretical ] and derive a canonical form for the dynamics around a torus canard . in section [ subsec : averagedcoefficients ] , we list ( algorithmically ) the averaged coefficients that appear in the canonical form . we consider four - dimensional singularly perturbed systems of ordinary differential equations of the form @xmath16 where @xmath3 measures the time - scale separation , @xmath17 is fast , @xmath18 is slow , @xmath6 and @xmath7 are sufficiently smooth functions , and @xmath19 and their derivatives are @xmath20 with respect to @xmath21 . [ ass : man ] the layer problem of system , given by @xmath22 possesses a manifold @xmath23 of limit cycles , parametrized by the slow variables . for each fixed @xmath24 , let @xmath25 denote the corresponding limit cycle and assume that @xmath25 has finite , non - zero period @xmath26 . that is , @xmath27 the floquet exponents of @xmath25 are given by @xmath28 where @xmath29 corresponds to a floquet multiplier equal to unity , which reflects the fact that @xmath25 is neutrally stable to shifts along the periodic orbit @xcite . the stability then , of the periodic orbit @xmath25 , is encoded in the floquet exponent @xmath30 . if @xmath31 , then @xmath25 is an asymptotically stable solution of and if @xmath32 , then @xmath25 is an unstable solution of . [ ass : fold ] the layer problem possesses a manifold @xmath33 of snpos given by @xmath34 moreover , we assume that the manifold of periodics is a non - degenerate folded manifold so that @xmath23 can be partitioned into attracting and repelling subsets , separated by the manifold of snpos . that is , @xmath35 where @xmath36 is the subset of @xmath23 along which @xmath31 , and @xmath37 is the subset of @xmath23 along which @xmath32 . we refer forward to section [ subsec : averagedcoefficients ] for a more precise formulation of the non - degeneracy condition that @xmath30 changes sign along the manifold of snpos . a schematic of our setup is shown in figure [ fig : setup ] . each point on the folded manifold @xmath23 corresponds to a limit cycle of the layer problem . ( b ) attracting ( blue ) and repelling ( red ) manifolds of limit cycles joined by the folded limit cycle @xmath38 ( which corresponds to the black marker in ( a ) ) shown in the cross - section @xmath39 . ( c ) the folded limit cycle @xmath38 , indicated by the black marker in ( a ) , shown in the cross - section @xmath40 , with unit tangent and normal vectors , @xmath41 and @xmath42 , respectively for some fixed @xmath43.,title="fig:",width=480 ] ( -360,340)(a ) ( -160,340)(b ) ( -284,182)(c ) [ ass : homoclinic ] if the layer problem has a critical manifold @xmath44 , then @xmath44 and @xmath33 are disjoint . assumption [ ass : homoclinic ] guarantees that the periodic orbits of the layer problem in a neighbourhood of the manifold of snpos have finite period . note that we are not eliminating the possibility of the manifold of limit cycles from intersecting the critical manifold @xmath44 , as would be the case near a set of hopf bifurcations of the layer problem . instead , we restrict the problem so that the snpos of stay a reasonable distance from the critical manifold . an important step in the analysis to follow is identifying unit tangent and unit normal vectors to the periodic orbit , @xmath25 , of the layer problem . one choice of unit tangent and normal vectors , @xmath45 and @xmath46 , to the periodic @xmath47 for fixed @xmath48 , is given by @xmath49 where @xmath50 denotes the standard euclidean norm and @xmath51 is the skew - symmetric matrix @xmath52 . the idea of averaging theory is to find a flow that approximates the slow flow on the family of periodic orbits of the layer problem @xcite . these averaging methods can be used to show that the effective slow dynamics on a family of asymptotically stable periodics are determined by an appropriately averaged system @xcite . that is , the slow drift on @xmath36 can be approximated by averaging out the fast oscillations , and the error in the approximation is @xmath10 . however , to our knowledge , there are currently no theoretical results about the slow drift near folded manifolds of periodics . the following theorem extends the averaging method from normally hyperbolic manifolds of limit cycles to folded manifolds of limit cycles . [ thm : averaging ] consider system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] , and let @xmath53 . then there exists a sequence of near - identity transformations such that the averaged dynamics of in a neighbourhood of @xmath54 are approximated by @xmath55 where an overbar denotes an average over one period of @xmath25 , and the coefficients in system can be computed explicitly ( see section [ subsec : averagedcoefficients ] ) . the fast variable @xmath56 in system can be thought of as the averaged radial perturbation to @xmath25 in the direction of @xmath46 , and the slow variables @xmath12 describe the averaged evolution of @xmath48 . we present the proof of theorem [ thm : averaging ] in section [ sec : arbitrarydimensions ] . the idea of the proof is to switch to a coordinate frame that moves with the limit cycles , apply a coordinate transformation that removes the linear radial perturbation , and then average out the rapid oscillations . the significance of theorem [ thm : averaging ] is that the averaged radial - slow dynamics described by system are autonomous , singularly perturbed , and occur in the neighbourhood of a folded critical manifold . as such , system falls under the framework of canard theory @xcite . theorem [ thm : averaging ] is a formal extension of the averaging method to folded manifolds of limit cycles . we defer the statement of asymptotic error estimates ( i.e. , the validity of this averaging method ) to section [ sec : arbitrarydimensions ] . here we list the averaged coefficients that appear in theorem [ thm : averaging ] ( and theorem [ thm : averagingarbitraryslow ] ) . we denote the period of @xmath25 by @xmath57 . the functions @xmath19 , and their derivatives are all evaluated at the limit cycle @xmath54 of the layer problem . recall that @xmath45 and @xmath46 denote unit tangent and unit normals to @xmath25 , respectively . we give the coefficients for the case of @xmath58-fast variables and @xmath15-slow variables , where @xmath59 . let @xmath60 be the fundamental solution defined by @xmath61 we will show in section [ sec : arbitrarydimensions ] that @xmath60 is @xmath57-periodic and bounded for all time ( lemma [ lemma : linear ] ) . the coefficients of the linear slow terms and the quadratic radial term in the radial equation are given ( component - wise ) by @xmath62 note that the coefficient @xmath63 of the quadratic @xmath56 term is a scalar . we compute the auxiliary quantities @xmath64 and @xmath65 as solutions of @xmath66 for @xmath67 . we refer forward to equation for the interpretation of @xmath64 and @xmath65 . using these auxiliary functions , we can compute the coefficients of the mixed terms in the radial equation according to @xmath68 where the @xmath69 matrix @xmath70 is given by @xmath71 and @xmath72 is the average of @xmath73 over one period of @xmath25 . the coefficients of the linear @xmath56-terms in the slow equations are @xmath74 finally , the @xmath75 matrix of coefficients of the linear @xmath12-terms in the slow equations is @xmath76 we can now simply list the averaged coefficients as @xmath77 for @xmath78 and @xmath67 , where @xmath79 , or @xmath80 . note that the non - degeneracy condition in assumption [ ass : fold ] is given by the requirement that the averaged coefficient of the quadratic radial term is non - zero ( i.e. , @xmath81 ) . we point out that the leading order terms in the averaged slow directions are simply given by the averages of the slow components of the vector field over one period of @xmath25 . we now study the dynamics of the averaged radial - slow system using geometric singular perturbation techniques . in section [ subsec : toralfoldedsing ] , we define the notion of a toral folded singularity a special limit cycle in the phase space from which torus canard dynamics can originate . we provide a topological classification of toral folded singularities and their associated torus canards in section [ subsec : toralclass ] . we then present the dynamics of the torus canards in the main cases , including those that exist near toral folded nodes ( section [ subsec : toralfn ] ) , toral folded saddles ( section [ subsec : toralfs ] ) , and toral folded saddle - nodes ( section [ subsec : toralfsn ] ) . we begin our geometric singular perturbation analysis of system by rewriting it in the more succinct form @xmath82 where @xmath83 correspond to the right - hand - sides of , and the prime denotes derivatives with respect to the fast time @xmath84 ( which is related to the slow time @xmath85 by @xmath86 ) . the idea is to decompose the dynamics of into its slow and fast motions by taking the singular limit on the slow and fast time - scales . the fast dynamics are approximated by solutions of the layer problem , @xmath87 where @xmath88 and @xmath89 are parameters , and was obtained by taking the singular limit @xmath90 in . the set of equilibria of , given by @xmath91 is called the critical manifold and is a key object in the geometric singular perturbations approach . assuming at least one of @xmath92 and @xmath93 is non - zero , the critical manifold has a local graph representation , @xmath94 , say . in the case of , the critical manifold is ( locally ) a parabolic cylinder in the @xmath95 phase space . linear stability analysis of the layer problem shows that the attracting and repelling sheets , @xmath96 and @xmath97 , of the critical manifold are separated by a curve of fold bifurcations , defined by @xmath98 note that @xmath96 and @xmath97 correspond to the attracting and repelling manifolds of limit cycles , @xmath36 and @xmath37 , respectively , introduced in . moreover , the fold curve @xmath99 corresponds to the manifold of snpos , @xmath33 . to describe the slow dynamics along the critical manifold @xmath44 , we switch to the slow time - scale ( @xmath100 ) in system and take the singular limit @xmath90 to obtain the reduced system @xmath101 where the overdot denotes derivatives with respect to @xmath85 . typically , to obtain a complete description of the flow on @xmath44 , we would need to compute the dynamics in an atlas of overlapping coordinate charts . in this case , and as is the case in many applications , we can use the graph representation of @xmath44 to project the dynamics of onto a single coordinate chart @xmath102 . the dynamics on @xmath44 are then given by @xmath103 where all functions and their derivatives are evaluated along @xmath44 . an important feature of the reduced flow highlighted by this projection is that the reduced flow is singular along the fold curve @xmath99 . that is , solutions of the reduced flow blow - up in finite time at the fold curve and are expected to fall off the critical manifold . to remove this finite - time blow - up of solutions , we introduce the phase space dependent time - transformation @xmath104 , which gives the _ desingularized system _ @xmath105 where we have recycled the overdot to denote derivatives with respect to @xmath106 , and @xmath94 . on the attracting sheets @xmath96 , the desingularized flow is ( topologically ) equivalent to the reduced flow . however , on the repelling sheets @xmath97 ( where @xmath107 ) , the time transformation reverses the orientation of trajectories , and the reduced flow is obtained by reversing the direction of the flow of the desingularized system . thus , the reduced flow can be understood by examining the desingularized flow and keeping track of the dynamics on both sheets of @xmath44 . the desingularized system possesses two types of equilibria : _ ordinary _ and _ folded_. the set of ordinary singularities @xmath108 consists of isolated points which are equilibria of both the reduced and desingularized flows and are @xmath10 close to equilibria of the fully perturbed problem provided they remain sufficiently far from the fold curve @xmath99 . the set of folded singularities @xmath109 consists of isolated points along the fold curve where the right - hand - side of the @xmath56-equation in ( and ) vanishes . whilst folded singularities are equilibria of the desingularized system , they are not equilibria of the reduced system . instead , folded singularities are points where the @xmath56-equation of the reduced system has a zero - over - zero type indeterminacy , which means trajectories may potentially pass through the folded singularity with finite speed and cross from one sheet of the critical manifold to another . that is , a folded singularity is a distinguished point on the fold curve where the reduced vector field may actually be regular . [ def : identification ] system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] , possesses a _ folded singularity of limit cycles _ , or a _ toral folded singularity _ for short , if it has a limit cycle @xmath53 , such that @xmath110 where @xmath111 for @xmath112 , are the averaged coefficients in theorem [ thm : averaging ] ( listed in section [ subsec : averagedcoefficients ] ) . in slow / fast systems with two slow variables and one fast variable , a folded singularity , @xmath45 , of the reduced flow on a folded critical manifold is a point on the fold of the critical manifold where there is a violation of transversality : @xmath113 geometrically , this corresponds to the scenario in which the projection of the reduced flow into the slow variable plane is tangent to the fold curve at @xmath45 . definition [ def : identification ] gives the averaged analogue for torus canards . more precisely , a toral folded singularity is a folded limit cycle @xmath114 such that the projection of the averaged slow drift along @xmath23 into the averaged slow variable plane is tangent to the projection of @xmath33 into the @xmath115-plane at the toral folded singularity ( see figure [ fig : phtoralfn ] for an example ) . we see that a toral folded singularity is a folded singularity of the @xmath95 system where solutions of the slow flow along the folded critical manifold can cross with finite speed from @xmath96 to @xmath97 ( or vice versa ) . that is , a toral folded singularity allows for singular canard solutions of the averaged radial - slow flow . a singular canard solution of the @xmath95 system corresponds , in turn , to a solution in the original @xmath116 system that slowly drifts along the manifold of periodics @xmath23 and crosses from @xmath36 to @xmath37 ( or vice versa ) via a toral folded singularity . based on this , we define singular torus canard solutions as follows . [ def : singtc ] suppose system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] has a toral folded singularity . a _ singular torus canard _ is a singular canard solution of the averaged radial - slow system . a _ singular faux torus canard _ is a singular faux canard solution of the averaged radial - slow system . canard theory classifies and characterizes singular canards based on their associated folded singularity . we have the following classification scheme for toral folded singularities . suppose system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] possesses a toral folded singularity @xmath117 . let @xmath118 denote the eigenvalues of the desingularized flow of , linearized about the origin ( i.e. , about @xmath117 ) . [ def : class ] the toral folded singularity is * a _ toral folded saddle _ if @xmath119 , * a _ toral folded saddle - node _ if @xmath120 and @xmath121 , * a _ toral folded node _ if @xmath122 , or a _ faux toral folded node _ if @xmath123 , * a _ degenerate toral folded node _ if @xmath124 , or * a _ toral folded focus _ if @xmath125 . thus , we may exploit the known results about the existence and dynamics of canards near classical folded singularities of the averaged radial - slow system in order to learn about the existence and dynamics of torus canards near toral folded singularities . folded nodes , folded saddles , and folded saddle - nodes are known to possess singular canard solutions . we examine their toral analogues in sections [ subsec : toralfn ] , [ subsec : toralfs ] , and [ subsec : toralfsn ] , respectively . folded foci possess no singular canards and so any trajectory of the slow drift along @xmath23 that reaches the neighbourhood of a toral folded focus simply falls off the manifold of periodics . in this section , we consider the toral folded node ( tfn ) and its unfolding in terms of the averaged radial - slow flow . fenichel theory guarantees that the normally hyperbolic segments , @xmath96 and @xmath97 , of @xmath44 persist as invariant slow manifolds , @xmath126 and @xmath127 , of respectively for sufficiently small @xmath21 . fenichel theory breaks down in neighbourhoods of the fold curve @xmath99 where normal hyperbolicity fails . the extension of @xmath126 and @xmath127 by the flow of into the neighbourhood of the tfn leads to a local twisting of the invariant slow manifolds around a common axis of rotation . this spiralling of @xmath126 and @xmath127 becomes more pronounced as the perturbation parameter @xmath21 is increased . moreover , there is a finite number of intersections between @xmath126 and @xmath127 . these intersections are known as ( non - singular ) maximal canards . the properties of the maximal canards associated to a tfn are encoded in the tfn itself in the following way . let @xmath128 be the eigenvalues of the tfn , where we treat the tfn as an equilibrium of , and let @xmath129 be the eigenvalue ratio . then , provided @xmath130 is bounded away from zero , the total number of maximal canards of that persist for sufficiently small @xmath21 is @xmath131 , where @xmath132 and @xmath133 is the floor function @xcite . this follows by direct application of canard theory @xcite to the averaged radial - slow system in the case of a folded node . the first or outermost intersection of @xmath126 and @xmath127 is called the primary strong maximal canard , @xmath134 , and corresponds to the strong stable manifold of the tfn . the strong canard is also the local separatrix that separates the solutions which exhibit local small - amplitude oscillatory behaviour from monotone escape . that is , trajectories on @xmath126 on one side of @xmath134 execute a finite number of small oscillations , whilst trajectories on the other side of @xmath134 simply jump away . the innermost intersection of @xmath126 and @xmath127 is the primary weak canard , @xmath135 , and corresponds to the weak eigendirection of the tfn . the weak canard plays the role of the axis of rotation for the invariant slow manifolds , which again follows directly from the results of @xcite applied to system . the remaining @xmath136 secondary canards , @xmath137 , @xmath138 , further partition @xmath126 and @xmath127 into sectors based on the rotational properties . that is , for @xmath139 , the segments of @xmath126 and @xmath127 between @xmath140 and @xmath141 consist of orbit segments that exhibit @xmath15 small - amplitude oscillations in an @xmath142 neighbourhood of the tfn , where @xmath143 . bifurcations of maximal canards occur at odd integer resonances in the eigenvalue ratio @xmath130 . when @xmath144 is an odd integer , there is a tangency between the invariant slow manifolds @xmath126 and @xmath127 . increasing @xmath144 through this odd integer value breaks the tangency between the slow manifolds resulting in two transverse intersections , i.e. , two maximal canards , one of which is the weak canard . thus , increasing @xmath144 through an odd integer results in a branch of secondary canards that bifurcates from the axis of rotation @xcite . thus , in the case of a tfn , the averaged radial - slow dynamics are capable of generating singular canards , which perturb to maximal canards that twist around a common axis of rotation . since a maximal canard of , by definition , lives at the intersection of ( the extensions of ) @xmath126 and @xmath127 in a neighbourhood of a folded node of , the corresponding trajectory in the original variables lives at the intersection of the extensions of @xmath145 and @xmath146 in the neighbourhood of the tfn ( see figure [ fig : phinvariantmanifolds ] ) . as such , we define a non - singular maximal torus canard as follows . [ def : maximaltc ] suppose system under assumptions [ ass : man ] , [ ass : fold ] , and [ ass : homoclinic ] has a tfn singularity . a _ maximal torus canard _ of is a trajectory corresponding to the intersection of the attracting and repelling invariant manifolds of limit cycles , @xmath145 and @xmath146 , respectively , in a neighbourhood of the tfn . the implication in moving from maximal canards of the averaged radial - slow system to maximal torus canards of the original system is that a torus canard associated to a tfn consists of three different types of motion working in concert . first , when the orbit is on @xmath145 , we have rapid oscillations due to limit cycles of the layer problem . the slow drift along @xmath145 moves the rapidly oscillating orbit towards the manifold of snpos and in particular , towards the tfn . in a neighbourhood of the tfn , we have canard dynamics occurring within the envelope of the waveform . combined , these motions ( rapid oscillations due to limit cycles of the layer problem , slow drift along the manifold of limit cycles , and canard dynamics on the radial envelope ) manifest as amplitude - modulated spiking rhythms . the maximal number of oscillations that the envelope of the waveform can execute is dictated by . there are two ways in which oscillations can be added to or removed from the envelope . first is the creation of an additional secondary torus canard via odd integer resonances in @xmath144 . we conjecture that this bifurcation of torus canards will correspond to a torus doubling bifurcation . the other method of creating / destroying oscillations in the envelope is by keeping @xmath21 and @xmath130 fixed , and varying the position of the trajectory relative to the maximal torus canards . when the trajectory crosses a maximal torus canard , it moves into a different rotational sector , resulting in a change in the number of oscillations in the envelope ( see figure [ fig : phsectors ] ) . in the case of a toral folded saddle , system possesses exactly one singular canard and one singular faux canard , which correspond to the stable and unstable manifolds of the toral folded saddle , respectively , when considered as an equilibrium of the desingularized flow . the singular canard in this case plays the role of a separatrix , which divides the phase space @xmath44 between those trajectories that encounter the fold curve @xmath99 , and those that turn away from it . the unfolding in @xmath21 of the toral folded saddle of shows that only the singular canard persists as a transverse intersection of the invariant slow manifolds . there is no oscillatory behaviour associated with this maximal canard , and only those trajectories that are exponentially close to the maximal canard can follow the repelling slow manifold . returning to the original @xmath116 variables , the toral folded saddle has precisely one maximal torus canard solution , which plays the role of a separatrix between those solutions that fall off the manifold of limit cycles at the manifold of snpos , and those that turn away from @xmath33 and stay on @xmath23 . we remark that the toral faux canard of a toral folded saddle plays the role of an axis of rotation for local oscillatory solutions of , analogous to the toral weak canard in the tfn case . however , these oscillatory solutions of the averaged radial - slow system start on @xmath127 and move to @xmath126 . that is , there is a family of faux torus canard solutions associated to the toral folded saddle that start on @xmath146 and move to @xmath145 . since these solutions are inherently unstable , we leave further investigation of their dynamics to future work . in canard theory , the special case in which one of the eigenvalues of the folded singularity is zero is called a folded saddle - node ( fsn ) . the fsn comes in a variety of flavours , each corresponding to a different codimension-1 bifurcation of the desingularized reduced flow . the most common types seen in applications are the fsn i @xcite and the fsn ii @xcite . the fsn i occurs when a folded node and a folded saddle collide and annihilate each other in a saddle - node bifurcation of folded singularities . geometrically , the center manifold of the fsn i is tangent to the fold curve . it has been shown that @xmath147 canards persist near the fsn i limit for sufficiently small @xmath21 . the fsn ii occurs when a folded singularity and an ordinary singularity coalesce and swap stability in a transcritical bifurcation of the desingularized reduced flow . in this case , the center manifold of the fsn ii is transverse to the fold curve and it has been shown that @xmath148 canards persist for sufficiently small @xmath21 @xcite . here we show that toral folded singularities may also be of the fsn types . we focus on the toral fsn ii and its implications for torus canards . in analogy with the classic fsn ii points , we define a _ toral fsn ii _ to be a fsn ii of the averaged radial - slow system . these occur when an ordinary singularity of crosses the fold curve , i.e. , under the conditions @xmath149 in which case the folded singularity automatically has a zero eigenvalue . in the classic fsn ii , there is always a hopf bifurcation at an @xmath10-distance from the fold curve @xcite . since the toral fsn ii is ( by definition ) a fsn ii of system , we have that system will possess a hopf bifurcation located at an @xmath10-distance from the fold curve . in terms of the original , non - averaged @xmath116 coordinates , the toral fsn ii is detected as a limit cycle , @xmath47 , of the layer problem such that @xmath150 along @xmath47 . that is , the toral fsn ii occurs when the averaged slow nullclines intersect the manifold of snpos . moreover , the averaged radial - slow dynamics undergo a singular hopf bifurcation @xcite , from which a family of small - amplitude limit cycles of the averaged radial - slow system emanate . this creation of limit cycles in the radial envelope corresponds to the birth of an invariant phase space torus in the non - averaged , fully perturbed problem . as such , we conjecture that the toral fsn ii unfolds in @xmath21 to a singular torus bifurcation of the fully perturbed problem . we provide numerical evidence to support this conjecture in sections [ subsec : phsingvsnonsing ] , [ subsubsec : mlttoralfs ] , [ subsec : tchr ] , and [ subsec : tcwci ] . we now demonstrate ( in sections [ sec : ph ] [ sec : tcimmo ] ) the predictive power of our analytic framework for generic torus canards , developed in sections [ sec : averaging ] and [ sec : classification ] , in the politi - hfer ( ph ) model @xcite . this model describes the interaction between calcium transport processes and the metabolism of inositol ( 1,4,5)-trisphosphate ( 3 ) , which is a calcium - releasing messenger . in section [ subsec : phmodel ] , we describe the ph model for intracellular calcium dynamics . in section [ subsec : phbifn ] , we investigate the bifurcation structure of the ph model , and report on a novel class of amplitude - modulated subcritical elliptic bursting rhythms . in the parameter space , these exist between the tonic spiking and bursting regimes . in section [ subsec : phlayer ] we formally show that the ph model is a 2-fast/2-slow system and follow in section [ subsec : phgeometry ] by showing that it falls under the framework of our torus canard theory . changes in the concentration of free intracellular calcium play a crucial role in the biological function of most cell types @xcite . in many of these cells , the calcium concentration is seen to oscillate , and an understanding of how these oscillations arise and determining the mechanisms that generate them is a significant mathematical and biological pursuit . the biological process in which calcium is able to activate calcium release from internal stores is known as calcium - induced calcium release . the sequence of events leading to calcium - induced calcium release is as follows . an agonist binds to a receptor in the external plasma membrane of a cell , which initiates a chain of reactions that lead to the release of 3 inside that cell . the 3 binds to 3 receptors on the endoplasmic reticulum , which leads to the release of calcium from the internal store through the 3 receptors . that there are calcium oscillations is indicative of the fact that there are feedback mechanisms from calcium to the metabolism of 3 at work . mathematical modelling of these feedback mechanisms is broadly split into three classes . class i models assume that the 3 receptors are quickly activated by the binding of calcium , and then slowly inactivated by slow binding of the calcium to separate binding sites . that is , class i models feature sequential ( fast ) positive and then ( slow ) negative feedback on the 3 receptor . class ii models assume that the calcium itself regulates the production and degradation rates of 3 , which provides an alternative mechanism for negative and positive feedback . the most biologically realistic scenario incorporates both mechanisms ( i.e. , calcium feedback on 3 receptor dynamics as well as on 3 dynamics ) , and such models are known as hybrid models . one hybrid model for calcium oscillations is the ph model @xcite , which includes four variables ; the calcium concentration @xmath151 in the cytoplasm , the calcium concentration @xmath152 in the endoplasmic reticulum stores , the fraction @xmath153 of 3 receptors that have not been inactivated by calcium , and the concentration @xmath45 of 3 in the cytoplasm . the calcium flux through the 3 receptors is given by @xmath154 where @xmath155 is the ratio of the cytosolic volume to the endoplasmic reticulum volume . the active transport of calcium across the endoplasmic reticulum ( via sarco / endoplasmic reticulum atp - ase or serca pumps ) and plasma membrane are given respectively by the hill functions @xmath156 the calcium flux into the cell via the plasma membrane is given by @xmath157 where @xmath158 is the leak into the cell , and @xmath159 is the steady - state concentration of 3 in the absence of any feedback effects of calcium on 3 concentration . the ph model equations are then given by @xmath160 here the @xmath151 and @xmath152 equations describe the balance of calcium flux across the plasma membrane ( @xmath161 ) of the cell and the endoplasmic reticulum ( @xmath162 ) . the parameter @xmath163 measures the relative strength of the plasma membrane flux to the flux across the endoplasmic reticulum . note that if @xmath164 , then this creates a closed cell model wherein the total calcium in the cell is conserved . the @xmath153-equation describes the inactivation of 3 receptors by calcium whilst the @xmath45-equation describes the balance of ( calcium - independent ) 3 production and calcium - activated 3 degradation . the kinetic parameters , their standard values , and their biological significance are detailed in table [ tab : phparams ] ( appendix [ app : ph ] ) . unless stated otherwise , all parameters will be fixed at these standard values . following @xcite , we take @xmath159 to be the principal bifurcation parameter , since it is relatively easy to manipulate in an experimental setting . the parameter @xmath159 represents the steady - state 3 concentration in the absence of calcium feedback . note that the maximal rate of 3 formation is given by @xmath165 . variations in @xmath159 can generate a wide array of different behaviours in the ph model . representative traces are shown in figure [ fig : phbifn ] . ( i.e. , @xmath166 ) , and ( a ) @xmath167 m , ( b ) @xmath168 m , ( c ) @xmath169 m , and ( d ) @xmath170 m . ( a ) the attractor is a stable equilibrium . ( b ) the system exhibits subcritical elliptic bursting . the inset shows the small oscillations due to slow passage through a delayed hopf bifurcation . ( c ) the model can also generate rapid spiking solutions . ( d ) the subcritical elliptic bursts here feature amplitude - modulation during the active burst phase . inset : magnified view of the oscillations in the envelope of the waveform . ( e ) bifurcation structure of with respect to @xmath159 . the equilibria ( black ) change stability at hopf bifurcations ( hb ) . emanating from the hopf bifurcations are families of limit cycles , which change stability at torus bifurcations ( tr ) . the torus bifurcations act as the boundaries between spiking ( red ) and bursting ( blue ) regions.,title="fig:",width=480 ] ( -364,372)(a ) ( -181,372)(b ) ( -364,262)(c ) ( -181,262)(d ) ( -364,150)(e ) for small 3 production rates ( i.e. , small @xmath159 ) , the negative feedback of calcium on 3 metabolism overwhelms the production of 3 . there are no calcium oscillations , and the system settles to a stable equilibrium ( figure [ fig : phbifn](a ) ) . with increased 3 production rate , the system undergoes a supercritical hopf bifurcation ( labelled hb@xmath171 ) at @xmath172 m , from which stable periodic orbits emanate . this family of periodics becomes unstable at a torus bifurcation ( tr@xmath171 ) at @xmath173 m and the stable spiking solutions give way to bursting trajectories ( figure [ fig : phbifn](b ) ) . these bursting solutions are in fact subcritical elliptic ( or subhopf / fold - cycle ) bursts @xcite . a subcritical elliptic burster has two primary bifurcations that determine its outcome . the active phase of the burst is initiated when the trajectory passes through a subcritical hopf bifurcation of the layer problem , and terminates when the trajectory reaches a snpo and falls off the manifold of limit cycles of the layer flow . these subcritical elliptic bursts persist in @xmath159 until there is another torus bifurcation ( tr@xmath174 ) at @xmath175 m , after which the system exhibits rapid spiking ( figure [ fig : phbifn](c ) ) . the branch of spiking solutions remains stable until another torus bifurcation ( tr@xmath176 ) at @xmath177 m is encountered . initially , for @xmath159 values @xmath10 close to tr@xmath176 , the system exhibits amplitude - modulated spiking ( not shown ) . the amplitude modulated spiking only exists on a very thin @xmath159 interval . moreover , the amplitude modulation becomes more dramatic as @xmath159 increases until @xmath178 m , after which the trajectory is a novel type of solution that combines features of amplitude - modulated spiking and bursting . these hybrid _ amplitude - modulated bursting _ ( amb ) solutions appear to be subcritical elliptic bursts with the added twist that there is amplitude modulation in the envelope of the waveform during the active burst phase ( compare figures [ fig : phbifn](b ) and ( d ) ) . we will carefully examine these amb rhythms in section [ sec : tcimmo ] . for sufficiently large @xmath159 , we recover subcritical elliptic bursting solutions like those shown in figure [ fig : phbifn](b ) , but with increasingly long silent phases . eventually these subcritical elliptic bursting solutions disappear in the torus bifurcation tr@xmath179 at @xmath180 m and the attractor of the system is a spiking solution , which disappears in a supercritical hopf bifurcation ( hb@xmath174 ) at @xmath181 m . the bifurcation structure of with respect to @xmath159 described above was computed using auto @xcite and is shown in figure [ fig : phbifn](e ) . two primary features of our bifurcation analysis here signal the presence of multiple time - scale dynamics in system . firstly , the presence of trajectories that have epochs of rapid spiking interspersed with silent phases ( i.e. , the bursting solutions ) indicates that there is an intrinsic slow / fast structure . second , the transition from rapid spiking to bursting via a torus bifurcation , together with the appearance of amplitude - modulated waveforms suggests the presence of torus canards , which naturally arise in slow / fast systems . motivated by this , we now turn our attention to the problem of understanding the underlying mechanisms that generate these novel bursting rhythms . we will use the analytic framework developed in sections [ sec : averaging ] and [ sec : classification ] as the basis of our understanding . to demonstrate the existence of a separation of time - scales in system , we first perform a dimensional analysis , following a procedure similar to that of @xcite . we define new dimensionless variables @xmath182 , and @xmath183 via @xmath184 where @xmath185 and @xmath186 are reference calcium and 3 concentrations , respectively , and @xmath187 is a reference time - scale . details of the non - dimensionalization are given in appendix [ app : ph ] . for the parameter set in table [ tab : phparams ] , natural choices for @xmath185 and @xmath186 are @xmath188 m and @xmath189 m . with these choices , a typical time scale for the dynamics of the calcium concentration @xmath151 is given by @xmath190 s. the @xmath152 dynamics evolve much more slowly with a typical time scale @xmath191 s. the @xmath153 dynamics have a typical time - scale @xmath192 s. the time - scale @xmath193 for the @xmath45 dynamics depends on @xmath159 and can range from @xmath194 s ( for @xmath195 m ) to @xmath196 s ( for @xmath197 m ) . setting the reference time - scale to be the slow time - scale ( i.e. , @xmath198 ) , and defining the dimensionless parameters @xmath199 leads to the dimensionless version of the ph model @xmath200 where the overdot denotes derivatives with respect to @xmath183 , and the functions @xmath201 , and @xmath202 are given in appendix [ app : ph ] . for the parameter set in appendix [ app : ph ] , we have that @xmath203 is small . thus , for a large regime of parameter space , the ph model is singularly perturbed , with two fast variables ( @xmath204 ) and two slow variables ( @xmath205 ) . we now show that the ph model satsfies assumptions [ ass : man ] [ ass : homoclinic ] . the first step is to examine the bifurcation structure of the layer problem @xmath206 where the prime denotes derivatives with respect to the ( dimensionless ) fast time @xmath207 , which is related ( for non - zero perturbations ) to the dimensionless slow time @xmath183 by @xmath208 . the geometric configuration of the layer problem is illustrated in figure [ fig : phlayerbifn ] . . the critical manifold ( red surface ) possesses a curve of subcritical hopf bifurcations @xmath209 ( red curve ) that separates the attracting and repelling sheets , @xmath96 and @xmath97 . also shown is the maximum @xmath151-value for the manifold of limit cycles ( blue surface ) emanating from @xmath209 . the manifold of limit cycles consists of attracting and repelling subsets , @xmath36 and @xmath37 , which meet in a manifold of snpos , @xmath33.,width=312 ] system has a critical manifold , @xmath44 , with a curve of subcritical hopf bifurcations , @xmath209 , that divide @xmath44 between its attracting and repelling sheets , @xmath96 and @xmath97 , respectively . the manifold , @xmath37 , of limit cycles that emerges from @xmath209 is repelling . the repelling family of limit cycles meets an attracting family of limit cycles , @xmath36 , at a manifold of snpos , @xmath33 . thus , the ph model has precisely the geometric configuration described in section [ subsec : assumptions ] . the bifurcation structure of contains many other features outside of the region shown here . the full critical manifold is cubic - shaped . only part of that lies in the region shown in figure [ fig : phlayerbifn ] , and outside this region , there are additional curves of fold and hopf bifurcations , cusp bifurcations , and bogdanov - takens bifurcations . in this work , we are only concerned with the region of phase space presented in figure [ fig : phlayerbifn ] . we now apply the results of section [ sec : classification ] to the ph model . in section [ subsec : phtoralfs ] , we show that the ph model possesses toral folded singularities . we carefully examine the geometry and maximal torus canards in the case of a tfn in section [ subsec : phmanifolds ] . in order to do so , we must compute the invariant manifolds of limit cycles , the numerical method for which is outlined in section [ subsec : phnumerical ] . we then show in section [ subsec : phtoralfsclass ] that the torus canards are generic and robust phenomena , and occur on open parameter sets . in this manner , we demonstrate the practical utility of our torus canard theory . we now proceed to locate and classify toral folded singularities of the ph model . for each limit cycle in @xmath33 , we numerically check the condition for toral folded singularities given in equation , @xmath210 where the overlined quantities are the averaged coefficients that appear in theorem [ thm : averaging ] . recall , that the condition @xmath211 corresponds geometrically to the scenario in which the projection of the averaged slow drift into the slow variable plane is tangent to @xmath33 ( figure [ fig : phtoralfn ] ) . for our computations , we take this condition to be satisfied if @xmath212 . for the representative parameter set given in table [ tab : phparams ] ( appendix [ app : ph ] ) , we find that the ph model possesses a toral folded singularity for @xmath213 m at @xmath214 where @xmath215 . m at @xmath216 . ( a ) projection of @xmath23 in a neighbourhood of the tfn into the @xmath217 phase space . the blue curves show the envelopes of the slow drift along @xmath23 for different initial conditions . ( b ) projection into the slow variable plane . the singular slow drift along @xmath23 is tangent to @xmath33 at the tfn.,title="fig:",width=480 ] ( -364,146)(a ) ( -160,146)(b ) once the toral folded singularity has been located , we simply compute the remaining averaged coefficients from theorem [ thm : averaging ] , which allows us to compute the eigenvalues of the toral folded singularity and hence classify it according to the scheme in definition [ def : class ] . for the toral folded singularity at @xmath216 , the eigenvalues are @xmath218 and @xmath219 , so that we have a tfn . the associated eigenvalue ratio of the tfn is @xmath220 , and so by , the maximal number of oscillations that the envelope of the waveform can execute for sufficiently small @xmath21 is @xmath221 ( compare with figure [ fig : phbifn](d ) where the envelope only oscillates 4 times ) . we point out that the type of toral folded singularity can change with the parameters ( see section [ subsec : phtoralfsclass ] ) . all other points on @xmath33 are regular folded limit cycles . that is , @xmath222 at all other points on @xmath33 ; and so , for @xmath223 m , there is only a single tfn . note that the tfn identified here is a simple zero of @xmath224 , i.e. , @xmath224 has opposite sign for points on @xmath33 on either side of the tfn . this tfn is the natural candidate mechanism for generating torus canard dynamics . we now examine the geometry of the ph model in a neighbourhood of the tfn away from the singular limit . recall that averaging theory @xcite together with fenichel theory @xcite guarantees that normally hyperbolic manifolds of limit cycles , @xmath36 and @xmath37 , persist as invariant manifolds of limit cycles , @xmath145 and @xmath146 , for sufficiently small @xmath21 . we showed in section [ subsec : toralfn ] that the extensions of @xmath145 and @xmath146 into a neighbourhood of a tfn results in a local twisting of these manifolds of limit cycles . and @xmath146 into a neighbourhood of the tfn projected into the @xmath217 phase space for @xmath223 m and @xmath225 . the attracting invariant manifold of limit cycles ( blue ) is computed until it intersects the hyperplane @xmath226 . similarly , the repelling invariant manifold of limit cycles ( red ) is computed up to its intersection with @xmath227 . the inset shows @xmath228 ( blue ) and @xmath229 ( red ) . their singular limit counterparts , @xmath230 and @xmath231 , are also shown for comparison . there are 13 intersections , @xmath232 , @xmath233 , of the invariant manifolds , each corresponding to a maximal torus canard ( with @xmath234).,width=480 ] figure [ fig : phinvariantmanifolds ] demonstrates that the attracting and repelling manifolds of limit cycles twist in a neighbourhood of the tfn , and intersect a countable number of times . for @xmath223 m , we find that there are 13 intersections , consistent with the prediction from section [ subsec : phtoralfs ] . these intersections of @xmath145 and @xmath146 are the maximal torus canards ( by definition [ def : maximaltc ] ) . the outermost intersection of @xmath145 and @xmath146 , denoted @xmath235 , is the maximal strong torus canard . the intersections , @xmath232 , @xmath236 , are the maximal secondary torus canards . the innermost intersection is the maximal weak torus canard , @xmath237 . the maximal strong torus canard , @xmath235 , is the local phase space separatrix that divides between rapidly oscillating solutions that exhibit amplitude modulation and those that do not . the maximal weak torus canard , @xmath237 , plays the role of a local axis of rotation . that is , the invariant manifolds twist around @xmath237 . the maximal secondary torus canards partition @xmath145 and @xmath146 into rotational sectors . every orbit segment on @xmath145 between @xmath238 and @xmath239 for @xmath240 , is an amplitude - modulated waveform where the envelope executes @xmath241 oscillations in a neighbourhood of the tfn . m and @xmath225 . left column : projection of @xmath145 and @xmath146 , onto the @xmath242 plane along with the maximal torus canards @xmath243 . also shown is the envelope of the transient solution , @xmath47 , of in the ( a ) sector bounded by @xmath235 and @xmath244 , ( b ) sector bounded by @xmath244 and @xmath245 , and ( c ) sector bounded by @xmath245 and @xmath246 . right column : corresponding time traces of the transient solution @xmath47 . note that time is given in seconds.,title="fig:",width=480 ] ( -364,452)(a ) ( -364,296)(b ) ( -364,140)(c ) figure [ fig : phsectors ] illustrates the sectors of amplitude - modulation formed by the maximal torus canards . for fixed parameters , it is possible to change the number of oscillations in the envelope of the rapidly oscillating waveform by adjusting the initial condition . more specifically , the trajectory of for an initial condition on @xmath145 between @xmath235 and @xmath244 is an amb with one oscillation in the envelope ( figure [ fig : phsectors](a ) ) . by changing the initial condition to lie in the rotational sector bounded by @xmath244 and @xmath245 ( figure [ fig : phsectors](b ) ) , the amplitude - modulated waveform exhibits two oscillations in its envelope . the deeper into the funnel of the tfn , the smaller and more numerous the oscillations in the envelope of the rapidly oscillating waveform ( figure [ fig : phsectors](c ) ) . moreover , each oscillation significantly extends the burst duration . figure [ fig : phinvariantmanifolds ] is the first instance of the numerical computation of twisted , intersecting , invariant manifolds of limit cycles of a slow / fast system with at least two fast and two slow variables . here , we outline the numerical method ( inspired by the homotopic continuation algorithms for maximal canards of folded singularities @xcite ) used to generate figure [ fig : phinvariantmanifolds ] . the idea of the computation of @xmath145 is to take a set of initial conditions on @xmath36 sufficiently far from both the tfn and manifold of snpos , and flow it forward until the trajectories reach the hyperplane @xmath247 where @xmath248 is the @xmath152 coordinate of the tfn identified in section [ subsec : phtoralfs ] . this generates a family of rapidly oscillating solutions that form a mesh of the manifold @xmath145 . the envelope of each of those rapidly oscillating solutions is then used to form a mesh of the projection of @xmath145 . and @xmath146 for the ph model for the same parameter set as figure [ fig : phinvariantmanifolds ] . the attracting invariant manifold of limit cycles , @xmath145 , is computed by taking the set @xmath249 and flowing it forward until it intersects the hyperplane @xmath227 . the blue curves illustrate the behaviour of the envelopes of the rapidly oscillating orbit segments that comprise @xmath145 . similarly , the repelling invariant manifold of limit cycles , @xmath146 , is computed by flowing @xmath250 backwards in time up to the hyperplane @xmath227 . the red curves illustrate the envelopes of these rapidly oscillating orbit segments that comprise @xmath146.,width=312 ] to initialise the computation , a suitable set of initial conditions must be chosen . note that the projection of @xmath33 into the slow variable plane is a curve , @xmath251 , say ( see figure [ fig : phtoralfn](b ) ) . we choose our initial conditions to be a manifold of attracting limit cycles , @xmath249 , such that the projection of @xmath252 into the @xmath253 plane is approximately parallel to @xmath251 , and is sufficiently far from @xmath251 ( figure [ fig : phnumericalmethod ] ) . similarly , to compute @xmath146 , we initialize the computation by choosing a set of repelling limit cycles , @xmath250 , where the projection of @xmath254 into the @xmath253 plane is approximately parallel to , and sufficiently distant from , the curve @xmath251 . we then flow that set of initial conditions @xmath254 backwards in time until the trajectory hits the hyperplane @xmath227 . the envelopes of these rapidly oscillating trajectories are then used to visualize @xmath146 . to locate the maximal torus canard @xmath255 , we locate the initial condition within @xmath252 that forms the boundary between those orbit segments with @xmath241 oscillations in their envelope and orbit segments with @xmath256 oscillations in their envelope . we point out that whilst numerical methods exist for the computation and continuation of maximal canards of folded singularities @xcite , these methods will not work for maximal torus canards . there are currently no existing methods to numerically continue @xmath145 and @xmath146 . consequently , there are no numerical methods that will allow for the numerical continuation of maximal torus canards in parameters , which is essential for detecting bifurcations of torus canards . we have now carefully examined the torus canards associated to a tfn for a single parameter set . however , the ph model has two slow variables and so it supports tfns on open parameter sets . the theoretical framework developed in sections [ sec : averaging ] and [ sec : classification ] allows to determine how those tfns and their associated torus canards depend on parameters . figure [ fig : pheigenvalues](a ) shows the eigenvalue ratio , @xmath130 , of the toral folded singularity as a function of @xmath159 , for instance . . ( a ) the eigenvalue ratio , @xmath130 , of the toral folded singularity as a function of @xmath159 . the black markers indicate odd integer resonances in the eigenvalue ratio , where secondary torus canards bifurcate from the weak torus canard . there is a toral fsn ii at @xmath257 m . for @xmath258 m , the toral folded singularity is a toral folded saddle . bottom row : @xmath145 and @xmath146 , in a hyperplane passing through the tfn for @xmath259 and ( b ) @xmath260 m where @xmath261 and ( c ) @xmath262 m where @xmath263 . in both cases , the invariant manifolds are shown in an @xmath142 neighbourhood of the tfn . also shown are the attracting and repelling manifolds of limit cycles , @xmath36 and @xmath37 , of the layer problem.,title="fig:",width=480 ] ( -360,274)(a ) ( -366,138)(b ) ( -180,138)(c ) we find that the ph model has tfns and hence torus canard dynamics for @xmath264 m @xmath265 m . the black markers in figure [ fig : pheigenvalues](a ) indicate odd integer resonances in the eigenvalue ratio , @xmath130 , of the tfn . these resonances signal the creation of new secondary canards in the averaged radial - slow system . as such , when @xmath144 increases through an odd integer , we expect additional torus canards to appear . figures [ fig : pheigenvalues](b ) and ( c ) illustrate the mechanism by which these additional torus canards appear . namely , as @xmath159 decreases and @xmath144 increases , the invariant manifolds of limit cycles become more and more twisted , resulting in additional intersections . thus , figure [ fig : pheigenvalues](a ) essentially determines the number of torus canards that exist for a given parameter value . an alternative viewpoint is that figure [ fig : pheigenvalues](a ) determines the maximal number of oscillations that the envelope of the rapidly oscillating waveform can execute . for example , for @xmath159 on the interval between @xmath266 and @xmath267 , the amplitude - modulated waveform can have , at most , one oscillation in the envelope . for @xmath159 on the interval between @xmath268 and @xmath269 , the amplitude - modulated waveform can have , at most , two oscillations in the envelope , and so on . the ph model supports other types of toral folded singularities . for @xmath258 m , system has toral folded saddles . the toral folded saddle has precisely one torus canard associated to it . this torus canard , however , has no rotational behaviour , and instead acts as a local phase space separatrix between trajectories that fall off the manifold of periodics at @xmath33 and those that turn away from @xmath33 and stay on @xmath36 . the other main type of toral folded singularity that can occur is the toral folded focus . in the ph model , we find a set of toral folded foci for @xmath270 m . as stated in section [ subsec : toralclass ] , toral folded foci have no torus canard dynamics . the transition between tfn and toral folded saddle occurs at @xmath257 m in a toral fsn of type ii ( corresponding to @xmath271 ) , in which an ordinary singularity of the averaged radial - slow system coincides with the toral folded singularity . having carefully examined the local oscillatory behaviour of the ph model due to tfns and their associated torus canards , we proceed in this section to identify the local and global dynamic mechanisms responsible for the amb rhythms . in section [ subsec : amb ] , we study the effects of parameter variations on the amb solutions . we then show in section [ subsec : phsingtc ] that the ambs are torus canard - induced mixed - mode oscillations . in section [ subsec : phsingvsnonsing ] , we examine where the spiking , bursting , and amb rhythms exist in parameter space . in so doing , we demonstrate the origin of the amb rhythm and show how it varies in parameters . in section [ subsec : phbifn ] , we reported on the existence of amb solutions in the ph model ( see figure [ fig : phbifn](d ) ) . the novel features of these ambs are the oscillations in the envelope of the rapidly oscillating waveform during the active phase , which significantly extend the burst duration . changes in the parameter @xmath159 have a measurable effect on the amplitude modulation in these amb rhythms . increasing @xmath159 causes a decrease in the number of oscillations that the profile of the waveform exhibits . that is , as @xmath159 increases , the envelope of the bursting waveform gradually loses oscillations and the burst duration decreases . this progressive loss of oscillations in the envelope continues until @xmath159 has been increased sufficiently that all of the small oscillations disappear . for instance , for @xmath272 m , we observe 5 oscillations in the envelope ( figure [ fig : phampmod](a ) ) . this decreases to 4 oscillations for @xmath223 m ( figure [ fig : phbifn](d ) ) , down to 3 for @xmath273 m ( figure [ fig : phampmod](b ) ) , and then to 2 for @xmath274 m ( figure [ fig : phampmod](c ) ) . further increases in @xmath159 result in just 1 oscillation in the envelope ( not shown ) until , for sufficiently large @xmath159 , the oscillations disappear . once the amplitude modulation disappears ( figure [ fig : phampmod](d ) ) , the trajectories resemble the elliptic bursting rhythms discussed previously . m , ( b ) @xmath273 m , ( c ) @xmath274 m , and ( d ) @xmath275 m . increasing @xmath159 decreases the number of oscillations that the envelope of the waveform executes , and consequently decreases the burst duration.,title="fig:",width=480 ] ( -365,252)(a ) ( -180,252)(b ) ( -364,116)(c ) ( -180,116)(d ) it is currently unknown what kinds of bifurcations , if any , occur in the transitions between amb waveforms with different numbers of oscillations in the envelope . we conjecture that these transitions occur via torus doubling bifurcations ( since they are associated with tfns ; see section [ subsec : phsingtc ] ) . further investigation of the bifurcations that organise these transitions is beyond the scope of the current article . we first concentrate on understanding the mechanisms that generate the amplitude - modualted bursting rhythm seen in figure [ fig : phbifn](d ) , corresponding to @xmath276 m . to do this , we construct the singular attractor of the ph model for @xmath276 m ( figures [ fig : phtoralmmo](a ) and ( b ) ) . the singular attractor is the concatenation of four orbit segments . starting in the silent phase of the burst , there is a slow drift ( black , single arrow ) along the critical manifold @xmath96 that takes the orbit up to the curve @xmath209 of hopf bifurcations , where the stability of @xmath44 changes . this initiates a fast upward transition ( black , double arrows ) away from @xmath209 towards the attracting manifold of limit cycles , @xmath36 . once the trajectory reaches @xmath36 , there is a net slow drift ( black , single arrow ) that moves the orbit segment along @xmath36 towards @xmath33 . this net slow drift along @xmath36 can be described by an appropriate averaged system ( theorem [ thm : averaging ] ) . we find that for @xmath223 m , the fast up - jump from @xmath209 to @xmath36 projects the trajectory into the funnel of the tfn . as such , the slow drift brings the trajectory to the tfn itself ( green marker ) . at the tfn , there is a fast downward transition ( black , double arrows ) that projects the trajectory down to the attracting sheet of the critical manifold , thus completing one cycle . m and @xmath277 ( black ) , @xmath278 ( purple ) , @xmath279 ( orange ) , and @xmath280 ( olive ) . ( a ) trajectories superimposed on the critical manifold , @xmath281 , and manifold of limit cycles , @xmath282 . the singular attractor alternates between slow epochs ( single arrows ) on @xmath96 and @xmath36 , with fast jumps ( double arrows ) between them . the @xmath21-unfoldings of the singular attractor ( coloured trajectories ) spend long times near the tfn ( green marker ) . inset : projection into the slow variable plane in an @xmath142 neighbourhood of the tfn . the associated time traces of are shown in ( b ) for @xmath277 , ( c ) for @xmath278 , ( d ) for @xmath279 , and ( e ) for @xmath280 . the coloured envelopes in ( b)(e ) correspond to the coloured trajectories in ( a).,title="fig:",width=480 ] ( -340,406)(a ) ( -364,180)(b ) ( -180,180)(c ) ( -364,81)(d ) ( -180,81)(e ) figure [ fig : phtoralmmo ] shows that the singular attractor perturbs to the amb rhythm for sufficiently small @xmath21 ( purple , orange , and olive trajectories ) . that is , for small non - zero perturbations , the silent phase of the orbit is a small @xmath10-perturbation of the slow drift on the critical manifold . note that the trajectory does not immediately leave the silent phase when it reaches the hopf curve . dynamic bifurcation theory shows that the initial exponential contraction along @xmath96 allows trajectories to follow the repelling slow manifold for @xmath20 times on the slow time - scale @xcite . however , there eventually comes a moment where the repulsion on @xmath97 overwhelms the accumulative contraction on @xmath96 and the trajectory jumps away to the invariant manifold of limit cycles @xmath145 . we have established that for the segments of @xmath36 that are an @xmath20-distance from @xmath33 , the slow drift along @xmath145 is a smooth @xmath10 perturbation of the averaged slow flow along @xmath36 . in a neighbourhood of the manifold of snpos , and the tfn in particular , we have shown in section [ subsec : phmanifolds ] that torus canards are the local phase space mechanisms responsible for oscillations in the envelope of the rapidly oscillating waveform . these oscillations are restricted to an @xmath142-neighbourhood of the tfn ( figure [ fig : phtoralmmo](a ) , inset ) . note that as @xmath21 increases the position of the amb trajectory changes relative to the maximal torus canards . for instance , the purple and orange trajectories in figure [ fig : phtoralmmo ] are closer to one of the maximal torus canards than the olive trajectory . in fact , the olive solution lies in a different rotational sector than the purple and orange solutions and hence has fewer oscillations . thus , the amb consists of a local mechanism ( torus canard dynamics due to the tfn ) and a global mechanism ( the slow passage of the trajectory through a delayed hopf bifurcation , which re - injects the orbit into the funnel of the tfn ) . consequently , the amb can be regarded as a _ torus canard - induced mixed - mode oscillation_. the ph model supports canard - induced mixed - mode dynamics @xcite , since it has a cubic - shaped critical manifold with folded singularities . in fact , careful analyses of the canard - induced mixed - mode oscillations in were performed in @xcite . the difference between our work and @xcite is that we are concentrating on the amb behaviour near the torus bifurcation tr@xmath176 ( see figure [ fig : phbifn](e ) ) , whereas @xcite focuses on the mixed - mode oscillations near hb@xmath171 . we have demonstrated the origin of the amb rhythm for the specific parameter value @xmath276 m . the other amb rhythms observed in the ph model ( such as in figure [ fig : phampmod ] ) can also be shown to be torus canard - induced mixed - mode oscillations . the number of oscillations that the envelopes of the ambs exhibit is determined by two key diagnostics : the eigenvalue ratio of the tfn which determines how many maximal torus canards exist , and the global return mechanism ( slow passage through the delayed hopf ) which determines how many oscillations are actually observed . whilst we have carefully studied the local mechanism in sections [ subsec : phtoralfs][subsec : phtoralfsclass ] and identified the global return mechanism , we have not performed any careful analysis of the global return or its dependence on parameters . in particular , the boundary @xmath283 , corresponding to the special scenario in which the singular trajectory is re - injected exactly on the singular strong torus canard marks the boundary between those trajectories that reach the tfn ( and exhibit torus canard dynamics ) and those that simply reach the manifold of snpos and fall off without any oscillations in the envelope . furthermore , the global return is able to generate or lose oscillations in the envelope by re - injecting orbits into the different rotational sectors formed by the maximal torus canards . we leave the investigation of the global return for these ambs to future work . the bursting and spiking rhythms shown in figures [ fig : phbifn](b ) and ( c ) , respectively , can also be understood in terms of the bifurcation structure of the layer problem . in the bursting case , the trajectory can be decomposed into four distinct segments , analogous to the amb rhythm . the only difference is that the bursting orbit encounters @xmath33 at a regular folded limit cycle instead of a tfn , and so it simply falls off @xmath23 without exhibiting torus canard dynamics . such a bursting solution , with active phase initiated by slow passage through a fast subsystem subcritical hopf bifurcation , and active phase terminated at an snpo , is known as a subcritical elliptic burster @xcite . note that in the subcritical elliptic bursting ( figure [ fig : phbifn](b ) ) and amb ( figure [ fig : phbifn](d ) ) cases , the averaged radial - slow flow possesses a repelling ordinary singularity ( i.e. , unstable spiking solutions ) . in the tonic spiking case , the trajectory of can be understood by locating ordinary singularities of the averaged radial - slow system . we find that the averaged radial - slow system of the ph model has an attracting ordinary singularity , which corresponds to a stable limit cycle @xmath47 of the layer problem . since @xmath47 is hyperbolic , the full ph model exhibits periodic solutions which are @xmath10 perturbations of the normally hyperbolic limit cycle @xmath47 for sufficiently small @xmath21 . in this spiking regime , the system possesses a toral folded saddle ( corresponding to the region of negative @xmath130 in figure [ fig : pheigenvalues](a ) ) . we are interested in the transitions between the different dynamic regimes ( spiking , bursting , and amb ) of . figure [ fig : phtr](a ) shows the two - parameter bifurcation structure of in the @xmath284 plane . continuation of the torus bifurcations tr@xmath176 and tr@xmath179 ( from figure [ fig : phbifn](e ) ) generates a single curve , which separates the spiking and bursting regimes . the region enclosed by the tr@xmath176/tr@xmath179 curve consists of subcritical elliptic bursting solutions ( including the amb ) . by similarly continuing the hopf bifurcation hb@xmath174 , we find that the spiking regime is the region bounded by the hb@xmath174 curve and the curve of torus bifurcations . note that the branch tr@xmath176 , which separates the rapid spiking and amb waveforms , converges to the toral fsn ii at @xmath257 m in the singular limit @xmath90 . this supports our conjecture from section [ subsec : toralfsn ] that the @xmath21-unfolding of the toral fsn ii is a singular torus bifurcation . and tr@xmath179 , and the hopf bifurcation , hb@xmath174 , from figure [ fig : phbifn](e ) in ( a ) the @xmath284 plane , and ( b ) the @xmath285 plane . ( a ) the torus bifurcation curve encloses the bursting region . in the limit as @xmath90 , tr@xmath176 converges to the toral fsn ii at @xmath286 m . the region between the hopf curve and the torus curve is the spiking region . ( b ) in the singular limit @xmath90 , the bursting region is enclosed by the toral fsn ii and tr@xmath179 curves . the tr@xmath176 curve unfolds ( in @xmath21 ) from the toral fsn ii curve . thus , for @xmath287 , the amb solutions exhibit more oscillations in their envelopes the closer the parameters are chosen to the tr@xmath176 boundary.,title="fig:",width=480 ] ( -364,140)(a ) ( -184,140)(b ) we provide further numerical evidence to support this conjecture in figure [ fig : phtr](b ) , where we compare the loci of the toral fsns of type ii and torus bifurcation tr@xmath176 in the @xmath285 parameter plane for various @xmath21 . the coloured curves correspond to the torus bifurcation tr@xmath176 for @xmath288 ( blue ) , @xmath289 ( red ) , and @xmath290 ( green ; inset ) . as demonstrated in figure [ fig : phtr](b ) , the tr@xmath176 curve converges to the toral fsn ii curve in the singular limit . also shown are the tr@xmath179 and hb@xmath174 curves , which remain close to each other in the @xmath285 plane and enclose a very thin wedge of spiking solutions in the parameter space . thus , in the singular limit , the bursting region is enclosed by the toral fsn ii and tr@xmath179 curves ( figure [ fig : phtr](b ) ; shaded region ) . moreover , the curve tr@xmath176 of torus bifurcations that unfolds from the toral fsn ii curve forms the boundary between amb and amplitude - modulated spiking rhythms . that is , the closer the parameters are to the tr@xmath176 curve , the more oscillations in the envelope of the amb trajectories and hence the longer the burst duration . similarly , the spiking solutions that exist near the toral fsn ii curve exhibit amplitude modulation . the numerical computation and continuation of the curve of toral fsns of type ii requires careful numerics ; it requires solutions of a periodic boundary value problem subject to phase and integral conditions . we outline the procedure in appendix [ app : tfscurve ] . we saw from figure [ fig : pheigenvalues](a ) that the ph model supports tfn - type torus canards for @xmath264 m @xmath265 m . figure [ fig : phtr ] , however , shows that the amplitude modulated bursting exists on a more restricted interval of @xmath159 . this indicates the importance of the global return mechanism in shaping the outcome of the torus canard - induced mixed - mode dynamics . the bifurcations that separate the different amplitude - modulated waveforms are currently unknown and left to future work . having established the predictive power of our analysis of torus canards in @xmath1 , we now examine the connection between our analysis and prior work on torus canards in @xmath0 , namely in 2-fast/1-slow systems . in section [ subsec : tcexpmodels ] , we carefully study the transition from tonic spiking to bursting via amplitude - modulated spiking in the morris - lecar - terman system for neural bursting . we show that the boundary between spiking and bursting is given by the toral folded singularity of the system . we further demonstrate the power of our theoretical framework by tracking the toral folded singularities in the hindmarsh - rose ( section [ subsec : tchr ] ) and wilson - cowan - izhikevich ( section [ subsec : tcwci ] ) models . we note that the analysis in this section relies on the results in section [ sec : arbitrarydimensions ] , namely theorem [ thm : averagingarbitraryslow ] , which extends the averaging method for folded manifolds of limit cycles to slow / fast systems with two fast variables and an arbitrary number of slow variables . our theoretical framework also allows one to determine the parameter values for which a torus canard explosion occurs in 2-fast/1-slow systems ( theorems [ thm:3daveraging ] and [ cor : explosion ] ) . this predictive power is illustrated in appendix [ app : tcexplosion ] in the case of the forced van der pol equation . the morris - lecar - terman ( mlt ) model @xcite is an extension of the planar morris - lecar model for neural excitability in which the constant applied current is replaced with a linear feedback control , @xmath48 . the ( dimensionless ) model equations are @xmath291 where @xmath9 is the ( dimensionless ) voltage , @xmath292 is the recovery variable , and @xmath48 is the ( dimensionless ) applied current . the steady - state activation functions are given by @xmath293 and the voltage - dependent time - scale , @xmath294 , of the recovery variable @xmath292 is @xmath295 following @xcite , we treat @xmath15 and @xmath296 as the principal control parameters , and fix all other parameters at the standard values listed in table [ tab : mlt ] . .standard parameter set for the mlt model . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ] using table [ tab : fvdp ] , we find that the expression for @xmath297 simplifies greatly and the condition for the torus canard explosion reduces to @xmath298 recall that the actual analytic result is @xmath299 ( plus an exponentially small correction ) . thus , in the case of the fvdp oscillator , theorem [ thm:3daveraging ] and corollary [ cor : explosion ] give the location of the torus canard explosion ( for @xmath300 ) correct up to exponentially small error . this research was partially supported by nsf - dms 1109587 . i would like to thank tasso kaper , mark kramer , jonathan rubin , and martin wechselberger for helpful discussions . i am particularly grateful to tasso kaper and mark kramer for their careful and critical reading of the manuscript . i am especially indebted to tasso kaper for being an excellent sounding board for my ideas throughout the development of this project . j. burke , m. desroches , a. granados , t. j. kaper , m. krupa , and t. vo , _ from canards of folded singularities to torus canards in a forced van der pol equation _ , j. nonlinear sci . , * 26 * ( 2016 ) , pp . 405451 . e. j. doedel , a. r. champneys , t. f. fairgrieve , y. a. kuznetsov , k. e. oldeman , r. c. paffenroth , b. sanstede , x. j. wang and c. zhang , _ auto-07p : continuation and bifurcation software for ordinary differential equations _ , available from : http://cmvl.cs.concordia.ca/ a. politi , l. d. gaspers , a. p. thomas , and t. hfer , _ models of 3 and ca@xmath302 oscillations : frequency encoding and identification of underlying feedbacks _ , biophys . j. , * 90 * ( 2006 ) , pp . 31203133 . roberts , j. rubin , and m. wechselberger , _ averaging , foided singularities and torus canards : explaining transitions between bursting and spiking in a coupled neuron model _ , siam j. appl . * 14 * ( 2015 ) , pp . 18081844 . r. roussarie , _ techniques in the theory of local bifurcations : cyclicity and desingularization _ , in `` bifurcations and periodic orbits of vector fields '' ( ed . d. szlomiuk ) , kluwer academic , dordrecht ( 1993 ) , pp . 347382 .

[ sec : abstract ] torus canards are special solutions of slow / fast systems that alternate between attracting and repelling manifolds of limit cycles of the fast subsystem . a relatively new dynamic phenomenon , torus canards have been found in neural applications to mediate the transition from tonic spiking to bursting via amplitude - modulated spiking . in @xmath0 , torus canards are degenerate : they require one - parameter families of 2-fast/1-slow systems in order to be observed and even then , they only occur on exponentially thin parameter intervals . the addition of a second slow variable unfolds the torus canard phenomenon , making them generic and robust . that is , torus canards in slow / fast systems with ( at least ) two slow variables occur on open parameter sets . so far , generic torus canards have only been studied numerically , and their behaviour has been inferred based on averaging and canard theory . this approach , however , has not been rigorously justified since the averaging method breaks down near a fold of periodics , which is exactly where torus canards originate . in this work , we combine techniques from floquet theory , averaging theory , and geometric singular perturbation theory to show that the average of a torus canard is a folded singularity canard . in so doing , we devise an analytic scheme for the identification and topological classification of torus canards in @xmath1 . we demonstrate the predictive power of our results in a model for intracellular calcium dynamics , where we explain the mechanisms underlying a novel class of elliptic bursting rhythms , called amplitude - modulated bursting , by constructing the torus canard analogues of mixed - mode oscillations . we also make explicit the connection between our results here with prior studies of torus canards and torus canard explosion in @xmath0 , and discuss how our methods can be extended to slow / fast systems of arbitrary ( finite ) dimension . * * keywords**torus canard , canard , geometric singular perturbation theory , folded singularity , averaging , bursting , spiking , amplitude - modulation , torus bifurcation * * ams subject classifications**34e17 , 34c29 , 34c15 , 37n25 , 34e15 , 37g15 , 34c20 , 34c45

the tucson - melbourne ( tm ) three - nucleon force due to two - pion exchange has a structure which , after an expansion of the invariant @xmath4n amplitudes in the inverse nucleon mass , was determined by the original implementation of chiral symmetry in the underlying @xmath4n scattering amplitude . given that structure , the strength constants ( the @xmath0 , @xmath1 , @xmath2 , and @xmath3 coefficients ) are then not free parameters but depend upon the @xmath4n scattering data base , which has improved greatly since the original determination of these coefficients . in this note , we review two recent developments in three - body force studies : i ) a critical analysis of the generic structure of a 2@xmath4 exchange three - body force ( tbf ) @xcite , and ii ) the new tm strength constants derived from invariant @xmath4n amplitudes @xcite corresponding to the contemporary data base which includes measurements taken at the meson factories since 1980 . we make updated tbf s of the tucson - melbourne type which reflect one or both developments , add them to a nn force , and calculate properties of the triton in order to see the effect of these developments in a simple nuclear system . to begin , we display the tucson - melbourne force ( leaving out an overall momentum conserving delta function ) : @xmath5 + ( i \vec { \tau}_3 \cdot \vec { \tau}_1 \times \vec { \tau}_2 ) d ( i \vec { \sigma}_3 \cdot { \vec q } \times { \vec { q ' } } ) \right\ } \ , , \label{eq : wpipi } \nonumber\\\end{aligned}\ ] ] where @xmath6 and @xmath7 and the pion rescatters from nucleon 3 . ( we refer the reader to refs . @xcite for diagrams , more extensive definitions , explanations of the other two cyclic terms , etc . needed for calculation but not directly relevant to the present discussion ) . now we review briefly the origin of this equation . the approach used in the tucson - melbourne ( tm ) family of forces is based upon applying the ward identities of current algebra to axial - vector nucleon scattering . the ward identities are saturated with nucleon and @xmath8 poles . then employing pcac ( partial conservation of the axial - vector current ) , one can derive expressions for the on - mass - shell pion - nucleon scattering amplitudes @xcite which map out satisfactorily the empirical coefficients of the hhler subthreshold crossing symmetric expansion based on dispersion relations @xcite and , after projection onto partial waves , describe the phase shifts reasonably well @xcite . the off - mass - shell extrapolation ( needed for the exchange of virtual , spacelike pions in a nuclear force diagram ) is trivial for the @xmath3 coefficient . it can be taken directly from the on - mass - shell theoretical or empirical amplitude @xmath9 since they coincide so closely ( see appendix a of ref . one can treat this coefficient more elaborately @xcite , but the result is the same . on the other hand , one really needs an off - shell @xmath10 amplitude for the important @xmath0 , @xmath1 , @xmath2 structure of eq . ( [ eq : wpipi ] ) . this structure relies on the fact that the off - pion - mass - shell amplitude @xmath11 can be written in a form which depends on measured on - shell amplitudes only . this rewriting of the pcac / current algebra amplitude exploits a convenient correspondence between the structure of the terms corresponding to spontaneously broken chiral symmetry and the structure of the model @xmath12 term . to see this , we note that the nonspin flip @xmath13-channel isospin even amplitude ( covariant nucleon pole term removed ) is @xmath14 where @xmath15 is the pion - nucleon @xmath15 term , @xmath16 mev , and the invariant amplitude @xmath17 is given in units of the charged pion mass ( 139.6 mev ) . the double divergence @xmath18 of the background axial vector amplitude denoted by @xmath19 contains the higher order @xmath12 isobar contribution . in general , @xmath20 must have the simple form @xcite @xmath21 on the other hand , the assumed form of the function @xmath22 , @xmath23 ( adapted @xcite for @xmath4n scattering from the @xmath24 generalization of the weinberg low energy expansion for @xmath25 scattering ) is such that @xmath26 satisfies soft pion theorems ( for a review see ref . @xcite ) , and ( with the aid of eq . ( [ eq : cexpa ] ) ) the constraint at the ( on - shell and measurable ) cheng - dashen point : @xmath27 the value of @xmath28 can be determined by taking the amplitude on - shell and comparing with on - shell data extrapolated into the subthreshold region @xcite , but it is not needed , as we will now demonstrate . neglecting the @xmath29 and @xmath30 terms in ( [ eq : fampli ] ) because they are of the order of @xmath31 or higher , the @xmath26 amplitude can be expanded in the three - vector pion momenta @xmath32 and @xmath33 as follows : @xmath34 the last equation explicitly exhibits the separation between the ( higher order in @xmath35 ) @xmath12 contribution contained in the @xmath36 term alone and the remaining chiral symmetry breaking terms . in ref . @xcite and subsequent discussions of the tm @xmath37 force , the @xmath36 and @xmath28 constants in the coefficient of the @xmath38 term were eliminated in favor of the on - shell ( measurable ) quantity @xmath39 @xmath40 from the expanded @xmath4n amplitude @xmath26 in conjunction with the @xmath4nn vertices @xmath41 and pion propagators , one constructs the three body force of eq . ( [ eq : wpipi ] ) . comparing eqs . ( [ eq : wpipi ] ) and ( [ eq : expan ] ) , ( @xmath42 and @xmath43 so that @xmath44 @xcite ) one sees that @xmath45 the @xmath12 constant @xmath36 contributes then to the overall coefficient b " that has been used in nuclear calculations ( @xmath46 ) @xmath47 \label{bcoef}\ ] ] finally the @xmath2-term of eq . ( [ eq : wpipi ] ) is given by @xmath48 the dominant part of @xmath2 comes from our ansatz eq.([eq : funf ] ) but a small part is due to the backward - propagating nucleon term @xmath49 ( z - graph " ) @xmath50 . this term ( which also appears in the @xmath3 coefficient ) is representation dependent and is the only local term of a consistent set of 15 terms derived some time ago @xcite . we note that the term proportional to @xmath51 did not appear before in eq . ( [ eq : expan ] ) . this term nevertheless is inserted in @xmath2 because both the backward - propagating part of the nucleon pole @xmath52 and the @xmath12 couple with the pion with a ( assumed the same ) form factor @xmath53 which is defined as @xmath54 . the chiral breaking @xmath15 term has no intrinsic @xmath55 dependence ( although it is multiplied by @xmath56 ) . it is convenient , if not necessary , however , since part of the amplitude is due to @xmath52 and @xmath19 , to multiply the final amplitude by form factors , dependent upon @xmath55 and @xmath57 . consequently , the constant term ( @xmath58 , labeled a " in the literature ) attains a spurious momentum dependence from the form factors . the term proportional to @xmath51 in eq . ( [ ccoef ] ) is inserted to correct for this spurious momentum dependence to the orders in @xmath55 and @xmath59 kept in the amplitude . the new development in the structure of a 2@xmath4 exchange tbf @xcite lies in another look at the decomposition of the @xmath2-term made originally @xcite to fourier transform eq . ( [ eq : wpipi ] ) , but true in general . begin with the schematic structure @xmath60 and rewrite it ( neglecting the isospin dependence in eq . ( [ wc ] ) ) as @xmath61 thus the @xmath2-term can be decomposed into a @xmath62-exchange term with the same operator structure as the @xmath0-term plus a short - range @xmath4-range term . without a form factor @xmath63 the short - range part would be a dirac delta function a zero - range or contact term . this operator structure is reflected in the coordinate space representations where one always finds the coefficient @xmath64 multiplying derivatives of two coordinate space yukawas " : see , for example , eqs . 3.9 - 3.11 of ref . @xcite or appendix a of ref . @xcite . without a form factor @xmath63 the short - range part would be a dirac delta function a zero - range or contact term . the tucson - melbourne force has an ( unadorned by @xmath65 ) @xmath2 coefficient multiplying a derivative of a product of a delta function and a coordinate space yukawa " as is easily seen in the same equations . it was the latter , rather singular , aspect of the tucson - melbourne force which made numerical work difficult in both coordinate space and momentum space ( the operator structure is the same ) . in addition , the recent trend toward a low mass cutoff @xmath66 in @xmath67 for pion exchange highlights the point already emphasized by the hokkaido group @xcite and , in the modern context , by the so paulo group @xcite . the contact terms ( those proportional to a coordinate - space @xmath68-function and its derivatives ) are spread out with increasing importance as @xmath66 becomes smaller and the ( strong interaction ) size of the nucleon grows . these groups contended that these contact terms , bringing the nucleon structure signature , should not be included in potential models . the subject of contact terms has been revived recently with the advent of effective field theories in which contact terms are used to emulate the short distance physics , and the long distance physics , including the physics of chiral symmetry , is retained explicitly . in these effective field theories ( chiral perturbation theory extended to two or more nucleons @xcite ) contact terms abound , both in the chosen chiral lagrangian and in the nucleon potentials . adapting a field redefinition technique first used in pion condensation @xcite , friar _ et al . _ @xcite were able to demonstrate , via a field theoretic calculation with an effective chiral lagrangian , why the contact term of eq . ( [ decomp ] ) does not appear in the @xmath62-three body force of chiral perturbation theory , even though that field theory can be transformed to emulate the soft pion theorems . in sum , although chiral symmetry in the form of pcac / current algebra motivated the ansatz eq . ( [ eq : funf ] ) which led to the operator structure of eq . ( [ decomp ] ) , chiral symmetry in the form of effective field theory dictates that only the @xmath62-exchange part ( @xmath69 for tm ) should be retained in a tbf from pion exchange . one moral which can be drawn from this new insight is that chiral constraints on the off - shell scattering amplitude are not enough to determine a three - nucleon force ; one must also satisfy chiral constraints on the on - shell three - nucleon @xmath70-matrix elements which are presumed to make up the force . this observation applies to other off - shell amplitudes embedded in nuclear force models @xcite . the removal of the spurious contact term from the tucson - melbourne force leaves a tbf with coefficients @xmath71 , @xmath1 , and @xmath3 which has been termed tm@xmath72 in ref . @xcite and subsequent works . in the following section we will examine the effects in the triton of the original tm tbf and the tm@xmath72 tbf . we consider tm and tm@xmath72 with the original strength constants and with strength constants from the current @xmath4n scattering data . we employ a variational monte carlo method developed for accurate numerical calculations of light nuclei @xcite . the urbana - type potentials " , suited to this variational approach , take the form of a sum of operators multiplied by functions of the interparticle distance . following our previous study of charge symmetry breaking in light hypernuclei @xcite , and in order to compare with other tbf studies @xcite , we use the reid soft core nucleon - nucleon potential in the form of the urbana - type reid @xmath73 potential @xcite . the reid @xmath73 is a simplified ( the sum of operators is truncated from a possible 18 @xcite to 8 operators ) @xmath74 force model which is equivalent to the original reid soft core nucleon - nucleon potential in the lower partial waves and can produce the dominant correlations in s - shell nuclei . to be specific , the reid @xmath73 is obtained from the reid soft core ( rsc ) potential in the singlet states @xmath75 and @xmath76 and the triplet states @xmath77 and @xmath78 . the binding energy of the triton , calculated with exact faddeev codes which include all partial waves @xmath79 ( 34 channels ) , is -7.59 mev for the reid @xmath73 ( as quoted in table iv of @xcite ) , to be compared with -7.35 mev obtained with the original rsc @xcite . this small discrepancy , presumably due to differences in the @xmath80-waves of the two potentials , should not affect our conclusions . the variational method we use , with monte carlo evaluations of the integrals , is described in ref . @xcite ( see also , ref . @xcite ) . here we specify only the _ differences _ from the equations in these references . in particular , the trial nuclear wave functions have the following structure : @xmath81 { \bf s}\left[\prod^{a}_{i < j}f_{ij}\right]\phi,\ ] ] where @xmath82 is an antisymmetric spin - isospin state , having appropriate values of total spin and isospin , with no spatial dependence , and @xmath83 is a symmetrization operator which makes 3 ! terms for the two - body correlation operator @xmath84 and one term for the three - body correlation operator @xmath85 . the nn correlation operator is @xmath86 and the triplet correlation induced by the three - body force has the usual linear form suggested by the first order perturbation theory @xcite @xmath87 where @xmath28 is a variational parameter . these pair correlations do not include the spin - orbit correlations described in ref . @xcite , nor do our triplet correlations include the more sophisticated three - body correlations introduced by arriaga _ et al . _ @xcite which reduce the difference between the variational upper bound and the faddeev binding energy of the triton to less than 2% . both improvements would be clearly desirable , but are beyond the scope of this preliminary investigation . we do , however , include the usual central three - body correlation @xmath88 multiplied by the correlation functions(@xmath89 , @xmath90 , @xmath91 , @xmath92 and @xmath93 ) : @xmath94\ ] ] with @xmath95 . with these correlations we get a binding energy of -7.28(3 ) mev with the reid @xmath73 alone , a number which compares favorably with variational results in table v of ref . @xcite , obtained with a slightly different trial wave function . we now demonstrate that our variational calculations track the faddeev results of ref . @xcite and suggest that the main outline of our results ( to be presented later ) will reflect the properties of the hamiltonians chosen , provided that the potentials are not too singular . the faddeev calculations we now examine used the rsc potential and the early parameters ( labeled tm(81 ) here ) of the tucson - melbourne tbf ( @xmath96a = + 1.13 , @xmath97b = -2.58 , @xmath97c = 1.00 , and @xmath97d = -0.753 in units of the charged pion mass : 139.6 mev ) obtained from an interior dispersion relation ( idr ) analysis of phase shifts circa 1973 @xcite . ten years ago there was little reason to look suspiciously at the @xmath2-term , and the goal of the exercise was to test the perturbative nature of the @xmath10 amplitude @xmath98-wave terms . to this end , a restricted model was chosen with @xmath99 , the faddeev eigenvalues calculated for a 34-channel solution for rsc / tm , and the solution tested by employing the resulting wave functions in a raleigh - ritz variational calculation . the variational result for this restricted hamiltonian coincided with the faddeev eigenvalue , indicating the high quality of the faddeev wave function . then the @xmath0 and @xmath2 terms were selectively set to their assigned values and the variational calculation was repeated . comparison of the results shows the non - perturbative role of the @xmath0 and the @xmath2 term on the triton wave function . to test our codes and to suggest that our methods can give insight into triton binding energy effects from the proposed redefinitions of the tm force , we made a parallel set of calculations with the reid @xmath73 and the old tm force , tm(81 ) , with the parameters given above . the results are shown in table 1 . we follow tradition and calculate the triton properties with the cutoff in the form factor @xmath100 . in the publications of the tucson - melbourne group @xmath101 has been recommended to match the goldberger - treiman discrepancy @xcite , another measure of chiral symmetry breaking @xcite . the value @xmath102 matches the goldberger - treiman discrepancy @xmath103 of the recent determinations of the @xmath4nn coupling constant @xmath104 @xcite . we do nt know the reason others have chosen @xmath105 as a test case but adopt it anyway . please notice from eq . ( [ ccoef ] ) that @xmath2 , and therefore @xmath106 , changes with different values of @xmath66 . from eq . ( [ ccoef ] ) , we see that @xmath107 , because the value of @xmath108 varies only between @xmath109 and @xmath110 as @xmath100 in eq . thus , the dependence of the value of @xmath106 with @xmath111 is slight , compared with the overall effect of the cutoff on the @xmath37 force . the results of our calculations are presented in figure 1 as the open circles and open squares . the plotted points include monte carlo error bars and the lines through the symbols are drawn to guide the eye . the open circles show the calculated triton binding energy with reid @xmath73/tm@xmath72(93 ) which has no short - range @xmath4-range term and the strength constants taken from twenty year old @xmath4n scattering data . the open squares indicate the results with the same nn potential and the updated strength constants of tm@xmath72(99 ) . each calculation was made variationally with the full hamiltonian with strength constants shown in table 2 . we indicate our calculated value of the binding energy of the triton with the reid @xmath73 alone ( @xmath112 mev ) by a horizontal ( sparse ) dotted line and the faddeev eigenvalue ( @xmath113 mev ) by the horizontal ( dense ) dotted line . our variational upper bounds are always above the corresponding faddeev eigenvalues . we compare our results with calculations in the literature with the old tbf tm(81 ) , where the lack of a prime means that the short - range @xmath4-range term is _ included_. we do not present our own variational estimates with this short - range @xmath4-range term included as they do not reflect the true situation ( see discussion of table 1 ) . the results of the combination rsc / tm(81 ) for the three cutoffs @xcite ( already quoted in table 1 for the cutoff @xmath114 ) are given by the points with an @xmath115 . another faddeev evaluation @xcite of the same hamiltonian ( rsc / tm(81 ) ) is shown as stars at the three values of @xmath116 and the short dashed line interpolates between the calculated values . the models tm@xmath72(93 ) and tm@xmath72(99 ) with the spurious short - range @xmath4-range tbf removed ( open circles and open squares ) give very similar binding energies in our calculation . the updating of the strength constants seems to have very little effect on the three nucleon bound state , once the spurious term is removed . it is difficult to estimate the effect of removing the short - range @xmath4-range force on the binding energy with the results available in figure 1 , because both the nn potential ( reid @xmath73 versus rsc ) and the tbf ( tm(81 ) and tm(93 ) ) are slightly different . however , once this spurious force is removed the two models tm@xmath72(93 ) and tm@xmath72(99 ) have a similar dependence upon @xmath66 ; those two curves are shifted vertically only slightly . it is noteworthy that the dependence upon @xmath66 is greater if the spurious short - range @xmath4-range term is included in the tbf @xcite ; and significantly greater for the momentum space calculations of ref . one would expect this as @xmath66 increases and the singular term ( in one nn separation ) becomes more like a delta function . it is a nice feature that removal of the spurious term makes the tucson - melbourne two - pion exchange force less sensitive to the cutoff .

we introduce new values of the strength constants ( i.e. , @xmath0 , @xmath1 , @xmath2 , and @xmath3 coefficients ) of the tucson - melbourne ( tm ) 2@xmath4 exchange three nucleon potential . the new values come from contemporary dispersion relation analyses of meson factory @xmath4n scattering data . we make variational monte carlo calculations of the triton with the original and updated three - body forces to study the effects of this update . we remove a short - range @xmath4-range part of the potential due to the @xmath2 coefficient and discuss the effect on the triton binding energy .

early - type stars are themselves modest x - ray emitters , with a mean output in the 0.110 kev band of only @xmath2 of their optical / uv flux . thus , even a population of a million ob stars , typical of that found in a galaxy undergoing a vigorous starburst , will only produce an x - ray luminosity of @xmath3 . a single young neutron star similar to that in the crab nebula or a single o - star binary with a neutron star or black hole companion will thus outshine the entire population of main sequence stars . in order to determine the expected contribution of a young stellar population to the x - ray luminosity of a galaxy , then , it is necessary to estimate accurately the specific x - ray luminosity per o star , most of which comes from the deceased segment of the population . such an exercise is of interest in light of the relatively high x - ray luminosities of starburst galaxies and the potential contribution of such objects to the cosmic x - ray background . we attempt here a systematic , empirical census of the direct contributions ob stars , their lighter siblings , and their stellar remnants make to the hard ( 210 kev ) x - ray luminosity of a starburst galaxy by calculating the specific x - ray luminosity per o - star in the solar neighborhood , the galaxy as a whole , and the other members of the local group . we begin ( 2 ) with a cautionary tale concerning the calculation of long - term , mean x - ray luminosities for high - mass x - ray binaries by surveying the literature on the most luminous such system in the local group , smc x-1 . in 3 , we compile a list of all high - mass , accretion - powered x - ray binaries that lie within 3 kpc of the sun and , using the rossi x - ray timing explorer ( _ rxte _ ) all - sky monitor database and other archival data , compute the total x - ray luminosity within this volume arising from such systems . using a recent census of ob stars in the solar vicinity , we then calculate the specific x - ray luminosity per o star from accreting systems . this local estimate is then compared to that for the galaxy as a whole . the following section ( 4 ) repeats this analysis for the other local group galaxies , concluding with a commentary on the reported dependence of this value on metallicity . in 5 , we consider the other contributions an ob star population makes to the integrated x - ray luminosity of a galaxy from associated t tauri stars , stellar winds , and supernovae . we then go on ( 6 ) to assess the fractional contribution these direct sources of x - ray emission make to the total hard x - ray luminosity of starburst galaxies . we conclude with a summary of our results , a brief discussion of additional possible contributions to the x - ray luminosity of starburst galaxies , and the implications of these results for the origin of the cosmic x - ray background . the high - mass x - ray binary ( hmxb ) smc x-1 , the only persistent , bright , accretion - powered source in the small magellanic cloud ( smc ) , consists of a b0 supergiant primary accompanied by a neutron star with a 0.7 s pulse period in a 3.9 d orbit . the system is the most luminous x - ray binary in the local group , and is frequently cited as `` superluminous , '' given that its nominal x - ray luminosity exceeds the eddington limit for a @xmath4 neutron star . but just what is the mean , integrated luminosity of smc x-1 as observed from earth ? the most complete recent catalog of x - ray binaries is that of van paradijs ( 1995 ) which lists , among other system parameters , a maximum and minimum ( if available ) reported flux density for each source . in the case of smc x-1 , these values are 57 and 0.5 @xmath5jy . the incautious reader might adopt either the maximum value ( especially for the majority of sources in the catalog for which only one value is given ) , or simply average the two numbers to estimate the mean x - ray luminosity of the source . in fact , to convert these values to a mean observed x - ray luminosity requires adoption of a distance to the smc , a mean spectral form for the x - ray emission , and a bandwidth over which the emission is integrated , as well as a description of the temporal behavior of the source . compiling the variety of assumptions actually adopted in the literature is instructive : 45 kpc @xmath6 70 kpc ( howarth 1982 ; seward & mitchell 1981 ) ; @xmath7 ; ( angelini , white , & stella 1991 ; coe et al . 1981 ) ; @xmath8 @xmath9 ( kahabka & pietsch 1996 ; davison 1977 ) ; and 0.22.4 kev to 2100 kev ( kahabka & pietsch 1996 ; coe et al . 1981 ) . as a consequence , quoted luminosities range from @xmath10 erg @xmath11 ( seward & mitchell 1981 ) to @xmath12 ergs @xmath11 ( price et al . 1971 ) ; far from all of this uncertainty results from the source s intrinsic variability . furthermore , some reports undertake systematic data editing that bias the flux estimates upward leaving out data during the 16% of the time the x - ray source is in eclipse , ignoring periods when the source is at an undetectable level for a given instrument , etc.which , while usually well - documented and appropriate for the task at hand , require reversal when attempting to define the source s mean contribution to its galaxy s x - ray luminosity . to continue with this example , we adopt a distance to the small cloud of 65 kpc and , for smc x-1 itself , we employ spectral parameters @xmath13 , @xmath14 ( angelini et al . 1991 ) and a cutoff energy of 6.5 kev ; while these parameters ignore a reported soft ( _ kt _ @xmath15 kev ) component , they suffice to illustrate our point . correcting the mean fluxes for smc x-1 reported over monitoring times of weeks to a decade by angelini et al . ( 1991 ) , wojdowski et al . ( 1998 ) , whitlock & lochner ( 1994 ) , gruber & rothschild ( 1984 ) , and levine et al . ( 1996 ) to the 210 kev band , we find _ all _ are consistent within two sigma with the value of @xmath16 ergs @xmath11 , or 17 @xmath17 , a factor of 5 below the highest value in the literature , and a factor of 3.5 below the cataloged flux . this value is below the eddington limit for a neutron star mass of @xmath18 even before correcting for the lower metallicity of the accreting material in the smc ; more importantly , however , it represents the appropriate value to adopt in any summation of the integrated x - ray luminosity of the smc ob - star population ( see 4.2 ) and illustrates the need for caution when undertaking such a task . we begin by examining in detail the populations of hmxbs and ob stars where our information is most complete within 3 kpc of the sun . in table 1 , we list all 57 accretion - powered x - ray binaries with high - mass stellar primaries ever reported in the literature as lying within this distance . much of the data are taken from the catalog of van paradijs ( 1995 ) ; a survey of the literature in the intervening five years has been used to bring the list up to date . the first column contains the original source name , followed by a vernacular name ( if any ) . columns 24 list the optical counterpart name and j2000 coordinates . these have been taken from the hipparcos / tycho catalogs ( perryman et al . 1997 ; hg et al . 1997 ) if available ( see also chevalier & ilovaisky 1998 ) , and elsewise from the guide star catalog , or from the highest precision position reported in the literature . for sources without confirmed optical identifications , the best available x - ray coordinates are given , and the star listed is the brightest object in the error circle . the source of the position is given in column 5 ; if the positional uncertainty is greater than @xmath19 and/or the identification is uncertain , its value is given in parentheses . columns 6 and 7 give the star s visual magnitude and spectral type ; these are taken from the hipparcos or tycho catalogs when available , and otherwise from the literature . the source distances follow ( cols . 810 ) ; they include the smallest distance reported in the literature , a best estimate ( based in a few cases on hipparcos parallaxes , but mostly on a qualitative assessment of the literature ) , and the maximum plausible distance . for stars with hipparcos observations , the 1.5 @xmath20 lower limit is given . if a distance upper limit is greater than 3.0 kpc , it is quoted as `` @xmath21 kpc '' , and the source s contribution to the local x - ray binary luminosity is computed as if it were at 3 kpc for purposes of calculating an upper limit to this value . clearly , if the source lies at a greater distance , its inferred luminosity would be higher , but its contribution to the quantity of interest is zero , making this a conservative approach to calculating an upper limit to the x - ray luminosity of the population as a whole . the remaining columns of the table report source fluxes and luminosity measurements . when spectral parameters are provided in the literature , they have been used to correct the observed flux to the 210 kev band . when only an instrumental flux and bandwidth are quoted , we have adopted a power - law spectral form with a photon index of 1.0 and a plausible column density for the adopted distance(s ) ( using @xmath22 unless excess extinction is indicated ) . our results are not sensitive to this specific choice of parameters : varying the power law index over the range @xmath23 and the column density from @xmath24 @xmath9 changes the inferred luminosities by @xmath25 . since our goal is to calculate the best available long - term mean integrated flux for each source , we have utilized the _ rxte _ all - sky monitor ( asm ; levine et al . 1996 ) light curves when available ; more than half the sources have such light curves with largely continuous coverage ( 80% to 98% ) over more than 1650 days . for each of these , we include the number of days in the light curve for which no flux is available , the global mean flux , the mean flux plus 2 @xmath20 ( since many sources are not detected on most days , this value is useful as a 2 @xmath20 upper limit ) , the number of days the source flux exceeded 4 @xmath20 , the mean flux on those days , and the first and tenth brightest daily mean fluxes in the 4.5-year interval ; the latter values are included in order to ascertain whether or not one , or a few , large outbursts dominate the time - integrated luminosity . the 26 sources not included in the asm database have never been significantly above the asm threshold at any time throughout the last 4.5 years ( r. remillard , private communication ) . for these sources , we adopt an upper limit of 1.0 asm ct @xmath26 and calculate the luminosity limit for each source as described above . this limit is very conservative . as shown below , the mean asm flux value for a truly absent source ( smc x-3 , for example ) over the asm monitoring interval is @xmath27 ct s@xmath1 . the fact that all of the 31 regularly monitored sources have mean values a factor of five or more above this limit suggests that they are often present just below the detection threshold . but there is no such evidence for those sources which have never crossed the asm detection threshold , suggesting that a reasonable upper limit to their contribution could be at least an order of magnitude lower . in addition , for all sources , we have searched the high energy astrophysics science archive research center ( heasarc ) x - ray binary catalog which archives all observations of x - ray binaries in the center s large collection of databases . for each source , we list the number of detections ( col . 18 ) , the maximum count rate and , in column 21 , the catalog from which the count rate was taken . in no case does the integrated luminosity in a major outburst exceed the integrated luminosity over 30 years derived from the asm 4.5-year averages or our conservative upper limits thereto . as can be seen from the final line in table 1 , our best estimate for the integrated , mean 210 kev luminosity of the hmxb population within 3 kpc of the sun is @xmath28 . roughly one - third of this total comes from the black hole system cyg x-1 , one third from a handful of neutron star binaries such as vela x-1 and 4u170037 , and the final third from the upper limits adopted for the 26 sources not detected in the asm . the estimate is conservative , since it includes all the sources not detected in the asm as contributing at 1 ct s@xmath1 . furthermore , nine of the systems ( contributing 11% of the total flux ) have nominal distances beyond 3.0 kpc , but are included because their distance uncertainties allow membership in our volume - limited sample . if we take the 2 @xmath20 upper limits for _ all _ objects , the integrated value only increases by 40% . adopting the additional extreme assumption that all sources are at their maximum allowed distances still does not raise the conservative best estimate by a factor of two . while examination of the 30-year history of all sources does show much higher luminosities in some cases for brief intervals , there is no evidence to suggest that the last 4.5 years of asm data is in any way atypical . thus , we conclude that the hmxb population within the 38 kpc@xmath29 volume surrounding the sun produces a 210 kev x - ray luminosity of 23 @xmath30 . the final step in calculating the specific x - ray luminosity per o - star in the solar neighborhood is to find the number of o - stars within 3.0 kpc . we use the recent ( unpublished ) compilation of k. garmany ( private communication ) . she finds a total of 351 spectroscopically confirmed stars of types o3 through o9 out to 1.95 kpc from the sun ; the number in bins of constant projected area ( excepting a local minimum ) is roughly constant out to this distance , suggesting incompleteness is not a problem . in addition , there are 1915 b0b2 stars , plus a total of 772 stars with the colors of ob stars which lack spectroscopic types . adopting the same o / b ratio as for the classified stars ( 18% ) suggests as many as @xmath31 140 additional o stars should be added to the total . extrapolating with a constant surface density out to 3.0 kpc , then , yields a total of 1165 o stars within the volume . the specific accretion - powered x - ray luminosity per o star is @xmath32 , with a conservative upper limit ( 2 @xmath20 x - ray source upper limits , maximum distances , and no o stars in the unclassified portion of the stellar sample ) of @xmath33 . the census conducted here allows an estimate of the fraction of early type stars that eventually form x - ray binary systems . for example , there are 3038 stars of types o3 through b2 in the garmany compilation , implying a total of @xmath34 stars within the 3 kpc distance ( correcting for a modest incompleteness evident in the b0b2 star counts ) ; in this same region ( using best - estimate distances ) , we currently know of 24 hmxbs with primaries of these types , as well as 22 other accreting systems for which the spectral class is too poorly established to include them unambiguously . thus , @xmath35 of all the early - type stars are currently active accretion - powered x - ray sources . since the hmxb phase lasts for a few percent of an ob star s lifetime ( portegies - zwart & verbunt 1996 ) , @xmath36 of all ob stars must produce an hmxb . this is roughly consistent with population synthesis studies ( dewey & cordes 1987 ; meurs & van den heuvel 1989 ; dalton & sarazin 1995 ; lipunov , postnov , & prokhorov 1997 ; terman , taam , & savage 1998 ; portegies - zwart & van den heuvel 1999 ) , although the predictions of such calculations are quite sensitive to the assumed kick velocity imparted to neutron stars at birth . in our limited sample , at least , the fraction of x - ray active o stars is similar to that for b stars ; given their shorter lifetimes , the fraction of hmxbs produced must be larger , consistent with the notion that kick velocities become increasingly sucessful at unbinding binaries as the mass of the companion star decreases . note that these statistics include hmxbs with luminosities as low as @xmath37 erg s@xmath1 ; the fraction of systems with persistent luminosities @xmath38 erg s@xmath1 is an order of magnitude smaller . the integrated lyman continuum luminosity of the milky way is @xmath39 ( van den bergh & tammann 1991 ) . using table 5 of vacca ( 1994 ) for solar metallicity , a salpeter mass - function slope of 2.35 , and a mass upper limit of 80 @xmath40 implies a total galactic population of o stars of @xmath41 . this is @xmath42 greater than a straightforward extrapolation from the local population discussed above to the full galactic disk ( r = 12 kpc ) , consistent with the observed enhancement of star formation activity in the inner galaxy . it is also consistent with the claim of ratnatunga and van den bergh ( 1989 ) that the total pop i content of the galaxy is @xmath43 times that found in a 1 kpc@xmath44 area of the disk centered on the sun , and with an estimate ( van den bergh & tammann 1991 ) based on counts of embedded o stars from iras observations ( wood & churchwell 1989 ) . the uncertainty in the number of o stars is probably less than 50% . the total number of hmxbs in the galaxy is less well constrained . eight persistent sources are known with luminosities greater than @xmath45 ( see dalton & sarazin 1995 ) ; these produce a total _ peak _ x - ray luminosity ( see 2 ) of @xmath46 . other bright , unidentified x - ray sources in the galactic plane could add to this population ; dalton & sarazin s population synthesis model predicts 12 sources with @xmath47 and 43 sources with @xmath48 . tripling the four known sources with @xmath47 to match this prediction would yield a luminosity contribution of @xmath49 . integrating the dalton and sarazin predicted luminosity function down to @xmath50 and adding the be star population at a mean luminosity of @xmath51 yields a nominal hmxb luminosity for the galaxy of @xmath52 . this value is dominated by the luminous sources , as is observed to be the case in the solar neighorhood . adding the predicted flux from sources with @xmath53 to the observed @xmath54 of the eight bright sources yields a lower limit of @xmath55 for the galaxy s total @xmath54 . dividing the nominal value by the o star population derived above gives @xmath56 , the same as the upper limit on locally derived value and , again , uncertain by a factor @xmath57 . the specific luminosities derived above depend on a number of factors which could well be different in environments such as the nuclear starbursts to which we ultimately wish to apply our results . for example , metallicity can introduce a variety of effects : lower metallicity ( 1 ) increases the eddington luminosity of accreting sources by decreasing the x - ray scattering cross section of the infalling material , ( 2 ) lowers the mass of a star of a given spectral type ( and thus changes the conversion factor between the number of lyman continuum photons and the number of o stars ) , and , possibly , ( 3 ) changes the ratio of black holes to neutron stars formed in stellar collapse ( hutchings 1984 ; helfand 1984 ) . in order to explore the range of specific x - ray luminosities in different galactic environments , we have repeated the exercise of counting hmxbs and o stars in the four largest external members of the local group . the first x - ray sources discovered in the large magellanic cloud were detected with non - imaging rocket - borne instruments thirty years ago ( price et al . since then , systematic imaging surveys have been carried out by the _ einstein _ observatory ( long , helfand , & grabelsky 1981 ; wang et al . 1991 ) , _ exosat _ ( pakull et al . 1995 ; pietsch et al . 1989 ) , and _ rosat _ ( see haberl & pietsch 1999 , although the complete results have yet to be published ) . in addition , ten x - ray binary candidates have been monitored by the _ rxte _ asm , allowing us to calculate accurate mean fluxes on timescales of years . of the four bright , persistent accreting binaries in the lmc , one ( lmc x-2 ) is a low - mass system and does not concern us here . lmc x-1 and lmc x-3 are both strong black hole candidates ( hutchings et al . 1987 ; cowley et al . 1983 ) with steep x - ray spectra ( @xmath58 ; white & marshall 1984 ) ; lmc x-4 is a hmxb pulsar with a flat power - law index of @xmath59 ( kelley et al . 1983 ) . using these spectral parameters and the asm mean count rates calculated as above , the integrated 210 kev luminosity of these three sources is @xmath60 for an adopted distance of 50 kpc . in addition to these persistent sources , the original imaging surveys , high resolution images of 30 doradus ( wang & helfand 1991 ; wang 1995 ) , studies of variable sources in the _ rosat _ data ( haberl & pietsch 1999 ) , and monitoring observations by _ ariel v _ , _ rxte _ , _ cgro _ , etc . have led to the detection of an additional fourteen hmxb candidates in the large cloud . several of these have been confirmed as be - pulsar systems through the detection of x - ray pulses , although the majority have neither firm optical or x - ray confirmation of their identity . seven have been monitored by the asm for periods ranging from @xmath61 to @xmath62 days . while most of these have been detected on a few occasions , none has a mean flux in excess of 0.15 asm ct s@xmath1 ; for nominal spectral parameters of @xmath63 and @xmath64 , this corresponds to an upper limit of @xmath65 . of the remaining non - asm sources , none has ever been reported above a luminosity of @xmath66 for more than a single day in outburst . finally , there remain several dozen unidentified x - ray point sources from the _ einstein _ survey . although the majority of these are background interlopers , some could be additional hmxbs ; however , the integrated luminosity of the brightest ten sources is @xmath67 . an even larger number of ( mostly fainter ) point sources are to be found in the _ rosat _ survey , but , again , the majority will be interlopers , and the integrated luminosity of any lmc hmxbs will not affect our sums by more than a few percent . thus , we estimate the total accretion luminosity of the ob population in the lmc to be @xmath68 , or roughly half that of the milky way . kennicutt & hodge ( 1986 ) have derived the total lyman continuum flux from the integrated h@xmath69 luminosity of the lmc : @xmath70 . this value should be regarded as a lower limit owing to leakage of some lyman continuum photons from regions . oey & kennicutt ( 1998 ) estimate the leakage fraction ranges from @xmath71 to @xmath72 for a sample of 12 bright lmc regions ; we adopt their median value of 25% to correct the kennicutt & hodge estimate . from the integrated radio continuum flux , israel ( 1980 ) derives a value for @xmath73 of @xmath74 , which should be regarded as an upper limit owing to the nonthermal continuum radiation which has not been subtracted . we adopt @xmath75 as a conservative estimate . for a mean metallicity of one - third solar , the measured upper imf slope of @xmath76 ( massey et al . 1995 ) , and an upper mass cutoff of 80 @xmath40 ( both of which will be assumed throughout ) , vacca s ( 1994 ) tables provide an estimate for the total number of o stars in the large cloud of 5530 with an estimated uncertainty of @xmath72 . thus , the specific 210 kev x - ray luminosity is @xmath77 , a factor of 1.5 to 3 greater than those derived for the solar neighborhood and the galaxy as a whole . smc x-1 is the most luminous hmxb in the local group . as discussed in detail in 2 , however , its mean observed luminosity is not quite as extraordinary as is often implied . to complement all of the long - term studies cited above , we have used the _ rxte _ asm database to calculate its mean flux over the past 4.5 years as we have for the lmc and galactic binaries . using the spectral parameters quoted in 2 and a distance of 65 kpc , we find @xmath78 , completely consistent with the value found above from monitoring studies over the past three decades . early studies of smc x-1 with sas-3 ( clark et al . 1978 ) also led to the discovery of two other putative hmxbs in the smc at flux levels only a factor of @xmath79 lower . one of these , smc x-3 , has never been seen again , despite sensitive searches by imaging missions which reached flux levels nearly @xmath80 times lower . the other source , smc x-2 which had also disappeared a few months after its discovery ( clark , li , & van paradijs 1979 ) has been detected once more in the intervening 23 years by _ rosat _ at a level of @xmath81 ( adopting the spectral parameters of smc x-1 ) , although a subsequent observation with the same instrument failed to detect it at a level 650 times lower ( kahabka & pietsch 1996 ) . both sources have been monitored for the past 4.5 years by the _ rxte _ asm and have been detected with 4 @xmath20 significance on only one and three days , respectively ( roughly consistent with the number of such detections expected by chance , especially considering that the crowded region in which they reside raises the systematic uncertainties in daily flux determinations ) . their mean values are both consistent with zero , with @xmath82 upper limits of 0.05 asm ct @xmath26 or luminosities of @xmath83 . as with the lmc , a number of surveys and targeted observations with _ einstein _ and _ rosat _ , as well as monitoring observations by _ rxte _ and _ cgro _ have revealed several additional hmxbs and hmxb candidates in the small cloud . one new source , xte j0111.2 - 7317 has had a mean asm flux of 0.4 ct @xmath26 over the last 2 years , contributing a luminosity of @xmath84 during this interval . however , the source was not detected in either the _ einstein _ or _ _ surveys of the cloud at flux levels more than 100 times lower , so this is unlikely to represent an accurate estimate of its long - term mean luminosity . the other two hmxbs included in the _ rxte _ monitoring together contribute less than 20% of this luminosity , and the remaining thirteen candidates reported in the literature all have mean fluxes far below this level . finally , as with the lmc , the total number of remaining unidentified cloud members from the _ einstein _ ( seward & mitchell 1981 ; wang & wu 1992 ) and _ rosat _ ( kahabka & pietsch 1996 ; haberl et al . 2000 ) surveys would , if identified as hmxbs , increase the integrated luminosity of the population by only a few percent . thus , we estimate the total accretion luminosity of the ob population of the smc to be @xmath85 , or roughly equal to that for the lmc . it is important to note , however , that more than two - thirds of this luminosity arises in the singular system smc x-1 which , in addition to being the most luminous persistent hmxb in the local group , also contains the most rapidly rotating x - ray pulsar ( @xmath86 = 0.71 s ) , an object with a spin - up timescale of only 2000 years . in their detailed study of the spin and orbital evolution of smc x-1 , levine et al . ( 1993 ) estimate that the current high - luminosity phase of the binary s evolution will last at most a few times the pulsar spin - up time , or @xmath87 yr . compared with the @xmath88 yr main sequence lifetime of this 20 @xmath40 star , we have a chance of @xmath89 of seeing the system at this x - ray luminosity . since only @xmath90 of massive stars end up as short - period hmxbs ( portegies - zwart & van den heuvel 1999 ) , the number of expected systems in the smc is @xmath91 . thus , while it is not enormously improbable that we see smc x-1 at this luminosity , the long - term integrated x - ray luminosity of this galaxy is likely to be overestimated by a factor of several as a consequence of this one source s current strut upon the stage . we pursue this matter further below in discussing the putative dependence of a galaxy s x - ray luminosity on metallicity . we can estimate the total o star population for the smc in a manner exactly analogous to that used for the lmc . kennicutt & hodge ( 1986 ) report @xmath92 photons s@xmath1 from h@xmath69 data , while israel ( 1980 ) derives @xmath93 from the radio continuum emission ; using the same considerations cited in the previous section , we adopt @xmath94 photons s@xmath1 . for a metallicity of 0.1 solar , the tables in vacca yield an estimate of 1300 o stars . the resulting specific luminosity , then , is @xmath95 at the present time , although given the lifetime of smc x-1 and the arguments presented above , it is likely to be lower by a factor of @xmath96 on long timescales , making it more similar to , but still significantly in excess of , the values derived for the solar neighborhood , the galaxy , and the lmc . given its distance ( 720 kpc ) , individual x - ray binaries in m33 are not detectable by non - imaging or asm instruments , leaving the _ einstein _ ( long et al . 1981 ; markert & rallis 1983 ; trinchieri , fabbiano , & peres 1988 ) and _ rosat _ ( schulman & bregman 1995 ; long et al . 1996 ) surveys as our only views of its x - ray source population . the deepest image is that from the _ rosat _ pspc obtained by long et al . ( 1996 ) : 50 sources were detected within @xmath97 of the nucleus above a luminosity threshold of @xmath98 ergs s@xmath1 ( for our adopted spectral form of a power - law spectrum with @xmath99 and @xmath100but see below ) . five of the sources are identified with foreground stars , one is a background agn , and ten are positionally coincident with optically identified supernova remnants ; since the latter sources have mainly soft x - ray spectra , these associations are thought to be mostly correct . over 60% of the soft x - ray luminosity of the galaxy comes from a nuclear source which is unresolved with the _ rosat _ hri ( fwhm = @xmath101 ; schulman & bregman 1995 ) . the origin of this emission is unknown . there was evidence from the _ einstein _ data that the source is variable on timescales of days to months ( markert & rallis 1983 ; peres et al . 1989 ) ; more recently , dubus et al . ( 1997 ) claim evidence for a 20% modulation with a 106 d periodicity , although their result is not significant at the 3 @xmath20 level and the periodicity is apparently inconsistent with the _ einstein _ measurement ( see their figure 3 ) . the object s high x - ray luminosity in the soft band of the imaging experiments is in part a consequence of the source s soft spectrum . the asca observations of makishima et al . ( 2000 ) provide the most detailed spectral data in the harder x - ray band , and produce a luminosity estimate of @xmath102 . the unusual stellar content of the m33 nucleus ( oconnell 1983 ) and the absence of obvious signs of an agn at other wavelengths , has led to a variety of speculative notions concerning the nature of this source : an anomalous agn , a single black - hole hmxb , a cluster of hmxbs , intermediate - mass ( her x-1type ) binaries , lmxbs , and ( predictably ) a `` new '' type of x - ray source . while _ chandra _ observations will soon eliminate many of these options , it is at present unclear whether some or all of this source s luminosity should be charged to the ob population s accretion account . we calculate the specific luminosity for m33 both including and excluding this contribution . as for the remaining 33 x - ray sources with @xmath103 , one ( the third brightest ) is known to be an eclipsing binary pulsar ( dubus et al . 1999 ) . in the somewhat unlikely event that all 32 remaining sources also are hmxbs , we can estimate the integrated 210 kev luminosity from the _ exosat _ observations reported in gottwald et al . the non - imaging me detector s field of view includes all the x - ray emission from m33 . the me count rate was @xmath104 ct s@xmath1 in the 16 kev band and , while not a good fit , the spectrum can be characterized for our purposes of estimating a 210 kev luminosity by their best - fit power law parameters of @xmath105 @xmath9 . we find a total x - ray luminosity of @xmath106 . some small fraction of this emission will be contributed by the soft foreground stars and m33 snrs , so we adopt a 210 kev luminosity of @xmath107 including the nuclear source , and @xmath108 if it is excluded . there is substantial disagreement between the estimated thermal radio continuum fluxes of m33 between israel ( 1980 ) and berkuijsen ( 1983 ) . however , more recent radio results from buczilowski ( 1988 ) and the h@xmath69 measurements of devereux , duric , and scowen ( 1997 ) agree quite closely with berkuijsen s estimate which we adopt here . the implied lyman continuum flux is , then , @xmath109 ph s@xmath1 ; for a metallicity of 1/3 solar , we derive a total o star population of 3460 for m33 . this yields a range for the specific x - ray luminosity of @xmath110 ergs s@xmath1 star@xmath1 , a value comparable to that for the smc if the nuclear emission is included . as the largest member of the local group , m31 has been studied by all the major x - ray satellite missions . the _ einstein _ survey ( van speybroeck et al . 1979 ) revealed a luminous population of lmxbs both in globular clusters and in the galactic bulge , plus a disk population presumably consisting of hmxbs and supernova remnants . the recent _ rosat _ pspc survey ( supper et al . 1997 ) brought the number of detected sources in the vicinity of the galaxy to nearly 300 , lowered the luminosity threshold to @xmath111 ergs s@xmath1 , and confirmed the general picture of two source populations outlined above . in addition to these soft x - ray images , _ ginga _ carried out a long pointing at the galaxy in the 220 kev band of direct interest here . makishima et al . ( 1989 ) , fitted the high signal - to - noise integrated spectrum with a composite model to represent the dominant lmxb and hmxb populations ; the derived fluxes were carefully corrected for collimator response using the distribution of resolved sources in the _ einstein _ images . they find an upper limit to the hmxb contribution , translated to the 210 kev band using their spectral assumptions ( the galactic foreground absorption of @xmath112 @xmath9 and a cutoff power law with @xmath113 and @xmath114 kev ) of @xmath115 ergs s@xmath1 ; they demonstrate that this is consistent with a value derived by summing the _ einstein _ sources in its softer bandpass . radio - continuum , far - infrared , and h@xmath69 images indicate that the bulk of the star - formation in m31 occurs in a thin ring in the galactic disk @xmath116 kpc from the nucleus ( e.g. , beck & grve 1982 ; devereux et al . 1994 ; walterbos & braun 1994 ; xu & helou 1996 ) . this star - forming ring is the region in m31 where high - mass binaries might be expected to reside . after eliminating x - ray sources identified with foreground stars , background agn , and globular clusters , a comparison of the pspc source catalog of supper et al . ( 1997 ) and the 60 @xmath5 m image of xu & helou ( 1996 ) reveals that 20 _ rosat _ sources are positionally coincident with the star - forming ring ; four additional sources are located close to the ring , four more are coincident with 60 @xmath5m bright features outside the ring ( excluding the bulge , which is not a site of massive star formation ; see devereux et al . 1994 ) , and six others are found in an outer spiral arm northeast of the ring where there is some low - surface brightness ir emission . only two of these 34 sources ( both of which are weak and in the ring ) are identified with supernova remnants , so the remainder could all be hmxbs . the pspc count rates of the hmxb candidates sum to 0.36 ct s@xmath1 , roughly 30% of the 0.12.4 kev flux associated with the galaxy . applying the makishima et al . spectral parameters to this count rate and extrapolating to the 210 kev band suggests a maximum hmxb luminosity of @xmath117 in m31 . assuming a somewhat softer spectrum with @xmath118 or assigning only half the sources to the hmxb population yields a result consistent with the _ ginga _ analysis : @xmath119 erg s@xmath1 . as with the other local group galaxies , the ionizing photon luminosity of m31 can be inferred from the thermal fraction of its radio continuum emission and its h@xmath69 luminosity . beck & grve ( 1982 ) estimate that within the central 20 kpc , the thermal radio flux density at 2.7 ghz is @xmath120 jy , which suggests @xmath121 photons s@xmath1 . the extinction - corrected h@xmath69 luminosity of @xmath122 ergs s@xmath1 measured by walterbos & braun ( 1994 ) gives , assuming case b recombination , @xmath123 @xmath124 , and @xmath125 k , a nearly identical value for @xmath73 . thus , adopting solar metallicity , we estimate that there are 3660 o stars in m31 . an estimated luminosity of @xmath126 ergs s@xmath1 for the hmxb population yields a specific luminosity of @xmath127 ergs s@xmath1 star@xmath1 , very similar to the value we obtained for the smc and nearly a factor of ten larger than that for the galaxy . while surprising , we can think of no plausible loopholes in our argument to eliminate this difference ( see 7 ) . in table 2 , we summarize data relevant to the o - star populations and x - ray emissivity of the local group galaxies . for each object , we give our adopted distance ( uncertain by less than 10% ) , the blue luminosity @xmath128 and the adopted metallicity . the next column lists the quantity @xmath129 from vacca ( 1994 ) , the ratio of the number of equivalent o7 stars needed to produce the observed lyman continuum flux to the total number of actual o stars in the galaxy . this quantity depends on metallicity , and on the assumed slope and mass cutoff of the upper part of the imf ; we have adopted the salpeter @xmath130 for the milky way and @xmath131 for the other galaxies . varying the imf slope from 2.0 to 3.0 ( e.g. , hill , madore , & freedman 1994 ) changes the o - star counts by 22% to + 60% for solar metallicity , and 22% to + 36% for a metallicity of 0.1 solar . likewise , changing @xmath132 from @xmath133 to @xmath134 produces changes in the estimated o - star population of roughly @xmath135 . thus , the imf parameters are not a major source of uncertainty in our estimates . the number of lyman continuum photons inferred from the observations described in the text , and the resulting number of o stars are found in columns 6 and 7 . we then include several quantities which depend on the massive star population : star formation rate , core - collapse supernova rate , lyman continuum flux and number of o - stars , all normalized to the value for the milky way . clearly these quantities are not all independent , but they are listed to demonstrate that , within a factor of two , these four quantities are consistent for each galaxy , giving us some confidence that the o - star numbers by which we normalize our specific x - ray luminosities are not in error by more than a factor of two . the final two columns contain our estimates for the 210 kev @xmath54 discussed above and the x - ray luminosities per o star which constitute our principal result . the range of specific luminosities spans an order of magnitude . in two cases , we list a range of values based on differing assumptions about the assignment of x - ray flux to pop i binaries : for m33 , we quote the value including and excluding the nuclear source , and for the smc , we include the current value , as well as one - third of that value based on our arguments about the lifetime of smc x-1 . in both cases , however , the entire range of allowed values falls within the extremes defined by the milky way and m31 . while it is somewhat curious that the galaxy for which we have the best information the milky way has the lowest value , the consistency of the results we obtain for the solar neighborhood and the galaxy as a whole adds to the robustness of this conclusion . bringing the value for the earlier type galaxy m31 down by a factor of @xmath116 to agree with m0 appears to lie outside of the range of the uncertainties involved . the implications of this result for the putative dependence of pop i x - ray luminosity on metallicity is discussed below . shortly after the discovery of highly luminous pop i x - ray binaries in the magellanic clouds , clark et al . ( 1978 ) discussed the apparent shift in the mean x - ray luminosity of hmxbs in the clouds with respect to that in the milky way , and attributed the higher luminosities of the cloud binaries to the lower metallicity of the accreting gas . the discovery that the metal - poor extragalactic region ngc 5408 has a very high x - ray luminosity ( stewart et al . 1982 ) reinforced the notion that the pop i x - ray luminosity of a galaxy and its metallicity are inversely correlated . alcock & paczynski ( 1978 ) calculated evolutionary tracks for low - metallicity massive stars , and pointed out that such stars spend more time in evolutionary phases with massive stellar winds that power much hmxb emission , offering a possible explanation for this trend . hutchings ( 1984 ) offered an alternative explanation , postulating that the fraction of compact objects in hmxbs that are black holes may be higher in late - type ( lower metallicity ) galaxies ; indeed , two of the three persistently bright lmc pop i binaries are among the best black hole candidates . without any quantitative analysis of its significance or cause , numerous studies on the contribution of starbursts to the x - ray background ( xrb ; e.g. , bookbinder et al . 1980 ; griffiths & padovani 1990 ) have adopted this @xmath136 relation . the results presented here , however , suggest caution . while our value for the specific x - ray luminosity per o star in the solar neighborhood is similar to the number of @xmath137 ergs s@xmath1 per o star quoted by stewart et al . ( 1982 ) , our values for the lmc and smc disagree by factors of 4 to 8 ; a similar table from bookbinder et al . ( 1980 ) contains values higher by yet another factor of 4 . since no details on the derivation of these numbers are given in this earlier work , it is difficult to pinpoint the causes of these discrepancies , although the common overestimation of the x - ray luminosities of specific sources , exemplified by our discussion in 2 , is a likely culprit . our use of the asm data to obtain long - term mean @xmath54 values and our detailed analysis of the imaging data for each galaxy ( as well as modern estimates for o star counts ) has , we hope , reduced the uncertainties in these estimates . our conclusion that m31 , the most metal - rich member of the local group , has a specific x - ray luminosity per o star very similar to that of the smc ( the lowest metallicity galaxy ) casts serious doubt on the widely adopted notion that these two quantities are anticorrelated . the recognition that smc x-1 , which dominates the value for the small cloud , may be sufficiently short - lived that the current luminosity of that galaxy is several times greater than the long - term average would actually reverse the trend . the detailed census of 210 kev point sources in m31 , the resolution of the nature of the m33 nuclear source , and the resolution of the point source populations in more distant galaxies with _ chandra _ should help to constrain further the values derived here and to clarify the dependence , if any , of x - ray luminosity on metallicity . while hmxbs are the most luminous individual x - ray sources arising from star formation , several other high energy phenomena associated with massive stars also produce hard x - rays . for completeness , we evaluate their contributions to the specific x - ray luminosity per o star here . since most of these phenomena are , like the hmxbs , associated with all stars down to 8 @xmath40 ( the approximate dividing line between stars which end their lives in core - collapse supernovae and those which end as white dwarfs ) , we include the integrated contributions from stars down to this mass cut . furthermore , since these phenomena are mostly short - lived compared to the main sequence lifetimes of ob stars , we calculate the expected contribution to the instantaneous x - ray luminosity of the population by dividing total x - ray luminosity produced by the mean main - sequence lifetime of the population , weighted by the initial mass function : @xmath138 where @xmath139 is the initial mass function ( and we adopt the salpeter slope of @xmath140 ) , and @xmath141 ( stothers 1972 ) . the result , adopting lower- and upper - mass limits of 8 @xmath40 and 80 @xmath40 , respectively , is @xmath142 myr . in the steady state , such as obtains today in the milky way , this provides the appropriate comparison to our empirical specific luminosity per o star from the hmxbs . in a galaxy undergoing a starburst with a duration comparable to this timescale , the relative contributions of these various additional sources of x - ray emission will be a function of the starburst age . however , for a population of such starbursting systems , the steady state value provides a valid approximation . as noted in the introduction , ob stars on the main sequence produce x - rays which are thought to originate from shocks that develop in unsteady wind outflows ( lucy & white 1980 ; cooper & owocki 1994 ; feldmeier et al . the typical ratio of @xmath143 ( pallavicini et al . the characteristic temperature of the emission is @xmath144 kev ( chlebowski , harnden , & sciortino 1989 ) , implying a 210 kev luminosity of @xmath145 ergs s@xmath1 for all spectral classes . the total is , then , @xmath146 of the binary contribution and can be safely ignored . harder emission , both thermal and nonthermal , can arise when winds from neighboring stars collide ( cooke , fabian , & pringle 1978 ; chen & white 1991 ; wills , schild , & stevens 1995 ) . in the orion trapezium region , the total 210 kev x - ray luminosity , not all of which can reasonably be associated with this phenomenon is @xmath147 ergs s@xmath1 in the 210 kev band ( yamauchi & koyama 1993 ; yamauchi et al . with at least several o stars participating , this yields a specific luminosity of @xmath148 that of binary systems . while it is possible that in the massive ob associations found in starburst nuclei wind collisions could be significantly enhanced , it seems highly improbable that they will compete with binaries as a significant source of an ob star population s hard x - ray luminosity . lower mass stars , formed in association with massive stars , undergo a t tauri phase prior to descending onto the main sequence during which significant hard x - ray emission is produced ( e.g. , koyama et al . 1996 ) . again , using the local example of orion as a template , the x - ray luminosity associated with t tauri stars in the 210 kev band is @xmath149 ergs s@xmath1 ( yamauchi & koyama 1993 ) . since there are @xmath150 o stars in the orion complex , this yields a specific luminosity of @xmath151 ergs s@xmath1 . if there is a discrepancy in the ratio of high- to low - mass stars in starburst galaxies versus the local sites of star formation , it is likely to be in the direction of a deficit of lower mass stars , reducing this contribution to an even smaller value . in any event , it appears unlikely that the contribution from pre - main sequence low - mass stars will exceed 1% that of the hmxbs . the violent deaths of massive stars in core - collapse supernovae provide several means of producing x - ray emission : thermal emission from shock - heated gas left by the passage of the sn blast wave , nonthermal emission from particles accelerated at the shock front , nonthermal emission from a synchrotron nebula generated by a young , rapidly rotating neutron star , and emission from a hot young neutron star s surface and magnetosphere . we examine each of these in turn , taking the galaxy s supernova remnant ( snr ) population as exemplary . the hot gas generated by the outward moving shock wave from the sn explosion , along with the stellar ejecta heated by the reverse shock , produce thermal x - ray emission with a temperature characteristic of the shock velocities ; for most of a remnant s life these range from 300 to 3000 km s@xmath1 , yielding nominal temperatures from 0.2 to 20 kev , although delayed equilibration between the protons and electrons , nonequilibrium ionization , and inhomogeneities in the ambient and ejected material conspire to produce observed temperatures for the bulk of the emitting material of @xmath152 kev . while more sophisticated models of remnant x - ray emission have been constructed over the past few decades , it suffices for our purposes of estimating the total 210 kev energy radiated to use the simple sedov equations ( e.g. , gorenstein & tucker 1976 ) . for typical snr parameters ( explosion energy @xmath153 ergs , ambient density @xmath154 @xmath124 ) , we have calculated the temperature , shock velocity , and radius , as well as the fraction of the radiated flux emitted in the 210 kev band , as a function of time . as the swept - up material decelerates the shock , the temperature falls and the x - ray luminosity rises . however , the fraction of the emission in the 210 kev band also falls once @xmath155 km s@xmath1 ( @xmath156 yr ) , such that , for @xmath157 yr , the 210 kev band luminosity is constant to within a factor of two , with an average value of @xmath158 erg s@xmath1 ; for later times , the emission in this band rapidily declines into insignificance . thus , the integrated contribution from thermal remnant emission is @xmath159 erg s@xmath160 erg ; dividing by our mean o - star lifetime @xmath161 gives @xmath162 erg s@xmath1 per o star or roughly 1 - 2% of the hmxb contribution . in addition to heating ambient gas and supernova ejecta , the shock wave sweeps up magnetic fields and accelerates particles to relativistic energies . the primary consequence of this is the bright radio emission associated with snrs . however , for young remnants at least , the particle spectrum extends to very high energies , producing detectable synchrotron radiation in the x - ray band . the 210 kev x - ray luminosity of the historical remnant sn1006 is dominated by such synchrotron emission ( koyama et al . 1995 ) , and evidence for such nonthermal radiation has recently been detected in several other young remnants ( petre et al . 1999 and references therein ) . indeed , petre et al . claim that there is evidence that _ all _ young remnants have an x - ray synchrotron component , and that we only see this as a dominant contributor to the remnant s x - ray emission when the sn takes place in a very low density region of the interstellar medium and thus can form no significant reverse shock to illuminate the ejecta . the synchrotron luminosities of these sources are typically @xmath163 of the thermal @xmath54 , and the timescale over which this component is significant is less than that for the thermal emission . thus , its overall contribution to the hard x - ray luminosity is almost certainly @xmath164 that of the hmxb contribution . one of the most luminous hard x - ray sources in the galaxy is the crab nebula , a remnant of the supernova of 1054 ad powered by rotational kinetic energy loss from the young neutron star created in the explosion ; in the 210 kev band , @xmath165 ergs s@xmath1 ( harnden & seward 1984 ) . while often characterized as the prototypical young neutron star , the crab is , in fact , not typical . for example , the sn of 1181 ad also produced a pulsar - powered synchrotron nebula ( 3c 58 ) , but its 210 kev x - ray luminosity is @xmath166 times lower at only @xmath167 ergs s@xmath1 , despite its slightly younger age ( helfand , becker , & white 1995 ) . furthermore , evidence for young pulsars in the remnants of other core collapse supernovae has been notoriously difficult to find , and while more than three dozen such cases of snr / neutron star associations have now been suggested , none produce x - ray luminosities within a factor of five of the crab pulsar ( see helfand 1998 for a review ) . broad distributions of initial spin period and magnetic field strength for newly born neutron stars are likely to be responsible for the wide range of properties observed . a firm upper limit on the contribution such objects can make to the hard x - ray luminosity of a young stellar population can be derived by assuming that all neutron stars are born with @xmath168 msec , yielding a total rotational kinetic energy of @xmath169 erg , where @xmath170 g cm@xmath44 ) is the star s moment of inertia and @xmath172 is the rotational frequency . for the crab , the fraction of the rotational kinetic energy loss rate @xmath173 emerging in the 210 kev band is @xmath174 ; other young crab - like remnants such as 0540 - 693 in the lmc and 1509 - 58 show similar ratios of @xmath175 . thus , the upper limit to the contribution of young pulsar nebulae to the 210 kev luminosity of an ob population is @xmath176 erg @xmath177 , where @xmath178 is the fraction of supernovae that produce neutron stars , and @xmath179 is the fraction of the crab spin - down luminosity of the average young neutron star . although the mass cut dividing black hole and neutron star remnants of core collapse events is unknown , @xmath178 is likely to be of order unity . the quantity @xmath179 is less well - determined , but is clearly much less than unity : for a core - collapse sn rate of one per century , there should be ten sources with @xmath180 in the galaxy . in fact , there is only one source at 0.2 @xmath181 ( g29.7 - 0.3 ; helfand & blanton 1996 ) and no other sources within an order of magnitude . we adopt @xmath182 , although we regard this as a conservative upper limit . thus , the x - ray luminosity contribution from pulsar synchrotron nebulae could be as high as @xmath183 ergs s@xmath1 per o star or roughly 10% of the hmxb contribution ; if , as seems to be the case in the galaxy , the median x - ray luminosity of young neutron stars is at least a factor of ten less than that of the crab , the contribution of synchrotron nebulae will be @xmath184 of the hmxb value . the final source of x - ray emission resulting from a sn explosion is the thermal emission from the hot surface of the young neutron star and the nonthermal emission from its magnetosphere . since rapid neutrino cooling reduces the surface temperature to under @xmath185 k within a few decades , the contribution of thermal emission in the 210 kev band is completely negligible . nonthermal pulsed emission in the crab accounts for only @xmath186 of the total nonthermal emission produced by the pulsar nebula . becker and trumper ( 1997 ) have shown that @xmath187 for a wide range of pulsar ages and magnetic field strengths ; thus , magnetospheric x - ray emission from rotation - powered pulsars is negligible compared to the hmxb contribution . the large mechanical energy input to the interstellar medium of a starburst galaxy from stellar winds and supernovae results in a pressure - driven wind of hot plasma . such superwinds " ( e.g. , heckman , armus , & miley 1990 ) have been observed to be characteristic of galaxies with high star formation rates , and diffuse x - ray emission associated with them has been detected in a number of galaxies ( e.g. , m82 , ngc 253 [ fabbiano 1988 ] ; ngc 3256 [ moran , lehnert , and helfand 1999 ] , etc . ) . the characteristic temperature of these winds , however is @xmath188 to @xmath189 kev , and their contribution to the galaxies emission above 2 kev is negligible . having characterized the dominant role of hmxbs in the production of hard x - rays in the milky way and other local group galaxies , we can now discuss the direct contribution of ob stars and their remnants to the total hard x - ray luminosities of galaxies undergoing bursts of star formation . to do this , we need to relate the specific x - ray luminosity per o star to observable quantities for nearby starbursts total hard x - ray flux and infrared luminosity . the integrated 210 kev x - ray luminosity of high - mass binaries in a star - forming galaxy can be expressed as @xmath190_{_{\rm hmxb } } \times n({\rm o})$ ] , where @xmath191_{_{\rm hmxb}}$ ] is an adopted value of the specific x - ray luminosity per o star for hmxbs , and @xmath192 is the actual number of o stars present . assuming an imf slope of 2.35 , an upper mass cutoff of 100 @xmath40 , and solar metallicity , the models of leitherer & heckman ( 1995 ) predict that a region producing stars at a constant rate of 1 @xmath40 yr@xmath1 for at least @xmath193 yr will have @xmath194 o stars and an associated bolometric luminosity of @xmath195 ergs s@xmath1 . provided that the young stellar population dominates the host galaxy s bolometric luminosity which is approximately equal to its total infrared luminosity @xmath196the number of o stars can be scaled for a system of arbitrary star - formation rate : @xmath197 . the binary luminosity expression then becomes @xmath198_{_{\rm hmxb}}\ > l_{_{\rm ir}}$ ] , or in terms of fluxes , @xmath199_{_{\rm hmxb}}\ > f_{_{\rm ir}}$ ] . using the latter equation , we have computed the range of hmxb x - ray fluxes expected at a given ir flux for the range of local group values of @xmath191_{_{\rm hmxb}}$ ] . these are represented by the shaded region in figure 1 . this region is bounded on the lower - right by the specific x - ray luminosity per o star derived from direct counts of hmxbs and o stars in the solar neighborhood ( @xmath200 ergs s@xmath1 star@xmath1 ) , and on the upper - left by the value of @xmath0 ergs s@xmath1 star@xmath1 obtained for the smc and m31 . the dashed line represents the upper limit derived for the solar neighborhood which is roughly equal to the global milky way value . also plotted in figure 1 are the locations in the @xmath201 plane of several nearby starburst galaxies that have been studied with _ asca_. the ir fluxes of these objects have been calculated from the highest reported _ iras _ flux densities using the @xmath202 prescription of sanders & mirabel ( 1996 ) . their 210 kev x - ray fluxes have been collected from published _ asca _ results ( references are provided in the figure caption ) . note that the x - ray luminosities of the starbursts span several orders of magnitude , from @xmath203 ergs s@xmath1 ( ngc 1569 and ngc 4449 ) to @xmath204 ergs s@xmath1 ( ngc 253 and ngc 2146 ) to @xmath205 ergs s@xmath1 ( ngc 3256 and ngc 3690 ) . several important conclusions can be drawn from figure 1 . first , there is a clear tendency for the 210 kev x - ray fluxes of starburst galaxies to increase with @xmath202 , indicating that their hard x - ray luminosities are largely governed by sources whose contributions are proportional to the star - formation rate . as discussed in the previous section , hmxbs are expected to dominate over all other such contributors . however , in order for hmxbs to account for _ all _ of the hard x - rays produced in starbursts , their typical output per o star must be significantly greater than that observed in the milky way or the lmc . even in the starburst galaxies with the lowest @xmath206 ratios ( ngc 1569 , ngc 3256 , m83 , and ngc 253 ) , hmxbs would have to exhibit @xmath191_{_{\rm hmxb}}$ ] values that are 5 times higher than the milky way s . both direct observations and population syntheses ( e.g. , dalton & sarazin 1995 ) indicate that the bulk of the x - ray emission of a binary population arises from the small fraction of objects with the highest individual luminosities . thus , if hmxbs produce most of the hard x - ray flux of starburst galaxies , we would expect such systems to have many more high - luminosity objects ( per o star ) than the milky way and the lmc . for the nearest starbursts ( including several of the objects in fig . 1 ) , this hypothesis is testable with high - resolution _ chandra _ observations . the good correlation between @xmath207 and @xmath208 stands in contrast to the large scatter in a plot of @xmath209 vs. @xmath210 , the blue optical flux , from these same galaxies . thus , unlike normal galaxies which show a tight correlation between @xmath209 and @xmath211 ( fabbiano 1989)i.e . , the x - ray luminosity is proportional to the light from the whole stellar population and is dominated by long - lived , low - mass x - ray binary emission in starbursts , the dominant x - ray production is associated with the young stellar population . two objects , m82 and ngc 3310 , deviate significantly from the @xmath212 trend exhibited by the other starburst galaxies in figure 1 . the x - ray fluxes of these two objects are clearly inconsistent with the level of emission expected from an hmxb population , even one similar to that of the smc , suggesting that each galaxy possesses an extra component of hard x - ray luminosity that is weak ( or absent ) in the other starbursts . images of ngc 253 ( strickland et al . 2000 ) and m82 ( griffiths et al . 1999 ) , which have similar ir luminosities but 210 kev x - ray luminosities that differ by at least a factor of 5 , reveal strikingly different hard x - ray morphologies . the hard x - ray flux of ngc 253 is produced almost entirely by discrete sources , whereas in the more luminous m82 , about half of the hard x - ray emission arises from a diffuse component coincident with the most active region of star formation . we have suggested previously ( moran & lehnert 1997 ; moran , lehnert , & helfand 1999 ) that inverse - compton scattered emission , resulting from the interaction of ir photons with supernova - generated relativistic electrons , may in some circumstances contribute appreciably to the hard x - ray fluxes of starburst galaxies . m82 and ngc 3310 thus represent the best sites for the investigation of this possibility . we have demonstrated that the hard ( 210 kev ) x - rays produced directly by a population of ob stars , their remnants , and their accompanying lower - mass brethren are dominated by the small fraction of massive stars that form x - ray binaries . despite our efforts to assess with care the long - term mean luminosities of such systems through our use of the asm database and our consistent methods for deducing o - star number counts , we find a range of an order of magnitude in the specific x - ray luminosity per o star among the galaxies of the local group . while the specific luminosities of m33 and the smc are each dominated by a single source , and , as we have argued , could plausibly have a long - term value within a factor of 2 of the milky way , m31 remains an outlier . it is possible that only a small fraction of the 34 luminous x - ray sources coincident with star - forming regions in that galaxy are hmxbs , although no other local group galaxy shows a similar population of bright , non - hmxb objects . alternatively , the o - star population of m31 could have been severely underestimated if a large fraction ( @xmath213 ) of its lyman continuum photons escape from the galaxy without ionizing a hydrogen atom . were either ( or both ) of these scenarios to hold , and were we to ignore the bright single sources in m33 and the smc ( a somewhat uncomfortable chain of assumptions ) , the specific x - ray luminosity per o star in local group members could all fall within a factor of two of @xmath214 erg s@xmath1 per o star . in this case , the starburst galaxies would all require a source of x - ray luminosity in addition to the direct contributions of the ob star population . even if we allow the full observed range of specific luminosities and brand the milky way as atypical , however , some starbursts still require an additional hard x - ray component . while a buried active nucleus is a plausible candidate , _ chandra _ observations have ruled this out in the case of m82 . the diffuse nature of a significant fraction of the hard x - ray flux from m82 is consistent with our suggestion of ic emission as the origin of this additional component . the fact that the less intense and more diffuse starburst in ngc 253 ( 1 ) shows no significant diffuse hard emission , and ( 2 ) falls within the @xmath215 band predicted from binaries alone , is also consistent with this picture , since the predicted ic luminosity from ngc 253 would be negligible . natarajan and almaini ( 2000 ) have recently used global energetics arguments to conclude that ob stars and their products ( hmxbs and snrs ) contribute at most @xmath216 of the x - ray background at energies above 2 kev . they ( reasonably ) assume the hmxb population tracks the global star formation rate , although their normalization assumes a milky way hard x - ray luminosity a factor of 2.5 lower than we derive in 3.2 , and , thus , a factor of @xmath116 below the mean value for the local group . in addition , as they note , extra contributions such as ic emission are not included in their calculation . thus , we conclude it remains plausible that a significant contribution to the hard x - ray background arises from starburst galaxies . it should be emphasized that such a conclusion is not inconsistent with existing deep - field x - ray source counts or faint x - ray source identifications . given the steep redshift dependence of the star formation rate , the vast majority of the xrb contribution from starbursts will arise at redshifts greater than 1 . to illustrate , we use the results of moran et al . ( 1999 ) in which we showed that , owing to the tight correlation between far ir and centimetric radio emission for starburst galaxies ( and the correlation shown in figure 1 between far - ir and x - ray luminosity in these same galaxies ) , faint radio source counts can be used to constrain the surface density of starburst xrb contributors . for a 75% starburst fraction ( richards 1998 ) in the @xmath217 radio flux density range , @xmath218 arcmin@xmath219 ( fomalont et al . the _ rosat _ deep survey in the lockman hole ( hasinger et al . 1998 ) had a 0.52.0 kev limit of @xmath220 erg @xmath9 s@xmath1 for their complete sample of 50 sources over @xmath221 deg@xmath44 . using the ratio of 5 kev to 5 ghz flux density for starbursts found in moran et al . ( 1999 ) , @xmath222 ergs @xmath9 s@xmath1 kev@xmath223 , this implies an equivalent radio flux density limit of @xmath224 mjy ; and @xmath225 sources in the _ rosat _ and _ chandra _ surveys , respectively . ] assuming an x - ray spectral index of @xmath226 as observed in ngc 3256 , we should then expect @xmath227 starbursts deg@xmath219 or 0.1 such sources in the survey . for the largest _ chandra _ deep survey published to date ( giacconi et al . 2000 ) , the 210 kev limit is @xmath228 erg @xmath9 s@xmath1 which corresponds to a 5.7 mjy radio flux density and an expected surface density of 2.2 sources deg@xmath219 , or @xmath229 sources in the 0.096 deg@xmath44 survey area . in summary , the deepest surveys yet performed have not gone deep enough to reveal the population of starbursts at their expected luminosities . the continued flattening of the agn - dominated x - ray log@xmath230-log@xmath231 seen by _ observations at 210 kev flux levels above @xmath232 erg @xmath9 s@xmath1 strengthens the requirement for a new population of objects at fainter fluxes in order to account for the remaining 20 - 25% of the x - ray background . starburst galaxies remain an attractive candidate , and deeper _ chandra _ surveys should begin to find them at a surface density of 30 deg@xmath219 ( @xmath96 per _ chandra _ field ) when a flux threshold of @xmath233 erg @xmath9 s@xmath1 is reached . djh is grateful for the support of the raymond and beverly sackler fund , and joins ecm in thanking the institute of astronomy of the university of cambridge for hospitality during much of this work . this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center . the work of ecm is supported by nasa through _ chandra _ fellowship pf8 - 10004 awarded by the _ chandra _ x - ray center , which is operated by the smithsonian astrophysical observatory for nasa under contract nas8 - 39073 . djh acknowledges support from nasa grant nag 5 - 6035 . this paper is contribution number 696 of the columbia astrophysics laboratory . alcock , c. , & paczyinski , b. 1978 , apj , 223 , 244 angelini , l. , white , n.e . , & stella , l. 1991 , apj , 371 , 332 awaki , h. , ueno , s. , koyama , k. , tsuru , t. , & iwasawa , k. 1996 , pasj , 48 , 409 beck , r. , & grve , r. 1982 , a&a 105 , 192 becker , r.h . , helfand , d.j . , & szymkowiak , a.e . 1982 , apj , 255 , 557 becker , w. , & trumper , j. 1997 , a&a , 326 , 682 bookbinder , j. , cowie , l.l . , ostriker , j.p . , krolik , j.h . , & rees , m.r . 1980 , apj , 237 , 647 chen , w. , & white , r.l . 1991 , apj , 366 , 512 chevalier , c. , & ilovaisky , s.a . 1998 , a&a , 330 , 201 chlebowski , t. , harnden , f.r . jr . , & sciortino , s. 1989 , apj , 341 , 427 clark , g. , doxsey , r. , li , f. , jernigan , j.g . , & van paradijs , j. 1978 , apj , 221 , l37 clark , g. , li , f. , & van paradijs , j. 1979 , apj , 227 , 54 coe , m.j . , burnell , s.j.b . , engel , a.r . , evans , a.j . , & quenby , j.j . 1981 , mnras , 197 , 247 cooke , b.a . , fabian , a.c . , & pringle , j.e . 1978 , nature , 273 , 645 cooper , r.g . , & owocki , s.p . 1994 , ap&ss , 221 , 427 cowley , a.p . , crampton , d. , hutchings , j.b . , remillard , r. , & penfold , j.e . 1983 , apj , 272 , 118 dalton , w.w . , & sarazin , c.l . 1995 , apj , 448 , 369 davison , p.j.n . 1977 , mnras , 179 , 15p della ceca , r. , griffiths , r.e . , heckman , t.m , & mackenty , j.w . 1996 , apj , 469 , 662 della ceca , r. , griffiths , r.e . , & heckman , t.m . 1997 , apj , 485 , 581 della ceca , r. , griffiths , r.e . , heckman , t.m , lehnert , m.d . , & weaver , k.a . 1999 , apj , 514 , 772 devereux , n.a . , price , r. , wells , r.a . , & duric , n. 1994 , aj , 108 , 1667 dewey , r.j . , & cordes , j.m . 1987 , apj , 321 , 780 dubus , g. , charles , p.a . , long , k.s . , & hakala , p.j . 1997 , apj , 490 , l50 dubus , g. , charles , p.a . , long , k.s . , hakala , p. , & kuulkers , e. 1999 , mnras , 302 , 731 fabbiano , g. 1988 , apj , 330 , 672 fabbiano , g. 1989 , araa , 27 , 87 feldmeier , a. , kudritzki , r .- , palsa , r. , pauldrach , a.w.a . , & puls , j. 1997 , a&a , 320 , 899 fomalont , e.b . , windhorst , r.a . , kristian , j.a . , & kellerman , k.i . 1991 , aj , 102 , 1258 giacconi , r. , et al . 2000 , preprint ( astro - 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we present an empirical analysis of the integrated x - ray luminosity arising from populations of ob stars . in particular , we utilize results from the all - sky monitor on _ rxte _ , along with archival data from previous missions , to assess the mean integrated output of x - rays in the 210 kev band from accreting early - type binaries within 3 kpc of the sun . using a recent ob star census of the solar neighborhood , we then calculate the specific x - ray luminosity per o star from accretion - powered systems . we also assess the contribution to the total x - ray luminosity of an ob population from associated t tauri stars , stellar winds , and supernovae . we repeat this exercise for the major local group galaxies , concluding that the total x - ray luminosity per o star spans a broad range from 2 to @xmath0 erg s@xmath1 . contrary to previous results , we do not find a consistent trend with metallicity ; in fact , the specific luminosities for m31 and the smc are equal , despite having metallicities which differ by an order of magnitude . in light of these results , we assess the fraction of the observed 210 kev emission from starburst galaxies that arises directly from their ob star populations , concluding that , while binaries can explain most of the hard x - ray emission in many local starbursts , a significant additional component or components must be present in some systems . a discussion of the nature of this additional emission , along with its implications for the contribution of starbursts to the cosmic x - ray background , concludes our report .

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

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

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

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

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

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

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

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

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

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

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

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

qcd analysis of deep - inelastic scattering ( dis ) data provides one with new knowledge of hadron physics and serves as a test of reliability of our theoretical understanding of the hard scattering of leptons and hadrons . at large momentum transfer @xmath3 gev@xmath4 we have the reliable description of dis that is based on the twist expansion and `` factorization '' theorems . at small ( moderate ) transfer @xmath5 ( a few ) gev@xmath4 this qcd description faces two main problems : ( i ) the high twist corrections to the leading twist contribution become important but remains poorly known ; ( ii ) perturbative qcd ( pqcd ) becomes unreliable due to the fact that the qcd running coupling @xmath6 grows and `` feels '' infra - red landau singularity appearing at the scale @xmath7 of a few tenth of gev . we discuss in this paper a solution of the last problem by applying to dis analysis a _ nonpower perturbative theory _ whose couplings have no singularity at @xmath8 and whose corresponding series possess a better convergence at low @xmath9 . a widely used approach to resolve the aforementioned problem is to apply the analytic perturbation theory ( apt ) developed by shirkov , solovtsov _ et al . _ @xcite . there , the running qcd coupling @xmath10 of pqcd is transformed into an analytic ( holomorphic ) function of @xmath9 , @xmath11 , apt coupling . this was achieved by keeping in the dispersion relation the spectral density @xmath12 unchanged on the entire negative axis in the complex @xmath9-plane ( i.e. , for @xmath13 ) , and setting it equal to zero along the unphysical cut @xmath14 . in the framework of apt the images @xmath15 of integer powers of the originals @xmath16 , @xmath17 following the same dispersion relations were also constructed . at low @xmath9 the couplings @xmath18 change slowly with @xmath9 in contrast with the original @xmath16 behaviour while at high @xmath9 @xmath19 . later , the correspondence @xmath20 was extended to noninteger powers / indices @xmath21 in @xcite and was called fractional apt ( fapt ) , which provides the basis for application to dis . in this respect let us mention a recent papers @xcite where the processing of the dis data has been performed in fapt in the one - loop approximation and the reasonable results for hadron characteristics has been obtained . various analytic qcd models can be constructed , and have been proposed in the literature , among them in refs . these models fulfill certain additional constraints at low and/or at high @xmath9 . for further literature on various analytic qcd models , we refer to review articles @xcite . some newer constructions of analytic models in qcd of @xmath22 include those based on specific classes of @xmath23 functions with nonperturbative contributions @xcite or without such contributions @xcite and those based on modifications of the the spectral density @xmath24 @xmath25 \equiv { \rm i m } \ ; { { \mathcal{a}}}_{1}(q^2=-\sigma - i \epsilon)/\pi\right]$ ] at low ( positive ) @xmath26 where @xmath27 is parameterized in a specific manner by adding two positive delta functions to @xmath28 , cf . @xcite . the possibility to extend the dis analysis formally in the whole @xmath9 range together with the effect of slowing - down of the fapt evolution of the parton distribution functions ( pdf ) in the low @xmath9 region are attractive phenomenological features of fapt . a number of works deal with this task in a naive form @xcite , where the authors show that at very low @xmath9 and bjorken variable @xmath29 apt agrees with experimental data . besides , the applicability of the apt approach was analyzed in the bjorken polarized sum rule @xcite confirming that the range of validity of apt is down to @xmath30 mev , as compared to experimental data . the common feature of these works was taking into consideration some nonperturbative effects against the background of apt , i.e. , higher twists in @xcite or an effective constant gluon mass in @xcite . the basis for applying fapt to low energies in this approach is the factorization theorem that allows one to shift the frontier between the perturbative and nonperturbative effects via the variation of the factorization scale . therefore , we shift the range where perturbation series is applicable in fapt , as it was demonstrated in @xcite ( see reviews of this issue in @xcite , where this phenomenon was also discussed for pion form factors ) . our goal here is to elaborate a general scheme of dis data processing in the framework of fapt taking as a pattern the dis analysis at nlo . in this respect the discussion here can be considered as an extension of the partonic results of the article @xcite on the higher - loop level . we shall focus on the specifics of the dokshitzer - gribov - lipatov - altarelli - parisi ( dglap ) evolution of the pdf @xmath31 in fapt . we involve into consideration the coefficient function @xmath32 of the process and compare the final result with a similar one in pqcd . an important problem of higher twist contribution remains untouched here , but higher twist effects can be taken as an unknown function @xmath33 , i.e. @xmath34 @xcite , or as a constant @xcite @xmath35 , or as an effective sum of all twists contributions in @xcite . we stress that ht effects are only indirectly affected by the analytization procedure . the behavior of ht will be given by the fit of experimental data together with the corresponding parton distribution functions @xcite . besides , in @xcite the authors included more terms in the ht expansion and demonstrated that they are essentially smaller and quickly decreasing . because of this ( theoretically ) unknown behavior we avoid this problem since we pretend to provide perturbation tools how to deal with fapt , while the pure phenomenological analysis is transferred to future investigations . let us recall that the dis analysis can be performed in a few different ways : one of them is provided by the mellin moments defined via inelastic structure functions ( sfs ) @xmath36 , @xmath37 the second approach is based on the direct application of the dglap integro - differential evolution equations @xcite to pdf @xmath38 , while the observable sf is the mellin convolution of the coefficient function and pdf , @xmath39 . the third approach makes use of the jacobi polynomial expansion method @xcite . just this method will be used in this work . the paper is organized as follows : in sec . [ tb ] , we present a theoretical background where we describe the jacobi polynomial ( jp ) method and how to calculate free parameters in order to obtain the nonsinglet structure function . in sec . [ fapt ] , we briefly describe the fapt approach and derive the dglap evolution for the moments @xmath40 in fapt . we present in sec . [ nr ] the free parameters obtained in the analysis of the so called mstw parameterization , see @xcite for details , and the nonsinglet sfs at lo . section [ nr ] contains the results of the analysis of numerical realization of the fapt evolution and the comparison with the results of analogous calculations in pqcd . finally , in sec . [ sec : summ ] we summarize our conclusions . important technical details including new findings are collected in four appendices . we shall focus here on nonsinglet ( ns ) structure functions , @xmath41 , with their corresponding mellin moments @xmath42 ( via eq . ( [ moments ] ) ) to avoid technical complications of the coupled system solution in the singlet case . the pdfs @xmath43 are universal process - independent densities explaining how the whole hadron momentum @xmath44 is partitioned in @xmath45 , i.e. , the momentum carried by the struck parton ( see , for instance @xcite ) . the @xmath29-dependence of pdf is formed at a hadron scale of an order of @xmath46 by nonperturbative forces , while its dependence on factorization / renormalization scale @xmath47 can be obtained within perturbation theory . a brief description of the evolution of the mellin moments in pqcd , up to nlo , is outlined in appendix [ app1 ] as well as the theoretical background with our notation and conventions . we consider the scale @xmath48 as a reference scale for the solution of the evolution equation ( [ mns ] ) where the pdfs are regarded as functions of @xmath29 and the parameters are fixed by comparison with dis data . in particular , we use here the data - based mstw pdfs ( see @xcite , where @xmath49 ) . namely , x u_v(x , q_0 ^ 2)&=&a_u x^_1(1-x)^_2(1+_u + _ u x ) , + x d_v(x , q_0 ^ 2)&=&a_d x^_3(1-x)^_4(1+_d + _ d x ) , [ mstw ] where the values of @xmath50 ( @xmath51 ) , @xmath52 and @xmath53 can be found in @xcite . we use only the valence quark pdfs because the ns pdf @xmath54 can be expressed as @xmath55 ( see appendix [ app1 ] for details ) . the ns sf @xmath56 is represented as the mellin convolution of coefficient function @xmath57 of the process and the corresponding pdf @xmath58 . the @xmath59 can be expanded in the jacobi polynomials @xmath60 , which was developed in refs . @xcite , in truncating the expansion at @xmath61 , where the method converges ( see , for review @xcite ) : @xmath62 here @xmath42 are the mellin moments of nonsinglet sf calculated explicitly in eq . ( [ mns ] ) ; @xmath63 is the weight function and the parameters @xmath64 will be obtained by fitting to the data . the jacobi polynomials @xmath65 are defined as an expansion series by means of @xmath66 they satisfy the orthogonality relation @xmath67 another way to obtain sf @xmath0 is to take the inverse mellin transform @xmath68 under the moments @xmath42 ( i.e. , the inverse of eq . ( [ moments ] ) ) . choosing a convenient path of integration one obtains for @xmath69 @xmath70 here we take the path along a vertical line @xmath71 . we perform the `` exact '' numerical inverse mellin transform , further comparing the results with the jacobi polynomial method , only at the one - loop level due to technical limitations . in this way , we estimate the accuracy of the applied polynomial method , the results of this numerical verification are outlined in appendix [ app3 ] . let us finally mention that one can take sf @xmath72 instead of the ns @xmath73 to consider , , the neutrino dis results of the ccfr collaboration , like it was started in @xcite . this replacement will lead to only minor changes of technical details in the procedure elaborated below . it is known that the perturbative qcd coupling suffers from unphysical ( landau ) singularities at @xmath74 . this prevents the application of perturbative qcd in the low - momentum spacelike regime and , in part , impedes the investigation of high twists in dis . our goal here is not to discuss the motivation and complete construction of fapt , which couplings @xmath75 are free of the aforementioned problems , but present to reader illustrations of the properties of this nonpower perturbation theory that are important for further dis analysis . application of the cauchy theorem to the running coupling @xmath76 , established in @xcite and developed in @xcite , gives us the following dispersion relation ( or klln - lehmann spectral representation ) for the images @xmath77 in the spacelike domain @xmath78}{\sigma+q^2}d\sigma = \int_{-\infty}^{\infty } \frac{\rho_\nu^{(l)}(l_\sigma)}{1+{\rm exp}(l - l_\sigma)}dl_\sigma\ , , \label{anusd}\ ] ] ( where @xmath79 ) that has no unphysical ( landau ) singularities . for the timelike regime analogous coupling reads @xmath80}{\sigma}d\sigma = \int_{l_s}^{\infty } \rho_\nu^{(l)}(l_\sigma ) dl_\sigma . \label{unusd}\ ] ] here , @xmath81 is the fapt image of the qcd coupling @xmath82 in the euclidean ( spacelike ) domain with @xmath83 and the label @xmath84 denotes running in the @xmath84-loop approximation , whereas in the minkowski ( timelike ) domain , we used in ( [ unusd ] ) @xmath85 . it is convenient to use the following representation for the spectral densities @xmath86 : & & _ ^(l)(l _ ) ( a_s(l)(l - i))^= , [ eq : rho ] + & & r_(l)(l)= |a_s(l)(l - i ) | ; _ ( l)(l)=arg(a_s(l)(l - i ) ) . from the definition ( [ anusd ] ) and eq.([eq : rho ] ) it follow that there is no standard algebra for the images @xmath75 , @xmath87 that justifies the name _ nonpower perturbative theory_. in the one - loop approximation , the @xmath88 has the simplest form , i.e. , @xmath89 substituting eq.([eq : rho-1 ] ) in eq.([eq : rho ] ) for @xmath90 and then the result @xmath91 in eq.([anusd ] ) , one reproduces at @xmath92 the well - known expression for maximum value of @xmath93 , @xmath94 @xcite , @xmath95 * * at the two - loop level , they have a more complicated form . to be precise , one gets a_s(2)=- , and r_(2)(l ) & = & c_1(n_f)| 1+w_-1(z_w ( l+i))| , + _ ( 2)(l)&= & arccos , with @xmath96 being the appropriate branch of the lambert function , @xmath97 , @xmath98 , where @xmath99 are the qcd @xmath23-function coefficients and @xmath100 is the number of active quarks , see the expressions in appendix [ app0 ] . for our purpose we use here only the two - loop couplings like @xmath101 . extensions up to four - loops can be found in @xcite . now we implement this formalism with the help of numerical calculation with the main module mathematica package fapt.m of @xcite ( confirmed by a recent program in @xcite ) . according to this and using the corresponding notation from @xcite in the rhs of eqs.([anufaptprog]-[alphafaptprog ] ) , we have _ ^(l)(l)&= & , ( l=14 ; n_f=3 6 ) [ anufaptprog ] + _ ^(l)(l_s)&= & , ( l=14 ; n_f=3 6 ) [ unufaptprog ] for the coupling in pqcd we obtain @xmath102}{4\pi } , \quad ( l=1\div 4).\label{alphafaptprog}\ ] ] the correspondence between the pqcd expansion and fapt one is based on the linearity of the transforms in eqs.([anusd ] ) and ( [ unusd ] ) , see @xcite . this can be illustrated for the simple case of a single scale quantity @xmath103 , calculated within minimal subtraction renormalization schemes and taken at the renormalization scale @xmath104 . the expansions for @xmath105 and for its image @xmath106 are written as d(q^2 ) & = & d_0a_s^(q^2 ) + _ n d_n a_s^n+(q^2 ) + d(q^2)&=&d_0_(q^2)+ _ n d_n _ ( n+)(q^2 ) [ eq : dtod ] at the _ same coefficients @xmath107 _ that are numbers at @xmath104 . we start with the well - known solution of dglap equation for the nonsinglet pdf @xmath108 in nlo approximation . this solution is combined with the corresponding coefficient function @xmath32 the parton cross - section taken at the parton momentum @xmath109 . this is presented in appendix [ app1 ] in the form of eq.([mns ] ) for the moments @xmath110 of the ns sf @xmath69 . rewriting eq . ( [ mns ] ) in the approximate form , i.e. , neglecting the @xmath111 terms in the two - loop evolution factor , one arrives at the commonly used relation @xmath112 the use of fapt will change in this scheme the sense of expansion parameters @xmath113 in accordance with ( [ eq : dtod ] ) . an analogous evolution relation for the analytic images of the moments @xmath110 , @xmath114 , can be obtained from eq.([appmpqcd ] ) by replacing the powers @xmath115 with the fapt couplings @xmath75 ( with @xmath21 being here an index rather than a power ) @xcite and reads @xmath116 the implementation of the proposed calculation in the form of ( [ anufaptprog],[alphafaptprog ] ) is quite direct . the fapt evolution relation ( [ mnsapt ] ) for the moments is the main result of the section . further , we shall use code ( [ anufaptprog ] ) from @xcite to obtain @xmath117 numerically . the last approximation was taken up to @xmath118 since the contribution of the next term in the fapt expansion in eq.([mnsapt ] ) is negligible in comparison with the previous one ( as we demonstrate in appendix [ app2 ] ) . this analytic version of the moment evolution does not face any problems at low energies due to the boundedness of couplings and rapid convergence of the fapt series . in the absence of a fit of experimental data for the fapt model we propose a relation for the initial moments at @xmath119 : f_(n , q^2_0)&= & + & = & , [ inmom ] where the moment of pdf ( see eq.([eq : nsmpdf ] ) ) in pqcd stands in the lhs , while the moment for pdf in fapt stands in the rhs of the second equation . in other words , we take the same initial pdf as in pqcd from the mstw data for these both cases ( in @xcite the parameters were taken the same since the difference between them was negligible ) . we can use either the jacobi polynomial expansion or directly the inverse mellin transform ( appendix [ app3 ] ) . the accuracy of the sf approximation by a finite number of jacobi polynomials ( truncated at @xmath120 ) depends on the choice of the weight - function parameters . therefore , we test the nonsinglet sf , given by the mstw data , by searching for the minimum of ( @xmath121 ) : @xmath122 where we have used eqs . ( [ moments ] ) and ( [ mns ] ) at @xmath121 . thus , we have @xmath123 and from eq . ( [ jpsf ] ) @xmath124 . then , we determine the values of @xmath125 and @xmath23 that provide the best fit to the data for different values of @xmath120 . at the one loop level we find : @xmath126 , @xmath127 , and @xmath128 for @xmath129 , whereas for two loops we get ( for even pdfs only ) : @xmath126 , @xmath130 , and @xmath131 for @xmath129 . to evolve nonsinglet moments , we need to fix the values of the qcd scale @xmath132 in the leading and next - to - leading order , taken in @xcite from the comparison with data , where @xmath133 and @xmath134 . in the case of fapt , the scales @xmath135 must be taken into account very carefully . the authors of @xcite fixed the @xmath136 value directly from the comparison with the data in the leading order ( where @xmath137 ) and obtained @xmath138 that corresponds to @xmath139 . we can see that the perturbative and the analytic values of @xmath136 are close to each other at least inside the margin of errors . for this reason , we will take @xmath140 for simplicity ( recalling that an appropriate value should be taken from the analysis of the experimental data but this goes beyond the scope of this work ) . the couplings in pqcd and in fapt were calculated with the mathematica package developed by bakulev and khandramai in @xcite where the heavy flavour thresholds were taken into account . taking into account the above estimates of the initial parameters , we substitute eqs . ( [ appmpqcd ] ) and ( [ mnsapt ] ) into eq . ( [ jpsf ] ) , and obtain the evolution of sfs up to nlo in pqcd or fapt , respectively . we show the final results of the evolution in figs . [ figsf1 ] , [ figsf2 ] using for dis the character interval @xmath141 . vs @xmath9 at ( a ) lo and ( b ) nlo . the bjorken @xmath142 for ( a.1 ) and ( b.1 ) , and @xmath143 for ( a.2 ) or ( b.2 ) . the solid line represents the fapt results and the dashed line the pqcd ones . ] vs @xmath29 at ( a ) lo and ( b ) nlo . the energy scale is @xmath144 in ( a.1 ) , ( b.1 ) , @xmath145 in ( a.2 ) , ( b.2 ) and @xmath146 in ( a.3 ) , ( b.3 ) . the solid line represents the fapt results and the dashed line the pqcd ones . ] in fig . [ figsf1 ] we fix @xmath29 at two different values : @xmath142 in ( a.1 ) , ( b.1 ) , and @xmath143 in ( a.2 ) , ( b.2 ) , where ( a ) and ( b ) represent the lo and nlo results , respectively . in fig . [ figsf2 ] , we fix @xmath9 at three different values : @xmath144 in ( a.1 ) , ( b.1 ) , with the initial point @xmath145 in ( a.2 ) , ( b.2 ) , and @xmath146 in ( a.3 ) , ( b.3 ) , where again ( a ) and ( b ) represent the lo and nlo , respectively . the main goal of this work is to propose a new theoretical tool for the dis analysis , based on fractional analytic perturbation theory , to the dis community . this approach allows one to analyze formally the leading - twist structure function in the whole @xmath9 range . this conclusion is explicitly shown in figs . [ figsf1 ] and [ figsf2 ] . the scheme of the approach is formulated in sec . [ fapt ] and applied for data processing in sec . our consideration is restricted to the leading twist . the higher twist contributions ( ht ) can be taken into account by a fit of experimental data together with pdfs . moreover , the role of the stability of apt for this fit was pointed out in @xcite ( and in introduction here ) . our investigation reveals the following main features of applying fapt : * structure function @xmath0 at fixed @xmath29 changes very slowly in the entire range of @xmath9 . * at high @xmath9 evolution ( @xmath147 ) the pqcd and fapt distributions become practically equal . * the evolution in fapt is more gradual ( i.e. , it evolves slower ) and smoother than in pqcd . * the new analytic ( fapt ) series converge faster than the pqcd series . from inspection of figs . [ figsf1 ] , [ figsf2 ] it is obvious that the one- and the two - loop fapt approximations do not differ significantly from each other ( the difference is less than @xmath148 ) . + in this work , we have analyzed only the nonsinglet part , the consideration of the singlet part can be performed along the same line but requires more complicated formulas and cumbersome numerical calculations . this is the task for forthcoming investigation . other important issues to complete this fapt approach as the reliable tool for dis is to add the target mass corrections ( tmc ) and the aforementioned ht contributions in our scheme of calculation . these improvements will help one to clarify in future the behavior at very low energies ( @xmath149 ) in more detail . it would be important to emphasize , that the fapt approach admits investigation of the ht contributions in the most sensitive regime of moderate / small @xmath9 due to the high stability of the radiative corrections . this investigation was started by the late a. p. bakulev to whom we dedicate this work . we are grateful to g. cveti for useful comments and to n. g. stefanis for careful reading of the paper and many valuable critical remarks . we thank a. v. sidorov and o. p. solovtsova for the useful remarks . this work was supported by the scientific program of the the russian foundation for basic research grant no . 14 - 01 - 00647 , belrffr jinr grant f14d-007 ( s.v.m ) , and by conicyt fellowship `` becas chile '' grant no.74150052 ( c.a ) . the renormalization group equation for @xmath150 at the expansion of the @xmath23-function up to the nlo approximation is given by [ eq : betaf ] a_s(l ) = -(a_s(l))= - _ 0 a_s^2(l ) - _ 1 a_s^3(l)+ , + where the first two beta coefficients are _ 0 = c_a - t_r n_f , _ 1 = c_a^2 - ( 4c_f + c_a)t_r n_f . [ eq : beta0&1 ] the anomalous dimensions of composite operators in lo , @xmath151 , nlo @xmath152 and the coefficient function @xmath153 are expressed by means of transcendental sums @xmath154 , see , e.g. , @xcite , [ eq : ad - lo - nlo ] _ ^(0)(n)&=&2c_f , _ ^(1)(n)&= & ( c_f^2-c_f c_a ) \{16s_1(n)+16 . + 64 ^(n)+24s_2(n)-3 - 8s_3^ ( ) . + & & .-816 } + & & + c_fc_a \ { s_1(n)-16s_1(n)s_2(n ) . + s_2(n ) --4 } + & & + c_f n_f t_r \{-s_1(n)+s_2(n)++16 } , [ ad ] c_^(1)(n)&=&c_f ( 2s_1 ^ 2(n)+3s_1(n)-2s_2(n)-+++ -9 ) . on the other hand , the series @xmath155 can be expressed via the generalized riemann @xmath156 functions , see @xcite , that are analytic functions in _ both variables _ @xmath157 : [ eq : s ] s_1 ( n)&=&(n+1)-(1 ) , + s_2 ( n)&=&(2)-(n+1)=(2)-(2,n+1 ) , + s_(n)&=&()-(,n+1 ) . ^(n)&=&s_-2,1= -(3 ) _ k=1^ ( ( k+n+1)- ( 1 ) ) . for the @xmath158 and @xmath159 series we use the notation given in @xcite . performing the analytic continuation from even @xmath160 , @xmath161 , and from odd @xmath160 , @xmath162 ( see for details @xcite ) one obtains [ eq : san ] [ eq : s+ ] & & s^+_(n/2)2 ^ -1= 2 ^ -1- ( ) , + & & s^-_(n/2)2 ^ -1= 2 ^ -1- ( ) , [ eq : s- ] where @xmath163 is the lerch transcendent function @xcite . the expressions on the r.h.s . of eqs.([eq : san ] ) are now _ analytic functions in both variables @xmath157 _ this is a new result . the pdfs are the nonsinglet @xmath164 and singlet @xmath165 parton distribution functions , f_ns(x , q^2)&=&u_v(x , q^2)-d_v(x , q^2 ) , + f_s(x , q^2 ) & = & u_v(x , q^2)+d_v(x , q^2 ) + s(x , q^2 ) v(x , q^2)+s(x , q^2 ) , [ nss ] whereas @xmath166 is the distribution of valence quarks and @xmath167 is the sea quark distribution . more generally , the ns pdf is a combination of the forms @xmath168 and @xmath169 but for our consideration we focus on the nucleon scattering provided by combination ( b1 ) . the moments representation for pdfs is defined as f_ns(n , q^2)&= & _ 0 ^ 1dx x^n-1f_ns(x , q^2 ) , [ eq : nsmpdf ] + f_s(n , q^2)&=&_0 ^ 1dx x^n-1f_s(x , q^2 ) . [ eq : mpdf]the moments @xmath170 for the structure function @xmath171 follow from the mellin convolution @xmath172 , [ eq : func - mom ] f_(z,^2)&=&(c _ f_)(z,^2)_0 ^ 1c_(y , a_s ) f_(x,^2 ) ( z - xy ) dydx , + m_(n,^2)&= & c_(n , a_s(^2 ) ) f_(n,^2 ) . here @xmath173 is the nonsinglet coefficient function of the process that can be presented as the perturbation series @xmath174 ; @xmath153 in appendix [ app0 ] is the moment of the @xmath175 . the qcd evolution of the moments @xmath110 up to nlo of is given by ( see ref @xcite ) m_(n , q^2 ) = ( ) ^p(n ) ^d_(n ) + m_(n , q_0 ^ 2 ) , [ mns ] where @xmath176 and : d_(n)=_^(0)(n)/2_0 , p(n ) = ( - ) . the coefficients of anomalous dimension in lo and nlo and the coefficient function in nlo are given in eqs.([eq : ad - lo - nlo ] ) , appendix [ app0 ] . in the case of the nucleon structure function @xmath0 , one needs to take into account only even values of @xmath160 in the nlo anomalous dimension . the accuracy of the evaluation of the structure functions depends on the method we use ; therefore , it is indispensable to verify it in our approach . the jacobi polynomial method promises us a good enough accuracy for the evolution , as was shown in previous works ( see @xcite ) . this method is applied directly to the terms of the bjorken variable @xmath29 , but it affects the @xmath9-dependence indirectly . therefore , it is necessary to confirm the @xmath29-range applicability of the jp method . to this end , we compare the results of the jp approach with the `` exact '' numerical calculations of inverse mellin moments following eq . ( [ exinvm ] ) but only in the one - loop approximation due to technical limitations . vs @xmath29 in lo ( a ) , at the energy scale @xmath177 gev@xmath4 and ( b ) @xmath178 gev@xmath4 . the solid ( red ) line represents the fapt result and the dashed ( blue ) line the pqcd one in the jp method . the thick squares ( red ) and spheres ( blue ) represent the result of the `` exact '' numerical inverse mellin transform.,width=566 ] the comparison of these two results in fig . [ figsf3 ] demonstrates a very good accuracy . so , in order to clarify it , we perform a zoom in @xmath29 , going to a lower @xmath29-region ( @xmath179 ) . we see in fig . [ figsf4 ] that the jp method gradually loses precision starting at @xmath180 . also , we can see that in this range , the difference between these two methods reaches @xmath181% . vs @xmath29 at lo , at energy scale ( a ) @xmath177 gev@xmath4 , and ( b ) @xmath178 gev@xmath4 . the solid ( red ) line represents the fapt outcome and the dashed ( blue ) line the pqcd one in the jp method . the thick squares ( red ) and spheres ( blue ) represent the result of the `` exact '' numerical inverse mellin transform.,width=566 ] here we investigate the accuracy of the rational approximation for the two - loop evolution factor combing the approximations in eqs . ( [ eq : c2 ] , [ eq : c3 ] ) in quantity @xmath185 we obtain the accuracy better than @xmath148 for any @xmath186 ( since jp expansion contains only 13 terms for good approximation ) , in both cases of pqcd and fapt for two different ranges of energy , i.e. , low @xmath187 and high @xmath188 . the results collected in table [ table1 ] .the accuracy in per cent of the difference of the approximations @xmath189 for pqcd : @xmath190 and for fapt : @xmath191 . the results are presented in two ranges of @xmath9 : low @xmath192 and high @xmath1 . 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we apply ( fractional ) analytic perturbation theory ( fapt ) to the qcd analysis of the nonsinglet nucleon structure function @xmath0 in deep inelastic scattering up to the next leading order and compare the results with ones obtained within the standard perturbation qcd . based on a popular parameterization of the corresponding parton distribution we perform the analysis within the jacobi polynomial formalism and under the control of the numerical inverse mellin transform . to reveal the main features of the fapt two - loop approach , we consider a wide range of momentum transfer from high @xmath1 to low @xmath2 where the approach still works .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

while the study of galaxy evolution has made important strides in recent years by being able to weigh individual galaxies ( i.e. , determine a mass ) , the field of quasar research is grappling with the issue of how to measure accurately the masses of a supermassive black holes ( smbhs ) for the distant quasar population . here the challenge is greater due to the fact that the sphere of influence of a smbh can only be resolved for a limited sample of nearby galaxies whereas the dynamical mass of a galaxy can easily be measured due to its large spatial extent . a significant leap forward in our ability to both accurately and efficiently measure the masses of smbhs , @xmath13 , at all redshifts will likely lead to new insights on questions such as how are black holes fueled , what it is the connection with its host galaxy , and how do smbhs evolve within a cosmological framework . spectroscopy enables us to probe the kinematics of ionized gas within the vicinity of a smbh in distant active galactic nuclei ( agns ) and luminous quasars to infer their black hole masses . traditionally , emission lines ( e.g. , , , h@xmath14 , and h@xmath3 ) detected in the optical and velocity - broadened between @xmath15 km s@xmath6 are used to probe the gravitational potential well of a smbh . this lower limit on the velocity width has been set somewhat arbitrarily since there exists a well - known population of both type 1 agns having narrower line widths ( i.e. , nls1 ; * ? ? ? * ) and those with intermediate - mass black holes @xcite . there are methods to determine the luminosity - weighted radial distance between the broad - line region ( blr ) and central source , @xmath16 , for agn ( @xmath17 ) through reverberation - mapping campaigns ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) based on balmer lines . even with the complex nature of the blr , this characteristic radius is tightly correlated with its luminosity @xcite , thus providing a means to infer such a distance to the blr in large quasar samples based solely on luminosity . then coupled with velocity information provides a viral mass estimate based on a single - epoch spectrum . such techniques have been applied to large quasar samples most notably the sloan digital sky survey ( sdss ) . a number of studies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) clearly demonstrate that such samples effectively probe smbhs above @xmath18 at @xmath12 ( an epoch of maximal black hole activity ) due to the wide area coverage and shallow depth . it is important to keep in mind that these black hole masses are based on calibrations using lower luminosity agns at low redshift ; their application to luminous quasars at high redshift is not well solidified with reverberation mapping @xcite . deep surveys , such as cosmos , goods and aegis , are effective probes of black hole accretion at lower masses . given that the black hole mass function is steeply declining at @xmath19 @xcite , studies of the global population with sdss are susceptible to large uncertainties when extrapolated to lower masses @xcite . while noble attempts have been made to characterize the low - mass end ( @xmath20 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , these studies have been based on luminosity and do not consider virial velocities . these deep survey fields have considerable x - ray coverage that can be utilized to select agns that mitigate biases incurred by host galaxy dilution and obscuration . such selection then has the potential to effectively probe the lower luminosity agn population that may be powered by lower mass black holes or those accreting at sub - eddington rates . followup optical spectroscopic observations are enabling single - epoch virial black hole mass estimates ( down to @xmath21 ) based on the properties of their broad emission lines and continuum luminosity @xcite . mass estimates for these higher redshift agns , which actually constitute the majority of the population in deep surveys such as cosmos , rely on or since the h@xmath14 line ( used to calibrate recipes based on local samples ) is no longer available in the optical window at @xmath22 . while the assumption that the line is produced from the same physical region as h@xmath14 may hold @xcite , there are studies that indicate that the physics of the broad - line region is not so simple @xcite , especially for the most luminous quasars that can have significant outflows possibly in response to a more intense radiation field . furthermore , it is non - trivial to disentangle the broad line from fe emission that sits at its base , especially for lower mass black holes and maybe even those at the high mass end . we present first results of a near - infrared spectroscopic survey of broad - line agns ( blagns ) primarily in the cosmos field using the fiber multi - object spectrograph ( fmos ) mounted on the subaru telescope . with fmos , we now have the capability to simultaneously acquire near - infrared spectra of @xmath23 targets over a field of view of 0.19 square degrees . in this study , we report on the comparison between the h@xmath3 emission line profile , detected in the near - infrared , with that of present in previously available optical spectra . we aim to establish how effective recipes ( established locally ) are to measure single - epoch black hole masses out to @xmath1 and for blagns of lower luminosity ( i.e. , lower black hole mass ) , as compared to those found in the sdss . our sample is supplemented with agns from the extended _ chandra _ deep field - south ( ecdf - s ) survey that reach fainter depths , in x - rays , than the _ chandra_/cosmos survey and improve our statistics at lower black hole masses . in section [ xxfmos ] , we fully describe the fmos observations including target selection , data reduction , and success rate with respect to detecting the h@xmath3 emission line . we describe our method for fitting broad emission lines in section [ xxxfit ] . our results are described in section [ result ] . throughout this work , we assume @xmath24 km s@xmath6 mpc@xmath6 , @xmath25 , @xmath26 , and ab magnitudes . the capability of fmos @xcite to simultaneously acquire near - infrared spectra for a large number of objects over a wide field offers great potential for studies of galaxies @xcite and agns @xcite at high redshift . over a circular region of 30@xmath27 in diameter , it is possible to place up to 400 fibers , each with a 12 aperture , across the field . to detect emission lines in agns over a wide range in redshift , we elect to use the low - resolution mode that effectively covers two wavelength intervals of @xmath28 ( j band ) and @xmath29 ( h band ) simultaneously . the spectral resolution is @xmath30 , thus a velocity resolution fwhm @xmath31 km s@xmath6 at @xmath32 , suitable for the study of broad emission lines of agn . unfortunately , this mode requires an additional optical element ( i.e. , vph grating ) that reduces the total throughput to @xmath33% at 1.3 and impacts the limiting depth reachable in a few hours of integration time . accurate removal of the bright sky background when observing from the ground with near - infrared detectors remains the primary challenge . with fmos , an oh - airglow suppression filter @xcite is built into the system that significantly reduces the intensity of strong atmospheric emission lines that usually plague the j band and h band . equally important , there is the capability ( cross - beam - switching ; cbs hereafter ) to dither targets between fiber pairs effectively measuring the sky background spatially close to individual objects and through the same fibers as the science targets . in this mode , two sequential observations are taken by offsetting the telescope by 60@xmath34 while keeping the target within one of the two fibers . the trade off is that only 200 fibers are available in this cbs mode . we refer the reader to @xcite for full details of the instrument and its performance . below , we briefly describe our observing program using fmos including the selection of type 1 agns , data reduction , and success rate with respect to the detection of broad emission lines . complete details of our program will be presented in silverman et al . ( in prep . ) along with the full catalog of emission - line properties of broad - line agns in the cosmos and ecdf - s . our primary selection of agns is based on their having x - ray emission detected by _ chandra _ @xcite within the central square degree of cosmos ( hereafter c - cosmos ) . the high surface density of agns ( @xmath35 per deg@xmath36 ) at the limiting depths ( @xmath37 ergs @xmath38 s@xmath6 ) of c - cosmos field ensures that we make adequate use of the multiplex capabilities of fmos . in addition , two fmos pointings were observed further out from the center of the cosmos field thus we relied upon the catalog of optical and near - infrared counterparts to _ xmm - newton _ sources @xcite for agn selection . we further require that optical spectroscopy @xcite is available for each source that yields a reliable redshift and detection of at least a single broad ( fwhm @xmath39 km s@xmath6 ) emission line , namely in many cases . we then specifically targeted those with spectroscopic redshifts that allows us to detect either h@xmath14 ( @xmath40 and @xmath41 ) , h@xmath3 ( @xmath42 and @xmath40 ) , or ( @xmath43 and @xmath44 ) in the observed fmos spectral windows in low - resolution mode , i.e. , @xmath28 and @xmath29 . fibers are assigned to blagns ( for which we can detect emission lines of interest ) with a limiting magnitude of @xmath45 . those at @xmath46 are given higher priority to ensure that this sample ( of lower density on the sky ) is well represented in the final catalog ; this also effectively improves upon our success rate of detection both continuum and line emission . due to the sensitivity of fmos and the low number density of agns at @xmath47 , our sample has very few detections of in the near infrared . in addition , we have acquired fmos observations of x - ray selected agns in the ecdf - s @xcite survey with the inclusion of those that are only detected in the deeper 2@xmath484 ms data @xcite in the central region that covers goods . this deeper x - ray field offers the potential to extend the dynamic range of our study in terms of black hole mass and eddington ratio . we specifically select agns , as mentioned above , based on their optical properties determined through deep spectroscopic campaigns @xcite . as with the cosmos sample , we place higher priority on the brighter agns ( @xmath49 ) while targeting the fainter cases with lower priority . we have acquired near - infrared spectra with subaru / fmos of broad - line agns from the cosmos and ecdf - s surveys . the majority of the data was obtained during open use time through naoj over three nights in december 2010 ( i d ; s10b-108 ) and two nights in december 2011 ( i d ; s11b-098 ) . additional targets were observed during other programs being carried out in the cosmos field through the university of hawaii in s10b@xmath48s11a . weather conditions were acceptable although clouds , mainly cirrus , reduced our observing efficiency . the typical seeing was @xmath50 with considerable variation across the nights . we elected to use the cbs mode while taking two sequential exposures each of 15 minutes for each position ( namely a and b positions hereafter ) . these pairs of exposure were repeated multiple times in order to reach an effective total integration time of @xmath51 hours on - source . some time is lost to refocusing and repositioning fibers at regular intervals during the full observation . in the early data , only one spectrograph ( irs1 ) was available thus @xmath52 fibers were available for science targets . we use the publicly available software fibre - pac ( fmos image - based reduction package ; * ? ? ? the reduction routines are based on iraf tasks although several steps are processed by additional tools written by the fmos instrument team . since our observation are carried out using an abab nodding pattern in cbs mode of the telescope , an effective sky subtraction ( a@xmath48b ) can be performed using the two different sky images : a@xmath53 b@xmath54 and a@xmath53 b@xmath55 taken before and after the @xmath56-th exposure . after the initial background subtraction , a cross talk signal is removed by subtracting 0.15% for irs1 and 1% for irs2 from each quadrant . the difference in the bias between the quadrants is corrected to make the average over each quadrant equal . we further apply a flat field correction using a dome lamp exposure . bad pixels are masked throughout the reduction procedure . additional steps include the distortion correction and the removal of residual airglow lines . this procedure is carried out for both positions a and b. individual frames are combined into an averaged image and an associated noise image . finally , the wavelength calibration is carried out based on a reference image of a th - ar emission spectrum . individual one - dimensional science and error spectra are extracted both to be used for the fitting of emission line profiles . we perform a first attempt at flux calibration by using spectra of bright stars , usually @xmath57 per spectrograph . a single stellar spectrum for each spectrograph is chosen to apply a correction based on the spectroscopic magnitude and photometry from 2mass . an improvement of the absolute flux level is required to account for aperture effects . therefore , we scale the flux of each fmos spectra of our agns to match the deep infrared photometry using total magnitudes available from the ultravista survey available over the cosmos field . while scale factors can reach as high as @xmath33 , the median value is 1.64 . we have observed over 100 type 1 agns in the combined cosmos and ecdf - s fields to date . in figure [ sample ] , we show the x - ray flux and nir magnitude distribution of the 108 agns ( originally identified as _ chandra _ x - ray sources ) in the cosmos field that have @xmath58 , a redshift interval where we are capable of detecting h@xmath3 in the fmos spectroscopic window . the distributions are shown for both the observed objects and the 56 having a significant detection of a broad h@xmath3 emission line ( at the expected wavelength ) that subsequently yielded a black hole mass estimate . prior to our observations , we had no empirical assessment of the performance of fmos thus agn were targeted to faint infrared magnitudes , now understood not to be feasible using the low - resolution mode for the faintest objects ( @xmath59 ) . as shown in figure [ sample](@xmath60 ) , we have a reasonable level of success ( 71% ) with the detection of broad emission lines at brighter magnitudes ( @xmath46 ) . unfortunately , the success rate is significantly lower at fainter magnitudes thus dropping to 50% for the entire sample with @xmath61 . it is worth highlighting that a survey depth of @xmath46 is about two magnitudes fainter than current near - infrared spectroscopic observations of sdss quasars @xcite at similar redshifts . in figure [ lbol_z ] , we demonstrate this by plotting the bolometric luminosity ( based on @xmath62 and a bolometric correction of 5.15 ; * ? ? ? * ) of our agn compared to those from sdss surveys . @xcite describe in detail the benefits of using h@xmath3 for black hole mass measurements . in particular , the h@xmath3 emission line is stronger ( @xmath63 ) thus more easily detected as compared to h@xmath14 . this is clearly evident from our observations . we successfully detect a broad h@xmath3 line in @xmath64% of the cases ( as mentioned above ) while the detection of h@xmath14 is very low ( 17% ) . even so , we do detect h@xmath14 in a fair number of cases up to @xmath65 that will be presented in the full emission - line catalog ( silverman et al . in prep . ) . any future fmos campaign designed to detect the h@xmath14 emission line ( both broad and narrow ) should increase the exposure time significantly and/or use the high - resolution mode that has roughly three times higher throughput in the h band as compared to low - resolution mode ( see figure 19 of * ? ? ? we measure the properties of broad emission lines present in optical and near - infrared spectra to derive single - epoch black hole mass estimates . for the emission lines of interest here ( i.e. , and h@xmath3 ) , we specifically measure the full width at half maximum ( fhwm ) , total luminosity of the emission line in the case of h@xmath3 , and continuum luminosity at 3000 . due to the moderate luminosities of the agn sample , there can be a non - negligible host galaxy contribution that impacts the estimate of the agn continuum at redder wavelengths ; therefore , we chose to use the h@xmath3 line flux rather than the continuum luminosity . fortunately , the multi - wavelength photometry of the cosmos , including the hst imaging , enables us to determine the level of such contamination that will be fully assessed in a future study . emission lines are fit using a procedure as outlined below that enables us to characterize the line shape for even those that have a considerable level of noise . our fitting procedure of the continuum and line emission utilizes mpfitfun , a levenberg - marquardt least squares minimization algorithm as available within the idl environment . even though this routine has well - known computational issues , this algorithm is widely used due to its ease of use and fast execution time . the routine returns best - fit parameters and their errors as well as measure of the goodness of the overall fit . we further describe the individual components required to successfully extract a parameterization of the broad component used in determining virial masses . it is worth recognizing that each line has its own advantages and disadvantages that need to be considered carefully especially when fitting data of moderate signal - to - noise ratio ( s / n ) . a final inspection of each fit by eye is performed to remove obvious cases where a broad component is not adequately determined almost exclusively due to spectra having low s / n . we perform a fit to the h@xmath3 emission line ( if detected within the fmos spectral window ) in order to measure line width and integrated emission - line luminosity . based on the spectroscopic redshift as determined from optical spectroscopy , we select the spectrum at rest wavelengths centered on the emission line and spanning a range that enables an accurate determination of the continuum characterized by a power law , @xmath66 . we employ multiple gaussian components to describe the line profile . it is common practice to make such an assumption on the intrinsic shape of individual components , even though it has been demonstrated that broad - emission lines in agn are not necessarily of such a shape . the h@xmath3 line is fit with two or three gaussians ( including a narrow component ) and the [ ] @xmath676548,6684 lines with a pair of gaussians . the ratio of the [ ] lines is fixed at the laboratory value of 2.96 . the narrow width of the [ ] lines is fixed to match the narrow component of h@xmath3 . the width of the narrow components is limited to @xmath68 km s@xmath6 ( a range not corrected for intrinsic dispersion ) . the velocity profile of the broad components is characterized by the fwhm , measured using either one gaussian or the sum of two gaussians . we then correct the velocity width for the effect of instrumental dispersion to achieve an intrinsic profile width . the h@xmath3 luminosity discussed throughout this work is the sum of the broad components . there are cases for which the fitting routine returns a solution with the width of the narrow component pegged at the upper bound of 800 km s@xmath6 . it is worth highlighting that this minimization routine stops the fitting procedure when a parameters hit a limit thus the returned values are not the true best - fit values . for these , we inspect all fits by eye and decide whether such an additional broad component is real . for many cases , we can use the [ ] @xmath675007 line profile , within the fmos spectral window , to determine whether such a fit to the narrow line complex is accurate . in addition , we can use the available optical spectra for such comparisons . when the level of significance of the narrow line is negligible , we rerun the fitting routine and fix the narrow line width to the spectral resolution of fmos , @xmath69 km s@xmath6 . in figure [ haexam ] , we show three examples of our fits to the h@xmath3 emission line that span a range of line properties . [ [ section ] ] we fit the emission line , as done in @xcite , observed in optical spectra primarily from zcosmos @xcite , magellan / imacs @xcite , and sdss @xcite . the emission line is modeled by a combination of one or two broad gaussian functions to best characterize the line shape . we first remove the continuum ( before attempting to deal with the emission lines ) by fitting the emission in a window surrounding the line . as with h@xmath3 , a power - law function is chosen to best characterize the featureless , non - stellar light attributed to an accretion disk . we further include a broad fe emission component based on an empirical template @xcite that is convolved by a gaussian of variable width and straddles the base of the emission line . a least square minimization is implemented to determine the best - fit parameters . when possible , we optimize residuals of the fits on a case - by - case basis by trying to minimize the number of components . absorption features are either masked out or interpolated across . the fit returns two parameters required for black hole mass estimates : fwhm and monochromatic luminosity at 3000 . examples of our fits to the line are presented in figure [ mgexam ] . we can determine how closely the parameters ( i.e. , luminosity and fwhm ) required to estimate single - epoch black hole masses agree between the h@xmath3 and emission lines . any systematic offset or inherent scatter may only add additional uncertainty to the derived masses . we essentially want to establish whether or not the kinematics of the blr is consistent with photoionized gas in virial motion around the smbh . we provide all measurements and derived masses in table [ catalg ] . our first concern is to determine whether the h@xmath3 emission - line luminosity scales appropriately with the uv continuum luminosity . for the following analysis , we do not correct for extinction due to dust and any contamination by the host galaxy ; the impact of these , thought to be small , will be quantified in a later study . in figure [ lumlum ] , the continuum luminosity , @xmath70 at 3000 , is plotted against the emission - line luminosity of h@xmath3 . our data ( as shown by the red points ) spans two decades in luminosity and exhibits a clear correspondence between continuum and line emission . based on our agn sample , we determine the best - fit linear relation to be @xmath71 . for the linear fitting , we adopt a fitexy method @xcite . the level of dispersion of the data about this fit is 0.20 dex that will contribute to the dispersion in the final mass estimates . rlrrlllllllll 178 & cosmos & 149.58521 & @xmath722.05114 & 1.350 & 43.76@xmath730.03 & 45.35@xmath730.03 & & 3.685@xmath730.013 & 3.811@xmath730.079 & & 8.68@xmath730.03 & 8.78@xmath730.16 + 5275 & cosmos & 149.59021 & @xmath722.77450 & 1.400 & 44.10@xmath730.13 & 45.50@xmath730.02 & & 3.835@xmath730.028 & 3.973@xmath730.072 & & 9.18@xmath730.09 & 9.18@xmath730.14 + 322 & cosmos & 149.62421 & @xmath722.18067 & 1.190 & 43.16@xmath730.14 & 45.02@xmath730.02 & & 3.440@xmath730.044 & 3.581@xmath730.081 & & 7.84@xmath730.12 & 8.17@xmath730.16 + 192 & cosmos & 149.66358 & @xmath722.08522 & 1.220 & 43.24@xmath730.14 & 45.05@xmath730.02 & & 3.339@xmath730.075 & 3.490@xmath730.035 & & 7.68@xmath730.17 & 8.00@xmath730.07 + 157 & cosmos & 149.67512 & @xmath721.98275 & 1.330 & 42.98@xmath730.13 & 44.84@xmath730.03 & & 3.701@xmath730.040 & 3.632@xmath730.017 & & 8.28@xmath730.11 & 8.19@xmath730.04 + we can compare our data set to more luminous quasars from the sdss . in particular , we identify 327 quasars from the sdss sample in @xcite with @xmath74 , that cover a similar luminosity range as our high redshift sample . these were selected from 1178 quasars at @xmath74 based on high - quality data determined by adopting the following criteria : error in @xmath75 , @xmath76 , and @xmath77 . in addition , we include data from a recent study by @xcite that provides the emission line properties including h@xmath3 line of high - luminosity quasars from the sdss with @xmath78 . these samples are added to our data shown figure [ lumlum ] . we clearly see that sdss quasars fall along the @xmath79-@xmath80 relation as established above . furthermore , the sdss quasars have similar dispersion at both low-@xmath81 and high-@xmath81 samples to our sample , @xmath82 and @xmath83 , respectively . we highlight that our agn sample nicely extends such comparisons between continuum luminosity and line emission at higher redshifts and to lower luminosities . we are able to effectively establish a wider dynamic range , not present in the high-@xmath81 sdss sample due to the limited luminosity range around @xmath84 . by merging all three samples , we find the following relation based on a linear fit : @xmath85 . while there is very good agreement between the uv continuum and emission line luminosity , there is a small difference that may impact , even slightly , our comparison of the masses . based on the fmos sample , the mean ratio @xmath86 is @xmath87 , slightly higher than that found for both the low-@xmath81 and high-@xmath81 sdss quasar samples mentioned above ; @xmath88 and @xmath89 , respectively . this can be seen in the inset histogram in figure [ lumlum ] . if this was due to the effect of dust extinction , one would find the opposite trend with reduced uv continuum relative to the h@xmath3 line emission . there may be other explanations such as an underlying sed that may be different for x - ray selected samples @xcite and has an impact on the response seen in photoionized gas , or the effect of the host galaxy on the aperture corrections that differs in each band . while this issue is of importance , we reserve a detailed investigation to subsequent work since it is beyond the scope of this paper to adequately demonstrate such effects . here , we are primarily concerned with the magnitude of a luminosity offset and whether it contributes to an offset between the masses . with black hole mass scaling with the square root of the luminosity ( see below ) , the offset in luminosity , as determined above , amounts to a very small offset in @xmath90 of 0.07 . a second pillar for the use of as a black hole mass indicator is that the emitting - line gas is located essentially within the same clouds that emit balmer emission . while there are claims that this is the case by comparing the velocity profile of with h@xmath14 ( e.g. , * ? * ; * ? ? ? * ) , there are reported differences and trends that are not well understood @xcite . for example , tends to be narrower than h@xmath14 with a difference significantly larger at higher velocity widths ( see figure 2 of * ? ? ? * ) . our aim here is to compare the fwhm of the and h@xmath3 emission lines using a sample not yet explored , namely the moderate - luminosity agns at high - redshifts in survey fields such as cosmos . in figure [ fwhfwh ] , we plot the emission - line velocity width between h@xmath3 and emission lines . based on the fmos sample only , a positive linear correlation is seen between the velocity width of the two emission lines with the mean ratio of @xmath91 ( @xmath92 ) . our data is in very good agreement with those from the sdss , i.e. , @xmath93 ( @xmath94 ) for the low-@xmath81 sample from @xcite , @xmath95 ( @xmath96 ) for the high-@xmath81 sample from @xcite , and recent fmos results from sxds ( see * ? ? ? based on a linear fit to the data shown in figure [ fwhfwh ] , we measure a slope that is consistent with unity ; @xmath97 for fmos only and @xmath98 for fmos@xmath72sdss , and that can not substantiate the claim by @xcite for a shallower value . these results are supportive of a scenario where the and the h@xmath3 emitting regions are essentially co - spatial with respect to the central ionizing source . there are a few noticeable outliers well outside the dispersion of the sample . these objects then appear as outliers when comparing their masses based on different lines in the next section . upon inspection , we find that these are the result of the fmos spectra having low s / n . in some cases , there may be a fit to the h@xmath3 emission line , based on different parameter constraints , that is equally acceptable to the original fit as assessed by a chi - square goodness of fit and has a velocity width in closer agreement with . although , we refrain from such selective fitting in order to present results that may be obtained from using similar fitting algorithms on larger data sets where such close inspection is not feasible . we now calculate black hole masses ( @xmath13 ) based on our single - epoch spectra using both ( i ) @xmath62 and fwhm@xmath99 , and ( ii ) @xmath80 and fwhm@xmath100 . this calculation can be expressed as follows : @xmath101 we explicitly use the recipes provided by @xcite and @xcite for the cases of h@xmath3 and lines , respectively : @xmath102 we note that the calibration of the relation for has been carried out by many studies ( e.g. , * ? ? ? * ; * ? ? ? * ) and there are known differences between them . in figure [ masmas ] , we show @xmath13 for our sample derived from h@xmath3 and . our sample spans a range of @xmath103 consistent with that reported by previous studies of type 1 agns in cosmos @xcite and is complementary to the higher-@xmath104 quasar sample at similar redshifts with @xmath105 @xcite . we find the average ( dispersion ) in the black hole mass ratio of @xmath106 is 0.17 ( @xmath11 ) for our fmos sample . these results are similar to that determined from the sdss sample ; the average ( dispersion ) in the black hole mass ratio is @xmath107 ( @xmath108 ) at low @xmath81 and @xmath109 ( @xmath82 ) at high @xmath81 . while an offset of 0.17 dex is seen in the fmos sample , we conclude that the recipes established using local relations give consistent results between - and h@xmath3-based estimates . we have investigated the emission line properties of agns in cosmos and ecdf - s to establish whether and h@xmath3 provide comparable estimates of their black hole mass . this study is the first attempt to do so for agns of moderate luminosity , hence lower black hole mass ( @xmath110 ) , at high redshift that complements studies of more luminous quasars . our results clearly show that the velocity profiles of and h@xmath3 are very similar when characterized by fwhm and the relation between continuum luminosity and line luminosity is tight . we then find that virial black hole masses based on and h@xmath3 have very similar values and a level of dispersion ( @xmath111 ) comparable to luminous quasars from sdss . it is important to keep in mind that these results pertain to specific calibrations @xcite for estimating black hole mass . the use of other recipes , such as provided by @xcite , will show a discrepancy larger than seen here . to conclude , the locally - calibrated recipes for black holes masses using and h@xmath3 are applicable for fainter agn samples at high redshift . these results further support a lack of evolution in the physical properties of the broad line region in terms of quantities such as @xmath112 ( e.g. , * ? ? ? * ; * ? ? ? * ) , emission - line strengths ( e.g. , * ? ? ? * ) , and the inferred metallicities . as a final word of caution , such estimates of black hole mass are likely to have inherent dispersion as discussed above and systematic uncertainties that are not yet well understood . for instance , recipes for estimating black hole mass depend on the assumption that the gas is purely in virial motion . this is unlikely to be true for all cases since both outflows and inflows are common in agns . even so , there is evidence that the virial product of mass and luminosity ( as a proxy for the radius to the blr ) is a useful probe of the central gravitational potential . in the very least , it is important to establish the level of dispersion in such relations since observed trends usually rely on offsets comparable to the dispersion such as the redshift evolution of the relation between black holes and their host galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? we thank kentaro aoki and naoyuki tamura for their invaluable assistance during our subaru / fmos observations . k.m . acknowledges financial support from the japan society for the promotion of science ( jsps ) . data analysis were in part carried out on common - 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we present an analysis of broad emission lines observed in moderate - luminosity active galactic nuclei ( agns ) , typical of those found in x - ray surveys of deep fields , with the aim to test the validity of single - epoch virial black hole mass estimates . we have acquired near - infrared ( nir ; @xmath0 ) spectra of agns up to @xmath1 in the cosmos and extended _ chandra _ deep field - south survey , with the fiber multi - object spectrograph ( fmos ) mounted on the subaru telescope . these low - resolution ( @xmath2 ) nir spectra provide a significant detection of the broad h@xmath3 emission line that has been shown to be a reliable probe of black hole mass at low redshift . our sample has existing optical spectroscopy ( through programs such as zcosmos ) which provides a detection of , a broad emission line typically used for black hole mass estimation at @xmath4 . we carry out a spectral - line fitting procedure using both h@xmath3 and to determine the virial velocity of gas in the broad line region , the monochromatic continuum luminosity at 3000 , and the total h@xmath3 line luminosity . with a sample of 43 agns spanning a range of two decades in luminosity ( i.e. , @xmath5 ergs s@xmath6 ) , we find a tight correlation between the rest - frame ultraviolet and emission - line luminosity with a distribution characterized by @xmath7 and a dispersion @xmath8 . there is also a close one - to - one relationship between the fwhm of h@xmath3 and of up to 10000 km s@xmath6 with a dispersion of 0.14 in the distribution of the logarithm of their ratios . both of these then lead to there being very good agreement between h@xmath3- and -based masses over a wide range in black hole mass ( i.e. , @xmath9 ) . we do find a small offset in -based masses , relative to those based on h@xmath3 , of @xmath10 dex and a dispersion @xmath11 . in general , these results demonstrate that local scaling relations , using or h@xmath3 , are applicable for agn at moderate luminosities and up to @xmath12 .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

galaxy - galaxy interactions lead to the redistribution of stars and gas about each system , and an infusion of material into the local intergalactic medium , promoting star formation ( @xcite ) . tidal tails are signatures of galactic mergers ( @xcite ) , illuminated by the ignition of star formation ( @xcite ) . turbulent energy injected into the local hi through these mergers compresses the gas , forming new stars ( @xcite ) ; this is observationally shown in @xcite , who were able to link the presence of star clusters candidates ( sccs ) to turbulent regions of hi . while the young , in - situ formed sccs of tidal tails have been studied in the past through imaging and spectroscopy ( e.g. @xcite ; @xcite ; @xcite ; @xcite ; @xcite ; @xcite ) , the composition of the underlying stellar material remains a mystery . we know that gas is easily extracted from the parent galaxies during interactions ( e.g. @xcite ) , but whether or not stars follow suit has not yet been established . the extracted gas can collapse to form stars in a clustered manner , yet at the same time , simulations have shown that clusters in tails can be easily disrupted ( @xcite ) . this sparks an obvious question : what are tidal tails made of ? what is the relative fraction of gas and old stars within the interaction ejecta ? the answer can inform dynamical simulations of interactions , and therefore help refine our understanding of the enrichment of the intergalactic medium . in fact , current dynamical simulations ( e.g. @xcite ) use an older stellar component as a gravitational ` anchor ' for the gas . is this requirement justified by the observations ? existing studies have focused on the young stellar components of tidal tails , and therefore do not have the capability to probe deep into these regions . we have developed a new observing program designed to do just this , using deep , photometric _ ugriz _ imaging . to derive an accurate age estimate for a tidal tail , we plan our exposure times to view across the stellar sequence in our diffuse tidal tail light . these tails are imaged in each filter to an adequate signal to noise ratio , allowing us to derive an average colour and age of their diffuse light . this has the additional benefit of allowing us to age date star clusters within the tail . we choose the twin tidal tails of ngc 3256 as a case study for our method . the system is relatively nearby at a distance of 38 mpc ( @xcite ) , with an interaction age of 400 myr ( @xcite ) - not so young that our observations will be drowned out by ob stars , yet young enough so that the tidal structure is still visible . it has been well studied in the past through spectroscopic ( @xcite ; @xcite ; @xcite ) , photometric ( @xcite ; @xcite ; @xcite ) and hi ( @xcite ) observations , giving us benchmarks to compare to . we will begin in section [ sec:2 ] by describing our data and our reduction process . in section [ sec:3 ] we describe our analysis methods , and in section [ sec:4 ] we show our results . in section [ sec:5 ] we discuss our findings , and conclude our paper with the main points of our research in section [ sec:6 ] . images were obtained from the gemini - south observatory from march 2013 to june 2013 , using the gmos imager . the gmos field of view is superimposed on an optical image from the digital sky survey in figure [ fig : gmos_fov ] , measuring @xmath10 arcmin . exposure details are shown in table [ table : exposures ] . [ cols="^,^,^,^,^ " , ] a colour - colour diagram with only these sccs is shown in figures [ fig : w_scc_colour ] and [ fig : e_scc_colour ] . overlaid in gray are data points for the diffuse light of the respective tidal tail . the ssp models from @xcite do not include nebular emission . however , the nebular continuum , as well as emission lines from h@xmath11 , h@xmath12 , [ o iii ] and [ o ii ] , can have strong effects on our colours for young objects , with ages < 10 myr . we use ` starburst99 ` ( @xcite ) to include a nebular continuum , as well as emission from h@xmath11 and h@xmath12 . following @xcite and @xcite , we find the strengths of the [ o iii ] and [ o ii ] lines from the kiss sample of nearby low - mass star - forming galaxies ( e.g. , @xcite ) . we use the median ratios of [ o iii]/h@xmath12 and [ o ii]/h@xmath12 , listed at 0.08 and 0.56 , respectively . h@xmath11 emission falls in the _ r _ filter , causing the @xmath13 colour to appear bluer . both h@xmath12 and [ o iii ] fall in the _ g_-band , while [ o ii ] is in the _ u_-band . depending on the relative strengths of the oxygen lines , this can cause the @xmath14 colour to become bluer or redder . to show a possible range of nebular emission , we also include a track setting [ o iii]/h@xmath12 to its 90th percentile value ( 0.66 ) and [ o ii]/h@xmath12 to its 10th percentile value ( 0.22 ) . this provides a better fit for several data points in the eastern tail , although the position of these clusters on the diagram is likely due to a combination of varying emission line strengths and dust extinction . @xmath15 and @xmath16 . gray points correspond to diffuse light . for comparison , we show colours of objects outside the tail region on the right . nebular tracks with emission from h@xmath12 , h@xmath11 , [ o iii ] , [ o ii ] , and continuum emission are included . dotted lines indicate median values of [ o iii]/h@xmath12 and [ o ii]/h@xmath12 emission , while dashed lines indicate the 90th and 10th percentile of [ o iii]/h@xmath12 and [ o ii]/h@xmath12 emission , respectively , from the kiss galaxy sample . 31 sccs are detected in the western tail , with a median age of 8.15 log yrs . ] but for the eastern tail . several young objects show strong nebular emission . 19 sccs are detected in the eastern tail , with a median age of 7.96 log yrs . ] in examining figure [ fig : w_scc_colour ] , we see a clear overdensity of sccs in the western tail compared to objects outside of the tail . in the eastern tail ( [ fig : e_scc_colour ] ) , a number of sccs exist at blue values of @xmath14 and @xmath13 which are distinct from those outside the tail . sccs in the western tail overlap with the diffuse light ( though skewed to the blue end ) , while those in the eastern tail are distinct from the diffuse light in the host tail . to quantitatively test this , we apply the kolmogorov smirnov ( ks ) test to the colour distribution of the diffuse light and sccs . this test allows us to determine the probability that two samples were drawn from independent distributions . p_-value of less than 0.013 indicates the populations are distinct from one another at more than a 2.5@xmath17 confidence level . for the @xmath14 colours , we find _ p_-values of 0.028 and @xmath18 for the western and eastern tails , respectively . for the @xmath13 colours , we have _ p _ values of @xmath19 and @xmath20 , for the western and eastern tails . these results show the diffuse light and scc colour distributions are distinct from each other in both tails . ages and masses of our sources are found using the 3def method ( @xcite ) . this uses a maximum likelihood estimator to find the ages and masses of each cluster , by adjusting the colour excess @xmath21 to fit the observed colour in each filter to a given ssp evolutionary model . results for the total masses and median ages of sccs are shown in table [ table : mass_all ] , with masses and ages shown in figure [ fig : mass_age ] ; dashed lines represent our scc detection limits based on our colour and magnitude criteria from section [ sec : cluster_colours ] . error bars show the maximum and minimum age and mass for each data point . at low masses , there are several points which do not fall within the detection band . these objects are subject to internal extinction , which dims them to fall within our scc colour and magnitude cutoffs . there is a gap in age between the young objects in the eastern tail and the main distribution with ages of 8.0 - 8.8 log yrs . this indicates a recent small burst of isolated star formation in the eastern tail , small enough that only low mass clusters are present . other similar bursts could have occurred in either tail between 7.0 and 8.0 log yrs , but would have faded from view by the present . it is clear , however that there is not continuous star formation at the level of that seen during the main interaction period , 8.0 - 8.8 log yrs ago . spatial maps of tail colours and sccs are shown in figure [ fig : colour_tail_sccs ] for both tails . measurement boxes from figures [ fig : boxes ] are colour coded to indicate their @xmath14 colours . scc positions and @xmath14 values are added in a similar manner . the difference in age between the tails is evident in the abundance of younger regions , represented by cyan and yellow , in the western tail , compared to older regions , represented by orange and red , in the eastern tail . the spatial location of sccs does not appear to influence the colour of the diffuse light . this is not too surprising , as these objects were masked out in section [ sec : masking ] . both of the tails show colour gradients across their lengths , with bluer colours near the galactic bulges and redder colours at the far tips . likewise , the edges of the tails appear redder than their interiors . such gradients were also seen among several tails in @xcite . distributions of @xmath14 colours for diffuse light and sccs are shown in figure [ fig : colour_hist ] . colour distribution for sccs and diffuse light in ngc 3256w ( dark green ) and e ( purple ) , with bin size of 0.1 . hashed histograms indicate diffuse light measurements , while solid histograms represent the sccs colour distributions for the separate diffuse structure in the western tail are shown in the inset graph in orange , plotted alongside those of the western tail itself ( dark green ) . the diffuse light colours between the two tails are clearly distinct from one another . the distribution of scc colour in the western tail overlaps its diffuse light colour distribution , suggesting a common origin . sccs in the eastern tail are much younger than any of the sccs in the western tail . ] we can summarize figure [ fig : colour_hist ] with the following points : 1 . the difference in @xmath14 colour between the eastern and western tail shows the total diffuse stellar light of the two tails are separated in age by over 500 myr , with a significantly larger contribution from a young population in the western tail . 2 . assuming the two tails were formed at the same time , we find the stellar masses of the eastern and western tails to be dominated by old ( @xmath2210 gyr ) populations drawn from the parent galaxies . 3 . while the western tail contains a significant old ( @xmath2210 gyr ) population , the stellar light is largely comprised of a population which formed soon after the interaction , with an age of 8.29 log yrs . the similarity of the @xmath14 colour distribution of sccs in the western tail compared to the diffuse light gives credence to the idea that the western tail is comprised of disrupted star clusters formed shortly after the formation of the tail , with the ones we see today being the ones that survived . the stark contrast in @xmath14 colours for sccs in the eastern tail as compared to the diffuse light colours suggests some very recent star formation as compared to the age of the tail . stellar populations in the two tails of ngc 3256 have different compositions . we base our conclusions on the assumption that the tails were formed at the same time . it s possible to interpret figure [ fig : lsbphot ] as due to the eastern tail having formed first , with the stellar light we observe being formed from a burst of star formation caused by that earlier interaction . however , prograde collisions between galaxies are required to form tidal tails ( @xcite ; @xcite ) , as in the case of ngc 3256 . in such a case , these tails form simultaneously , which indicates a common age between ngc 3256 s tails . @xcite found three nuclei within the centre of ngc 3256 , leading them to construct a merger scenario involving two separate mergers and three galaxies . the eastern tail could be explained to have formed during the first merger , while the western tail formed later with a prograde interaction between the merger remnant and the third galaxy . however , @xcite suggests a more likely scenario with the two major galaxies initially merging , and a smaller satellite galaxy merging after the major interaction , supported in further work by @xcite . additionally , @xcite only observed two broad hi tails , leading them to conclude that ngc 3256 is likely created by a two galaxy prograde merger . given these works , it is very probable that the two tails were formed at the same time . we also find possible evidence for the small satellite merger in the separate diffuse structure , seen in the western tail . this structure is younger than either tail , supporting the theory of a late minor merger , occurring after the major merger . the diffuse light in the western tail is dominated by a younger population , formed soon after the formation of the tail . this population is also present in the eastern tail , although at a lower concentration . the source of the young population is uncertain , as stars may form in a variety of methods : in bound clusters , unbound stellar associations , down to near individual stellar formation . the fraction of stars forming in clusters can be up to 70% in regions of large gas density ( @xcite ) . however , these clusters are subject to frequent tidal shocks which will preferentially destroy low mass clusters ( @xcite ) , dispersing their material into the diffuse tidal light and making clusters an ideal candidate for the source of our young population . the destruction of star clusters is modeled in two parts : number loss ( removal of stars in a cluster ) and mass loss ( removal of mass in a cluster ) . number loss can be understood as effects from stellar feedback , such as stellar winds and supernovae , which expel gas in a cluster , causing it to become unbound . mass loss will remove stars from a cluster via two - body interactions . number loss , also known as infant mortality , will only be important for the first @xmath2210 myr , as the hot and massive stars evolve and explode . the remnants of these clusters are thus seen as the diffuse light of the tail . a similar effect is seen in ngc 7714 ages of hii regions within the tidal tails have been shown to be older than star clusters residing within them , suggesting previously formed clusters have been dispersed and surround the newly conceived clusters ( @xcite ) . the coincidence of peaks in histograms of @xmath14 colour between the diffuse light of the western tail and its sccs ( figure [ fig : colour_hist ] ) strongly suggests the two are intertwined . results of ks tests show the populations are distinct from each other , however , this is likely due to the presence of an old stellar population in the diffuse light of the tail . the timescale between tail formation and cluster formation is separated by @xmath22200 myr . we compare this to simulations of ngc 4038/9 , a similar merger , which found a peak in star formation @xmath2225 myr after the interaction ( @xcite ) . individual masses of sccs in the western tail are on average more massive than in the eastern tail , as shown in figure [ fig : mass_age ] , although the eastern tail has several very young , low mass objects absent in the western tail . the greater number of sccs for the western tail can be attributed to its larger hi mass . this can be related back to the diffuse light as well , as the western tail s higher abundance of gas led it to form more star clusters which could be disrupted and dispersed in the tail . we compare the ages of our clusters to those found in the nuclei from previous studies . @xcite spectroscopically studied 23 star clusters inside the centres of the galaxies , finding an average age of @xmath2210 myr . @xcite performed photometry on several hundred objects within the centre finding bright and blue objects . our sccs are substantially older than those found within the nuclei , suggesting the star formation in the tails was cut off earlier relative to the interior . this is consistent with spectroscopic observations of star clusters in the western tail by @xcite and photometric analysis of sccs in the eastern tail by @xcite , both of whom found ages of clusters to be older than those in the interior . a similar viewpoint is shared in the antennae simulations , which show a cessation of star formation in the tails , while star formation in the interiors is ongoing ( @xcite ) . star formation in the interiors can be stimulated as material previously thrown out during the initial encounters falls back into the centre of the potential well . of particular interest are the very young objects found in the eastern tails , with @xmath23 . these objects indicate relatively recent star formation in a tail whose diffuse light suggests a limited star formation history . it is possible these objects are now being formed as material falls back through the tail to the interiors , creating turbulence in the hi gas and sparking small bursts of star formation . we do not see these very young objects in the western tail . however , note that they are low mass , with masses @xmath24 . they will fade from detectability as they age ( see figure [ fig : mass_age ] ) ; additionally , they may subject to tidal shocks from regions of dense gas which can disrupt them . this suggests that there may be small star formation events periodically in both tails , but the evidence of these events will rapidly fade away . @xcite found that tidal tails with large hi line of sight velocity dispersion @xmath25 and high hi column densities were ideal locations for sccs . both the 3256 tails fit these criteron , and the majority of their sccs , as determined by @xcite using _ hst _ wfpc2 data , lie in hi pixels with these characteristics . however , the tails can be distinguished by measurements of shear ( @xmath26 ) , with the western tail s shear about one - third that of the eastern s . @xcite found that sccs were preferentially located in regions of low shear , which is suggestive that these areas can be ideal for star formation . we reproduce spatial maps of hi shear from figures 4.18 and 4.19 in @xcite , and compare them to our spatial @xmath14 maps in figure [ fig : e_shear ] for the eastern tail and figure [ fig : w_shear ] for the western tail . from figure [ fig : e_shear ] , two pockets of low shear are visible in yellow . the first of these , closest to the bulge , corresponds to the location of the youngest sccs in the eastern tail , visualized as beige circles . the second low shear region does not contain any sccs , although the diffuse light near this area is younger than its surroundings . the western tail has a ridge of low shear seen in figure [ fig : w_shear ] , running from the middle of the tail to the tip . we find the majority of sccs within the hi field of view reside in this region , with few ( @xmath223 out of 23 ) outside . the diffuse light appears to follow the shear as well . at the tip of the tail , the low hi shear seen in blue matches the yellow diffuse light boxes . as the shear increases to yellow and orange , the diffuse light reddens and the boxes shift to orange and red . the effect of dust on the diffuse light in these tails is similar to that of an older population ; in either case the colours are reddened . to investigate this effect , we redden the eastern tail colour to match the western s , by minimizing the distance between the two values . we find the amount of extinction needed for this is @xmath27 , which we find unrealistic , as tidal tails are regions of relatively low extinction ( @xcite ; @xcite ) . additionally , examination of archival _ galex _ nuv data ( @xcite ) show similar brightnesses for the two tails , suggesting similar levels of low extinction . extinction in the eastern tail would dim the _ galex _ data , with respect to the western tail , but this is not seen . @xcite examined the @xmath28 colour distribution within the eastern tail , finding negligible levels of reddening , supporting the lack of dust within the tails . we finally look at extinction in the three star clusters spectroscopically studied in the western tail from @xcite , which are listed at 0.0 , 0.3 , 0.5 @xmath29 mag , showing minimal extinction . additionally , as mentioned in section [ sec : cluster_mass ] , several of our young sccs are subject to internal extinction . extinction values of these objects range from 1.1 to 0.7 @xmath29 mag . dust can be expected to associate with star clusters , as dust is needed for star formation , so we expect to see the most extinction in these regions . however , such levels of extinction exist for a small handful of objects , and as these are masked and do not factor into our lsb measurements , we are lead to believe extinction does not significantly affect our analysis of the stellar age and mass distribution . our observational program has proven successful in allowing us to characterize the stellar populations of our tidal tails . we find ngc 3256w to be bluer than its twin tail , ngc 3256e . measured colours indicate that diffuse light in the western tail has a large contribution from a young population formed after the interaction , perhaps from dispersed star clusters , as compared to the eastern tail , which is primarily illuminated by an old population derived from the host galaxy . both tails exhibit colour gradients along their lengths , suggesting a gradient in the time scale of star formation . despite these colour differences , both tails appear to be dominated in mass by an old , underlying population , originating from the interacting galaxies . analysis of sccs shows a lack of old objects in either tail ( > @xmath30 yr ) , but a clustering of objects below 400 myr in the western tail and eastern tail . the eastern tail shows an interesting clustering of young objects , with ages < @xmath31 yr . these objects are low mass structures and are not likely to be detected as they age , disappearing as they fade beyond our detection limits , or are dispersed into the tail . the @xmath14 colour distribution of the western sccs is proven to be distinct from the diffuse stellar light in the western tail through ks probability tests , however the peaks of the colour distributions of the diffuse light and sccs match well , suggesting these objects and regions are intertwined . the fact that the ks test shows separate populations can be explained by the addition of an old , underlying stellar population to the diffuse light . ngc 3256 has been shown in past studies to contain a large number of sccs compared to other systems ( @xcite ; @xcite ) . we plan to apply our current observational program to additional tidal tail systems with varying amounts of sccs and hi properties , particularly those with low numbers of sccs . the presence of a tidal dwarf galaxy may also play a role in determining the composition of tidal tail diffuse light . _ galex _ observations of the antennae galaxy reveal a gradient in colour along the tail , with bluer colours near the tip of the southern tail , where two tidal dwarf galaxies reside ( @xcite ) . @xcite found that tails with tidal dwarfs did not contain as many sccs as those without them . have these structures already dispersed their clusters into their tails , or has there been a complete absence of star cluster formation ? this remains to be seen . we would like to thank the anonymous referee for helpful comments which have improved the quality and content of this paper . based on observations obtained at the gemini observatory ( program i d gs-2013a - q-57 , processed using the gemini iraf package ) , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the national research council ( canada ) , conicyt ( chile ) , ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) , and ministrio da cincia , tecnologia e inovao ( brazil ) . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . the images of these surveys are based on photographic data obtained using the oschin schmidt telescope on palomar mountain and the uk schmidt telescope . the plates were processed into the present compressed digital form with the permission of these institutions . scg thanks the natural science and engineering research council of canada for support . kk is supported by an nsf astronomy and astrophysics postdoctoral fellowship under award ast-1501294 . the institute for gravitation and the cosmos is supported by the eberly college of science and the office of the senior vice president for research at the pennsylvania state university .

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

complex systems with interacting constituents are ubiquitous in nature and society . to understand the microscopic mechanisms of emerging statistical laws of complex systems , one records and analyzes time series of observable quantities . these time series are usually nonstationary and possess long - range power - law cross - correlations . examples include the velocity , temperature , and concentration fields of turbulent flows embedded in the same space as joint multifractal measures @xcite , topographic indices and crop yield in agronomy @xcite , temporal and spatial seismic data @xcite , nitrogen dioxide and ground - level ozone @xcite , heart rate variability and brain activity in healthy humans @xcite , sunspot numbers and river flow fluctuations @xcite , wind patterns and land surface air temperatures @xcite , traffic flows @xcite and traffic signals @xcite , self - affine time series of taxi accidents @xcite , and econophysical variables @xcite . a variety of methods have been used to investigate the long - range power - law cross - correlations between two nonstationary time series . the earliest was joint multifractal analysis to study the cross - multifractal nature of two joint multifractal measures through the scaling behaviors of the joint moments @xcite , which is a multifractal cross - correlation analysis based on the partition function approach ( mf - x - pf ) @xcite . over the past decade , detrended cross - correlation analysis ( dcca ) has become the most popular method of investigating the long - range power - law cross correlations between two nonstationary time series @xcite , and this method has numerous variants @xcite . statistical tests can be used to measure these cross correlations @xcite . there is also a group of multifractal detrended fluctuation analysis ( mf - dcca ) methods of analyzing multifractal time series , e.g. , mf - x - dfa @xcite , mf - x - dma @xcite , and mf - hxa @xcite . the observed long - range power - law cross - correlations between two time series may not be caused by their intrinsic relationship but by a common third driving force or by common external factors @xcite . if the influence of the common external factors on the two time series are additive , we can use partial correlation to measure their intrinsic relationship @xcite . to extract the intrinsic long - range power - law cross - correlations between two time series affected by common driving driving forces , we previously developed and used detrended partial cross - correlation analysis ( dpxa ) and studied the dpxa exponents of variable cases , combining the ideas of detrended cross - correlation analysis and partial correlation @xcite . in ref . @xcite , the dpxa method has been proposed independently , focussing on the dpxa coefficient . here we provide a general framework for the dpxa and mf - dpxa methods that is applicable to various extensions , including different detrending approaches and higher dimensions . we adopt two well - established mathematical models ( bivariate fractional brownian motions and multifractal binomial measures ) in our numerical experiments , which have known analytical expressions , and demonstrate how the ( mf-)dpxa methods is superior to the corresponding ( mf-)dcca methods . consider two stationary time series @xmath0 and @xmath1 that depend on a sequence of time series @xmath2 with @xmath3 . each time series is covered with @xmath4 $ ] non - overlapping windows of size @xmath5 . consider the @xmath6th box @xmath7 $ ] , where @xmath8 . we calibrate the two linear regression models for @xmath9 and @xmath10 respectively , @xmath11 where @xmath12^{\mathrm{t}}$ ] , @xmath13^{\mathrm{t}}$ ] , @xmath14 and @xmath15 are the vectors of the error term , and @xmath16 is the matrix of the @xmath17 external forces in the @xmath6th box , where @xmath18 is the transform of @xmath19 . equation ( [ eq : xy : z : rxy : betas ] ) gives the estimates @xmath20 and @xmath21 of the @xmath17-dimensional parameter vectors @xmath22 and @xmath23 and the sequence of error terms , @xmath24 we obtain the disturbance profiles , i.e. , @xmath25 where @xmath26 . we assume that the local trend functions of @xmath27 and @xmath27 are @xmath28 and @xmath29 , respectively . the detrended partial cross - correlation in each window is then calculated , @xmath30\left[r_{y , v}(k)-\widetilde{r}_{y , v}(k)\right],\ ] ] and the second - order detrended partial cross - correlation is calculated , @xmath31^{1/2}.\ ] ] if there are intrinsic long - range power - law cross - correlations between @xmath32 and @xmath33 , we expect the scaling relation , @xmath34 there are many ways of determining @xmath28 and @xmath29 . the local detrending functions could be polynomials @xcite , moving averages @xcite , or other possibilities @xcite . to distinguish the different detrending methods , we label the corresponding dpxa variants as , e.g. , px - dfa and px - dma . when the moving average is used as the local detrending function , the window size of the moving averages must be the same as the covering window size @xmath5 @xcite . to measure the validity of the dpxa method , we perform numerical experiments using an additive model for @xmath32 and @xmath33 , i.e. , @xmath35 where @xmath36 is a fractional gaussian noise with hurst index @xmath37 , and @xmath38 and @xmath39 are the incremental series of the two components of a bivariate fractional brownian motion ( bfbms ) with hurst indices @xmath40 and @xmath41 @xcite . the properties of multivariate fractional brownian motions have been extensively studied @xcite . in particular , it has been proven that the hurst index @xmath42 of the cross - correlation between the two components is @xcite @xmath43 this property allows us to assess how the proposed method perform . we can obtain the @xmath44 of @xmath32 and @xmath33 using the dcca method and the @xmath45 of @xmath38 and @xmath39 using the dpxa method . our numerical experiments show that @xmath46 . we use @xmath47 for theoretical or true values and @xmath48 for numerical estimates . in the simulations we set @xmath49 , @xmath50 , @xmath51 , and @xmath52 in the model based on eq . ( [ eq : dpxa : model ] ) . three hurst indices @xmath40 , @xmath41 , and @xmath37 are input arguments and vary from 0.1 to 0.95 at 0.05 intervals . because @xmath38 and @xmath39 are symmetric , we set @xmath53 , resulting in @xmath54 triplets of @xmath55 . the bfbms are simulated using the method described in ref . @xcite , and the fbms are generated using a rapid wavelet - based approach @xcite . the length of each time series is 65536 . for each @xmath56 triplet we conduct 100 simulations . we obtain the hurst indices for the simulated time series @xmath38 , @xmath39 , @xmath57 , @xmath32 , and @xmath33 using detrended fluctuation analysis @xcite . the average values @xmath58 , @xmath59 , @xmath60 , @xmath61 , and @xmath62 over 100 realizations are calculated for further analysis , which are shown in fig . [ fig : dpxa : dhxyz ] . a linear regression between the output and input hurst indices in fig . [ fig : dpxa : dhxyz](a c ) yields @xmath63 , @xmath64 , and @xmath65 , suggesting that the generated fbms have hurst indices equal to the input hurst indices . figure [ fig : dpxa : dhxyz](d ) shows that when @xmath66 , @xmath61 is close to @xmath60 . when it is not , @xmath67 . figure [ fig : dpxa : dhxyz](e ) shows that @xmath68 . because @xmath69 and @xmath70 [ see fig . [ fig : dpxa : dhxyz](a)(b ) ] , we verify numerically that @xmath71 note also that @xmath72 , and that @xmath73 is a function of @xmath58 , @xmath59 and @xmath60 . a simple linear regression gives @xmath74 which indicates that the dpxa method can be used to extract the intrinsic cross - correlations between the two time series @xmath32 and @xmath33 when they are influenced by a common factor @xmath57 . we calculate the average @xmath75 over different @xmath37 and then find the relative error @xmath76 figure [ fig : dpxa : dhxyz](f ) shows the results for different combinations of @xmath58 and @xmath59 . although in most cases we see that @xmath77 , when both @xmath58 and @xmath59 approach 0 , @xmath78 increases . when @xmath79 , @xmath80 , and when @xmath81 and @xmath82 , @xmath83 . for all other points of @xmath84 , the relative errors @xmath78 are less than 0.10 . in a way similar to detrended cross - correlation coefficients @xcite , we define the detrended partial cross - correlation coefficient ( or dpxa coefficient ) as @xmath85 as in the dcca coefficient @xcite , we also find @xmath86 for dpxa . the dpxa coefficient indicates the intrinsic cross - correlations between two non - stationary series . ( color online . ) detrended partial cross - correlation coefficients . ( a ) performance of different methods by comparing three cross - correlation coefficients @xmath87 , @xmath88 and @xmath89 of the mathematical model in eq . ( [ eq : dpxa : model ] ) . ( b ) estimation and comparison of the cross - correlation levels between the two return time series ( @xmath90 ) and two volatility time series ( @xmath91 ) of crude oil and gold when including and excluding the influence of the usd index.,title="fig : " ] ( color online . ) detrended partial cross - correlation coefficients . ( a ) performance of different methods by comparing three cross - correlation coefficients @xmath87 , @xmath88 and @xmath89 of the mathematical model in eq . ( [ eq : dpxa : model ] ) . ( b ) estimation and comparison of the cross - correlation levels between the two return time series ( @xmath90 ) and two volatility time series ( @xmath91 ) of crude oil and gold when including and excluding the influence of the usd index.,title="fig : " ] we use the mathematical model in eq . ( [ eq : dpxa : model ] ) with the coefficients @xmath92 and @xmath93 to demonstrate how the dpxa coefficient outperforms the dcca coefficient . the two components @xmath38 and @xmath39 of the bfbm have very small hurst indices @xmath94 and their correlation coefficient is @xmath95 , and the driving fbm force @xmath57 has a large hurst index @xmath96 . figure [ fig : dpxa : rho](a ) shows the resulting cross - correlation coefficients at different scales . the dcca coefficients @xmath88 between the generated @xmath38 and @xmath39 time series overestimate the true value @xmath95 . because the influence of @xmath57 on @xmath38 and @xmath39 is very strong , the behaviors of @xmath32 and @xmath33 are dominated by @xmath57 , and the cross - correlation coefficient @xmath97 is close to 1 when @xmath5 is small and approaches 1 when @xmath5 us large . in contrast , the dpxa coefficients @xmath89 are in good agreement with the true value @xmath95 . note that the dpxa method better estimates @xmath38 and @xmath39 than the dcca method , since the @xmath88 curve deviates more from the horizontal line @xmath95 than the @xmath89 curve , especially at large scales . to illustrate the method with an example from finance , we use it to estimate the intrinsic cross - correlation levels between the futures returns and the volatilities of crude oil and gold . it is well - documented that the returns of crude oil and gold futures are correlated @xcite , and that both commodities are influenced by the usd index @xcite . the data samples contain the daily closing prices of gold , crude oil , and the usd index from 4 october 1985 to 31 october 2012 . figure [ fig : dpxa : rho](b ) shows that both the dcca and dpxa coefficients of returns exhibit an increasing trend with respect to the scale @xmath5 , and that the two types of coefficient for the volatilities do not exhibit any evident trend . for both financial variables , fig . [ fig : dpxa : rho](b ) shows that @xmath98 for different scales . although this is similar to the result between ordinary partial correlations and cross - correlations @xcite , the dpxa coefficients contain more information than the ordinary partial correlations since the former indicate the partial correlations at multiple scales . an extension of the dpxa for multifractal time series , notated mf - dpxa , can be easily implemented . when mf - dpxa is implemented with dfa or dma , we notate it mf - px - dfa or mf - px - dma . the @xmath99th order detrended partial cross - correlation is calculated @xmath100^{1/q}\ ] ] when @xmath101 , and @xmath102~.\ ] ] we then expect the scaling relation @xmath103 according to the standard multifractal formalism , the multifractal mass exponent @xmath104 can be used to characterize the multifractal nature , i.e. , @xmath105 where @xmath106 is the fractal dimension of the geometric support of the multifractal measure @xcite . we use @xmath107 for our time series analysis . if the mass exponent @xmath104 is a nonlinear function of @xmath99 , the signal is multifractal . we use the legendre transform to obtain the singularity strength function @xmath108 and the multifractal spectrum @xmath109 @xcite @xmath110 to test the performance of mf - dpxa , we construct two binomial measures @xmath111 and @xmath112 from the @xmath17-model with known analytic multifractal properties @xcite , and contaminate them with gaussian noise . we generate the binomial measure iteratively @xcite by using the multiplicative factors @xmath113 for @xmath38 and @xmath114 for @xmath39 . the contaminated signals are @xmath115 and @xmath116 . figures [ fig : mfpx : pmodel](a)(c ) show that the signal - to - noise ratio is of order @xmath117 . figures [ fig : mfpx : pmodel](d)(f ) show a power - law dependence between the fluctuation functions and the scale , in which it is hard to distinguish the three curves of @xmath118 . figure [ fig : mfpx : pmodel](g ) shows that for @xmath119 and @xmath120 , the @xmath121 function an approximate straight line and that the corresponding @xmath122 spectrum is very narrow and concentrated around @xmath123 . these observations are trivial because @xmath119 and @xmath120 are gaussian noise with the hurst indices @xmath124 , and the multifractal detrended cross - correlation analysis @xcite fails to uncover any multifractality . on the contrary , we find that @xmath125 and @xmath126 . thus the mf - dpxa method successfully reveals the intrinsic multifractal nature between @xmath127 and @xmath128 hidden in @xmath119 and @xmath120 . in summary , we have studied the performances of dpxa exponents , dpxa coefficients , and mf - dpxa using bivariate fractional brownian motions contaminated by a fractional brownian motion and multifractal binomial measures contaminated by white noise . these mathematical models are appropriate here because their analytical expressions are known . we have demonstrated that the dpxa methods are capable of extracting the intrinsic cross - correlations between two time series when they are influenced by common factors , while the dcca methods fail . the methods discussed are intended for multivariate time series analysis , but they can also be generalized to higher dimensions @xcite . we can also use lagged cross - correlations in these methods @xcite . although comparing the performances of different methods is always important @xcite , different variants of a method can produce different outcomes when applied to different systems . for instance , one variant that outperforms other variants under the setting of certain stochastic processes is not necessary the best performing method for other systems @xcite . we argue that there are still a lot of open questions for the big family of dfa , dma , dcca and dpxa methods . this work was partially supported by the national natural science foundation of china under grant no . 11375064 , fundamental research funds for the central universities , and shanghai financial and securities professional committee . 69ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1017/s0022112075000304 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1140/epjb / e2009 - 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5075/94/18007 [ * * , ( ) ] link:\doibase 10.1103/physreve.77.036104 [ * * , ( ) ] link:\doibase 10.1088/1742 - 5468/2009/03/p03037 [ ( ) ] in link:\doibase 10.1109/icassp.2009.4960233 [ _ _ ] ( ) pp . link:\doibase 10.1088/1367 - 2630/12/4/043057 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.11.011 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2013 - 40705-y [ * * , ( ) ] link:\doibase 10.1016/j.physa.2014.03.015 [ * * , ( ) ] link:\doibase 10.1142/s0218348x14500078 [ * * , ( ) ] link:\doibase 10.1103/physreve.91.022802 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2009 - 00310 - 5 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2010.10.022 [ * * , ( ) ] link:\doibase 10.1103/physreve.84.066118 [ * * , ( ) ] link:\doibase 10.1103/physreve.77.066211 [ * * , ( ) ] link:\doibase 10.1103/physreve.84.016106 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/95/68001 [ * * , ( ) ] link:\doibase 10.1155/2009/249370 [ * * , ( ) ] link:\doibase 10.1140/epjb / e2009 - 00384-y [ * * , ( ) ] link:\doibase 10.1371/journal.pone.0015032 [ * * , ( ) ] link:\doibase 10.1111/j.1467 - 842x.2004.00360.x [ * * , ( ) ] @noop master s thesis , ( ) , link:\doibase 10.1038/srep08143 [ * * , ( ) ] link:\doibase 10.1103/physreve.49.1685 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreve.58.6832 [ * * , ( ) ] link:\doibase 10.1140/epjb / e20020150 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1016/j.physa.2007.02.074 [ * * , ( ) ] link:\doibase 10.1016/j.physa.2011.07.008 [ * * , ( ) ] link:\doibase 10.1103/physreve.82.011136 [ * * , ( ) ] link:\doibase 10.1016/j.spl.2009.08.015 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1006/acha.1996.0030 [ * * , ( ) ] link:\doibase 10.1016/s0378 - 4371(01)00144 - 3 [ * * , ( ) ] link:\doibase 10.1016/j.resourpol.2010.05.003 [ * * , ( ) ] link:\doibase 10.1016/j.econmod.2012.09.052 [ * * , ( ) ] link:\doibase 10.1080/14697688.2014.946660 [ * * , ( ) ] link:\doibase 10.1016/s0378 - 4371(02)01383 - 3 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physreve.74.061104 [ * * , ( ) ] link:\doibase 10.1103/physreve.76.056703 [ * * , ( ) ] link:\doibase 10.1209/0295 - 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when common factors strongly influence two power - law cross - correlated time series recorded in complex natural or social systems , using classic detrended cross - correlation analysis ( dcca ) without considering these common factors will bias the results . we use detrended partial cross - correlation analysis ( dpxa ) to uncover the intrinsic power - law cross - correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces . the dpxa method is a generalization of the detrended cross - correlation analysis that takes into account partial correlation analysis . we demonstrate the method by using bivariate fractional brownian motions contaminated with a fractional brownian motion . we find that the dpxa is able to recover the analytical cross hurst indices , and thus the multi - scale dpxa coefficients are a viable alternative to the conventional cross - correlation coefficient . we demonstrate the advantage of the dpxa coefficients over the dcca coefficients by analyzing contaminated bivariate fractional brownian motions . we calculate the dpxa coefficients and use them to extract the intrinsic cross - correlation between crude oil and gold futures by taking into consideration the impact of the us dollar index . we develop the multifractal dpxa ( mf - dpxa ) method in order to generalize the dpxa method and investigate multifractal time series . we analyze multifractal binomial measures masked with strong white noises and find that the mf - dpxa method quantifies the hidden multifractal nature while the mf - dcca method fails .

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

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

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

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

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

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

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

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

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

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

@xmath0 nuclei ( nuclear systems with a bound anti - kaon ) have recently become a hot topic in hadron and nuclear physics . with a phenomenological @xmath2 potential , it was suggested that the @xmath0 nuclei could exist as deeply bound states with small width @xcite . experiments performed in search for such states have so far been inconclusive @xcite . an important prototype is the @xmath1 system , the simplest @xmath0-nuclear cluster . recently this system has been studied using faddeev @xcite and variational @xcite approaches with @xmath2 interactions constrained by scattering data and properties of the @xmath7 . an essential ingredient to study @xmath0 nuclei is the @xmath2 interaction below threshold , which is only accessible through the subthreshold extrapolation of the amplitude adjusted to @xmath2 scattering data . theoretical guidance is required for this extrapolation . here we report on the study of a variational calculation of @xmath1 system @xcite using the effective @xmath2 interaction based on chiral su(3 ) dynamics @xcite . the present variational investigation focuses on the @xmath1 system with spin and parity @xmath9 and isospin @xmath10 , where the parity assignment includes the intrinsic parity of the antikaon . our model wave function for this @xmath1 state , @xmath11 , has two components : @xmath12_{t_n=1 } \ , \bar{k } \ , \right]_{t=1/2 , t_z=1/2 } \right\rangle , \label{phi+ } \\ & & |\phi_- \rangle \equiv \phi_- ( \bm{r}_1 , \bm{r}_2 , \bm{r}_k ) \ ; \left| s_n = 0 \right\rangle \times \ ; \left| \ , \left [ \ , [ nn]_{t_n=0 } \ , \bar{k } \ , \right]_{t=1/2 , t_z=1/2 } \right\rangle , \label{phi-}\end{aligned}\ ] ] where @xmath13 is a normalization factor . the first , second and third terms in eqs . ( [ phi+ ] ) and ( [ phi- ] ) correspond to the spatial wave function , the spin wave function of the two nucleons ( assuming @xmath14 ) , and the isospin wave function of the total system , respectively . we consider two different isospin states of the two nucleons ( @xmath15 in @xmath16 and @xmath17 in @xmath18 ) , while the @xmath19 system in both cases has total isospin and third component @xmath10 . the dominant contribution is the @xmath15 component corresponding to the leading @xmath1 configuration . the mixing with the @xmath17 component is caused by the difference between the @xmath2 interactions in @xmath20 and @xmath21 . the spatial part of the wave functions are products of single particle wave packets and two - particle correlation functions . the @xmath22 correlation function permits an adequate treatment of a realistic @xmath22 potential with its strong short - range repulsion . the parameters in the model wave function are determined by minimization of the energy . the hamiltonian used in the present study is of the form @xmath23 where @xmath24(@xmath25 ) stands for the @xmath22(@xmath2 ) interaction . here @xmath26 is the total kinetic energy . the energy of the center - of - mass motion , @xmath27 , is subtracted . as a realistic nucleon - nucleon interaction @xmath24 we choose the argonne v18 potential ( av18 ) @xcite . we employ the central , @xmath28 and spin - spin parts of the av18 potential for the singlet - even ( @xmath29 ) and singlet - odd ( @xmath30 ) channel , since the total spin of the two nucleons is restricted to zero in our model . we use the @xmath2 interaction @xmath25 derived from chiral su(3 ) dynamics @xcite . this complex and energy - dependent interaction is parametrized by a gaussian spatial distribution : @xmath31 , \nonumber\end{aligned}\ ] ] where @xmath32 is the isospin projection operator for the @xmath2 pair . the interaction strength @xmath33 is a function of the center - of - mass energy variable @xmath34 of the @xmath2 subsystem . the strength @xmath35 and the range parameter @xmath36 are systematically determined within the chiral coupled - channel approach . the energy dependence of the @xmath2 interaction requires the self - consistency in the variational procedure @xcite . we introduce an auxiliary ( non - observable ) antikaon binding energy " @xmath37 to control the energy @xmath34 of the @xmath2 subsystem within the @xmath1 cluster . this @xmath37 is defined as @xmath38 where @xmath39 is the nucleonic part of the hamiltonian . the relation between the @xmath2 two - body energy @xmath34 and @xmath37 within the three - body system is not @xmath40 @xmath41 fixed . in general , @xmath34 can take values @xmath42 , where @xmath43 is a parameter describing the balance of the antikaon energy between the two nucleons of the @xmath19 three - body system . one expects @xmath44 . the upper limit ( @xmath45 ) corresponds to the case in which the antikaon field collectively surrounds the two nucleons , a situation encountered in the limit of static ( infinitely heavy ) nucleon sources . in the lower limit ( @xmath46 ) the antikaon energy is split symmetrically half - and - half between the two nucleons . we investigate both cases and label them type i " and type ii " , respectively : @xmath47 our calculation is then carried out such that self - consistency for @xmath34 is achieved , namely , the @xmath34 used in the effective @xmath2 potential is made to coincide with the @xmath34 evaluated with the finally obtained wave function . due to the elimination of the @xmath48 and @xmath49 channels , the effective @xmath2 potential is complex . we perform the variational calculation with the real part of the potential to obtain the wave function . the decay width is then calculated perturbatively by taking the expectation value of the imaginary part of the potential : @xmath50 , which represents the mesonic decay channels ( @xmath51 ) . the dispersive effect induced by the imaginary part of the potential and the non - mesonic absorption width for @xmath52 are treated separately . relative density in @xmath1 for a chiral model @xcite with the type i ansatz . solid ( dashed ) line shows @xmath20 ( @xmath21 ) @xmath2 density . solid line with diamond shows the @xmath2 density of @xmath7 in the same model . all densities are displayed with @xmath53-multiplied . , title="fig:",scaledwidth=50.0% ] relative density in @xmath1 for a chiral model @xcite with the type i ansatz . solid ( dashed ) line shows @xmath20 ( @xmath21 ) @xmath2 density . solid line with diamond shows the @xmath2 density of @xmath7 in the same model . all densities are displayed with @xmath53-multiplied . , title="fig:",scaledwidth=45.0% ] here we present the results of the variational calculation . for an estimate of theoretical uncertainties , we have used four effective @xmath2 potentials derived from different versions of chiral models , and employed both the type i and the type ii ansatz , as described in ref . @xcite . in all cases the @xmath1 system turns out to be rather weakly bound as compared to previous calculations . as shown in the left panel of fig . [ fig:1 ] , the total binding energies range from 17 mev to 23 mev , and the mesonic decay width @xmath54 ( @xmath55 ) lies between 40 and 70 mev . the different versions of the chiral models give similar results within a relatively small window of uncertainties , while type ii ansatz gives slightly deeper binding than type i by a few mev . the reason for the shallow binding is found in the relatively weak @xmath2 potentials based on chiral dynamics . the chiral low energy theorem in the su(3 ) meson - baryon sector dictates strong @xmath48 attraction , and the coupled - channel dynamics locates the resonance structure in the @xmath2 amplitude at 1420 mev , displaced from the 1405 mev measured in the @xmath48 spectrum . the binding energy of the isolated @xmath6 system is about 12 mev measured from @xmath2 threshold . table [ tab:1 ] shows a typical result of @xmath1 calculated with a chiral model @xcite and the type i ansatz . the results of the other cases under study are essentially the same . the mean distance between two nucleons , @xmath56 , is about 2.2 fm which is smaller than that of the deuteron ( about 4 fm ) and close to the @xmath22 distance in normal nuclei , but the system is obviously not much compressed . it is interesting to compare the @xmath6 component in @xmath1 with the @xmath7 as the @xmath6 two - body quasibound state . the mean distance of the @xmath2 pair in @xmath1 is found to be close to that for @xmath7 , namely @xmath57 fm and @xmath58 fm . calculating the expectation value of the relative @xmath2 orbital angular momentum , it turns out that the @xmath6 pair is dominated by @xmath59-wave , just as the @xmath2 pair forming the @xmath7 . the structure of the @xmath6 pair in the @xmath1 system is thus similar to that of the @xmath7 . the right panel in fig . [ fig:1 ] shows the @xmath2 relative density distribution of @xmath60 and 1 ) components extracted from @xmath1 which are normalized to compare with that of @xmath7 . apparently , the distribution of @xmath6 pair in @xmath1 is very similar to that of the @xmath2 two - body quasibound state . cc|cc|cc||cc b. e. ( @xmath1 ) & @xmath61 & @xmath56 & @xmath62 & @xmath63 & @xmath64 & b. e. ( @xmath65 ) & @xmath66 + 16.9 & 47.0 & 2.21 & 1.97 & 1.82 & 2.33 & 11.5 & 1.86 + based on the wavefunction obtained above , we estimate the following contributions to the results which have not been taken into account so far : 1 ) dispersive corrections by the imaginary part of the potential , 2 ) effect of the @xmath8-wave @xmath2 interaction , and 3 ) decay width from the two - nucleon absorption process @xcite . first , we consider the dispersive correction induced by the imaginary part of the @xmath2 potential . this effect can be calculated explicitly for the two - body @xmath2 system , by comparing the bound state solution of the real part of the potential with the resonance structure observed in the scattering amplitude with original complex potential . examining four chiral models , we find an attractive shift of the binding energy @xmath67 mev in the two - body @xmath2 system . we therefore estimate that the dispersive correction would add another @xmath68 mev to the binding energy of the @xmath1 system . secondly , the contribution of the @xmath8-wave @xmath2 interaction is estimated perturbatively with the @xmath8-wave @xmath2 interaction : @xmath69\ , \nabla~.\end{aligned}\ ] ] the coefficient @xmath70 is complex and a detailed expression is given in ref . a prominent feature in the @xmath8-wave interaction is the @xmath71 resonance below the threshold . since the @xmath1 system is weakly bound and the energy variable @xmath34 lies slightly above the @xmath71 resonance , the @xmath8-wave contribution to the binding energy is repulsive , about @xmath72 mev . the decay width is increased by 10 @xmath73 35 mev , because of the large imaginary part around the @xmath71 resonance structure . next , we estimate the contribution of the two - nucleon absorption process ( non - mesonic decay width , @xmath74 ) . the width is calculated with the correlated three - body density @xmath75 as @xmath76 this is a generalization of the formula for @xmath77 absorption on proton pairs in a heavy nucleus , where the coupling constant is constrained by a global fit to the kaonic atom data @xcite . for the application to the few - body system , we modify the delta function type interaction to the finite range gaussian form . this procedure is necessary to account for short range correlations of the nucleons in the few - body system , and physically motivated by the underlying mechanism of the meson - exchange picture . using the correlation density obtained from the wave function of the @xmath1 , the two nucleon absorption width is estimated to be 4 - 12 mev . we have investigated the @xmath1 system with a variational method , employing a realistic @xmath22 potential ( av18 potential ) and an effective @xmath2 potential based on chiral su(3 ) dynamics . with theoretical uncertainties in the model , the binding energy and decay width of the @xmath1 turns out to be @xmath78 as a consequence of the strong @xmath79 interaction in chiral scheme , the strength of the @xmath2 interaction is reduced and therefore we find a weakly bound state . the @xmath6 pair in the obtained wave function of the @xmath1 system exhibits a similar structure as the @xmath2 two - body quasibound state in vacuum . we have estimated corrections , such as dispersive correction , the @xmath8-wave @xmath2 potential , and the two - nucleon absorption process . taking these effects into account , the total binding energy increases slightly and the total decay width becomes as large as 60 - 120 mev . our result should be compared with another three - body faddeev calculation with chiral interaction @xcite , where the @xmath1 state was found with 80 mev binding energy . while faddeev approach treats the coupled channels explicitly , our variational calculation works by eliminating the @xmath80 channel . although the two - body @xmath48 dynamics is fully incorporated in the effective @xmath2 interaction , the dynamics of @xmath81 three - body system may generate additional attraction ( see also ref . @xcite ) . in the coupled - channel framework , the obtained state is the mixture of the @xmath19 and @xmath80 components , as the @xmath7 resonance in @xmath2-@xmath48 system . in this sense , our strategy is to focus on the @xmath19 component , and the present framework may not be sensitive to the @xmath80 component . based on a recent experimental analysis , a broad structure at about 100 mev below the @xmath19 threshold is reported @xcite , the maximum of which coincides with the @xmath80 threshold . in the chiral framework , such a broad state in the deep subthreshold region would be interpreted in terms of @xmath80 dynamics , driven by the strong @xmath48 attraction . the present investigation is however not capable to deal with the @xmath80 component , since we have eliminated this channel . while the new report @xcite is an interesting observation , more careful analysis is needed to answer the question about its detailed structure . this project is partially supported by bmbf , gsi , by the dfg excellence cluster origin and structure of the universe . 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the prototype of a @xmath0 nuclear cluster , @xmath1 , has been investigated using effective @xmath2 potentials based on chiral su(3 ) dynamics . variational calculation shows a bound state solution with shallow binding energy @xmath3 mev and broad mesonic decay width @xmath4 - @xmath5 mev . the @xmath6 pair in the @xmath1 system exhibits a similar structure as the @xmath7 . we have also estimated the dispersive correction , @xmath8-wave @xmath2 interaction , and two - nucleon absorption width . example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore

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

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

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

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

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

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

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

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

few star forming regions surpass the resplendent beauty of ngc2264 , the richly populated galactic cluster in the mon ob1 association lying approximately 760 pc distant in the local spiral arm . other than the orion nebula cluster , no other star forming region within one kpc possesses such a broad mass spectrum and well - defined pre - main sequence population within a relatively confined region on the sky . estimates for the total stellar population of the cluster range up to @xmath01000 members , with most low - mass , pre - main sequence stars having been identified from h@xmath4 emission surveys , x - ray observations by _ rosat _ , _ chandra _ , and _ xmm - newton _ , or by photometric variability programs that have found several hundred periodic and irregular variables . the cluster of stars is seen in projection against an extensive molecular cloud complex spanning more than two degrees north and west of the cluster center . the faint glow of balmer line emission induced by the ionizing flux of the cluster ob stellar population contrasts starkly with the background dark molecular cloud from which the cluster has emerged . the dominant stellar member of ngc2264 is the o7 v star , s monocerotis ( s mon ) , a massive multiple star lying in the northern half of the cluster . approximately 40 south of s mon is the prominent cone nebula , a triangular projection of molecular gas illuminated by s mon and the early b - type cluster members . ngc2264 is exceptionally well - studied at all wavelengths : in the millimeter by crutcher et al . ( 1978 ) , margulis & lada ( 1986 ) , oliver et al . ( 1996 ) , and peretto et al . ( 2006 ) ; in the near infrared ( nir ) by allen ( 1972 ) , pich ( 1992 , 1993 ) , lada et al . ( 1993 ) , rebull et al . ( 2002 ) , and young et al . ( 2006 ) ; in the optical by walker ( 1956 ) , rydgren ( 1977 ) , mendoza & gmez ( 1980 ) , adams et al . ( 1983 ) , sagar & joshi ( 1983 ) , sung et al . ( 1997 ) , flaccomio et al . ( 1999 ) , rebull et al . ( 2002 ) , sung et al . ( 2004 ) , lamm et al . ( 2004 ) , and dahm & simon ( 2005 ) ; and in x - rays by flaccomio et al . ( 2000 ) , ramirez et al . ( 2004 ) , rebull et al . ( 2006 ) , flaccomio et al . ( 2006 ) , and dahm et al . ( 2007 ) . ngc2264 was discovered by friedrich wilhelm herschel in 1784 and listed as h viii.5 in his catalog of nebulae and stellar clusters . the nebulosity associated with ngc2264 was also observed by herschel nearly two years later and assigned the designation : h v.27 . the roman numerals in herschel s catalog are object identifiers , with ` v ' referring to very large nebulae and ` viii ' to coarsely scattered clusters of stars . one of the first appearances of the cluster in professional astronomical journals is wolf s ( 1924 ) reproduction of a photographic plate of the cluster and a list of 20 suspected variables . modern investigations of the cluster begin with herbig ( 1954 ) who used the slitless grating spectrograph on the crossley reflector at lick observatory to identify 84 h@xmath4 emission stars , predominantly t tauri stars ( tts ) , in the cluster region . herbig ( 1954 ) postulated that these stars represented a young stellar population emerging from the dark nebula . walker s ( 1956 ) seminal photometric and spectroscopic study of ngc2264 discovered that a normal main sequence exists from approximately o7 to a0 , but that lower mass stars consistently fall above the main sequence . this observation was in agreement with predictions of early models of gravitational collapse by salpeter and by henyey et al . walker ( 1954 , 1956 ) proposed that these stars represent an extremely young population of cluster members , still undergoing gravitational contraction . walker ( 1956 ) further noted that the ttss within the cluster fall above the main sequence and that they too may be undergoing gravitational collapse . walker ( 1956 ) concluded that the study of ttss would be `` of great importance for our understanding of these early stages of stellar evolution . '' square false - color iras image ( 100 , 60 & 25 @xmath5 m ) of the mon ob1 and mon r1 associations . ngc2264 lies at the center of the image with several nearby iras sources identified , including the reflection nebulae ngc2245 and ngc2247 , and ngc2261 ( hubble s variable nebula ) . south of ngc2264 is the rosette nebula and its embedded cluster ngc2244 , lying 1.7 kpc distant in the perseus arm . [ f1 ] ] the molecular cloud complex associated with ngc2264 was found by crutcher et al . ( 1978 ) to consist of several cloud cores , the most massive of which lies roughly between s mon and the cone nebula . throughout the entire cluster region , oliver et al . ( 1996 ) identified 20 molecular clouds ranging in mass from @xmath6 to @xmath7 m@xmath3 . with ngc2264 these molecular clouds comprise what is generally regarded as the mon ob 1 association . active star formation is ongoing within ngc2264 as evidenced by the presence of numerous embedded protostars and clusters of stars , as well as molecular outflows and herbig - haro objects ( adams et al . 1979 ; fukui 1989 ; hodapp 1994 ; walsh et al . 1992 ; reipurth et al . 2004a ; young et al . 2006 ) . two prominent sites of star formation activity within the cluster are irs1 ( also known as allen s source ) , located several arcminutes north of the tip of the cone nebula , and irs2 , which lies approximately one - third of the distance from the cone nebula to s mon . new star formation activity is also suspected within the northern extension of the molecular cloud based upon the presence of several embedded iras sources and giant herbig - haro flows ( reipurth et al . 2004a , c ) . from 60 and 100 @xmath5 m iras images of ngc2264 , schwartz ( 1987 ) found that the cluster lies on the eastern edge of a ring - like dust structure , 3@xmath8 in diameter . shown in figure 1 is a 125@xmath1125 false - color iras image ( 100 , 60 , and 25 @xmath5 m ) centered near ngc2264 . the reflection nebulae ngc2245 and ngc2247 , members of the mon r1 association , are on the western boundary of this ring ( see the chapter by carpenter & hodapp ) . other components of the mon r1 association include the reflection nebulae ic446 and ic2169 , lkh@xmath4215 , as well as several early type ( b3b7 ) stars . it is generally believed that the mon r1 and mon ob 1 associations are at similar distances and are likely related . the rosette nebula , ngc2237 - 9 , and its embedded young cluster ngc2244 lie 5@xmath8 southwest of ngc2264 , 1.7 kpc distant in the outer perseus arm ( see the chapter by romn - ziga & lada ) . several arcs of dust and co emission have been identified in the region , which are believed to be supernovae remnants or windblown shells . many of these features are apparent in figure 2 , a wide - field h@xmath4 image of ngc2264 , ngc2244 , and the intervening region obtained by t. hallas and reproduced here with his permission . it is possible that star formation in the mon ob1 and r1 associations was triggered by nearby energetic events , but it is difficult to assess the radial distance of the ringlike structures evident in figure 2 , which may lie within the perseus arm or the interarm region . shown in figure 3 is a narrow - band composite image of ngc2264 obtained by t.a . rector and b.a . wolpa using the 0.9 meter telescope at kitt peak . s mon dominates the northern half of the cluster , which lies embedded within the extensive molecular cloud complex . image of ngc2264 ( upper center ) , the rosette nebula and ngc2244 ( lower right ) , and the numerous windblown shells and supernova remnants possibly associated with the mon ob1 or mon ob2 associations . the cone nebula is readily visible just above and left of image center as is s mon . also apparent in the image is the dark molecular cloud complex lying to the west of ngc2264 . this image is a composite of 16 20-minute integrations obtained by t. hallas using a 165 mm lens and an astrodon h@xmath4 filter . [ f2 ] ] to summarize all work completed over the last half - century in ngc2264 would be an overwhelming task and require significantly more pages than alloted for this review chapter . the literature database for ngc2264 and its members has now grown to over 400 refereed journal articles , conference proceedings , or abstracts . here we attempt to highlight large surveys of the cluster at all wavelengths as well as bring attention to more focused studies of the cluster that have broadly impacted our understanding of star formation . the chapter begins with a review of basic cluster properties including distance , reddening , age , and inferred age dispersion . it then examines the ob stellar population of the cluster , the intermediate and low - mass stars , and finally the substellar mass regime . different wavelength regions are examined from the centimeter , millimeter , and submillimeter to the far- , mid- , and near infrared , the optical , and the x - ray regimes . we then review many photometric variability studies of the cluster that have identified several hundred candidate members . finally , we consider future observations of the cluster and what additional science remains to be reaped from ngc2264 . the cluster has remained in the spotlight of star formation studies for more than 50 years , beginning with the h@xmath4 survey of herbig ( 1954 ) . its relative proximity , low foreground extinction , large main sequence and pre - main sequence populations , the lack of intense nebular emission , and the tremendous available archive of observations of the cluster at all wavelengths guarantee its place with the orion nebula cluster and the taurus - auriga molecular clouds as the most accessible and observed galactic star forming region . ( red - orange ) , and [ s ii ] ( blue - violet ) . the field of view is approximately 0.75@xmath8@xmath11@xmath8 . s mon lies just above the image center and is believed to be the ionizing source of the bright rimmed cone nebula . ngc2264 is ideally suited for accurate distance determinations given its lack of significant foreground extinction and the abundant numbers of early - type members . difficulty in establishing the cluster distance , however , arises from the near vertical slope of the zero age main sequence ( zams ) for ob stars in the color - magnitude diagram and from the depth of the cluster along the line of sight , which has not been assessed . the distance of ngc2264 is now widely accepted to be @xmath0760 pc , but estimates found in the early literature vary significantly . walker ( 1954 ) used photoelectric observations of suspected cluster members earlier than a0 to derive a distance modulus of 10.4 mag ( 1200 pc ) by fitting the standard main sequence of johnson & morgan ( 1953 ) . herbig ( 1954 ) adopted a distance of 700 pc , the mean of published values available at the time . for his landmark study of the cluster , walker ( 1956 ) revised his earlier distance estimate to 800 pc using the modified main sequence of johnson & hiltner ( 1956 ) . prez et al . ( 1987 ) redetermined the distances of ngc2264 and ngc2244 ( the rosette nebula cluster ) assuming an anomalous ratio of total - to - selective absorption . their revised distance estimate for ngc2264 was @xmath9 pc . included in their study is an excellent summary of distance determinations for ngc2244 and ngc2264 found in the literature from 1950 to 1985 ( their table xi ) . the mean of these values for ngc2264 is @xmath10 pc , significantly less than their adopted distance . prez ( 1991 ) provides various physical characteristics of the cluster including distance , total mass , radius , mean radial velocity , and age , but later investigations of the cluster have since revised many of these estimates . from modern ccd photometry of ngc2264 , sung et al . ( 1997 ) , determined a mean distance modulus of @xmath11 or 760 pc using 13 b - type cluster members , which have distance moduli in the range from @xmath12 mag . this value is now cited in most investigations of the cluster . the projected linear dimension of the giant molecular cloud associated with ngc2264 including the northern extension is nearly 28 pc . if a similar depth is assumed along the cluster line of sight , an intrinsic uncertainty of nearly 4% is introduced into the distance determination . interstellar reddening toward ngc2264 is recognized to be quite low . walker ( 1956 ) found @xmath13 or @xmath14 assuming the normal ratio of total - to - selective absorption , @xmath15 . prez et al . ( 1987 ) found a similar reddening value with @xmath16 , but used @xmath17 to derive their significantly greater distance to the cluster . in their comprehensive photometric study , sung et al . ( 1997 ) found the mean reddening of 21 ob stars within the cluster to be @xmath18 , in close agreement with the estimates of walker ( 1956 ) and prez et al . no significant deviation from these values has been found using the early - type cluster members . individual extinctions for the suspected low - mass members , however , are noted to be somewhat higher . rebull et al . ( 2002 ) derive a mean @xmath19 or @xmath20 from their spectroscopic sample of more than 400 stars , only 22% of which are earlier than k0 . dahm & simon ( 2005 ) find that for the h@xmath4 emitters within the cluster with established spectral types , a mean a@xmath21 of 0.71 mag follows from @xmath22 . some of these low - mass stars are suffering from local extinction effects ( e.g. circumstellar disks ) or lie within deeply embedded regions of the molecular cloud . the mean extinction value derived for the ob stellar population , which presumably lies on the main sequence and possesses well - established intrinsic colors , should better represent the distance - induced interstellar reddening suffered by cluster members . table 1 summarizes the distance and extinction estimates of the cluster found or adopted in selected surveys of ngc2264 . lcccccc + + authors & age ( myr ) & m@xmath21 range & isochrone & e(@xmath23 ) & distance ( pc ) & notes + walker ( 1956 ) & 3.0 & @xmath24 & henyey et al . ( 1955 ) & 0.082 & 800 & pe and pg + & & & & & & + mendoza & gmez ( 1980 ) & 3.0 & @xmath25 & iben & talbot ( 1966 ) & 0.06 & 875 & pe + & & & & & & + adams et al . ( 1983 ) & 3.0 - 6.0 & @xmath26 & cohen & kuhi ( 1979 ) & [ 0.06 ] & [ 800 ] & pg + & & & & & & + sagar & joshi ( 1983 ) & 5 & @xmath27 & cohen & kuhi ( 1979 ) & @xmath280.12 & 794 & pe + & & & & & & + prez et al . ( 1987 ) & ... & @xmath29 & ... & 0.06 & 950 & pe + & & & & & & + feldbrugge & van genderen ( 1991 ) & @xmath30 & @xmath31 & ... & 0.04 & 700 & pe + & & & & & & + neri et al . ( 1993 ) & ... & @xmath32 & ... & 0.05 & 910 & pe + & & & & & & + sung et al . ( 1997 ) & 0.88.0 & @xmath33 & s94 , bm96 & 0.071 & 760 & ccd + & & & & & & + flaccomio et al . ( 1999 ) & 0.110.0 & @xmath34 & dm97 & [ 0.06 ] & [ 760 - 950 ] & ccd + & & & & & & + park et al . ( 2000 ) & 0.94.3 & @xmath35 & dm94 , s94 , b98 & 0.066 & 760 & ccd + & & & & & & + rebull et al . ( 2002 ) & 0.16.0 & @xmath36 & dm94 , sdf00 & 0.15 & [ 760 ] & ccd + & & & & & & + sung et al . ( 2004 ) & 3.1 & @xmath37 & sdf00 & 0.070.15 & ... & ccd + + + + + + the age of ngc2264 has long been inferred to be young given the large ob stellar population of the cluster and the short main sequence lifetimes of these massive stars . walker ( 1956 ) derived an estimate of 3 myr , based upon the main sequence contraction time of an a0-type star ( the latest spectral type believed to be on the zams ) from the theoretical work of henyey et al . iben & talbot ( 1966 ) , however , directly compared theoretical time - constant loci or isochrones with the color - magnitude diagram of ngc2264 , concluding that star formation began within the cluster more than 65 myr ago . they further suggested that the star formation rate has been increasing exponentially with time , and that the average mass of each subsequent generation of stars has also increased exponentially . strom et al . ( 1971 ) and strom et al . ( 1972 ) , however , in their insightful investigations of balmer line emission and infrared excesses among a and f - type members of ngc2264 , concluded that dust and gas shells ( spherical or possibly disk - like ) were common among pre - main sequence stars , making strict interpretation of age dispersions from color - magnitude diagrams difficult if not impossible . strom et al . ( 1972 ) conclude that an intrinsic age dispersion of 1 to 3 myr is supported by the a and f - type members of ngc2264 . adams et al . ( 1983 ) revisited the question of age dispersion within ngc2264 in their deep photometric survey of the cluster . from the theoretical hr diagram of probable cluster members , a significant age spread of more than 10 myr is inferred . adams et al . ( 1983 ) further suggest that sequential star formation has occurred within the cluster , beginning with the low - mass stars and continuing with the formation of more massive cluster members . from the theoretical models of cohen & kuhi ( 1979 ) , they derive a mean age of the low - mass stellar population of 45 myr . modern ccd investigations of ngc2264 have yielded similar ages , but they are strongly dependent upon the pre - main sequence models adopted for use . sung et al . ( 1997 ) use the pre - main sequence models of bernasconi & maeder ( 1996 ) and swenson et al . ( 1994 ) to find that the ages of most suspected pre - main sequence members of ngc2264 are from 0.8 to 8 myr , while the main sequence stars range from 1.4 to 16 myr . park et al . ( 2000 ) compare ages and age dispersions of ngc2264 from four sets of pre - main sequence models : those of swenson et al . ( 1994 ) , dantona & mazzitelli ( 1994 ) , baraffe et al . ( 1998 ) , and the revised models of baraffe et al . ( 1998 ) , which incorporate a different ratio of mixing length to pressure scale height . from the distribution of pre - main sequence candidates , the median ages and age dispersions ( respectively ) from the models of swenson et al . ( 1994 ) are 2.1 and 8.0 myr ; dantona & mazzitelli ( 1994 ) 0.9 and 5.5 myr ; baraffe et al . ( 1998 ) 4.3 and 15.3 myr , and for the revised models of baraffe et al . ( 1998 ) 2.7 and 10 myr . figure 4 is a reproduction of the hr diagrams from park et al . ( 2000 ) , with the evolutionary tracks for the various models superposed . the cluster ages and age spreads are represented by the solid and dashed lines , respectively . these surveys used photometry alone to place stars on the hr diagram . ] rebull et al . ( 2002 ) compare the derived ages and masses from dantona & mazzitelli ( 1994 ) and siess et al . ( 2000 ) for a spectroscopically classified sample of stars in ngc2264 , finding systematic differences between the models of up to a factor of two in mass and half an order of magnitude in age . dahm & simon ( 2005 ) use the evolutionary models of dantona & mazzitelli ( 1997 ) to determine a median age of 1.1 myr and to infer an age dispersion of @xmath05 myr for the nearly 500 h@xmath4 emission stars identified within the immediate cluster vicinity . star formation , however , continues within the cluster as evidenced by _ observations of the star forming cores near irs1 and irs 2 . young et al . ( 2006 ) suggest that a group of embedded , low - mass protostars coincident with irs2 exhibits a velocity dispersion consistent with a dynamical age of several @xmath7 yr . the presence of this young cluster as well as other deeply embedded protostars ( allen 1972 ; castelaz & grasdalen 1988 ; margulis et al . 1989 ; thompson et al . 1998 ; young et al . 2006 ) among the substantial number of late b - type dwarfs in ngc2264 implies that an intrinsic age dispersion of at least 35 myr exists among the cluster population . soderblom et al . ( 1999 ) obtained high resolution spectra for 35 members of ngc2264 in order to determine li abundances , radial velocities , rotation rates , and chromospheric activity levels . their radial velocities indicate that the eight stars in their sample lying below the 5 myr isochrone of the cluster are non - members , implying that the age spread within the cluster is only @xmath04 myr . the hierarchical structure of the cluster would indicate that star formation has occurred in different regions of the molecular cloud over the last several myr . we can speculate that from the large quantities of molecular gas remaining within the various cloud cores , star formation will continue in the region for several additional myr . the cluster ages adopted or derived by selected previous investigations of ngc2264 are also presented in table 1 . s mon ( 15 mon ) dominates the northern half of the cluster and is believed to be the ionizing source for the cone nebula ( schwartz et al . 1985 ) as well as many of the observed bright rims in the region including sharpless 273 . in addition to exhibiting slight variability ( hundredths of a magnitude ) , gies et al . ( 1993 , 1997 ) determined s mon to be a visual and spectroscopic binary from speckle interferometry , _ hst _ imaging , and radial velocity data . with a semi - major axis of over 27 au ( assuming a distance of 800 pc ) , the 24-year orbit of the binary is illustrated in figure 5 , taken from gies et al . ( 1997 ) and based upon an orbital inclination of 35@xmath8 to the plane of the sky . the mass estimates for the primary component of s mon ( o7 v ) , 35 m@xmath3 , and the secondary ( o9.5 v ) , 24 m@xmath3 , assume a distance of 950 pc ( prez et al . 1987 ) , which is significantly greater than the currently accepted cluster distance . using their derived orbital elements and with the adopted cluster distance ( 800 pc ) , a mass ratio of q@xmath380.75 is derived , leading to primary and secondary masses 18.1 and 13.5 m@xmath3 , respectively . these mass estimates , however , are inconsistent with the spectral type of s mon and its companion . over the last 13 years , observations of the star using the fine guidance sensor onboard hst have revised earlier estimates of the orbital elements , and the period of the binary now appears to be significantly longer than previously determined ( gies , private communication ) . s mon also exhibits uv resonance line profile variation as well as fluctuation in soft x - ray flux ( snow et al . 1981 ; grady et al . both of these phenomenon are interpreted as being induced by variations in mass loss rate . , width=288 ] the ob - stellar population of ngc2264 consists of at least two dozen stars , several of which were spectroscopically classified by walker ( 1956 ) . morgan et al . ( 1965 ) list 17 early - type stars within the cluster , ranging in spectral type from o7 to b9 . most of these stars are concentrated around s mon or the small rosette - shaped nebulosity lying to its southwest , however , several do lie in the southern half of the cluster . table 2 lists the suspected early - type members of ngc2264 with their positions , available photometry , spectral types , and any relevant notes of interest . several members of the greater mon ob 1 association are also included in table 1 for completeness , some obtained from the compilation of turner ( 1976 ) . the hipparcos survey of de zeeuw et al . ( 1999 ) was unable to provide any rigorous kinematic member selection for mon ob1 members given the distance of the association . many of the early - type stars are known binaries including s mon , hd 47732 , and hd 47755 ( gies et al . 1993 , 1997 ; morgan et al . 1965 ; dahm et al . 2007 ) . the b3 v star hd 47732 is identified by morgan et al ( 1965 ) as a double - line spectroscopic binary and is examined in detail by beardsley & jacobsen ( 1978 ) and koch et al . another early - type cluster member , hd 47755 , was classified by trumpler ( 1930 ) as a b5-type spectroscopic binary , but assigned a low probability of membership by the proper motion survey of vasilevskis et al . koch et al . ( 1986 ) derive a 1.85 day period for the shallow 0.1 mag eclipse depth . dahm et al . ( 2007 ) list several early - type adaptive optics ( ao ) binaries and give their separations and position angles ( see their table 2 ) . other massive cluster members include several mid - to - late b - type stars clustered near s mon including bd@xmath3910@xmath81222 ( b5v ) , hd 261938 ( b6v ) , w 181 ( b9v ) , hd 261969 ( b9iv ) , and @xmath07 to the east , hd 47961 ( b2v ) . although on the cloud periphery , hd 47961 is surrounded by several x - ray and h@xmath4 emission stars and its derived distance modulus , m@xmath40m@xmath21@xmath389.74 , matches that of the cluster . the proper motion data of vasilevskis et al . ( 1965 ) also support its cluster membership status . within the rosette - shaped emission / reflection nebula southwest of s mon is a second clustering of massive stars including w 67 ( b2v ) , hd 47732 ( b3v ) , hd 47755 ( b5v ) , w 90 ( b8e ) , and hd 261810 ( b8v ) . between this grouping and hd 47887 ( w 178 ) , the b2 iii binary just north of the cone nebula , there are just a handful of candidate early - type members : hd 47777 ( b3v ) , hd 262013 ( b5v ) , and hd 261940 ( b8v ) . hd 47887 was assigned a low probability of membership by vasilevskis et al . ( 1965 ) , but if assumed to lie on the zams , the distance modulus of this b2 binary equals that of the cluster . the other component of hd 47887 is the b9 dwarf , hd 47887b , several arcseconds southwest of the primary . hd 262042 , the b2 dwarf directly south of the cone nebula , is a probable background star given its distance modulus ( @xmath41 ) . far to the west is the b5 star hd 47469 , a probable non - member lying @xmath0200 pc in front of the cluster . most of the ob stellar population of the cluster lies concentrated within the boundaries of the molecular cloud , matching the distribution of suspected low - mass members ( i.e. x - ray detected sources and h@xmath4 emitters ) . walker ( 1956 ) notes the presence of five yellow giants within the cluster region and suggests they might lie in the foreground . he assigns to these potential interlopers spectral types and luminosity classes of g5 iii to w73 , k2 ii - iii to w237 , k3 ii - iii to w229 , k3 ii - iii to w69 , and k5 iii to w37 . underhill ( 1958 ) presents radial velocities of these stars , finding that only one ( w73 ) is consistent with the mean radial velocities of the early - type stars . ccccccc + + identifier & ra ( j2000 ) & @xmath42 ( j2000 ) & @xmath43 & @xmath23 & sp t & notes + hd 44498 & 06 22 22.5 & + 08 19 36 & 8.82 & @xmath44 & b2.5v & mon ob1 + hd 45789a & 06 29 55.9 & + 07 06 43 & 7.10 & @xmath45 & b2.5iv & + hd 45827 & 06 30 05.5 & + 09 01 46 & 6.57 & 0.11 & a0iii & + hd 46300 & 06 32 54.2 & + 07 19 58 & 4.50 & 0.00 & a0ib & 13 mon + hd 46388 & 06 33 20.3 & + 04 38 58 & 9.20 & 0.15 & b6v & mon ob1 + hd 261490 & 06 39 34.3 & + 08 21 01 & 8.91 & 0.18 & b2iii & mon ob1 + hd 261657 & 06 40 04.8 & + 09 34 46 & 10.88 & 0.03 & b9v & + hd 47732 & 06 40 28.5 & + 09 49 04 & 8.10 & @xmath46 & b3v & bin + bd+09 1331b & 06 40 37.2 & + 09 47 30 & 10.79 & 0.62 & b2v & + hd 47755 & 06 40 38.3 & + 09 47 15 & 8.43 & @xmath45 & b5v & eb + hd 47777 & 06 40 42.2 & + 09 39 21 & 7.95 & @xmath47 & b3v & + hd 261810 & 06 40 43.2 & + 09 46 01 & 9.15 & @xmath45 & b8v & var + v v590 mon & 06 40 44.6 & + 09 48 02 & 12.88 & 0.15 & b8pe & w90 + bd+10 1220b & 06 40 58.4 & + 09 53 42 & 7.6 & ... & ob & s mon b + hd 261902 & 06 40 58.5 & + 09 33 31 & 10.20 & 0.06 & b8v & + hd 47839 & 06 40 58.6 & + 09 53 44 & 4.66 & @xmath48 & o7ve & s mon a + bd+10 1222 & 06 41 00.2 & + 09 52 15 & 9.88 & @xmath49 & b5v & var + hd 261938 & 06 41 01.8 & + 09 52 48 & 8.97 & @xmath49 & b6v & var + hd 261903 & 06 41 02.8 & + 09 27 23 & 9.16 & @xmath50 & b9v & + hd 261940 & 06 41 04.1 & + 09 33 01 & 10.0 & 0.04 & b8v & + hd 261937 & 06 41 04.5 & + 09 54 43 & 10.36 & 0.35 & b8v & bin + hd 47887 & 06 41 09.6 & + 09 27 57 & 7.17 & @xmath51 & b2iii & bin + hd 47887b & 06 41 09.8 & + 09 27 44 & 9.6 & ... & b9v & + irs 1 & 06 41 10.1 & + 09 29 33 & ... & ... & b2b5 & + hd 261969 & 06 41 10.3 & + 09 53 01 & 9.95 & 0.00 & b9iv & var + w 181 & 06 41 11.2 & + 09 52 55 & 10.03 & @xmath52 & b9vn & bin + hd 262013 & 06 41 12.9 & + 09 35 49 & 9.34 & @xmath46 & b5vn & + hd 262042 & 06 41 18.7 & + 09 12 49 & 9.02 & 0.03 & b2v & + hd 47934 & 06 41 22.0 & + 09 43 51 & 8.88 & @xmath53 & b9v & + hd 47961 & 06 41 27.3 & + 09 51 14 & 7.51 & @xmath47 & b2v & bin + hd 48055 & 06 41 49.7 & + 09 30 29 & 9.00 & @xmath53 & b9v & var + + the full extent of the intermediate and low - mass ( @xmath54 m@xmath3 ) stellar population of ngc 2264 and its associated molecular cloud complex has not been assessed , but several hundred suspected members exhibiting h@xmath4 emission , x - ray emission , or photometric variability have been cataloged principally in the vicinity of the known molecular cloud cores . the traditional means of identifying young , low - mass stars has been the detection of h@xmath4 emission either using wide - field , low - resolution , objective prism imaging or slitless grating spectroscopy . in pre - main sequence stars , strong h@xmath4 emission is generally attributed to accretion processes as gas is channeled along magnetic field lines from the inner edge of the circumstellar disk to the stellar photosphere . weak h@xmath4 emission is thought to arise from enhanced chromospheric activity , similar to that observed in field dme stars . herbig ( 1954 ) conducted the first census of ngc2264 , discovering 84 h@xmath4 emission stars concentrated in two groups in a @xmath55 rectangular area centered roughly between s mon and the cone nebula . the larger of the two groups lies north of hd 47887 , the bright b2 star near the tip of the cone nebula . the second group is concentrated southwest of s mon , near w67 and the prominent rosette of emission / reflection nebulosity . marcy ( 1980 ) re - examined ngc2264 using the identical instrumentation of herbig ( 1954 ) and discovered 11 additional h@xmath4 emitters , but was unable to detect emission from a dozen of the original lh@xmath4 stars . the implication was that h@xmath4 emission varied temporally among these low - mass stars , and that a reservoir of undetected emitters was present within the cluster and other young star forming regions . narrow - band filter photometric techniques have also been used to identify strong h@xmath4 emitters in ngc2264 . the first such survey , by adams et al . ( 1983 ) , was photographic in nature and capable of detecting strong emitters , i.e. classical t tauri stars ( ctts ) . their deep @xmath56 photographic survey of the cluster found @xmath0300 low - luminosity ( 7.5@xmath5712.5 ) pre - main sequence candidates . ogura ( 1984 ) conducted an objective prism survey of the mon ob1 and r1 associations , focusing upon regions away from the core of ngc2264 that had been extensively covered by herbig ( 1954 ) and marcy ( 1980 ) . this kiso h@xmath4 survey discovered 135 new emission - line stars , whose distribution follows the contours of the dark molecular cloud complex . more than three dozen of these h@xmath4 emitters lie within or near ngc2245 to the west . in the molecular cloud region north of ngc2264 , another two dozen emitters were identified by ogura ( 1984 ) . reipurth et al . ( 2004b ) , however , noticed a significant discrepancy with ogura s ( 1984 ) emitters , many of which were not detected in their objective prism survey . it is possible that many of ogura s ( 1984 ) sources are m - type stars which exhibit depressed photospheric continua near h@xmath4 caused by strong tio band absorption . the resulting `` peak '' can be confused for h@xmath4 emission . sung et al . ( 1997 ) conducted a ccd narrow - band filter survey of ngc2264 , selecting pre - main sequence stars based upon differences in @xmath58 color with that of normal main sequence stars . using this technique , sung et al . ( 1997 ) identified 83 pre - main sequence stars and 30 pre - main sequence candidates in the cluster . of these 83 pre - main sequence stars , dahm & simon ( 2005 ) later established that 61 were cttss , 12 were weak - line t tauri stars ( wtts ) and several could not be positively identified . while capable of detecting strong h@xmath4 emission , the narrow - band filter photometric techniques are not able to distinguish the vast majority of weak - line emitters in the cluster . lamm et al . ( 2004 ) also obtained narrow - band h@xmath4 photometry for several hundred stars in their periodic variability survey of ngc2264 . more recently , reipurth et al . ( 2004b ) carried out a 3@xmath8@xmath13@xmath8 objective prism survey of the ngc2264 region using the eso 100/152 cm schmidt telescope at la silla , which yielded 357 h@xmath4 emission stars , 244 of which were newly detected . reipurth et al . ( 2004b ) ranked emission strengths on an ordinal scale from 1 to 5 , with 1 indicating faint emission and strong continuum and 5 the inverse . comparing with the slitless grism survey of dahm & simon ( 2005 ) , the technique adopted by reipurth et al . ( 2004b ) is capable of detecting weak emission to a limit of w(h@xmath4)@xmath59 . shown in figure 6 ( from reipurth et al . 2004b ) is the distribution of their 357 detected h@xmath4 emitters , superposed upon a @xmath60co ( 10 ) map of the region obtained with the at&t bell laboratories 7 m offset cassegrain antenna . the main concentrations of h@xmath4 emitters were located within or near the strongest peaks in co emission strength . a halo of h@xmath4 emitters in the outlying regions does not appear to be associated with the co cloud cores and may represent an older population of stars that have been scattered away from the central molecular cloud . reipurth et al . ( 2004b ) speculate that this may be evidence for prolonged star formation activity within the molecular cloud hosting ngc2264 . the slitless wide - field grism spectrograph ( wfgs ) survey of the cluster by dahm & simon ( 2005 ) increased the number of h@xmath4 emitters in the central @xmath61 region between s mon and the cone nebula to nearly 500 stars . capable of detecting weak emission with _ w_(h@xmath4 ) @xmath62 , the wfgs observations added more than 200 previously unknown h@xmath4 emitters to the growing number of suspected cluster members . combining the number of known emitters outside of the survey of dahm & simon ( 2005 ) identified by ogura ( 1984 ) and reipurth et al . ( 2004b ) , we derive a lower limit for the low - mass ( 0.22 m@xmath3 ) stellar population of the cluster region of more than 600 stars . from a compilation of 616 candidate cluster members with @xmath63 , ( from two deep _ chandra _ observations ) , flaccomio ( private communication ) finds that 202 do not exhibit detectable h@xmath4 emission . accounting for lower masses below the detection threshold of the various h@xmath4 surveys , x - ray selected stars that lack detectable h@xmath4 emission , and embedded clusters of young stars still emerging from their parent molecular cloud cores , the stellar population of the cluster could exceed 1000 members . emitters in ngc2264 from a 3@xmath8@xmath13@xmath8 objective prism survey of the region by reipurth et al . ( 2004b ) , superposed upon a grayscale map of @xmath60co ( 10 ) by john bally . the emitters are concentrated along the broad ridgeline of the molecular cloud and near s mon . north and west of the cluster , h@xmath4 emitters are scattered around and within the molecular cloud perimeter . the stars that appear unassociated with molecular emission may have been scattered from the central cluster region , implying that star formation has been occurring with the molecular cloud complex for several myr . [ f6],scaledwidth=95.0% ] x - ray emission is another well - established discriminant of stellar youth , with pre - main sequence stars often exhibiting x - ray luminosities ( @xmath64 ) 1.5 dex greater than their main sequence counterparts . the importance of x - ray emission as a youth indicator was immediately recognized in young clusters and associations where field star contamination is significant . the ability to select pre - main sequence stars from field interlopers , most notably after strong h@xmath4 emission has subsided , led to the discovery of a large population of weak - line t tauri stars associated with many nearby star forming regions . among field stars , however , the e - folding time for x - ray emission persists well into the main sequence phase of evolution . consequently , some detected x - ray sources are potentially foreground field stars not associated with a given star forming region . for ngc2264 , flaccomio ( private communication ) estimates the fraction of _ chandra _ detected x - ray sources that are foreground interlopers to be @xmath07% . detailed discussion of x - ray observations made to date with rosat , _ xmm - newton _ , or _ chandra _ is given in section 9 , but here we summarize these results in the context of the low - mass stellar population of ngc2264 . patten et al . ( 1994 ) used the high resolution imager ( hri ) onboard rosat to image ngc2264 and identified 74 x - ray sources in the cluster , ranging from s mon to late - k type pre - main sequence stars . this shallow survey was later incorporated by flaccomio et al . ( 2000 ) who combined six pointed rosat hri observations of the cluster for a total integration time of 25 ks in the southern half and 56 ks in the northern half of the cluster . in their sample of 169 x - ray detections ( 133 with optical counterparts ) , @xmath65 were gkm spectral types , including many known cttss and wttss . ramirez et al . ( 2004 ) cataloged 263 x - ray sources in a single 48.1 ks observation of the northern half of the cluster obtained with the advanced ccd imaging spectrometer ( acis ) onboard _ chandra_. flaccomio et al . ( 2006 ) added considerably to the x - ray emitting population of the cluster in their analysis of a single 97 ks _ acis - i observation of the southern half of the cluster , finding an additional 420 sources . of the nearly 700 x - ray sources detected by _ chandra _ in the cluster region , 509 are associated with optical or nir counterparts . of these , the vast majority are still descending their convective tracks . dahm et al . ( 2007 ) identify 316 x - ray sources in ngc2264 in two 42 ks integrations of the northern and southern cluster regions made with the european photon imaging camera ( epic ) onboard _ xmm - newton_. the vast majority of these sources have masses @xmath66 2 m@xmath3 . the positions of these x - ray selected stars as well as several embedded sources lacking optical counterparts are shown in figure 7 , superposed upon the red dss image of the cluster . square red image of ngc2264 obtained from the digitized sky survey ( dss ) showing the field of view for each of the _ xmm _ epic exposures from the survey of dahm et al . the black circles mark 300 detected x - ray sources having optical counterparts , presumably cluster members , while the white circles denote embedded sources or background objects observed through the molecular cloud . the distribution of x - ray sources closely matches the distribution of h@xmath4 emitters seen in figure 6 . [ f7],width=480 ] photometric variability is among the list of criteria established by joy ( 1945 ) for membership in the class of t tauri stars . young , low - mass stars are believed to experience enhanced , solar - like magnetic activity in which large spots or plage regions cause photometric variability at amplitudes typically @xmath670.2 mag . if active accretion is still taking place from a circumstellar gas disk , hot spots from the impact point on the stellar surface are believed to induce large amplitude , irregular variations . wolf ( 1924 ) first examined ngc2264 for photometric variables , identifying 24 from nine archived plates of the cluster taken between 1903 and 1924 . herbig ( 1954 ) noted that 75% ( 18 of 24 ) of these variables also exhibited h@xmath4 emission . herbig ( 1954 ) further suggested that given the variable nature of h@xmath4 emission , continued observation of the cluster would likely remove some of the variables from the non - emission group of stars . nandy ( 1971 ) found that among the h@xmath4 emitters in ngc2264 that are also photometric variables , a positive correlation exists between infrared ( out to @xmath68band ) and ultraviolet excesses . nandy & pratt ( 1972 ) show that the range of variability among the t tauri population in ngc2264 is less in @xmath68band than in other filters ( @xmath69 ) and also demonstrate a technique of studying the extinction properties of dust grains enveloping t tauri stars from their color variations . breger ( 1972 ) undertook an extensive photometric variability study of ngc2264 , finding that only 25% of the pre - main sequence a and f - type stars exhibit short - period variations . breger ( 1972 ) also found a strong correlation between `` shell indicators '' and variability , particularly among the t tauri stars . w90 ( lh@xmath4 25 ) was also found to have brightened by @xmath70 mag in @xmath43 since 1953 , an interesting trend for this peculiar herbig aebe star . koch & perry ( 1974 ) undertook an extensive photographic study of the cluster , identifying over 50 new variables , 16 of which were suspected eclipsing binaries , but no periods could be established . koch & perry ( 1974 ) conclude that many more low - amplitude ( @xmath670.2 mag ) variables must remain unidentified within the cluster , a prescient statement that would remain unproven until the coming of modern ccd photometric surveys . variability studies of individual cluster members include those of rucinski ( 1983 ) for w92 , koch et al . ( 1978 ) for hd 47732 , and walker ( 1977 ) also for w92 . ccd photometric monitoring programs have proven highly effective at identifying pre - main sequence stars from variability analysis . kearns et al . ( 1997 ) reported the discovery of nine periodic variables in ngc2264 , all g and k - type stars . an additional 22 periodic variables were identified by kearns & herbst ( 1998 ) including the deeply eclipsing ( 3.5 mag . in @xmath71 ) , late k - type star , kh-15d . two large - scale photometric variability campaigns have added several hundred more periodic variables to those initially identified by kearns et al . ( 1997 ) and kearns & herbst ( 1998 ) . the surveys of makidon et al . ( 2004 ) and lamm et al . ( 2004 ) were completed in the 2000 - 2001 and 1996 - 1997 observing seasons , respectively . their derived periods for the commonly identified variables show extraordinary agreement , confirming the robustness of the technique . the makidon et al . ( 2004 ) survey used the 0.76 m telescope at mcdonald observatory imaging upon a single 2048@xmath12048 ccd . the short focal length of the set - up resulted in a total field of view of approximately 46square . their observations were made over a 102-day baseline with an average of 5 images per night on 23 separate nights . makidon et al . ( 2004 ) identified 201 periodic variables in a period distribution that was indistinguishable from that of the orion nebula cluster ( herbst et al . this surprising result suggests that spin - up does not occur over the age range from @xmath671 to 5 myr and radii of 1.2 to 4 r@xmath3 ( rebull et al . makidon et al . ( 2004 ) also found no conclusive evidence for correlations between rotation period and the presence of disk indicators , specifically @xmath72 , @xmath73 , and @xmath74 colors as well as strong h@xmath4 emission . the lamm et al . ( 2004 ) survey was made using the wide - field imager ( wfi ) on the mpg / eso 2.2 m telescope at la silla . the wfi is composed of eight 2048@xmath14096 ccds mosaiced together , resulting in a field of view of @xmath033 square . this @xmath75 survey was undertaken over 44 nights with between one and 18 images obtained per night . the resulting photometric database from the lamm et al . ( 2004 ) survey is one of the most extensive available for ngc2264 and includes spectral types from rebull et al . ( 2002 ) as well as other sources . over 10,600 stars were monitored by the program , of which 543 were found to be periodic variables and 484 irregular variables with 11.4@xmath67@xmath76@xmath6719.7 . of these variables , 405 periodic and 184 irregulars possessed other criteria that are indicative of pre - main sequence stars . lamm et al . ( 2004 ) estimate that 70% of all pre - main sequence stars in the cluster ( @xmath76@xmath6718.0 , @xmath77@xmath671.8 ) were identified by their monitoring program ( implying a cluster population of @xmath0850 stars ) . lamm et al . ( 2005 ) use this extensive study to conclude that the period distribution in ngc2264 is similar to that of the orion nebula cluster , but shifted toward shorter periods ( recall that no difference was noted by makidon et al . 2004 ) . for stellar masses less than 0.25 m@xmath3 , the distribution is unimodal , but for more massive stars , the period distribution is bimodal . lamm et al . ( 2005 ) suggest that a large fraction of stars in ngc2264 are spun up relative to their counterparts in the younger orion nebula cluster . no significant spin up , however , was noted between older and younger stars within ngc2264 . lamm et al . ( 2005 ) also find evidence for disk locking among 30% of the higher mass stars , while among lower mass stars , disk locking is less significant . the locking period for the more massive stars is found to be @xmath08 days . for less massive stars a peak in the period distribution at 23 days suggests that these stars while not locked have undergone moderate angular momentum loss from star - disk interaction . other photometric surveys of the low - mass stellar population of ngc2264 include those of prez et al . ( 1987 , 1989 ) and fallscheer & herbst ( 2006 ) , who examined @xmath78 photometry of 0.41.2 m@xmath3 members and found a significant correlation between @xmath72 excess and rotation such that slower rotators are more likely to exhibit ultraviolet excess . numerous spectroscopic surveys of the intermediate and low - mass stellar populations of ngc2264 have been completed including those of walker ( 1954 , 1956 , 1972 ) , young ( 1978 ) , barry et al . ( 1979 ) , simon et al . ( 1985 ) , king ( 1998 ) , soderblom et al . ( 1999 ) , king et al . ( 2000 ) , rebull et al . ( 2002 ) , makidon et al . ( 2004 ) , dahm & simon ( 2005 ) , and furesz et al . walker ( 1972 ) examined the spectra of 25 uv excess stars in the orion nebula cluster and in ngc2264 having masses between 0.20.5 m@xmath3 and bolometric magnitudes ranging from @xmath79 to @xmath80 . from the radial velocities of the h i and ca ii lines , walker ( 1972 ) concluded that nine of these stars were experiencing accretion either from an enveloping shell of gas or from a surrounding disk . the rate of infall was estimated to be @xmath28 10@xmath81 m@xmath3 yr@xmath82 . young ( 1978 ) presented spectral types for 69 suspected cluster members ranging from b3v to k5v . combining the spectra with broad - band photometry , he concluded that differential reddening is present within the cluster , likely induced by intracluster dust clouds . king ( 1998 ) examined li features among six f and g - type stars within the cluster , finding the mean non - lte abundance to be log n(li)@xmath83 . this is identical to meteoritic values within our solar system , implying that no galactic li enrichment has occurred over the past 4.6 gyr . from rotation rates of 35 candidate members of ngc2264 , soderblom et al . ( 1999 ) conclude that the stars in ngc2264 are spun - up relative to members of the orion nebula cluster . furesz et al . ( 2006 ) present spectra for nearly 1000 stars in the ngc2264 region from the hectochelle multiobject spectrograph on the mmt . of these , 471 are confirmed as cluster members based upon radial velocities or h@xmath4 emission . furesz et al . ( 2006 ) also find that spatially coherent structure exists in the radial velocity distribution of suspected members , similar to that observed in the molecular gas associated with the cluster . the substellar population of ngc2264 remains relatively unexplored given the distance of the cluster . several deep imaging surveys have been conducted of the region , including that of sung et al . ( 2004 ) and kendall et al . sung et al . ( 2004 ) observed ngc2264 with the cfht 12k array in the broadband @xmath84 filters and a narrowband h@xmath4 filter . their deep survey extends to the substellar limit . the optical photometry was combined with _ chandra _ x - ray observations of the cluster to derive the cluster imf . using the ( @xmath85 ) color - color diagram to remove background giants , sung et al . ( 2004 ) conclude that the overall shape of the cluster imf is similar to that of the pleiades or the orion nebula cluster . kendall et al . ( 2005 ) identified 236 low - mass candidates lying redward of the 2 myr isochrones of the dusty evolutionary models generated by baraffe et al . ( 1998 ) . of these substellar candidates , 79 could be cross - correlated with the 2mass database , thereby permitting dereddening from nir excesses . most of these candidates range in mass between that of the sun and the substellar limit . the reddest objects with the lowest a@xmath21 values are possible brown dwarfs , but deeper optical and nir ( @xmath86 ) surveys are needed to further probe the substellar population of ngc2264 . for the brown dwarf candidates identified thus far within the cluster , spectroscopic follow - up is needed for confirmation . shown in figure 8 is the ( @xmath87 ) color - magnitude diagram from kendall et al . ( 2005 ) , which shows all 236 low - mass candidates identified in the cluster . , @xmath71 ) color - magnitude diagram of ngc2264 from the survey of kendall et al . ( superposed are the solid isochrones of the dusty models of baraffe et al . for ages of 2 and 5 myr . the dashed isochrones are the nextgen stellar models for the same ages . the 236 substellar candidates ( squares ) were selected to be redward of the sloping straight line . distance assumed for the isochrones is 760 pc . the mass scale on the left is for the 2 myr models . [ f8],scaledwidth=90.0% ] among the earliest radio surveys of ngc2264 is that of menon ( 1956 ) who observed the region at 21 cm and found a decrease in neutral hydrogen intensity toward the molecular cloud , leading to speculation that the formation of molecular hydrogen depleted the region of neutral atoms . minn & greenberg ( 1975 ) surveyed the dark clouds associated with the cluster at both the 6 cm line of h@xmath88co and the 21 cm line of h i. minn & greenberg ( 1975 ) discovered that the h@xmath88co line intensity decreased dramatically outside of the boundaries of the dark cloud , implying that the molecule was confined within the visual boundaries of the cloud . depressions within the 21 cm h i profiles observed along the line of sight were also found to be well - correlated with the h@xmath88co line velocities , but no further quantitative estimates were made . molecular gas dominates the mon ob1 association , with stars accounting for less than a few percent of the total mass of the cluster - cloud complex . zuckerman et al . ( 1972 ) searched for hcn and cs molecular line emission in the cone nebula , but later discovered their pointing was actually 4 north of the cone . emission , however , was serendipitously discovered from the background molecular cloud , presumably the region associated with irs1 . riegel & crutcher ( 1972 ) detected oh emission from several pointings within ngc2264 including a position near that observed by zuckerman et al . ( 1972 ) . from the relative agreement among line radial velocities , riegel & crutcher ( 1972 ) concluded that the oh emission arises from the same general region as that of the other molecular species . mayer et al . ( 1973 ) also detected nh@xmath89 emission at the position reported by zuckerman et al . ( 1972 ) , finding comparable velocity widths among the nh@xmath89 , hcn and cs lines . the measured velocity widths , however , were larger than the thermal doppler broadening inferred from the nh@xmath89 kinetic temperatures . from this , mayer et al . ( 1973 ) suggested that a systematic radial velocity gradient exists over the region where irs1 is located . rickard et al . ( 1977 ) mapped 6 cm continuum emission and h@xmath88co absorption over an extended region of ngc2264 , finding that h@xmath88co absorption toward the cluster was complex and possibly arises from multiple cloud components . blitz ( 1979 ) completed a co ( 10 ) 2.6 mm survey of ngc2264 at a resolution of 8and identified two large molecular clouds in the region with a narrow bridge of gas between them . one of the clouds is centered upon ngc2264 and the other lies 2@xmath8 northwest of the cluster and contains several reflection nebulae ( ngc2245 and ngc2247 ) . crutcher et al . ( 1978 ) mapped the region in the j@xmath3810 lines of @xmath90co and @xmath60co at somewhat better resolution ( half - power beam diameter of @xmath916 ) . shown in figure 9 are their resulting maps for @xmath90co and @xmath60co . the primary cloud core lying approximately 8 north of the cone nebula is elongated with a position angle roughly parallel to the galactic plane . the peak antenna temperature of 22 k occurs over an extended region , which includes irs1 ( allen s source ) but is not centered upon this luminous infrared source . two less prominent peaks were identified by crutcher et al . ( 1978 ) about 4 south and 8 west of s mon . co and @xmath60co of the ngc2264 region from crutcher et al . the coordinate scale is for equinox and epoch b1950 . the contour intervals are 4 and 2 k for the @xmath90co and @xmath60co maps , respectively , with the lowest contours representing the 8 k isotherm . the cross , plus symbol , and square denote the positions of s mon , irs1 , and the cone nebula . [ f9],width=288 ] the kinematic structure of the molecular clouds is somewhat complex with a 2 km s@xmath82 velocity gradient across the primary cloud and three distinct velocity components near s mon . individual clouds identified by crutcher et al . ( 1978 ) are listed in table 3 , reproduced from their table 1 . included with their identifiers are the positions ( b1950 ) , radii in pc , lsr velocity for @xmath90co , antenna temperatures for @xmath90co , and mass . summing these individual fragments , a total cloud mass of @xmath07@xmath110@xmath92 m@xmath3 is estimated . this , however , is a lower limit based upon co column densities . crutcher et al . ( 1978 ) also determined a virial mass of 3@xmath110@xmath2 m@xmath3 for the molecular cloud complex from the velocities of the individual cloud fragments . assuming this to be an upper limit , crutcher et al . ( 1978 ) adopt a middle value of 2@xmath110@xmath2 m@xmath3 for the total cloud mass associated with the cluster , half of which is contained within the primary cloud core north of the cone nebula . considering the energetics of the cloud , they further suggest that the luminous stars in ngc2264 are not capable of heating the core of this massive cloud . this conclusion is challenged by sargent et al . ( 1984 ) on the basis of a revised cooling rate . shown in figure 10 is an optical image of the cloud core c region from crutcher et al . ( 1978 ) , which lies just west of s mon . 15 , with north up and east left . image obtained at the cfht . courtesy j .- c . cuillandre and g. anselmi . [ f10],scaledwidth=90.0% ] cccccccc + + identifier & @xmath4 & @xmath42 & r & v & t@xmath93 & n(co ) & mass + & ( b1950 ) & ( b1950 ) & ( pc ) & ( km / s ) & ( k ) & 10@xmath94 @xmath95 & ( m@xmath3 ) + a & 6 37.6 & + 9 37.0 & 1.7 & 4.9 & 12 & 8 & 1200 + b & 6 37.2 & + 9 56.0 & 0.9 & 8.8 & 19 & 8 & 400 + c & 6 37.7 & + 9 56.0 & 0.9 & 10.4 & 22 & 6 & 300 + d & 6 37.9 & + 9 49.0 & 0.4 & 11.4 & 14 & 2 & 20 + e & 6 38.2 & + 9 53.0 & 0.8 & 8.9 & 23 & 10 & 400 + f & 6 38.2 & + 9 40.0 & 0.8 & 5.5 & 22 & 53 & 1800 + g & 6 38.4 & + 9 32.0 & 0.9 & 7.5 & 22 & 54 & 2300 + h & 6 38.9 & + 9 22.0 & 0.9 & 4.5 & 11 & 5 & 300 + schwartz et al . ( 1985 ) completed co ( 10 ) , c@xmath96o , and cs ( 32 ) cs observations of the irs1 and irs2 regions with the 11 meter nrao antenna on kitt peak as well as nh@xmath89 observations using the nrao 43 meter antenna at green bank . the nh@xmath89 and cs observations were used to examine high density gas while co and c@xmath96o were employed as tracers of h@xmath88 column density . schwartz et al . ( 1985 ) combined these molecular gas observations with far - infrared ( 35250 @xmath5 m ) data obtained with the multibeam photometer system onboard the kuiper airborne observatory . the general morphology of the co and cs maps of the regions were found to be similar , but two key differences were noted : first , the irs2 core appears as two unresolved knots in the cs map and second , the high density gas near irs1 ( allen s source ) lies east of the far - infrared peak and appears extended along the northeast to southwest axis . schwartz et al . ( 1985 ) conclude that allen s source is an early - type star ( @xmath0 b3v ) , which is embedded within a dense molecular cloud . they further speculate that the molecular cloud associated with irs1 is ring - shaped , lying nearly edge - on with respect to the observer ( see their figure 7 ) . this conclusion was drawn from the gas - density and velocity structure maps of the region . krugel et al . ( 1987 ) completed a more spatially resolved survey of the irs1 region in both nh@xmath89 and co , mapping a 10@xmath110 area . two distinct cloud cores were identified , each of roughly 500 m@xmath3 . kinetic gas temperatures were found by the authors to be 18 k in the north and 25 k in the south with a peak of 60 k around irs1 . complex structure was noted in the southern cloud , which exhibits multiple subclouds with differing temperatures and velocities . krugel et al . ( 1987 ) find that the smooth velocity gradient across the region observed at lower resolution disappears completely in their higher resolution maps . they also failed to detect high velocity wings toward irs1 that might be associated with a molecular outflow . tauber et al . ( 1993 ) observed one of the bright rims in ngc2264 west of s mon in the @xmath90co and @xmath60co ( 32 ) transitions with the caltech submillimeter observatory . in their high spatial resolution maps ( 20 ) , they find a morphology which suggests interaction with the ionizing radiation from s mon . the @xmath60co maps reveal a hollow shell of gas broken into three components : two eastern clumps with their long axes pointing toward s mon and a southern clump , which exhibits kinematic structure indicative of an embedded , rotating torus of dense gas . oliver et al . ( 1996 ) completed a sensitive , unbiased co ( 10 ) line emission survey of the mon ob1 region using the 1.2 meter millimeter - wave radio telescope at the harvard smithsonian center for astrophysics . the survey consisted of over 13,400 individual spectra and extended from @xmath97 , with individual pointings uniformly separated by 3.75 . oliver et al . ( 1996 ) find that the molecular gas along the line of sight of the mon ob1 association possesses radial velocities consistent with the local spiral arm and the outer perseus arm . within the local arm , they identify 20 individual molecular clouds ranging in mass from @xmath0100 to 5.2@xmath110@xmath2 m@xmath3 . their table 5 ( reproduced here as table 4 ) summarizes the properties of these molecular clouds including position , v@xmath98 , distance , radius , mass , and associated clusters , molecular clouds , or bright nebulae . the most massive of these molecular clouds hosts ngc2264 , but their derived mass estimates assume the distance of prez et al . ( 1987 ) , 950 pc , which probably overestimates the actual cluster distance by a factor of 1.2 . the revised cloud mass assuming a distance of 800 pc is 3.7 @xmath1 10@xmath2 m@xmath3 , which is in somewhat better agreement with the value derived by crutcher et al . ( 1978 ) . oliver et al . ( 1996 ) also identify six arc - like molecular structures in the mon ob1 region , which may be associated with supernova remnants or wind - blown shells of gas . these structures , if associated with the local spiral arm and at an appropriate distance , may imply that star formation within the region was triggered by an energetic supernova event . c@c@c@c@c@c@c@c@c + + cloud & _ l _ & _ b _ & v@xmath99 & @xmath42v & d & r & mass ( co ) & associations + & @xmath8 & @xmath8 & km / s & km / s & kpc & kpc & m@xmath3 & + 1 & 196.25 & @xmath400.13 & + 4.7 & 4.6 & 0.9 & 9.4 & 3.7@xmath110@xmath92 & + 2 & 196.75 & + 1.50 & + 5.3 & 2.3 & 0.9 & 9.7 & 9@xmath110@xmath92 & + 3 & 196.88 & @xmath401.13 & + 5.3 & 3.3 & 0.9 & 9.4 & 1.1@xmath110@xmath92 & + 4 & 198.88 & + 0.00 & @xmath406.9 & 3.4 & 0.8 & 9.3 & 1.0@xmath110@xmath92 & + 5 & 199.31 & @xmath400.50 & + 3.3 & 2.6 & 0.9 & 9.4 & 6.7@xmath110@xmath100 & + 6 & 199.56 & @xmath400.44 & + 6.4 & 4.9 & 0.9 & 9.4 & 2.0@xmath110@xmath92 & + 7 & 199.81 & + 0.94 & + 6.9 & 4.6 & 0.9 & 9.4 & 2.4@xmath110@xmath2 & l1600 , l1601 + & & & & & & & & l1604 , lbn886 + & & & & & & & & lbn889 + 8 & 200.19 & + 3.44 & @xmath406.8 & 3.4 & 0.8 & 9.3 & 1.9@xmath110@xmath100 & + 9 & 200.81 & + 0.13 & @xmath4011.0 & 5.2 & 0.8 & 9.3 & 1.5@xmath110@xmath92 & vy mon , lbn895 + 10 & 201.38 & + 0.31 & @xmath401.0 & 3.2 & 0.9 & 9.3 & 4.3@xmath110@xmath92 & mon r1 , l1605 + & & & & & & & & vdb76,77,78 + & & & & & & & & lbn898,903 + 11 & 201.44 & + 0.69 & + 5.1 & 4.4 & 0.9 & 9.3 & 2.1@xmath110@xmath2 & mon r1 , vy mon + & & & & & & & & ngc2245 , ngc2247 + & & & & & & & & l1605 , lbn901/904 + 12 & 201.44 & + 2.56 & + 0.7 & 3.1 & 0.9 & 9.4 & 2.0@xmath110@xmath92 & ngc2259 + 13 & 201.50 & + 2.38 & + 4.6 & 6.9 & 0.95 & 9.4 & 6.3@xmath110@xmath92 & ngc2259 , lbn899 + 14 & 202.06 & + 1.44 & + 1.1 & 2.9 & 0.95 & 9.4 & 1.5@xmath110@xmath92 & l1609 + 15 & 202.25 & + 1.69 & + 4.7 & 2.5 & 0.95 & 9.4 & 9@xmath110@xmath100 & l1610 + 16 & 203.25 & + 2.06 & + 6.9 & 5.0 & 0.95 & 9.4 & 5.2@xmath110@xmath2 & ngc2264 , irs1 , l1613 + & & & & & & & & s mon , lbn911/912/922 + 17 & 203.75 & + 1.25 & + 8.7 & 2.8 & 0.80 & 9.2 & 1.0@xmath110@xmath92 & ngc2261 + & & & & & & & & hh 39(a f ) + & & & & & & & & r mon , lbn920 + 18 & 204.13 & + 0.44 & + 5.2 & 2.8 & 0.80 & 9.2 & 1.9@xmath110@xmath92 & ngc2254 , lbn929 + 19 & 204.44 & @xmath400.13 & @xmath403.1 & 2.6 & 0.80 & 9.2 & 1.1@xmath110@xmath100 & ngc2254 + 20 & 204.81 & + 0.44 & + 9.4 & 2.6 & 0.80 & 9.2 & 8.2@xmath110@xmath100 & ngc2254 + wolf - chase & gregersen ( 1997 ) analyzed observations of numerous transitions of cs and co in the irs1 region . taken as a whole , the observations suggest that gravitational infall is taking place . schreyer et al . ( 1997 ) mapped the region around irs1 in various molecular transitions of cs , co , methanol , and c@xmath96o . complementing the millimeter survey , schreyer et al . ( 1997 ) also imaged the region in the nir ( @xmath101 ) , revealing the presence of several suspected low - mass stars surrounding irs1 . to the southeast , a small , deeply embedded cluster was noted in the @xmath102band mosaic of the field , which coincided with a second cloud core identified by the millimeter survey . extending northwest from irs1 , schreyer et al . ( 1997 ) noted a jet - like feature in all nir passbands that connects to the fan - shaped nebula evident on optical images of the field . two bipolar outflows were also detected in their cs mapping of the irs1 region , one originating from irs1 itself and another from the millimeter source lying to the southeast . thompson et al . ( 1998 ) examined irs1 with nicmos onboard hst and discovered six point sources at projected separations of 26 to 49 from allen s source ( irs1 ) . from the nir colors and magnitudes of these sources , thompson et al . ( 1998 ) suggest that these faint stars are pre - main sequence stars whose formation was possibly triggered by the collapse of irs1 . ward - thompson et al . ( 2000 ) observed the irs1 region in the millimeter and submillimeter continuum ( 1.3 mm 350 @xmath5 m ) , using the 7-channel bolometer array on the iram 30 meter telescope and the ukt14 detector on the james clerk maxwell telescope ( jcmt ) on mauna kea . the resulting maps revealed a ridge of bright millimeter emission as well as a clustering of five sources with masses ranging from 1050 m@xmath3 . table 4 of ward - thompson et al . ( 2000 ) summarizes the properties of these five massive condensations , which are assumed to be the progenitor cores of intermediate - mass stars . the third millimeter source ( mms3 ) of ward - thompson et al . ( 2000 ) was identified as the source of one of the bipolar outflows discovered by schreyer et al . ( 1997 ) in their millimeter survey of the region . williams & garland ( 2002 ) completed 870 @xmath5 m continuum emission maps and ( 32 ) line surveys of hco@xmath103 and h@xmath60co@xmath103 of the irs1 and irs2 regions of ngc2264 . the submillimeter continuum emission was used to trace dust around the young clusters of protostars , while line emission was used as a diagnostic of gas flow within the region . shown in figure 11 is the 870 @xmath5 m continuum emission map of williams & garland ( 2002 ) , their figure 1 , which clearly demonstrates the fragmented nature of irs2 . although irs1 possesses a much higher peak flux density , the masses of both regions are found to be comparable , @xmath010@xmath92 m@xmath3 . williams & garland ( 2002 ) find evidence for large - scale collapse for both irs1 and irs2 with infall velocities of @xmath00.3 km s@xmath82 . from their derived virial mass of the irs2 protostellar cluster , they conclude that the system is very likely gravitationally unbound . m continuum emission map of the irs1 and irs2 regions from williams & garland ( 2002 ) . the intensity scale and beamsize are annotated in the lower right and left corners of the figure , respectively . the elongated shape of irs1 exhibits signs of substructure , while irs2 is more fragmented and therefore suggestive of a more evolved cluster of protostars . [ f11],width=336 ] nakano et al . ( 2003 ) completed high resolution h@xmath60co@xmath103 ( 10 ) and 93 ghz continuum observations of the irs1 region using the nobeyama millimeter array and the nobeyama 45-meter telescope . four sources were identified in the resulting map , three of which were coincident with sources identified by ward - thompson et al . nakano et al . ( 2003 ) conclude that a dense shell of gas @xmath104 pc in diameter envelops irs1 , the interior of which has been evacuated on timescales of @xmath105 myr . schreyer et al . ( 2003 ) examined irs1 at high resolution using the iram plateau de bure interferometer at 3 mm and in the cs ( 21 ) transition . they complement this data with 2.2 , 4.6 , and 11.9 @xmath5 m imaging to interpret the immediate environment around irs1 . no circumstellar disk was found around irs1 . this young b - type star and several low - mass companions appear to be within a low - density cavity of the remnant cloud core . the source of the large bipolar outflow is also identified as a deeply embedded sources lying 20 north of irs1 . shown in figure 12 is the hst nicmos image of the irs1 field with several embedded sources identified , including source 8 , a binary which exhibits a centrosymmetric polarization pattern consistent with circumstellar dust emission . m image ( blue ) , the mean of 1.6 and 2.2 @xmath5 m images ( green ) , and the 2.2 @xmath5 m image ( red ) . numerous embedded sources are evident around irs1 including source 8 , a binary which exhibits a centrosymmetric polarization pattern consistent with dust emission . [ f12],width=288 ] hedden et al . ( 2006 ) used the heinrich hertz telescope and the aro 12 meter millimeter - wave telescope to map several outflows in the northern cloud complex of the mon ob1 association in @xmath90co ( 32 ) , @xmath60co ( 32 ) , @xmath90co ( 10 ) , and 870 @xmath5 m continuum . several continuum emission cores were identified and the seds of these sources were constructed to derive their column densities , masses , luminosities , and temperatures . hedden et al . ( 2006 ) conclude that the molecular cloud complexes are maintaining their integrity except along the axes of outflows . the outflows are found to deposit most of their energy outside of the cloud , leading to a weak correlation between outflow kinetic energy and turbulent energy within the clouds . peretto et al . ( 2006 ) surveyed the two massive cloud cores associated with irs1 and irs2 in dust continuum and molecular line emission , finding 12 and 15 compact millimeter continuum sources within each core , respectively . the millimeter sources have typical diameters of @xmath00.04 pc and range in mass from @xmath0241 m@xmath3 . although similar in size to cores within the @xmath106 oph star forming region , the millimeter sources in ngc2264 exhibit velocity dispersions two to five times greater than those of the @xmath106 oph main cloud and the isolated cores in the taurus - auriga complex . as many as 70% of the sources within the irs1 cloud core host class 0 protostars that are associated with jets of shocked h@xmath88 . in the irs2 cloud , only 25% of the millimeter sources are associated with 2mass or mid - infrared point sources . peretto et al . ( 2006 ) also find evidence for widespread infall within both cloud cores and suggest that the irs1 core is collapsing along its long axis in a free - fall timescale of @xmath01.7@xmath107 years . this is consistent with the findings of williams & garland ( 2002 ) . within this core , peretto et al . ( 2006 ) conclude that a high - mass ( 1020 m@xmath3 ) protostar is currently forming . reipurth et al . ( 2004c ) completed a 3.6 cm radio continuum survey of young outflow sources including ngc2264 irs1 using the very large array in its a configuration . their map reveals 8 sources in the general region of irs1 , including a source that is coincident with irs1 itself . the sources vla2 ( mms4 ) and vla7 appear extended and show significant collimation . three arcminutes southeast of irs1 , reipurth et al . ( 2004c ) find a bipolar radio jet with a 3.6 cm flux density of @xmath013 mjy . the eastern portion of this jet is comprised of at least 8 well - resolved knots that appear flattened perpendicular to the axis of the flow . the extent of this well - collimated jet is estimated to be 28 . the western lobe of the outflow exhibits only one large clump . deep optical and @xmath102band imaging of the central region of the jet reveal no evidence for a source , suggesting that it may be extragalactic in origin . trejo & rodrguez ( 2008 ) compare 3.6 cm observations of this non - thermal radio jet obtained in 2006 with archived data from 1995 and find no evidence for proper motion or polarization changes . flux density variations were found in one knot , which is tentatively identified as the core of a quasar or radio galaxy . teixeira et al . ( 2007 ) present high angular resolution ( 1 ) 1.3 mm continuum observations of the core d - mm1 in the spokes cluster obtained using the submillimeter array ( sma ) . they find a dense cluster of 7 class 0 objects within a 20@xmath120 region with masses ranging from 0.4 to 1.2 m@xmath3 . teixeira et al . ( 2007 ) conclude that the 1.3 mm continuum emission arises from the envelopes of the class 0 sources , which are found to be @xmath0600 au in diameter . the sources within the d - mm1 cluster have projected separations that are consistent with hierarchical fragmentation . one of the earliest infrared surveys of the cluster by allen ( 1972 ) identified several sources that were correlated with known stars , but a single bright source was identified near the tip of the cone nebula that lacked an optical counterpart . this source , now referred to as allen s infrared source or ngc2264 irs1 , is recognized to be an embedded early - type ( b spectral class ) star . allen ( 1972 ) suggested that this source was the most massive and luminous star within the cluster , a claim that has since been refuted by subsequent infrared observations . harvey et al . ( 1977 ) surveyed ngc2264 and ngc2244 in the mid- and far - infrared ( 53175 @xmath5 m ) using the kuiper airborne observatory . irs1 remained unresolved in the far infrared , but possessed relatively cool ( 53175 @xmath5 m ) colors . from their luminosity estimate for irs1 of 10@xmath92 @xmath108 , harvey et al . ( 1977 ) concluded that relative to compact hii regions believed to be the progenitors of massive o - type stars , irs1 was significantly less luminous , implying that it was an embedded intermediate - mass star of 510 m@xmath3 . sargent et al . ( 1984 ) completed a balloon - borne , large - scale mapping of ngc2264 at 70 and 130 @xmath5 m , identifying a number of far infrared sources within the cluster . the luminosity of irs1 was found to be 3.8@xmath110@xmath92 @xmath108 , which was consistent with the earlier higher resolution observations of harvey et al . ( 1977 ) . warner et al . ( 1977 ) obtained @xmath109 band observations of 66 members of ngc2264 using an indium antimonide detector on the mount lemmon 1.5 meter telescope . they confirmed the existence of infrared excesses for a significant fraction ( 30% ) of stars with spectral types later than a0 . margulis et al . ( 1989 ) identified 30 discrete iras sources in the mon ob1 molecular cloud complex , 18 of which were found to have class i spectral energy distributions . from the large population of class i sources , margulis et al . ( 1989 ) concluded that active star formation is still taking place within the molecular cloud complex hosting the cluster . neri et al . ( 1993 ) presented @xmath110 and @xmath111 band photometry for @xmath112 peculiar stars within the cluster , from which they determined a revised distance estimate and mean extinction value . the authors also examined optical and infrared variability among the sample stars and derive effective temperatures and log g values for each . they found no evidence for a or b - type stars with infrared fluxes lower than expected for their observed optical magnitudes . such stars had been reported previously in the ori i ob association . the first extensive nir imaging surveys of ngc2264 were completed by pich ( 1992 , 1993 ) and lada et al . ( 1993 ) , who mapped most of the cluster region in the @xmath113bands . lada et al . ( 1993 ) detected over 1650 @xmath102band sources in their survey and concluded that 360@xmath114130 were probable cluster members . of these , 50@xmath11420% possessed infrared excess emission , possibly implying the presence of circumstellar disks . rebull et al . ( 2002 ) undertook an optical and nir survey of the cluster , presenting photometry for over 5600 stars and spectral types for another 400 . three criteria were used to identify circumstellar disk candidates within the cluster : excess ( @xmath72 ) emission , excess nir emission ( @xmath115 and @xmath74 ) , and large h@xmath4 equivalent widths , if spectra were available . rebull et al . ( 2002 ) established an inner disk fraction ranging from 21% to 56% . no statistically significant variation was found in the disk fraction as a function of age , mass , @xmath68band mag , or ( @xmath116 ) color . mass accretion rates were derived from @xmath117band excesses with typical values on the order of 10@xmath118 m@xmath3 yr@xmath82 . wang et al . ( 2002 ) completed narrowband h@xmath88 , @xmath119 s(1 ) imaging of the irs1 region and identified at least four highly collimated jets of emission as well as several isolated knots of h@xmath88 emission . some of the jets are associated with millimeter and submillimeter sources identified by ward - thompson et al . aspin & reipurth ( 2003 ) imaged the irs2 region in the nir ( @xmath101 ) and thermal @xmath120 and @xmath121bands , finding two stars with spectra similar to those characteristic of fu ori type stars . the stars form a close ( 28 ) binary and exhibit arcuate reflection nebulae . aspin & reipurth ( 2003 ) compile the ( @xmath122 ) magnitudes for @xmath032 stars in the irs2 region . _ spitzer _ space telescope is revolutionizing our understanding of the star formation process and circumstellar disk evolution . three - color infrared array camera ( irac ) and multiband imaging photometer for _ spitzer _ ( mips ) images of ngc2264 have been released that reveal significant structure within the molecular cloud cores as well as embedded clusters of class i sources . teixeira et al . ( 2006 ) identify primordial filamentary substructure within one cluster such that the protostars are uniformly spaced along cores of molecular gas in a semi - linear fashion and at intervals consistent with the jeans length . shown in figure 13 is the three - color composite image of the embedded `` spokes '' cluster from teixeira et al . ( 2006 ) constructed from irac 3.6 @xmath5 m ( blue ) , irac 8.0 @xmath5 m ( green ) , and mips 24.0 @xmath5 m ( red ) images of the region . several quasi - linear structures appear to be coincident with dust emission from dense cores of molecular gas traced at 850 @xmath5 m with scuba by wolf - chase et al . figure 14 , obtained from teixeira et al . ( 2006 ) , compares the spatial distribution of the dust emission cores with 24 @xmath5 m point sources . the sizes of the stars representing 24 @xmath5 m sources are proportional to their magnitudes . m data as well as mips 24 @xmath5 m band imaging were used to create the image . [ f13 ] ] when complete , the _ spitzer _ irac and mips surveys of ngc2264 will unambiguously identify the disk - bearing population of the cluster and provide tentative characterizations of disk structure based upon the stellar optical to mid - infrared seds . when merged with extant rotational period data , the _ spitzer _ results may resolve long - standing questions regarding the impact of circumstellar disks upon stellar rotation and angular momentum evolution . population statistics for the embedded clusters will also add to the stellar census of ngc2264 and the greater mon ob1 association . m dust emission from wolf - chase et al . ( 2003 ) with 24 @xmath5 m point sources superposed as five - pointed stars . the squares mark the positions of the two brightest 24 @xmath5 m sources . the size of the stars and squares are proportional to the magnitudes of the sources . the beam size for the 850 @xmath5 m data is indicated in the lower right . [ f14],width=384 ] the earliest x - ray survey of ngc2264 was undertaken by simon et al . ( 1985 ) with the imaging proportional counter ( ipc ) aboard the _ einstein _ observatory . these early x - ray observations had 1 spatial resolution and an energy bandwidth of 0.44.0 kev . all three images obtained were centered upon s mon and had short integration times of 471 , 1660 , and 1772 s. in addition to s mon , seven x - ray sources were identified by the _ einstein _ program with x - ray luminosities ranging from 2.45.2@xmath110@xmath123 ergs s@xmath82 . these sources were among the most x - ray luminous of all young cluster stars observed with _ einstein_. rosat observed the cluster in 1991 march and 1992 september using the high resolution imager , hri . exposure times were significantly longer , 19.4 and 10.9 ks , respectively , resulting in the detection of 74 x - ray sources with s / n @xmath124 3.0 ( patten et al . 1994 ) . with the exception of a through early f - type stars , the hri observations detected cluster members over a range of spectral types from o7v to late - k . patten et al . ( 1994 ) also compared x - ray surface fluxes of solar analogs in the pleiades , ic 2391 , and ngc2264 , finding all three to be comparable . flaccomio et al . ( 2000 ) combined three archived rosat images of ngc2264 with three new observations made approximately 15 southeast of the earlier epochs of data . from these images , 169 distinct x - ray sources were identified : 133 possessed single optical counterparts , 30 had multiple counterparts and six had no optical counterparts . flaccomio et al . ( 2000 ) used optical ( @xmath125 ) photometry from flaccomio et al . ( 1999 ) and sung et al . ( 1997 ) to construct an hr diagram of the x - ray emission population of the cluster . ages and masses were also derived from the models of dantona and mazzitelli ( 1997 ) using the 1998 updates . comparing the ages and masses of the x - ray population with those derived from their earlier optical survey of the southern half of the cluster , flaccomio et al . ( 2000 ) concluded that the x - ray sample was representative of the entire pre - main sequence population of the cluster . flaccomio et al . ( 2000 ) also found x - ray luminosities of known cttss and wttss to be comparable , implying that accretion was not a significant source of x - ray emission . using the same data set , but with an improved determination of x - ray luminosities and a better reference optical sample , flaccomio et al . ( 2003 ) did find that cttss have on average lower x - ray luminosities with respect to wttss . nakano et al . ( 2000 ) used the advanced satellite for cosmology and astrophysics ( asca ) to observe ngc2264 in 1998 october with the gas - imaging spectrometer ( gis ) and the solid state imaging spectrometer ( sis ) . the field center of the observation was near w157 , several arcmin northwest of the cone nebula . given the moderate resolution of sis ( 30 ) , establishing optical or infrared counterparts of the x - ray emission sources was difficult . a dozen x - ray sources within the cluster were identified including two class i sources , irs1 and irs2 , as well as several known h@xmath4 emitters . nakano et al . ( 2000 ) suggest that most of the detected hard x - ray flux originates from intermediate mass class i sources , similar to allen s source . the improved spatial resolution ( @xmath01 ) and sensitivity of _ xmm - newton _ and _ chandra _ have revolutionized x - ray studies of young clusters and star forming regions . optical and nir counterparts of most x - ray sources can now be unambiguously identified , even in clustered regions . simon & dahm ( 2005 ) used deep ( 42 ks ) _ xmm - newton _ epic observations of the northern and southern halves of ngc2264 to probe sites of active star formation . the resulting integrations revealed strong x - ray emission from three deeply embedded ysos near irs1 and irs2 . the brightest x - ray source was located 11 southwest of allen s source and had a x - ray luminosity of 10@xmath126 ergs s@xmath82 and a temperature of 100 mk . follow - up 14 @xmath5 m , moderate - resolution spectra of the sources revealed deep water ice absorption bands at 3.1 @xmath5 m as well as many emission and absorption features of hi , co , and various metals . the nir images of the irs1 and irs2 regions with the _ xmm _ contours superposed are shown in figure 15 . within the _ xmm _ epic frames , over 316 confirmed x - ray sources were identified , 300 of which have optical or nir counterparts . dahm et al . ( 2007 ) find that most of these x - ray sources lie on or above the 3 myr isochrone of siess et al . ( 2000 ) . given the estimated low - mass population of ngc2264 from variability studies and h@xmath4 emission surveys , the _ xmm _ sample represents only the most x - ray luminous members of the cluster . ) false color images of the embedded x - ray sources near irs1 . contours for the brightest x - ray emission levels are overlaid . the strongest x - ray source in the cluster is exs-1 , which is identifiable with the infrared source 2mass j06410954 + 0929250 , which lies 11 southwest of allen s source . ( _ b _ ) in the irs2 field , the x - ray sources are from left to right : w164 , w159 , rno - e ( exs-26 ) , and 2mass j06405767 + 0936082 ( exs-10 ) . x - ray contours are superposed in green and white . [ f15],title="fig:",scaledwidth=80.0% ] ) false color images of the embedded x - ray sources near irs1 . contours for the brightest x - ray emission levels are overlaid . the strongest x - ray source in the cluster is exs-1 , which is identifiable with the infrared source 2mass j06410954 + 0929250 , which lies 11 southwest of allen s source . ( _ b _ ) in the irs2 field , the x - ray sources are from left to right : w164 , w159 , rno - e ( exs-26 ) , and 2mass j06405767 + 0936082 ( exs-10 ) . x - ray contours are superposed in green and white . [ f15],title="fig : " ] the advanced ccd imaging spectrometer ( acis ) onboard the _ chandra _ x - ray observatory was used by ramirez et al . ( 2004 ) to observe the northern half of ngc 2264 in 2002 february . the field of view of the imaging array ( acis - i ) is approximately 17@xmath117 and the total integration time of the observation was 48.1 ks . the pipeline reduction package detected 313 sources , 50 of which were rejected as cosmic ray artifacts or duplicate detections , leaving 263 probable cluster members . of these sources , 41 exhibited flux variability and 14 were consistent with flaring sources ( rapid rise followed by slow decay of x - ray flux ) . of the confirmed sources , 213 were identified with optical or nir counterparts . the deepest x - ray survey of ngc2264 to date is that of flaccomio et al . ( 2006 ) , who obtained a 97 ks _ acis - i integration of the southern half of the cluster in 2002 october . within the field of view of acis , 420 x - ray sources were detected , 85% of which have optical and nir counterparts . flaccomio et al . ( 2006 ) found that the median fractional x - ray luminosity , @xmath127/@xmath128 , of the sample is slightly less than 10@xmath129 . cttss were found to exhibit higher levels of x - ray variability relative to wttss , which was attributed to the stochastic nature of accretion processes . flaccomio et al . ( 2006 ) also found that cttss for a given stellar mass exhibit lower activity levels than wttss , possibly because accretion modifies magnetic field geometry resulting in mass loading of field lines and thus damping the heating of plasma to x - ray temperatures ( preibisch et al . 2005 ; flaccomio et al . flaccomio et al . ( 2006 ) also find , however , that the plasma temperatures of cttss are on average higher than their wtts counterparts . rebull et al . ( 2006 ) combine both of these _ chandra _ observations of ngc2264 in order to examine the x - ray properties of the full cluster population . they find that the level of x - ray emission is strongly correlated with internal stellar structure , as evidenced by an order of magnitude drop in x - ray flux among 12 m@xmath3 stars as they turn onto their radiative tracks . among the sample of x - ray detected stars with established rotation periods , rebull et al . ( 2006 ) find no correlation between @xmath127 and rotation rate . they also find no statistically significant correlation between the level of x - ray flux and the presence or absence of circumstellar accretion disks or accretion rates as determined by ultraviolet excess . herbig - haro ( hh ) objects and outflows are regarded as indicators of recent star formation activity . early surveys of the ngc2264 region by herbig ( 1974 ) identified several candidate hh objects . adams et al . ( 1979 ) conducted a narrow - band h@xmath4 emission survey of the cluster and identified 5 additional hh condensations that they conclude are associated with the molecular cloud complex behind the stellar cluster . walsh et al . ( 1992 ) discovered two additional hh objects in ngc2264 ( hh 124 and hh 125 ) using narrow - band imaging and low - dispersion spectra . hh 124 lies north of the cluster region and emanates from the cometary cloud core brc 25 , which contains iras 06382 + 1017 . hh 124 is composed of at least 6 knots of emission with the western condensations exhibiting negative high velocity wings and the eastern components positive ( up to + 100 km s@xmath82 ) . from a large - scale co ( 32 ) map of brc 25 , reipurth et al . ( 2004a ) report a significant molecular outflow along the axis of hh 124 with no source identified . ogura ( 1995 ) identified a giant ( 1 pc ) bow - shock structure associated with hh 124 using narrowband ( [ s ii ] @xmath130@xmath1306717 , 6731 ) ccd imaging and slit spectroscopy . the large projected distance of the shock from its proposed source implies a dynamical age in excess of 10,000 yrs . image of the hh 576 and 577 region with the co ( 32 ) emission contours superposed from reipurth et al . ( 2004a ) . the extent of the region mapped in co is shown by the light dashed white line . solid black contours represent blueshifted ( -3 to + 3 km s@xmath82 ) co emission , and dashed black contours redshifted ( + 13 to + 19 km s@xmath82 ) emission . [ f16],width=377 ] walsh et al . ( 1992 ) found that hh 125 is composed of at least 16 knots of emission and lies near other known hh objects identifed previously by adams et al . the angular extent of the hh object implies a projected linear dimension of @xmath00.78 pc . walsh et al . ( 1992 ) suggested that iras 06382 + 0945 is the exciting source for hh 125 . wang et al . ( 2003 ) completed an extensive 2@xmath8 wide - field [ s ii ] narrowband imaging survey of the mon ob1 region . in the northern part of the molecular cloud , two new hh objects were discovered ( hh 572 and hh 575 ) . reipurth et al . ( 2004a ) identify 15 additional hh objects in the region , some of which have parsec - scale dimensions . one these is the giant ( @xmath05.2 pc ) bipolar flow hh 571/572 which also originates from a source within brc 25 , possibly iras 06382 + 1017 . the co map of reipurth et al . ( 2004a ) revealed two large molecular outflows with position angles similar to those of hh 576 and 577 , suggesting a physical association . figure 16 shows the co contours overlaying an h@xmath4 image of hh 576 and 577 . the southwestern quadrant of the brc 25 cloud core also shows optical features that suggest significant outflow activity , but no co counterpart was identified . reipurth et al . ( 2004a ) suggested that hh 125 , 225 , and 226 form a single giant outflow , possibly originating in the irs2 region . table 5 summarizes known hh objects in the ngc2264 region . ccccc + + identifier & ra ( j2000 ) & @xmath42 ( j2000 ) & notes & ref + hh 39 & 06 39 07 & + 08 51.9 & ngc2261 & h74 + hh 575a & 06 40 31.6 & + 10 07 56 & brightest knot & r04 + hh 576 & 06 40 35.9 & + 10 39 48 & bow shock ( west ) & r04 + hh 577 & 06 40 36.6 & + 10 34 02 & brightest knot & r04 + hh 571 & 06 40 46.5 & + 10 05 15 & brightest point & r04 + hh 580 & 06 40 56.7 & + 09 32 52 & tip of bow & r04 + hh 582 & 06 40 56.9 & + 09 31 20 & western knot & r04 + hh 581 & 06 41 00.5 & + 09 32 56 & southern knot & r04 + hh 573a & 06 41 01.9 & + 10 14 51 & diffuse knot in brc 25 & r04 + hh 226 & 06 41 02.2 & + 09 39 49 & & w03 + hh 124 & 06 41 02.7 & + 10 15 03 & bow shock & o95 + hh 225 & 06 41 02.7 & + 09 44 16 & & w03 + hh 125 & 06 41 02.8 & + 09 46 07 & & w92 + hh 583 & 06 41 06.5 & + 09 33 16 & middle knot & r04 + hh 574 & 06 41 07.9 & + 10 16 19 & brightest knot & r04 + hh 578 & 06 41 10.6 & + 10 20 50 & star at end of jet & r04 + hh 579 & 06 41 14.4 & + 09 31 10 & tip of bow & r04 + hh 585 & 06 41 25.3 & + 09 24 01 & middle of bow & r04 + hh 572 & 06 41 28.8 & + 10 23 45 & eastern knot in bow & r04 + hh 584 & 06 41 38.8 & + 09 28 28 & central knot & r04 + + + * irs1 ( allen s infrared source ) : * first discovered by allen ( 1972 ) in his near infrared survey of ngc2264 , irs1 ( iras 06384 + 0932 ) is now recognized as a deeply embedded , early - type ( b2b5 ) star lying within a massive molecular cloud core . critical investigations into the nature of irs1 include those of allen ( 1972 ) , thompson & tokunaga ( 1978 ) , schwartz et al . ( 1985 ) , schreyer et al . ( 1997 , 2003 ) , and thompson et al . at least one molecular outflow is associated with irs1 ( schreyer et al . 1997 ; wolf - chase & gregersen 1997 ) as well as a jet - like structure detected in the near infrared . hst nicmos imaging has revealed several point sources surrounding irs1 assessed as solar - mass , pre - main sequence stars ( thompson et al . the millimeter continuum and molecular line observations and mid - infrared imaging of schreyer et al . ( 2003 ) suggest that irs1 is not associated with a circumstellar disk of primordial gas and dust . + * w90 ( lh@xmath425 ) : * no discussion of ngc2264 would be complete without addressing the remarkable herbig aebe star , w90 ( lh@xmath4 25 ) . herbig ( 1954 ) lists the star as an early - a spectral type with bright h@xmath4 emission and possible evidence for photometric variation ( @xmath00.1 mag ) relative to earlier brightness estimates by trumpler ( 1930 ) . he further noted that for a normal a2a3 type star , lh@xmath4 25 is three mag fainter than expected for the adopted distance of the cluster . walker ( 1956 ) found the star to lie well below the zams of ngc2264 . he further states that the star appears unreddened , but that structure within the balmer lines and the presence of several [ fe ii ] lines in its spectrum argue for a weak shell enveloping the star . herbig ( 1960 ) revised the spectral type of w90 to b8pe + shell , but noted that the balmer line wings are weak relative to other late b - type stars . poveda ( 1965 ) suggested that stars below the zams similar to w90 are surrounded by optically thick dust and gas shells , which induce neutral extinction . strom et al . ( 1971 ) estimated the surface gravity of w90 using its balmer line profiles , finding log g @xmath131 , consistent with giant atmospheres . they further speculated that w90 is surrounded by a dust shell that absorbs 95% of the radiated visible light . strom et al . ( 1972 ) confirmed the presence of dust around w90 , finding an extraordinary infrared ( @xmath132 ) excess of 3 mag . rydgren & vrba ( 1987 ) presented an sed for the star spanning from 0.3520 @xmath5 m , and concluded that w90 is observed through an edge - on dust disk . dahm & simon ( 2005 ) presented a nir spectrum of the star ( 0.852.4 @xmath5 m ) , which reveals strong brackett and paschen series line emission as well as he i and fe ii emission . w90 remains something of an enigma , but is a key representative of the herbig aebe population . + * kh-15d ( v582 mon ) : * this deeply eclipsing ( @xmath03.5 mag in @xmath71 ) k7 tts lies just north of the cone nebula and the b2 star hd 47887 . the eclipse profile of kh-15d ( kearns & herbst 1998 ) suggests that an inclined knife - edge screen is periodically occulting the star ( herbst et al . 2002 ) . during eclipse , herbst et al . ( 2002 ) find that the color of the star is bluer than when outside of eclipse . these observations and the polarization measurements of agol et al . ( 2004 ) support the conclusion that the flux received during eclipse is scattered by large dust grains within the obscuring screen ( hamilton et al . 2005 ) . by combining archived observations with recent photometry , hamilton et al . ( 2005 ) were able to analyze a 9-year baseline of eclipse data . their finding suggest that the eclipse is evolving rapidly , with its duration lengthening at a rate of 2 days per year . johnson et al . ( 2004 ) present results of an high resolution spectroscopic monitoring program for kh-15d and find it to be a single - line spectroscopic binary with a period of 48.38 days , identical to the photometric period . they conclude that the periodic dimming of kh-15d is caused by the binary motion that moves the visible stellar component above and behind the edge of an obscuring cloud . estimates for the eccentricity and mass function are given . johnson et al . ( 2005 ) present historical @xmath133 photometric observations of kh-15d obtained between 1954 and 1997 from multiple observatories . they find that the system has been variable at the level of 1 mag since at least 1965 . no evidence for color variation is found , and johnson et al . ( 2005 ) conclude that kh-15d is being occulted by an inclined , precessing , circumbinary ring . winn et al . ( 2006 ) use radial velocity measurements , ccd and photographic photometry obtained over the past 50 years to examine whether a model of kh-15d that incorporates a circumbinary disk can successfully account for its observed flux variations . after making some refinements such as the inclusion of disk scattering , they find that the model is successful in reproducing the observed eclipses . + , width=453 ] * ngc2261 ( hubble s variable nebula ) : * over a degree south of s mon lies the small reflection nebula ngc2261 , first noted by friedrich wilhelm herschel in his catalog of nebulae and stellar clusters . shown in figure 17 is a composite image of this object that reveals some of the extraordinary detail associated with the nebulosity . hubble ( 1916 ) describes ngc2261 as a cometary nebula in the form of an equilateral triangle with a sharp stellar nucleus at the extreme southern point . " what drew hubble s attention to the object , however , were indisputable changes in the outline and structure of the nebula that occurred between march 1908 and march 1916 . within the nucleus of ngc2261 is the irregular variable and herbig aebe star , r mon , which ranges in brightness from @xmath1349.5 to 13 mag . the changes within the nebula , however , did not appear to coincide with the brightness variations of r mon . slipher ( 1939 ) obtained a spectrum of ngc2261 that revealed bright hydrogen emission lines superposed upon a faint continuum . the nova - like spectrum of the nebula was identical to that of r mon , even in its outlying regions . lampland ( 1948 ) examined several hundred photographic plates of ngc2261 obtained over more than two decades and concluded that changes in the nebula were caused by varying degrees of veiling or obscuration , not physical motion . several polarization studies of the nebula have been made including those of hall ( 1964 ) , kemp et al . ( 1972 ) , aspin et al . ( 1985 ) , menard et al . ( 1988 ) , and close et al . ( 1997 ) . spectroscopic studies of ngc2261 include those of slipher ( 1939 ) , herbig ( 1968 ) , and stockton et al . herbig ( 1968 ) discovered the presence of hh 39 lying 8 north of the apex of ngc2261 . co ( 10 ) observations of ngc2261 by cant et al . ( 1981 ) identified an elongated molecular cloud centered upon r mon that is interpreted as being disk - shaped in structure . stellar winds from r mon are proposed to have created a bipolar cavity within the cloud , the northern lobe of which is the visible nebulosity of ngc2261 . brugel et al . ( 1984 ) identify both components of the highly collimated bipolar outflow associated with r mon . the measured velocities for the two flows are @xmath135 and @xmath136 km / s , respectively . brugel et al . ( 1984 ) argue that the flow collimation occurs within 2000 au of the star . walsh & malin ( 1985 ) obtained deep @xmath137 and @xmath138band ccd images of hh 39 , revealing a knot of nebulosity ( hh 39 g ) that has varied in brightness and has a measured proper motion . a filament is also observed between hh 39 and r mon that may be related to a stellar wind - driven flow . movsessian et al . ( 2002 ) determined radial velocities for several knots in the hh 39 group and find that the kinematics of the system as a whole suggest the precession of the outflow . polarization maps of ngc2261 by aspin et al . ( 1985 ) identified small lobes close to r mon that support the bipolar model proposed by cant et al . ( 1981 ) . among recent investigations of r mon and ngc2261 is the adaptive optics @xmath139band imaging polarimetry survey of close et al . ( 1997 ) who find that r mon is a close binary ( 0.69 separation ) . the companion is believed to be a 1.5 m@xmath3 star that dereddens to the classical t tauri star locus . r mon itself appears to be an unresolved point source , but exhibits a complex of twisted filaments that extend from 1000 - 100,000 au from the star and possibly trace the magnetic field in the region . weigelt et al . ( 2002 ) use near infrared speckle interferometry to examine structure within the immediate vicinity of r mon at 55 mas ( in @xmath140band ) scales . the primary ( r mon ) appears marginally extended in @xmath102band and significantly extended in @xmath140band . weigelt et al . ( 2002 ) also identify a bright arc - shape feature pointing away from r mon in the northwesterly direction , which is interpreted as the surface of a dense structure near the circumstellar disk surrounding r mon . their images confirm the presence of the twisted filaments reported by close et al . ( 1997 ) . m ( blue ) , 5.8 @xmath5 m ( cyan ) , 8 @xmath5 m ( green ) , and 24 @xmath5 m ( red ) emission . [ f18],width=458 ] ngc2264 has remained a favored target for star formation studies for more than half a century , but significant work remains unfinished . analysis of extensive _ spitzer _ irac and mips surveys of ngc2264 is nearing completion and will soon be available . the small sampling of _ spitzer _ data presented by teixeira et al . ( 2006 ) of the star forming core near irs2 provides some insight into the details of the star formation process that will be revealed . from these datasets the disk - bearing population of the cluster will be unambiguously identified , and tentative classifications of disk structure will be possible by comparing observed seds with models . a preview of the final _ spitzer _ image is shown in figure 18 , which combines irac and mips data to create a composite 5-color image . the spokes cluster is readily apparent near image center . ngc2264 is perhaps best described not as a single cluster , but rather as multiple sub - clusters in various stages of evolution spread across several parsecs . other future observations of the cluster that are needed include : high resolution sub / millimeter maps of all molecular cloud cores within the region ; deep optical and near infrared photometry for a complete substellar census of the cluster ; and a modern proper motion survey for membership determinations . also of interest will be high fidelity ( hst wfpc2/wfc3 ) photometric studies that may be capable of reducing the inferred age dispersion in the color - magnitude diagram of the cluster . because of its relative proximity , significant and well - defined stellar population , and low foreground extinction , ngc2264 will undoubtedly remain a principal target for star formation and circumstellar disk evolution studies throughout the foreseeable future . + * acknowledgments . * i wish to thank the referee , ettore flaccomio , and the editor , bo reipurth , for many helpful comments and suggestions that significantly improved this work . i am also grateful to t. hallas for permission to use figure 2 , t.a . rector and b.a . wolpa for figure 3 , and to j .- c . cuillandre and g. anselmi for figure 10 , and to carole westphal and adam block for figure 17 . sed is supported by an nsf astronomy and astrophysics postdoctoral fellowship under award ast-0502381 . adams , m. t. , strom , k. m. , & strom , s. e. 1979 , apj , 230 , l183 adams , m. t. , strom , k. m. , & strom , s. e. 1983 , apjs , 53 , 893 agol , e. , barth , aaron j. , wolf , s. , & charbonneau , d. 2004 , apj , 600 , 781 aspin , c. , mclean , i. s. , & coyne , g. v. 1985 , a&a , 149 , 158 aspin , c. & reipurth , b. 2003 , aj , 126 , 2936 allen , d. a. 1972 , , 172 , l55 baraffe , i. , chabrier , g. , allard , f. , & hauschildt , p. h. 1998 , a&a , 337 , 403 barry , d. c. , cromwell , r. h. , & schoolman , s. a. 1979 , apjs , 41 , 119 beardsley , w. r. & jacobsen , t. s. 1978 , apj , 222 , 570 bernasconi , p. a. & maeder , a. 1996 , a&a , 307 , 829 blitz , l. 1979 , ph.d . thesis columbia univ . , new york , ny breger , m. 1972 , apj , 171 , 267 brugel , e. w. , mundt , r. , & buehrke , t. 1984 , apj , 287 , 73 cant , j. , rodriguez , l. f. , barral , j. f. , & carral , p. 1981 , apj , 244 , 102 castelaz , m. w. , & grasdalen , g. 1988 , apj , 335 , 150 close , l. m. , roddier , f. , hora , j. l. , graves , j. e. , northcott , m. , et al . 1997 , apj , 489 , 210 cohen , m. & kuhi , l. v. 1979 , apjs , 41 , 743 crutcher , r. m. , hartkopf , w. i. , & giguere , p. t. 1978 , apj , 226 , 839 dahm , s. e. , & simon , t. 2005 , aj , 129 , 829 dahm , s. e. , simon , t. , proszkow , e. m. , & patten b. m. 2007 , aj , 134 , 999 dantona , f. , & mazzitelli , i. 1994 , apjs , 90 , 467 dantona , f. , & mazzitelli , i. 1997 , in _ cool stars in clusters and associations _ , ed . g. micela & r. pallavicini ( firenze : soc . astron . italiana ) , 807 de zeeuw , p. t. , hoogerwerf , r. , & de bruijne , j. h. j. 1999 , 117 , 354 fallscheer , c. & herbst , w. 2006 , apj , l155 feigelson , e. d. & montmerle , t. 1999 , ara&a , 37 , 363 feldbrugge , p. t. m. & van genderen , a. m. 1991 , a&as , 91 , 209 flaccomio , e. , micela , g. , sciortino , s. , favata , f. , corbally , c. , & tomaney , a. 1999 , a&a , 345 , 521 flaccomio , e. , micela , g. , sciortino , s. , damiani , f. , favata , f. , harnden , f. r. , jr . , & schachter , j. 2000 , a&a , 355 , 651 flaccomio , e. , micela , g. , & sciortino , s. 2003 , a&a , 402 , 277 flaccomio , e. , micela , g. , & sciortino , s. 2006 , a&a , 455 , 903 fukui , y. 1989 , in _ proceedings of the eso workshop on low mass star formation and pre - 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ngc2264 is a young galactic cluster and the dominant component of the mon ob1 association lying approximately 760 pc distant within the local spiral arm . the cluster is hierarchically structured , with subclusters of suspected members spread across several parsecs . associated with the cluster is an extensive molecular cloud complex spanning more than two degrees on the sky . the combined mass of the remaining molecular cloud cores upon which the cluster is superposed is estimated to be at least @xmath03.7@xmath110@xmath2 m@xmath3 . star formation is ongoing within the region as evidenced by the presence of numerous embedded clusters of protostars , molecular outflows , and herbig - haro objects . the stellar population of ngc2264 is dominated by the o7 v multiple star , s mon , and several dozen b - type zero - age main sequence stars . x - ray imaging surveys , h@xmath4 emission surveys , and photometric variability studies have identified more than 600 intermediate and low - mass members distributed throughout the molecular cloud complex , but concentrated within two densely populated areas between s mon and the cone nebula . estimates for the total stellar population of the cluster range as high as 1000 members and limited deep photometric surveys have identified @xmath0230 substellar mass candidates . the median age of ngc2264 is estimated to be @xmath03 myr by fitting various pre - main sequence isochrones to the low - mass stellar population , but an apparent age dispersion of at least @xmath05 myr can be inferred from the broadened sequence of suspected members . infrared and millimeter observations of the cluster have identified two prominent sites of star formation activity centered near ngc2264 irs1 , a deeply embedded early - type ( b2b5 ) star , and irs2 , a star forming core and associated protostellar cluster . ngc2264 and its associated molecular clouds have been extensively examined at all wavelengths , from the centimeter regime to x - rays . given its relative proximity , well - defined stellar population , and low foreground extinction , the cluster will remain a prime candidate for star formation studies throughout the foreseeable future .

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

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

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

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

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

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

polarimetry is a powerful diagnostic of specific phenomena at work in cosmic sources in the radio - wave and optical energy bands , but very few results are available at high photon energies : the only significant observation in the x - gamma energy range , to date , is the measurement of a linear polarisation fraction of @xmath0 of the 2.6kev emission of the crab nebula by a bragg polarimeter on board oso-8 @xcite . at higher energies , hard - x - ray and soft - gamma - ray telescopes that have flown to space in the past ( comptel @xcite , batse@xcite ) were not optimized for polarimetry , and their sensitivity to polarisation was poor . presently active missions ( integral ibis@xcite and spi @xcite ) have provided some improvement , with , in particular , mildly significant measurements of @xmath1 ( 130 to 440 kev @xcite ) and @xmath2 ( 200 to 800 kev @xcite ) for the crab nebula . a number of compton polarimeter / telescope projects have been developed , some of which also propose to record photon conversions to @xmath3 pairs . a variety of technologies have been considered , such as scintillator arrays ( pogo @xcite , grape @xcite , polar @xcite ) , si or ge microstrip detectors ( mega @xcite , astrogam @xcite ) or combinations of these ( si @xmath4 labr@xmath5 for grips @xcite , si @xmath4 csi(tl ) for tigre @xcite ) , semiconductor pixel detectors ( cipher @xcite ) and liquid xenon ( lxegrit @xcite ) time projection chambers ( tpc ) . in most compton telescopes the reconstruction of the direction of the incident photon provides an uncertainty area which has the shape of a thin cone arc . the tracking of the recoil electron from the first compton interaction with a measurement of the direction of the recoil momentum , as is within reach with a gas tpc , allows to decrease the length of the arc and therefore to improve dramatically the sensitivity of the detector ( @xcite and references therein ) . some of these telescopes are sensitive to photon energies up to tens of mev in the compton mode , but their sensitivity to polarisation above a few mev is either nonexistent or undocumented . as is well known , the sensitivity to polarisation of compton scattering is excellent at low energies ( thomson scattering ) , as the polarisation asymmetry @xmath6 , also known as the modulation factor and defined by the phase - space dependence of the differential cross section , @xmath7 } \right ] , \label{eq : def : diff : cross : section : compton}\ ] ] reaches @xmath8 at a polar angle @xmath9 of @xmath10 ( fig . 1 of ref . @xcite ) . in this expression , @xmath11 is the azimuthal angle , that is the angle between the scattering plane and the direction of polarisation of the incident photon . unfortunately , @xmath6 is decreasing with energy , and as the precision of the measurement scales as @xmath12 when the background noise is negligible and where @xmath13 is the number of signal event , the sensitivity of compton polarimetry decreases at high energies . with the goal of a quantitative assessment of this sensitivity , in this paper we compute the average polarisation asymmetry @xmath14 from the klein - nishina differential cross section on free electrons at rest @xcite . @xmath14 is defined from the differential cross section in @xmath11 , that is after the full differential cross section ( eq . ( [ eq : def : diff : cross : section : compton ] ) ) has been integrated over the other variables that describe the final state : @xmath15 } \right ] . \label{eq : def : diff : cross : section : compton : int}\ ] ] following heitler @xcite , the doubly differential cross section for linear polarised radiation reads : @xmath16 , \label{eq : diff : cross : section : compton}\ ] ] where @xmath17 , @xmath18 and @xmath19 are the energy of the incident and scattered @xmath20s , respectively ; @xmath9 is the scattering angle , that is the polar angle of the direction of the scattered @xmath20 with respect to the direction of the incident @xmath20 . the differential element @xmath21 is @xmath22 as usual . in the case of partially polarised emission with polarisation fraction @xmath23 , the differential cross section becomes : @xmath24 \sin\theta { \mbox{d}}\theta { \mbox{d}}\phi.\ ] ] the minus sign reflects the fact that photons compton scatter preferentially into the direction perpendicular to the orientation of the electric field of the incoming radiation . the energy of the scattered @xmath20 is related to @xmath9 from energy - momentum conservation : @xmath25 $ ] , @xmath26 $ ] , @xmath27 , @xmath28 and @xmath29 - \left[1/(x k_0 ) - 1/k_0 \right]^2 $ ] . we then obtain @xcite : @xmath30 \left(\cos{(2\phi ) } p + 1 \right ) \right ] { \mbox{d}}x { \mbox{d}}\phi.\ ] ] @xmath19 varies in a range such that @xmath31 , that is @xmath32 . the distributions of these kinematic variables are shown in fig . [ fig : compton : spectra ] . after an elementary integration over @xmath33 , we obtain : @xmath34 { \mbox{d}}\phi , \nonumber\end{aligned}\ ] ] that is a total cross section of @xcite : @xmath35 . \nonumber\end{aligned}\ ] ] equating the constant term and the term proportional to @xmath36 in eqs . ( [ eq : def : diff : cross : section : compton : int ] ) and ( [ eq : phi : diff : cross : section ] ) , we obtain for the average polarisation asymmetry : @xmath37 [ ht ] spectra of the azimuthal and polar angles @xmath11 and @xmath9 , of @xmath38 , of the fraction @xmath33 of the incident photon energy carried away by the scattered photon , and of the 1d and 2d weights @xmath39 and @xmath40 , for incident photon energies @xmath41 , @xmath42 , and @xmath43 , all for a fully polarized beam . , title="fig : " ] [ ht ] absolute value of the average asymmetry in compton scattering on free electrons at rest , as a function of the incident photon energy in electron rest - mass units . thick solid line : full expression ( eq . ( [ eq : compton : asymmetry : full ] ) ) . dashed line : high - energy approximation ( eq . ( [ eq : compton : asymmetry : high ] ) ) . thin solid line : high - energy asymptote . ] we now examine two limiting cases : * at low energies , @xmath44 , eq . ( [ eq : phi : diff : cross : section ] ) reduces to : @xmath45 \frac{k_0}{3 } { \mbox{d}}\phi,\ ] ] which results in a total cross section of @xmath46 , i.e. , the thomson cross section . the low - energy average asymmetry is @xmath47 . * at high energies , @xmath48 { \mbox{d}}\phi,\ ] ] which results in a total cross section of @xmath49 . the high - energy average asymmetry is @xmath50 the average asymmetry decreases at high energies , asymptotically approaching @xmath51 . the variation of the average polarisation asymmetry of photon compton scattering on free electrons at rest ( eq . ( [ eq : compton : asymmetry : full ] ) ) is compared to its high - energy approximation ( eq . ( [ eq : compton : asymmetry : high ] ) ) in fig . [ fig : compton : asymmetry ] . the absence of sensitivity of compton polarimeters at high energies @xcite is due to this strong decrease of @xmath52 . the value of the polarisation fraction @xmath23 is classically obtained by a fit to the @xmath11 distribution . a way to improve the polarisation sensitivity is to make an optimal use of the information contained in the multi - dimensional probability density function ( pdf ) through the use of an optimal variable ( @xcite and references therein ) , that is , of a weight @xmath53 such that the @xmath23 dependence of the expectation value @xmath54 of @xmath39 allows a measurement of @xmath23 , and that the variance of such a measurement is minimal . the solution , up to a multiplicative factor , is ( eg . @xcite ) : @xmath55 in the present case of a polarisation measurement : @xmath56 with @xmath57 and @xmath58 , we obtain : @xmath59 that is , if @xmath60 is small compared to @xmath61 , @xmath62 the @xmath63 moment of @xmath40 is @xmath64 { \mbox{d}}\omega = p \int { \displaystyle\frac{g^2(\omega)}{f(\omega ) } } { \mbox{d}}\omega $ ] , which is proportional to @xmath23 . the expressions for @xmath65 and @xmath66 are obtained from the measured values of @xmath11 and @xmath9 ( and therefore of @xmath33 ) by equating the constant term and the term proportional to @xmath36 in eqs . ( [ eq : p(omega)f+pg ] ) and ( [ eq : def : diff : cross : section : compton ] ) . the spectrum of @xmath40 is shown in fig . [ fig : compton : spectra ] for a fully polarised beam . we can see that @xmath67 is most often much smaller than unity ( beware the vertical log scale ) , so that our neglecting @xmath60 in the expression of @xmath68 was legitimate . the asymmetry , the non - evenness of the @xmath40 distribution makes the non - zero average due to the beam polarisation explicit . moment s methods are equivalent to a likelihood analysis in the case where the pdf is a linear function of the variables that one aims to measure , as is the case here , but they are much simpler to instantiate as one just has to compute @xmath69 , and average it over the whole statistics . although the analysis of experimental data is beyond the scope of this paper , the following considerations apply : * background subtraction reduces to a simple subtraction in computing the average of @xmath40 . their @xmath70-dimensional parametrization is not needed . * likelihood methods need the use of a @xmath70-dimensional parametrization of the acceptance , or efficiency , for correction . this is pretty simple in the case of compton scattering for which the final state is described by only two variables , but for higher - dimensional systems , producing enough monte carlo ( mc ) statistics and determining a parametrization becomes a nightmare : in that case the use of a moments - based efficiency correction becomes mandatory ( for a real - case presentation see eg . , section iv.a , eqs . ( 18)-(24 ) and vi.b eqs . ( 47)-(49 ) of @xcite ) . in the `` reduced '' 1d case of eq . ( [ eq : def : diff : cross : section : compton : int ] ) , @xmath71 becomes @xmath72 and the estimator for @xmath73 is @xmath74 @xcite . the uncertainty then reads : @xmath75 that is , in the case of thomson scattering ( @xmath76 ) , @xmath77 . needless to say , in the case where the direction of the polarisation of the emission of a particular cosmic source `` on the sky '' is unknown , a combined use of @xmath78 and of @xmath79 should be used . [ ht ] ratio @xmath80 of the figures of merit of the 2d to 1d estimators of the linear polarisation fraction , as a function of the incident photon energy in electron rest - mass units . ] the performance of the 2d estimator @xmath81 is compared to that of the 1d @xmath74 by the comparison of the ratios of the rms width normalized to the mean value : @xmath82 in contrast with polarimetry performed with @xmath83 telescopes , for which an improvement in the precision of the measurement of the linear polarisation fraction by a factor of larger than two is at hand ( fig . 21 right of @xcite ) , in the case of compton polarimeters the improvement is found to be marginal , varying from @xmath84 at low energy to @xmath85 at high energy ( fig . [ fig : figure : of : merit : ratio ] ) . these results are in qualitative agreement with those obtained at 100 kev by a likelihood analysis of the doubly differential cross section @xcite . in summary , we have obtained the expression for the average polarisation asymmetry , or modulation factor , of compton scattering on free electrons at rest , eq . ( [ eq : compton : asymmetry : full ] ) , fig . [ fig : compton : asymmetry ] . we have then obtained a simple optimal estimator of the polarisation fraction @xmath23 that makes use of all the information ( azimuthal and polar angles of the scatter ) , avoiding the technicalities of a maximum likelihood analysis but with the same performance . it a pleasure to acknowledge the support by the french national research agency ( anr-13-bs05 - 0002 ) and the scrutiny and the suggestions of referee # 1 of nuclear instruments and methods in physics research a. 99 `` a precision measurement of the x - ray polarization of the crab nebula without pulsar contamination '' m. c. weisskopf _ _ , astrophysical journal * 220 * ( 1978 ) l117 . `` characteristics of comptel as a polarimeter and its data analysis '' f. lei _ et al . _ , astronomy and astrophysics supplement , * 120 * ( 1996 ) 695 . `` evidence of polarisation in the prompt gamma - ray emission from grb 930131 and grb 960924 , '' d. r. willis _ et al . _ , astron . astrophys . * 439 * ( 2005 ) 245 , [ astro - ph/0505097 ] . `` polarization of the crab pulsar and nebula as observed by the integral / ibis telescope , '' m. forot _ et al . _ , astrophys . j. * 688 * ( 2008 ) l29 , [ arxiv:0809.1292 [ astro - ph ] ] . `` status of the integral / ibis telescope modeling and of the response matrices generation '' p. laurent _ et al . _ , a&a * 411 * ( 2003 ) l185 . `` polarimetry in the hard x - ray domain with integral spi , '' m. chauvin _ et al . _ , astrophys . j. * 769 * ( 2013 ) 137 , [ arxiv:1305.0802 [ astro-ph.im ] ] . `` beam test of a prototype detector array for the pogo astronomical hard x - ray / soft gamma - ray polarimeter , '' t. mizuno _ et al . _ , nucl . instrum . a * 540 * , 158 ( 2005 ) [ astro - ph/0411341 ] . `` calibration of the gamma - ray polarimeter experiment ( grape ) at a polarized hard x - ray beam '' , p.f . et al . _ , instrum . a * 600 * ( 2009 ) 424 . `` response of the compton polarimeter polar to polarized hard x - rays '' , s. orsi _ et al . _ , instrum . meth . a * 648 * ( 2011 ) 139 . `` polarization measurements with the mega telescope '' , a. zoglauer _ et al . _ , proceedings of the 5th integral workshop on the integral universe ( esa sp-552 ) . 16 - 20 february 2004 , munich , germany . `` astrogam '' , proposal submitted for the esa m4 mission programme january 15 2015 , http://astrogam.iaps.inaf.it/ grips - gamma - ray imaging , polarimetry and spectroscopy , j. greiner _ et al . _ , exper.astron . * 34 * ( 2012 ) 551 . `` tracking , imaging and polarimeter properties of the tigre instrument '' , t. j. oneill _ et al . _ , astronomy and astrophysics supplement , * 120 * ( 1996 ) 661 . 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we compute the average polarisation asymmetry from the klein - nishina differential cross section on free electrons at rest . as expected from the expression for the asymmetry , the average asymmetry is found to decrease like the inverse of the incident photon energy asymptotically at high energy . we then compute a simple estimator of the polarisation fraction that makes optimal use of all the kinematic information present in an event final state , by the use of `` moments '' method , and we compare its statistical power to that of a simple fit of the azimuthal distribution . in contrast to polarimetry with pair creation , for which we obtained an improvement by a factor of larger than two in a previous work , here for compton scattering the improvement is only of 1020% . hard x - ray , gamma - ray , compton scattering , polarimeter , polarisation asymmetry , optimal variable

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

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

semiconductors exhibit a multitude of nonlinear optical responses for resonant as well as non - resonant excitation.@xcite one of the most prominent nonlinear features is the generation of higher harmonics of the exciting frequency . when the frequency of the incoming field is tripled one speaks of third harmonic generation ( thg ) . such thg can be employed in spectroscopy and provides important insights into biological processes@xcite or even for palaeontology.@xcite in semiconductors , thg has , for example , been studied in coupled quantum wells,@xcite quantum cascade structures,@xcite quantum wires and dots,@xcite while it is also of interest in newly developed materials like graphene@xcite and atomically thin semiconductors.@xcite in order to understand thg one requires a description of the optical fields and the material which is excited by them and generates the nonlinear interaction . here we focus on the photointeraction of semiconductor quantum wells ( qw ) with ultrashort light pulses . to this end , we employ an auxiliary differential equation finite difference time domain ( fdtd ) approach to describe the dynamics of the light field along with the dynamics of the carriers in the qw . this approach goes beyond rotating wave approximation and slowly - varying envelope approximation , allowing to treat fundamental and third harmonic on the same footing and describe photonic structures that vary on scales much smaller than the wavelength . the combination of fdtd with density matrix models through auxiliary differential equations includes not only the effect of the field on the material but also self - consistently describes the effect of the material on the field . this feature allows , for example , to describe propagation of sit solitons in 1 and 2 dimensions@xcite and to study loss compensation and lasing dynamics in metamaterials@xcite or plasmonic stopped - light lasers.@xcite however , the few - level models employed in those studies can not describe the complicated behavior of an interacting electron gas excited in semiconductor qws.@xcite on the other hand more complex wave - vector resolved semiconductor models have been developed that also consider coulomb interaction between excited carriers within different levels of approximation@xcite or spatially resolved quantum kinetics calculations.@xcite such models have been used to investigate various non linear effects such as the two - band mollow triplet in thin gaas films,@xcite the carrier - wave rabi flopping in bulk gaas@xcite and thg from carbon nanotubes both in the perturbative and non - perturbative regime.@xcite these approaches , however , do not include the self - consistent , spatially resolved resolution of electromagnetic fields . combining a spatially dependent full time - domain ( fdtd ) approach with a description of semiconductor qws containing a wave - vector resolved , many - level density matrix description of the qw in a two - band approximation , has been pioneered in @xcite to describe the spatio - temporal dynamics of semiconductor lasers and recently to describe lasing of semiconductor nanowires.@xcite here , we extend the previous description by taking into account coulomb interaction in hartree - fock approximation , which allows us to describe the excitonic nature of the qw absorption . in this work , we are going to consider specifically the ultrashort pulse excitation of a qw embedded in a bragg mirror structure typical for a semiconductor saturable absorber mirror ( sesam ) . we obtain the carrier dynamics associated with excitation of the qw exciton and study the intensity dependence of thg in this qw . we find that the power - law exponent of the intensity dependence of the thg strongly varies with excitation frequency . for far off - resonant pulses the expected cubic behavior is found , while for pulses resonant with the exciton energy the exponent is reduced due to saturation effects . similar findings have been reported in theoretical and experimental studies on the excitation of carbon nanotubes with ultrashort laser pulses.@xcite the hamiltonian describing the semiconductor structure is given by the three parts@xcite @xmath0 with the free carrier part @xmath1 , the carrier - carrier interaction @xmath2 and the carrier - light field interaction @xmath3 . we assume a two - band structure with one conduction and one valence band , such that the free carrier hamiltonian reads @xmath4.\ ] ] @xmath5 and @xmath6 are the electron / hole creation and annihilation operators with wave - vector @xmath7 and @xmath8 are the corresponding energies . we consider a qw , where the energy is quantized in the @xmath9-direction with a fixed @xmath10 , while @xmath7 always refers to the two - dimensional inplane wave vector @xmath11 . the confinement along @xmath9 is included by applying the envelope function approximation,@xcite while the inplane bands are assumed to have parabolic dispersion . the electron and hole energies are @xmath12 with the effective masses @xmath13 and the band gap @xmath14 . the carrier - carrier interaction is given by the coulomb potential @xmath15,\end{aligned}\ ] ] with the coulomb matrix elements @xmath16 obtained by multiplying the ideal @xmath17 coulomb matrix elements by a band - dependent form - factor obtained from the envelope function approximation . we consider the plasmon - pole@xcite approximation to the screening of the coulomb potential , where the inverse screening length is kept constant at the initial value , @xmath18 , as screening typically builds up on timescales longer than those considered here.@xcite we treat the carrier - light field interaction in dipole approximation resulting in @xmath19,\ ] ] with dipole matrix element @xmath20 for the transition from valence to conduction band . the classical light field @xmath21 is assumed to be spatially constant over the region of the qw , denoted by the parametric dependence of the light field on @xmath22 . to calculate the dynamics of the system we set up the equations of motion for the occupations @xmath23 and @xmath24 and the polarization @xmath25 via the heisenberg equation of motion @xmath26 - \gamma_{p } p_{{\textbf{k } } } , \nonumber \\ \partial_{t } n^{e}_{{\textbf{k } } } & = & i \ , [ \omega_{{\textbf{k } } } p^{*}_{{\textbf{k } } } - \omega^{*}_{{\textbf{k } } } p_{{\textbf{k } } } ] , \nonumber \\ \partial_{t } n^{h}_{{\textbf{k } } } & = & i \ , [ \omega_{{\textbf{k } } } p^{*}_{{\textbf{k } } } - \omega^{*}_{{\textbf{k } } } p_{{\textbf{k}}}].\end{aligned}\ ] ] here @xmath27 is the transition frequency , @xmath28 is a phenomenological dephasing rate and @xmath29 is the rabi frequency beyond rotating wave approximation , i.e. , calculated with the time dependent electric field @xmath30 . due to the homogeneity of the problem , we only take into account the @xmath7-diagonal elements of the density matrix . the off - diagonal element are known to play a crucial role for spatially inhomogeneous problems.@xcite the equations of motion ( eq . [ eq : bloch ] ) already include coulomb interaction under hartree - fock approximation , which is justified for ultra short time scales . within this approximation the interaction leads to a renormalization of the transition energies @xmath31 with the coulomb hole self - energy @xmath32 and the bare ( unscreened ) coulomb potential @xmath33 . also the light - matter coupling becomes renormalized due to the coulomb interaction leading to the renormalized rabi frequency @xmath34 the integration of the equations of motion ( eq . [ eq : bloch ] ) is performed on a grid of @xmath35 @xmath36-points , homogeneously distributed between @xmath37 and @xmath38 , where @xmath39 is the bohr radius in the bulk material . the integration algorithm is runge - kutta of order 4 , where the rabi frequency at the midpoints is obtained by interpolation of the electric field.@xcite in the simulation we are not only interested in the light field acting on the carriers in the qw , but also on the back - action on the field itself . we model the dynamics of the electric field @xmath40 in the whole structure as well as the ingoing and outgoing field through a one dimensional fdtd simulation.@xcite in the one - dimensional case , with the field propagating along @xmath9 , maxwell equations can be reduced to @xmath41,\end{aligned}\ ] ] where @xmath42 and @xmath43 are the electric and magnetic fields and @xmath44 is the background permittivity . the dynamic material polarization @xmath45 is zero everywhere but at the position of the qw . @xmath46 can be calculated from the microscopic polarizations as @xmath47 the spatial grid used to describe the system has a step of @xmath48 . due to fdtd stability constraints this results in a time step of @xmath49 , which has been used for the simultaneous resolution of the semiconductor equations of motion and maxwell equations . the injection of field inside the simulation domain is performed through the total field scattered field ( tfsf ) technique.@xcite the open boundaries of the system are simulated through perfectly matched layers ( pml ) boundary conditions . to test our model we will start by investigating a qw in a homogeneous background . we will then study the field and semiconductor dynamics for a qw embedded in a multilayered structure . in the simulation different passive materials are defined by a constant refractive index and different structures can be modeled by defining a space dependent refractive index profile . the active medium we chose to investigate with our model is a @xmath50 qw . the parameters required for the simulation are listed in table [ tab : qw_parameters ] . the system is probed with pulses having a hyperbolic secant shape @xmath51 with the pulse energy @xmath52 . the full width half maximum ( fwhm ) of the pulse is @xmath53 . as function of energy @xmath54 of a single qw immersed in an infinitely extended background of gaas with ( black ) and without ( red ) coulomb interaction.[fig : onlyqwabsorption ] ] .parameters for the carrier dynamics in the qw , with the free electron mass @xmath55.[tab : qw_parameters ] [ cols="<,^,^ " , ] after studying the electron dynamics of an isolated qw in a homogeneous background we proceed by introducing a more realistic optical environment . for this we choose a sesam , which is a well established structure for ultra short pulse generation.@xcite the whole structure is included in our simulations as a spatially varying background permittivity @xmath56 , as shown in fig . [ fig : structure ] , and is surrounded by @xmath57 of air on each side . table [ tab : refractive_indices ] contains the refractive index and background permittivity of all materials included in the structure . in order for the structure to be effective , most of the layer thickness need to be proportional to the central wavelength of the incoming pulse , @xmath58 . we coated both ends of the structure with a sin layer of optical length @xmath59 which minimizes reflection of the pulse coming from air . this allows for a more efficient in - coupling of the light . the mirror is composed by a set of alternating gaas and alas layers , each with an optical length of @xmath59 . this basic two - layered module is repeated @xmath60 times in order to achieve a very high reflectivity around @xmath58 , as shown in fig . [ fig : qw_absorption](a ) , where the pulse is resonant with the exciton energy @xmath61 . due to the presence of the mirror , a standing wave is created inside the gaas layer between the mirror itself and the anti - reflective layer . such an interference pattern has zeros at even integer multiples of @xmath62 and maxima at odd ones . the qw is then located in such a maximum , in order to take advantage of the field enhancement provided by this interference pattern , and is thus at an optical distance @xmath59 from the start of the mirror . two further layers of arbitrary size are used to isolate the anti - reflection coatings from the mirror on one side and the qw on the other . ] we start again by analyzing the linear regime where we use a pulse with @xmath63 and a fwhm of @xmath64 . from the reflected and transmitted field we obtain the spectra of transmittance ( @xmath65 ) and reflectance ( @xmath66 ) of the structure , shown in fig . [ fig : qw_absorption](a ) . we see that the mirror used in the simulation reflects almost perfectly in a broad spectral region around the central wavelength of the pulse . we further calculate the absorption of the structure as @xmath67 , shown in fig . [ fig : qw_absorption](b ) . the absorption is mostly determined by the qw and thus we find a similar behavior as in fig . [ fig : onlyqwabsorption ] with a resonance at the exciton energy . the resonance height is increased by a factor of @xmath68 in comparison with the isolated qw , as a consequence of the almost four - fold enhancement in intensity introduced by the structure , while the in - band absorption is now decreasing the further we go from the band edge . the difference is due to the mirror which is optimized for the central wavelength of the pulse and whose reflectance decreases with the distance from the exciton . [ fig : thg_spectra ] ] next , we shine a set of subsequently stronger pulses on the sesam structure to investigate the non - linear regime . the pulses are resonant with the exciton ( @xmath69 ) and have a fwhm of @xmath70 . figure [ fig : thg_spectra ] shows the spectra of the reflected intensity for the same peak intensities used in fig . [ fig : onlyqwspectra ] , namely @xmath71 . similarly to what happened for the isolated qw , increasing the intensity of the exciting pulse brings a second spectral peak above the background . this is located at three times the pulse energy and is due to thg in the semiconductor layer of the structure . by comparing the spectra with fig . [ fig : onlyqwspectra ] we see that the third harmonic is more intense in the sesam structure than it is for an isolated qw excited with the same pulse . this is due to the structure enhancing the field at the qw position . the oscillations appearing in fig . [ fig : thg_spectra ] , particularly evident in the fundamental peak , are due to the fabry - prot resonance associated with the whole structure . as a function of the spectrally integrated intensity @xmath72 of the incoming pulse on a log - log scale for @xmath69 ( black squares ) and @xmath73 ( red circles ) . the dashed lines are the best fit of the data according to eq . [ eq : intensity_dependence_linear].[fig : thg_intensity ] ] a more quantitative analysis of the intensity is given in fig . [ fig : thg_intensity ] , where we plot the integrated intensity of the thg as function of the incoming pulse integrated intensity . the two sets of data correspond to different excitation energies , where one is obtained with pulses resonant with the exciton @xmath69 ( squares ) and one with off - resonant pulses with @xmath73 ( circles ) . the spectrally integrated intensity of the fundamental and its third harmonic are defined as the integral of the intensity over the corresponding spectral peak , @xmath74 where the integration is carried out over the width of the highest intensity peak . we find that the intensity of the thg has a power law behavior as function of the incoming field , @xmath75 , as @xmath76 in the log - log plot this is seen as a linear curve , where we can get the value of the exponent by performing a linear fit of the logarithm of the data according to @xmath77 which gives the dashed lines in fig . [ fig : thg_intensity ] . the values obtained for the exponents ( slope of the lines ) are @xmath78 for the off - resonant configuration and @xmath79 for the pulses resonant with the exciton energy . , obtained by fitting eq . [ eq : intensity_dependence_linear ] , as function of the pulse energy @xmath80 . the inset shows the residual density as a function of the spectrally integrated intensity for resonant and off - resonant excitations.[fig : thg_pulse ] ] figure [ fig : thg_pulse ] shows the value of the exponent @xmath81 as a function of the central wavelength of the pulse . similarly to fig . [ fig : thg_intensity ] we have performed a linear fit of eq . [ eq : intensity_dependence_linear ] to different sets of data . we see that when the excitation is enough off - resonant , i.e. , the pulse energy lies in the band gap of the semiconductor , the value of @xmath81 approaches an asymptotic value of @xmath82 which is consistent with a phenomenological description in terms of the non - linear susceptibility of third order , @xmath83.@xcite conversely , as the excitation gets closer to be resonant we observe a decrease in @xmath81 down to a value of about @xmath84 . a similar non cubic dependence has been observed by @xcite while performing four wave mixing experiments on znse qws with the central laser energy close to resonance with the exciton . we want to stress that we are able to uncover this unusual behavior only due to a self - consistent combination of the bloch equations with a description of the light field using a full time - domain ( fdtd ) code with spatial resolution on sub - wavelength scales . to further investigate the origin of this subcubic dependence , we analyzed the density of carriers generated by pulses with different detuning from the excitonic resonance . the inset of fig . [ fig : thg_pulse ] shows the density remaining in the semiconductor after the pulse has left the simulation domain as a function of the spectrally integrated intensity of the exciting pulse . we see that the amount of population generated in the qw is significantly higher under resonant excitation , even for the lowest intensity generating a thg signal , than the population for the off - resonant excitation via more intense pulses . also the amount of population excited in the qw rises linearly with the pulse intensity for off - resonant excitation , while it shows a marked saturation behavior under resonant excitation . in order to test whether the power law changes for smaller intensities our numerical simulation allows us to repeat the same analysis analyzing the transmitted field which has a lower level of background noise . we observe a crossover from @xmath85 for low intensities , to @xmath86 at higher intensities . we find that the minimum intensity for which an exponent different from 3 is obtained is @xmath87 . we have also checked that the crossover position is independent of the dephasing time . because of this and the correlation with the density of carriers in the qw , we attribute the change of the power law exponent @xmath81 to the presence of optically excited carriers and to the saturation of the total density in the semiconductor . in summary , we have studied the emergence of third harmonic signals in semiconductor quantum wells ( qw ) , photo - excited by intense femtosecond optical pusles . for this , we have introduced a general model combining a full time and space dependent finite - difference time - domain ( fdtd ) description of the light field , i.e. , a discretization of maxwell s equations without the inherent limitations of the slowly - varying envelope approximation , with a wave - vector resolved many level and many - body density matrix approach for the charge carrier dynamics . for a qw embedded in a homogeneous background we studied the interplay of light field dynamics and carrier dynamics , demonstrating the emergence of non - linear optical effects such as third harmonic generation ( thg ) . we further analyzed the intensity dependence of the generated non - linear response for a qw embedded in a many - layer semiconductor saturable absorber mirror ( sesam ) structure and show that the intensity dependence of the thg signal strongly varies with excitation frequency . for an excitation well below the band gap of the qw , we found that the intensity of the thg signal follows a cubic dependence on the intensity of the exciting pulse . this is in direct agreement with a description based on an expansion in powers of the field with non - linear susceptibilities as constant coefficients . for a resonant excitation at the excitonic frequency , however , the intensity dependence still follows a power - law , now with an exponent that is reduced to @xmath84 , clearly deviating from the cubic behavior . although a non - cubic dependence can also be obtained with a more phenomenological approach of an intensity dependent @xmath83 coefficient,@xcite this can only be fit to existing data rather than emerge from a more fundamental model . the simultaneous description of the light field and carrier dynamics not only allows for a deeper understanding of non - linear optical effects but is also readily expandable to other 2-dimensional semiconductor systems such as graphene , transition metal dichalcogenides@xcite or more complex structures like combined plasmonic - semiconductor structures.@xcite der gratefully acknowledges support from the german academic exchange service ( daad ) within the p.r.i.m.e . this study was partially support by the air force office of scientific research ( afosr ) , and the european office of aerospace research and development ( eoard ) is also acknowledged . 45ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/revmodphys.70.145 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1038/ncomms5972 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , vol . ( , ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1063/1.4821158 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.92.217403 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.87.057401 [ * * , ( ) ] link:\doibase 10.1016/j.carbon.2006.02.035 [ * * , ( ) ] link:\doibase 10.1063/1.1521508 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop _ _ , nanoscience and technology ( , , ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop _ _ ( , ) @noop _ _ ( , ) @noop * * , ( ) http://dx.doi.org/10.1103/physrevb.90.245423 [ * * , ( ) ] http://dx.doi.org/10.1103/physrevb.92.235307 [ * * , ( ) ] link:\doibase 10.1126/science.1258479 [ * * , ( ) ]

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

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

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

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

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

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

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

complex networks have become a natural abstraction of the interactions between elements in complex systems @xcite . when the type of interaction is essentially identical between any two elements , the theory of complex networks provides with a wide set of tools and diagnostics that turn out to be very useful to gain insight in the system under study . however , there are particular cases where this classical approach may lead to misleading results , e.g. when the entities under study are related with each other using different types of relations in what is being called multilayer interconnected networks @xcite . representative examples are multimodal transportation networks @xcite where two geographic places may be connected by different transport modes , or social networks @xcite where users are connected using several platforms or different categorical layers . here , we focus our study on the transportation congestion problem in multiplex networks , where each node is univocally represented in each layer and so the interconnectivity pattern among layers becomes a one - to - one connection ( i.e. , each node in one layer is connected to the same node in the rest of the layers , thus allowing travelling elements to switch layer at all nodes ) . this representation is an excellent proxy of the structure of multimodal transportation systems in geographic areas @xcite . the particular topology of each layer is conveniently represented as a spatial network where nodes correspond to a certain coarse grain of the common geography at all layers @xcite . transportation dynamics on networks can be , in general , interpreted as the flow of elements from an origin node to a destination node . when the network is facing a number of simultaneous transportation processes , we find that many elements travel through the same node or link . this , in combination with the possible physical constraints of the nodes and links , can lead to network congestion , in which the amount of elements in transit on the network grows proportional to time @xcite . usually , to analyze the phenomenon , a discrete abstraction of the transportation dynamics in networks is used @xcite . multimodal transportation can also be mathematically abstracted as transportation dynamics on top of a multiplex structure . note that routings on the multilayer transportation system are substantially different with respect to routings on single layer transportation networks . in the multilayer case , each location of the system ( e.g. geographical location ) has different replicas that represent each entry point to the system using the different transportation media . thus , each element with the intention of traveling between locations @xmath0 and @xmath1 have the option to choose between the most appropriate media to start and end its traversal . we assume that elements traverse the network using shortest paths , so each element chooses the starting / ending media that minimizes the distance between the starting / ending locations . as we will show in this work , this `` selfish '' behavior provokes an unbalance in the load of the transportation layers inducing congestion , similarly to what is presented in the classical counterintuitive result of the braess paradox @xcite . note that in a multiplex network we can have two types of shortest paths : paths that only use a single layer ( intra - layer paths ) and paths that use more than one layer ( inter - layer paths ) . hereafter , we develop the analysis of transportation in multiplex networks , consisting of @xmath2 locations ( nodes per layer ) and @xmath3 layers , and quantify when this structure will induce congestion . to this aim , we describe , with a set of discrete time balance equations , ( one for each node at each layer ) , the increment of elements , @xmath4 , in the queue of each node @xmath0 on layer @xmath5 : @xmath6 where @xmath7 is the average number of elements injected at node @xmath0 in layer @xmath5 ( also called the injection rate , which can be assimilated to an external particle reservoir ) , @xmath8 is the average number of elements that arrive to node @xmath0 in layer @xmath5 from the adjacent links of that node ( ingoing rate ) , and @xmath9 $ ] corresponds to the average number of elements that finish their traversal in node @xmath0 in layer @xmath5 or that they are forwarded to other neighboring nodes . the control parameter is @xmath7 : small values of it correspond to low density of elements in the network and high values to high density of elements . a graphical explanation of the variables of the model is shown in fig . [ fig : modelexplanation ] . and @xmath10 of the node . ] before reaching congestion , the amount of elements in the queue of each node is constant in average , @xmath11 and consequently , @xmath12 , where @xmath13 is the maximum processing rate of the node . a node @xmath0 on layer @xmath5 becomes congested when it is requested to process more elements than its maximum processing rate , @xmath14 , and therefore , its onset of congestion is achieved when @xmath15 . we are interested on computing the maximum injection rate @xmath7 for which the network is congestion free . in the non - congested phase , as well as on the onset of congestion , the amount of ingoing elements to each node @xmath8 can be obtained in terms of the node s effective betweenness , see @xcite . our scenario is slightly different since we need to account for the effective betweenness of the multiplex . in addition to the intra - layer and inter - layer paths , our definition of the dynamics also accounts for the number of shortest paths that start ( @xmath16 ) and end ( @xmath17 ) at node @xmath0 on layer @xmath5 ( this can be computed using any classical shortest path algorithm ) . note that @xmath18 . these factors are essential to understand the unbalance of loads between layers in the multiplex network , and only depend on the distribution of shortest path in the full structure . in the following , we assume a constant injection rate , @xmath19 , being @xmath20 the common injection rate at all locations @xmath0 . in addition we also suppose , without loss of generality , that the maximum processing rate is the same for all nodes of the multiplex network , @xmath21 . these hypothesis simplify the analysis but are not crucial to develop it . to obtain the critical injection rate of the multiplex , we require expressions for @xmath7 and @xmath8 . the injection rate of node @xmath0 on layer @xmath5 can be obtained as the product of the amount of elements that enter the network using location @xmath0 , @xmath22 , and the fraction of multiplex shortest paths that start on node @xmath0 on layer @xmath5 , @xmath23 : @xmath24 the ingoing rate of each node , @xmath8 , depends on the fraction of shortest paths that pass through or end in it @xcite . thus , @xmath8 can be obtained as the number of generated elements overall the network at each time step , @xmath25 , times the fraction of them that arrive ( @xmath17 ) or traverse it ( @xmath26 , the topological betweenness ) : @xmath27 when the network is already congested , eq . ( [ multiplexsigmabeforethreshold ] ) does not generally holds since elements traversing congested paths stack in intermediate nodes resulting in a cascade effect not captured by the betweenness . therefore , our analysis only covers the onset of congestion and it can not be directly applied to the congested regime . an efficient algorithm to compute the betweenness on multiplex structures can be found in @xcite for shortest paths dynamics and in @xcite for random walk dynamics . the computation of @xmath16 and @xmath17 for shortest paths dynamics can be obtained modifying the previously cited algorithm to account for the amount of paths that reach the source and destination nodes . the onset of congestion of the multiplex is attained when a node @xmath0 in layer @xmath5 is required to process elements at its maximum processing rate , i.e. @xmath28 . therefore , the critical injection rate of the system , @xmath29 , becomes @xmath30 where @xmath31 and @xmath32 . in the following we call @xmath33 the interconnected betweenness . note that @xmath33 depends on intra - layer paths , inter - layer paths , and on the migration of shortest paths between layers ( more efficient layers contain a larger proportion of the starting and ending routes ) . we test the validity of eq . ( [ eq : criticalgeneratiorate ] ) against monte carlo simulations on top of erds - rnyi multiplex networks , see fig . [ fig : corr_criticgenrate ] . given by eq . ( [ eq : criticalgeneratiorate ] ) predicting the actual onset of congestion in experimental simulations on 500 random multiplex networks formed by two erds - rnyi networks ( of 500 nodes ) as layers . inset ( * a * ) shows the correlation between the experimentally obtained critical injection rate and the analytical approximation in eq . ( [ eq : approximatio_rho_c ] ) where @xmath34 is approximated by @xmath35 . @xmath36 is the coefficient of determination for linear fits.,scaledwidth=45.0% ] in the following , we investigate the role of the topology of the individual layers on the multiplex congestion . first of all , note that in the definition and computation of the multiplex betweenness ( see @xcite ) , the shortest paths ( possibly degenerated ) between all pair of multiplex locations , @xmath37 , are considered . the multiplex structure unbalances , in a highly non - linear way , the distribution of shortest paths among the layers . however , some approximations are possible to grasp the effect of the different contributions to the onset of congestion in multiplex structures . as stated before , an important parameter of traffic dynamics in multiplex networks is the fraction of inter - layer shortest paths , i.e. the fraction of shortest paths that contain , at least , one inter - layer edge . experiments with multiplex networks composed of two layers , each one being a different random erds - rnyi network , show that most of the shortest paths are fully contained within a layer , see fig . [ fig : percentpathusemultiplex ] . this effect becomes more evident as the degree of the layers increases . therefore , the fraction of shortest paths fully contained within layers , @xmath34 , is basically 1 , and the main factor influencing the traffic dynamics is the migration of shortest paths from the less efficient layer ( the one with larger shortest paths ) to the most efficient one . under this situation we can approximate the interconnected betweenness of node @xmath0 in layer @xmath5 , @xmath38 , in terms of the betweenness of node @xmath0 of layer @xmath5 , @xmath39 , when layer @xmath5 is considered as a single layer network : @xmath40 where @xmath41 is the fraction of shortest paths using only layer @xmath5 , satisfying @xmath42 . the effect of the product of @xmath43 is to precisely account for the fraction of all shortest paths that traverse only layer @xmath5 in the multiplex . note that the approximation in eq . ( [ eq : approx_betweennessmultiplex ] ) does not account for the betweenness contribution of the paths that use inter - layer edges . however , the high value @xmath44 , indicates that they are usually negligible , and we can even further approximate @xmath45 . of paths fully contained within layers . each multiplex network is formed by two erds - rnyi layers of 500 nodes each . we plot 100 random realizations for each pair of mean degrees @xmath46 and @xmath47.,scaledwidth=45.0% ] taking advantage of eq . ( [ eq : approx_betweennessmultiplex ] ) , the critical injection rate of the multiplex can be obtained by rescaling the critical injection rate of the individual layers : @xmath48 where @xmath49 is the critical injection rate of the most efficient layer @xmath50 . fractions @xmath51 and @xmath34 are genuine properties of the multiplex network structure that can be obtained by means of the multiplex extension of the brandes betweenness algorithm @xcite . figure [ fig : corr_criticgenrate ] * ( a ) * shows the accuracy of this approximation in the calculation of @xmath52 . the high accuracy obtained in the approximation evidences that the critical injection rate of the multiplex crucially depends on the migration of shortest paths between layers , which is captured in @xmath53 . as an example , consider a multiplex structure composed by two identical layers . in this case , there are no shortest paths using inter - layer edges since they would be longer than the ones fully included in one layer , thus @xmath54 . since paths in both layers are identical , there is a multiplex path degeneration : for each shortest path in layer 1 there is an equivalent shortest path in layer 2 . as a consequence , nodes on the paths only obtain @xmath55 of the betweenness contribution they would obtain if layers were separated , which results in @xmath56 . eventually , we see that for identical layers the multiplex betweenness is @xmath55 of the betweenness computed on any of the layers . [ cols= " < , < " , ] on the other side , consider a multiplex network in which most of the paths in layer @xmath35 have length @xmath57 and most of the paths in layer @xmath57 have length @xmath58 . again , there are very few shortest paths using inter - layer edges since their minimum length is @xmath58 ( i.e. one intra - layer edge , followed by a change of layer through an inter - layer link , and finally another intra - layer edge ) , therefore @xmath59 . moreover , most of the shortest paths make use of layer @xmath35 , where the lengths are shorter , so @xmath60 and @xmath61 . substitution in eq . [ eq : approx_betweennessmultiplex ] shows that the interconnected betweenness of the multiplex is equivalent to the betweenness of the most efficient layer , that in this case is layer @xmath35 . we can compute the congestion induced by a multiplex as the situation in which a multiplex network reaches congestion with less load than the worst of its layers when operating individually . in a multiplex with two layers @xmath35 and @xmath57 ( being @xmath57 the most efficient ) , this limiting situation is obtained when @xmath62 , and consequently : @xmath63 figure [ fig : probabilitymorelessresilient]*(a ) * shows the regions where the multiplex structure induces congestion for sets of erds - rnyi multiplex networks . in each experiment , two erds - rnyi networks with different mean degree are coupled to form a multiplex network . for each pair of mean degrees we have evaluated 100 random realizations of the multiplex network and for each realization we have computed the onset of congestion of the multiplex network and of the individual layers . we have then obtained the fraction of times that the multiplex network reaches congestion before both layers . the boundaries approximated by equation [ eq : congestioninducedbymultiplex_lambdamu ] determine accurately the regions where the multiplex induces congestion . as expected , the approximation using only @xmath64 works well except when both mean degrees are low since on these cases the amount of shortest paths using the multiplex structure is more relevant . surprisingly , for larger degrees ( in the diagonal ) the er networks generated present small fluctuations on the average degree that eventually make a node in one layer to have a maximum degree a little bit larger than in the other layer . this asymmetry , for such dense networks , is enough to provoke a load unbalance that is reflected in the simulations . we have used homogenous random networks multiplexes to demonstrate the use of the analytical approach , however the theory is general for any other multiplex network structure . to conclude this letter , we have also used a different type of topology , random geometric graphs , more akin to represent transportation networks @xcite . to this end , we propose a simple configuration of a random geometric multiplex . we assume each random geometric multiplex is composed of two types of transportation media : short range ( e.g. the bus network ) and long range ( e.g. the subway ) , see fig . [ fig : geometricmultiplex ] . our construction method allows to generate very extreme geometric multiplexes ; from configurations where the long range layer only contains some of the longer edges of the short range layer ( @xmath65 ) , to a long range layer that only contains edges larger than the ones in the short range layer ( @xmath66 ) . however , we usually obtain configurations where the long range layer have some degree of edge overlap with the short range layer . the test set where we have performed the experiments has been constructed by creating @xmath67 random geometric multiplex networks choosing uniformly at random the parameters of the model . figure [ fig : probabilitymorelessresilient]*(b ) * shows that eq . [ eq : congestioninducedbymultiplex_lambdamu ] accurately predicts the region where the multiplex structure induces congestion . in summary , we have analyzed the congestion phenomena on multiplex transportation networks . we have developed an standardized model of how elements traverse those networks and we have provided analytical expression for the onset of congestion . then , we have shown that the multiplex structure induces congestion and derived analytical expressions to determine the network parameters that raise this phenomena . all analytical expressions have been assessed on erds - rnyi and geometric multiplex networks , and showing a perfect agreement with the empirical results . the reason behind this phenomenology is the unbalance of shortest paths between layers . the flow follows the shortest path , increasing the load of the most efficient ( in terms of shortest paths ) layer , and eventually congesting it . theory and experiments developed in this paper are specially useful to understand transportation dynamics on multilayer networks and might help on the development of more efficient transportation networks and routing algorithms . ^2 $ ] ; these are our node locations . we then generate the first layer by adding edges between all locations @xmath0 and @xmath1 separated by a distance @xmath68 lower than a certain radius @xmath69 $ ] . the second layer is generated by adding edges between all node pairs with distance @xmath70 . the values of @xmath71 $ ] force minimum overlapping between both layers . the value of @xmath72 with @xmath73 $ ] ensures the range @xmath74 $ ] does not exceed the radius of first layer.,scaledwidth=35.0% ] this work has been supported by ministerio de economa y competitividad ( grant fis2012 - 38266 ) and european comission fet - proactive projects plexmath ( grant 317614 ) . a.a . also acknowledges partial financial support from the icrea academia and the james s. mcdonnell foundation . 26ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ ( , , ) @noop * * , ( ) @noop ( ) @noop * * , ( ) , @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) in link:\doibase 10.1109/asonam.2011.114 [ _ _ ] ( , ) pp . @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * ( ) @noop * * , ( ) http://dblp.uni-trier.de/db/journals/transci/transci39.html#braessnw05 [ * * , ( ) ] in link:\doibase 10.1145/2615569.2615687 [ _ _ ] ( ) pp . @noop ( ) @noop * * ( )

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

experiments at lep , slc and tevatron have provided a large number of high - precision data , which , being supplemented by detailed studies of higher - order corrections , allow to probe the standard model at the loop level and subsequently to predict the mass of the higgs boson . in this context , the leptonic effective weak - mixing angle , @xmath3 , plays the most crucial role . it can be defined through the effective vector and axial - vector couplings , @xmath4 and @xmath5 , of the @xmath6 boson to leptons ( @xmath7 ) at the @xmath6-boson pole , @xmath8 the effective weak - mixing angle can be related to the on - shell weinberg angle , @xmath9 , as @xmath10 where @xmath11 and @xmath12 . at tree level , @xmath13 and @xmath14 . the form factor @xmath15 incorporates the higher - order loop corrections . usually , the @xmath16-boson mass , @xmath17 , is not treated as an input parameter but it is calculated from the fermi constant , @xmath18 , which is precisely known from the muon lifetime . the relation between @xmath17 and @xmath18 can be cast in the form @xmath19 where the quantity @xmath20 @xcite contains all higher - order corrections . the presently most accurate calculation of the @xmath16-boson mass includes full two - loop and leading higher - order corrections @xcite . on the other hand , the quantity @xmath21 in eq . ( [ eq : sin ] ) incorporates all corrections to the form factors of the @xmath22 vertex . recently , the calculation of the two - loop electroweak corrections has been completed @xcite . the uncertainty on @xmath3 due to unknown higher orders has been estimated to be 0.000047 , which is substantially smaller than the error of the current experimental value @xmath23 @xcite , but still larger than the expected precision , @xmath24 , of a future high - luminosity linear collider running at the @xmath6-boson pole @xcite . the experimental value for @xmath3 is determined from six asymmetry measurements , @xmath25 , @xmath26 , @xmath27 , @xmath28 , @xmath29 , and @xmath30 . of those , the average leptonic and hadronic measurements differ by 3.2 standard deviations , which is one of the largest discrepancies within the standard model . the main impact stems from two measurements , the left - right asymmetry with a polarised electron beam at sld , @xmath31 , and the forward - backward asymmetry for bottom quarks at lep , @xmath28 . on the experimental side , the only possible source of this discrepancy are uncertainties in external input parameters , in particular parameters describing the production and decay of heavy - flavoured hadrons ; see section 5 of ref . @xcite for a discussion . however , the interpretation of the asymmetry measurements in terms of @xmath3 requires also some theoretical input . the leptonic asymmetries depend on lepton couplings only and can be translated straightforwardly into the leptonic effective weak - mixing angle , with small corrections due to @xmath32- and @xmath33-channel photon exchange . by contrast , the hadronic observables , @xmath29 , @xmath28 and @xmath30 , depend on the quark couplings , @xmath34 . these couplings are associated with a flavour - dependent hadronic effective weak - mixing angle , @xmath35 , @xmath36 the forward - backward pole asymmetry of a quark @xmath37 , @xmath38 , is related to the effective couplings , @xmath39 and @xmath40 , and the effective weak - mixing angle , @xmath35 , by vertex also has a scalar part , besides the vector and axial - vector parts . we checked explicitly that the contribution of this scalar form factor to @xmath28 is more than a factor 1000 smaller than the current experimental uncertainty and thus truly negligible . ] @xmath41 with @xmath42 at tree level , @xmath35 and @xmath3 are identical , but the relations between these quantities receive sizable radiative corrections that need to be included in the analysis . note that , due to the small electric charge of the bottom quark , @xmath43 , the parameter @xmath44 is close to 1 , and @xmath28 is only weakly sensitive to @xmath0 . therefore , it seems unlikely that the discrepancy between @xmath45 and @xmath28 could be explained by radiative corrections . nevertheless , the theoretical prediction for @xmath0 enters in the standard - model fits through several observables , so that a precise prediction of this quantity is important for a robust analysis . for all fermions except bottom quarks , the known radiative corrections to @xmath46 include at least two - loop fermionic electroweak contributions and some leading higher - order corrections ; see ref . @xcite for details . however , for the @xmath47 vertex only one - loop corrections , leading two - loop corrections for large values of the top - quark mass of @xmath48 , and two- and three - loop qcd corrections have been calculated @xcite and included in the zfitter program @xcite ( see also the new program gfitter @xcite ) , which is widely used for global standard - model fits . the remaining two - loop electroweak corrections beyond the @xmath49 contributions are still unknown , although they are expected to be larger than the @xmath48 term , based on experience from @xmath3 . as a result , the present treatment of higher - order electroweak corrections leads to inconsistencies , for example in @xmath28 , since the corrections to @xmath3 and @xmath50 include two - loop and leading three - loop corrections that are absent for @xmath0 and @xmath44 ( see recent discussion in ref . @xcite ) . in this paper , the part of the missing two - loop corrections to @xmath0 with closed fermion loops is presented . we begin by explaining the techniques employed for the calculation in the next section . in section [ results ] , numerical results for @xmath0 are given before the summary in section [ concl ] . we work in the standard model and adopt the on - shell renormalisation scheme , which relates the renormalised masses and couplings to physical observables . details on the renormalisation scheme and explicit expressions for the relevant counterterms can be found in refs . @xcite . for the loop integrations , we employ dimensional regularisation . the problem of @xmath51 matrices in two - loop vertex diagrams with fermion triangle sub - loops is treated in the same way as in refs . @xcite , by evaluating the finite non - anticommutative contribution from @xmath51 to the vertex diagrams in four dimensions . most aspects connected with the calculation of the effective weak - mixing angle for the @xmath52 vertex are the same as for the leptonic effective weak - mixing angle and are discussed in detail in ref . @xcite . the contributions for the two - loop renormalisation terms are identical to the case of @xmath3 , with the exception of the two - loop bottom - quark wave - function counterterm , which involves new self - energy diagrams with internal top - quark propagators ; the first terms of this quantity are given in ref . @xcite . for the two - loop @xmath47 vertex corrections , on the other hand , a number of new three - point diagrams need to be computed . in general , electroweak two - loop corrections can be divided into two groups , which are separately finite and gauge invariant : fermionic corrections ( with at least one closed fermion loop ) and bosonic corrections ( without any closed fermion loops ) . in this article , we focus on the fermionic diagrams as a first step . for the purpose of this calculation , all light - quark masses are neglected in the two - loop diagrams , including the bottom - quark mass . as a result , for many diagrams , known results from the @xmath3 calculation can be used @xcite . the loop integrals for diagrams with closed massless - fermion loops are given in analytical form , while large - mass expansions were employed for diagrams with top quarks in the loops . however , the two - loop corrections to @xmath0 include a new group of integrals that were not covered in previous calculations of @xmath3 , stemming from diagrams with internal @xmath16-boson and top - quark propagators ; see fig . [ diags ] . the computation of these diagrams will be discussed in detail in the following subsections . the two - loop diagrams are computed with several independent methods , so that cross checks can be performed . the first method , based on the observation that all new diagrams in fig . [ diags ] include internal top - quark propagators , uses asymptotic expansions for large top - quark mass . this method was already employed successfully for the calculation of @xmath3 @xcite . for references on the subject , we refer the reader to ref . @xcite . secondly , we develop a code for the evaluation of feynman diagrams with a semi - numerical method , based on the bernstein - tkachov ( bt ) method of ref . this method had already been used previously for one - loop problems @xcite . in a recent series of papers @xcite , it was extended to general two - loop vertices , and some applications to two - loop problems are already known : the leptonic effective weak - mixing angle was presented in ref . @xcite and corrections to the @xmath53 decay width in ref . @xcite . finally , we use another semi - numerical method based on dispersion relations @xcite , which was also used previously for @xmath3 @xcite . this method allows us to evaluate all self - energy diagrams , the vertex diagrams in figs . [ diags](a)(d ) , as well as the scalar integrals with the topology of figs . [ diags](e)(g ) . however , due to problems with the complex tensor structure , the complete diagrams in figs . [ diags](e)(g ) can not be checked with this technique . in the next subsections , we explain the applications of these methods for our purposes and present a comparison between them . we perform an expansion in a parameter @xmath54 , where @xmath55 for any two - loop problem , there are four regions to consider . let @xmath56 and @xmath57 represent the internal momenta in the loops and @xmath58 stand for any external momentum , while @xmath59 generically denotes all masses that are small compared to @xmath60 , @xmath61 . in our case , @xmath62 . then the four regions can be identified as follows : [ cols= " < , < " , ] following earlier publications on two - loop electroweak corrections , we express our results in terms of fitting formulas . the form factor @xmath63 , which contains the fermionic two - loop electroweak corrections to @xmath0 according to eq . , can be approximated as @xmath64 where @xmath65 is the one - loop result , and @xmath66 \delta_z & = \frac{m_z}{91.1876~\mathrm{gev } } -1 , & \delta_w & = \frac{m_w}{80.404~\mathrm{gev } } -1 . \end{aligned}\ ] ] fitting this formula to the exact result , we obtain @xmath67 this parametrisation reproduces the exact calculation with maximal and average deviations of @xmath68 and @xmath69 , respectively , as long as the input parameters stay within their @xmath70 ranges of the experimental errors quoted in table [ tab : input ] and the higgs - boson mass is in the range 10 gev @xmath71 1 tev . if the top - quark mass and the @xmath16-boson mass vary within 4@xmath72 ranges , the formula is still accurate to @xmath73 . we also present a simple parametrisation for the currently best prediction for @xmath0 , including all known corrections to @xmath74 and @xmath20 ( for the calculation of @xmath17 from @xmath18 see refs . @xcite ) . for @xmath74 , in addition to the one - loop and fermionic two - loop electroweak corrections , we include qcd corrections of @xmath75 @xcite and @xmath76 @xcite to the one - loop contribution , as well as universal corrections for large top - quark mass , of @xmath77 and @xmath78 @xcite . moreover , leading four - loop qcd correction to the @xmath79 parameter , which arise from top- and bottom - quark loops , are taken into account @xcite . we use the parametrisation @xmath80 with @xmath81 the best - fit numerical values for the coefficients are @xmath82 this parametrisation approximates the full result with maximal and average deviations of @xmath83 and @xmath84 , respectively , for 10 gev@xmath85 tev and the other input parameters in their @xmath70 ranges . in this paper , the calculation of the two - loop electroweak fermionic corrections to the effective weak - mixing angle for the @xmath52 vertex , @xmath0 , was presented . such an accurate theoretical prediction for @xmath0 is necessary for the interpretation of the bottom - quark asymmetry measurements at the @xmath6-boson pole . compared to the previously known corrections to @xmath0 , the new electroweak two - loop result turns out to be sizable , of order @xmath86 for a higgs - boson mass near 100 gev . the calculation was performed by using methods that had been used earlier for the computation of the leptonic effective weak - mixing angle , as well as a newly developed code based on the bt algorithm . the results of the different methods were checked against each other . although we did not perform a detailed analysis of the error from unknown high - order corrections , in particular the missing bosonic two - loop corrections and terms of order @xmath87 , we expect those to be of similar order as for the leptonic effective weak - mixing angle . the main difference between the leptonic and bottom - quark effective weak - mixing angles are the vertex diagrams with internal @xmath16-boson and top - quark propagators . while leading to numerical differences between @xmath3 and @xmath0 , these diagrams do not introduce special enhancement or suppression factors . therefore , we expect the theoretical uncertainty to our result for @xmath0 to be about @xmath88 , similar to ref . @xcite . the work of m.a . and b.a.k . was supported in part by the german research foundation ( dfg ) through grant no . kn 365/3 - 1 and through the collaborative research centre 676 _ particles , strings and the early universe the structure of matter and space time_. the work of m.c . was supported in part by the sofja kovalevskaja award of the alexander von humboldt foundation and by the tok program _ algotools _ ( mtkd - cd-2004 - 014319 ) . a.f . is grateful for warm hospitality at argonne national laboratory and the enrico fermi institute of the university of chicago , where part of his work on this project was performed . m. awramik , m. czakon , a. freitas , g. weiglein , phys . d 69 ( 2004 ) 053006 , arxiv : hep - ph/0311148 . m. awramik , m. czakon , a. freitas , g. weiglein , phys . lett . 93 ( 2004 ) 201805 , arxiv : hep - ph/0407317 ; 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we present the first calculation of the two - loop electroweak fermionic correction to the flavour - dependent effective weak - mixing angle for bottom quarks , @xmath0 . for the evaluation of the missing two - loop vertex diagrams , two methods are employed , one based on a semi - numerical bernstein - tkachov algorithm and the second on asymptotic expansions in the large top - quark mass . a third method based on dispersion relations is used for checking the basic loop integrals . we find that for small higgs - boson mass values , @xmath1 gev , the correction is sizable , of order @xmath2 . keywords : electroweak radiative corrections , effective weak - mixing angle , bernstein - tkachov algorithm pacs : 12.15.lk , 13.38.dg , 13.66.jn , 14.70.hp

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

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

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

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

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

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

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

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

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

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

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

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

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

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