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Proceed to summarize the following text: the invention of scanning tunneling microscopy ( stm ) @xcite was a milestone in experimental surface physics . moreover , it became possible to tailor and analyze small nanostructures on various conducting surfaces @xcite . perhaps the most spectacular and pioneering examples are the quantum corral experiments , in which closed atomic structures were assembled with help of atomic manipulations @xcite . as the stm is a real space technique , and is very sensitive to the local atomic and electronic structures , it allows to study the properties of various , not necessarily periodic , structures with atomic resolution . those include various surface reconstructions @xcite and low dimensional structures , like single adatoms @xcite , islands @xcite or one - dimensional monoatomic chains @xcite-@xcite . in particular , the one - dimensional structures are very interesting from a scientific point of view , as they exhibit extremely rich phenomena , very often different from those in two and three dimensions @xcite . however , in reality all the one - dimensional chains always stay in contact with their neighborhood ( substrate , external electrodes , etc . ) , thus usually preventing the observations of the exotic physics . moreover , very often they contain various imperfections , like impurities , dislocations or lacks of atoms . such a situation likely takes place when those monoatomic chains are fabricated in self - assembly processes . the typical examples are all one - dimensional structures on vicinal si surfaces @xcite . the electron transport through a chain in two terminal geometry , in which the end atoms of the chain were coupled to external electrodes has been extensively studied , both experimentally and theoretically ( see ref . @xcite for a review ) . a number of experiments has revealed many interesting phenomena , like conductance quantization in units of @xmath0 @xcite , deviations from that ( @xmath1 ( @xmath2 ) anomaly ) @xcite , spin - charge separation ( luttinger liquid ) @xcite , oscillations of the conductance as a function of the chain length @xcite or spontaneous spin polarization @xcite . the problem of impurities or disorder in one - dimensional wires has also been studied both experimentally @xcite and theoretically @xcite-@xcite . the theoretical studies revealed that even a single impurity can lead to a dramatic modifications of the low energy physics . in particular , the conductance of a wire with interacting impurity shows a power law behavior with the scaling exponent depending on the strength of the impurity @xcite . the problem of impurities ( single atoms or clusters of atoms ) , but coupled sideways to a wire , has also been extensively studied recently @xcite-@xcite . for a single impurity with a strong coulomb interaction many authors predicted a suppression of the wire conductance due to the fano interference between ballistic ( wire ) channel and the impurity channel @xcite-@xcite . to be more precise , the fano interference with the impurity reverses the gate voltage dependence of the conductance compared to the case when the impurity is embedded in a wire . the fano effect in this case can have a classical nature , if the interference comes from the impurity single particle channel , or a many - body nature , when the resonant channel is formed by the kondo effect at the impurity . the fano effect in a wire with a side coupled quantum dot has been recently confirmed experimentally @xcite . it is the purpose of the present paper to see what modifications of the transport properties of a chain introduce a side coupled impurity . here , however , we shall study transport properties in different geometry , in which all the atoms in the chain as well as the impurity are coupled to one electrode and the second electrode is attached to one particular chain atom or the impurity via additional probing atom . this geometry simply corresponds to stm tunneling through monoatomic chain with a side attached impurity , and can be related to previously mentioned stm study of self - assembled monoatomic chains on vicinal surfaces @xcite . to our knowledge , the problem in this geometry has not been studied experimentally so far , except the effects of impurities on the length distribution of atomic chains @xcite . our system is described by tight binding model with no electron correlations , as we are not interested in many body effects like kondo effect , metal - insulator transition , spin - charge separation , etc . so our model can be applied to the wires where the interaction energy is smaller than the kinetic energy associated with the hopping along the wire . for example , this could describe situation on various vicinal surfaces ( like si(335 ) , si(557 ) , etc . ) with monoatomic au chains grown on them , where the correlation effects are negligible due to small carrier concentration @xcite . moreover , the problem can be solved exactly in this case . in order to calculate the tunneling current we have adapted a non - equilibrium keldysh green s function technique . the organization of the rest of the paper is as follows . in sec . [ model ] we introduce our model , and the results of the calculations and discussion are presented in sec . [ results ] . we end up with summary and conclusions . our model system is shown schematically in fig . [ fig1 ] and is composed of a wire ( @xmath3 ) with atomic energies @xmath4 and hopping integrals @xmath5 between nearest neighbor atoms , impurity ( @xmath6 ) with single energy level @xmath7 , which is side - coupled to the wire via hopping @xmath8 . both the wire and the impurity interact with the surface ( @xmath9 ) via @xmath10 and @xmath11 , respectively . the surface is treated as a reservoir for electrons with the wave vector @xmath12 , the spin @xmath13 and single particle energies @xmath14 . above the wire there is a stm tip ( @xmath15 ) modeled by a single atom with the energy level @xmath16 attached to another reservoir ( @xmath17 ) ( with electron energies @xmath18 ) via parameter @xmath19 . tunneling of the electrons between stm tip and one of the atoms in a wire is described by the parameter @xmath20 . the whole system is described by the following model hamiltonian @xmath21 where , as usually @xmath22 ( @xmath23 ) stands for the creation ( annihilation ) electron operator in the stm lead ( @xmath24 ) , tip atom ( @xmath25 ) , wire ( @xmath26 ) , impurity ( @xmath27 ) and the surface ( @xmath28 ) . in order to calculate the tunneling current from the stm electrode to the surface we follow the standard derivations @xcite and get @xmath29 \ ; , \label{current}\end{aligned}\ ] ] where @xmath30 is the fermi distribution function , @xmath31 ( @xmath32 ) is the bias voltage , i.e. the difference between chemical potentials in the stm ( @xmath33 ) and the surface ( @xmath34 ) reservoirs , and the transmittance @xmath35 is given in the form @xmath36 with the coupling parameter latexmath:[$\gamma_{t(s)}(\omega ) = 2 \pi \sum_{{\bf k } \in s(t ) } the stm electrode ( @xmath17 ) and the tip atom ( @xmath15 ) and between the surface ( @xmath9 ) and the wire ( @xmath3 ) . note that , to get the above expression for the transmittance we have assumed the same values of the coupling of the wire and the impurity with the surface , i.e. @xmath10 = @xmath38 . @xmath39 ( @xmath40 ) is the fourier transform of the retarded green s function ( gf ) @xmath41_+ \rangle$ ] , i.e. the matrix element ( connecting the tip atom @xmath15 with @xmath42th atom in the chain or with the impurity @xmath6 ) of full gf , obtained from the solution of the equation @xmath43 the full gf @xmath44 is a @xmath45 matrix ( @xmath46 atoms in a chain , the impurity and the tip atom ) , which is obtained by inverting the matrix @xmath47 . before the presentation of the numerical results , it is worthwhile to comment on choice of the model parameters used in the present work . in numerical calculations we have assumed equal and energy independent coupling parameters ( @xmath48 ) , which reflects constant energy bands in both electrodes , and chosen @xmath49 as an energy unit . the other parameters have been chosen in order to satisfy the realistic situation in many experiments . the hopping integral within a wire is @xmath50 , @xmath51 , @xmath52 , and @xmath53 . for example , taking @xmath54 ev , we get @xmath55 ev , @xmath56 ev , and @xmath57 ev . such a value of the parameter @xmath20 together with the typical value of the work function @xmath58 ev , gives a tip - surface distance @xmath59 @xcite , and the stm current stays in the range of a few na . these are typical conditions in real stm experiments . note that , with such a value of @xmath20 , the modifications of the wire density of states due to the stm tip are negligible . in the following we will discuss the properties of wires containing even and odd number of atoms , as we expect different behavior for them , showing the results of the calculations for two representative examples , namely , for @xmath60 and @xmath61 atom wires . however the conclusions drawn from consideration of @xmath60 ( @xmath61 ) atom wire remain valid for any even ( odd ) @xmath46 atom wire , provided the system is in ballistic regime . moreover , we use the convention , in which the index @xmath42 labels the wire atoms , while @xmath62 indicates the position of the tip with respect to the wire atoms . let us first discuss the modifications of the wire density of states ( dos ) due to the side - coupled impurity . the local dos of @xmath42th atom in a wire is related to the corresponding diagonal element of the retarded gf , i.e. @xmath63 . similarly , for the impurity @xmath64 . figure [ fig2 ] shows a local dos of a wire consisted of @xmath60 ( left panels ) and @xmath61 atoms ( right panels ) in various impurity - wire configurations , indicated in the insets to the panels . the top panels show the wire local densities of states in the case when there is no impurity attached , i.e. for clean wire . in this case , for @xmath46 atom wire the local dos on @xmath42th atom is the same as on @xmath65 atom ( @xmath66 ) , i.e. there is a symmetry with respect to the middle of the wire , thus the only non - equivalent @xmath67 are plotted . for @xmath60 atom wire the @xmath67 has similar structure ( @xmath68 - thin solid line and @xmath69 - thicker solid line ) featuring small dos at the fermi energy @xmath70 and resonances corresponding to the wire atomic energies @xmath53 split by the hopping @xmath50 . on the other hand , for @xmath61 atom wire there are large dos at the @xmath71 , and the resonances are eventually split by @xmath5 . such a behavior is similar to the case of two terminal geometry in which the end wire atoms are connected to electrodes @xcite , i.e. a local minimum at the fermi energy @xmath70 for even number of atoms in a wire and local maximum at @xmath71 for odd atom wire and can be explained in terms of bonding , antibonding and nonbonding states @xcite . in case of even atom wire there are always bonding ( @xmath72 ) and antibonding ( @xmath73 ) states . when @xmath46 is odd , there is additional nonbonding state , which is situated at exactly the same position as the original atomic level ( @xmath74 ) , thus giving large dos at @xmath71 , as @xmath75 in our case . interestingly , for the second ( or second from the end ) atom in an odd atom wire ( thicker solid line in the top right panel ) , the situation is quite different , namely , the local dos has very small value at the @xmath71 . it turns out that this is a general tendency in odd atom wires , i.e. every second atom in an odd atom wire shows small dos at the fermi energy . this could be understood in the following way . let us forget for a moment about the stm tip , as we mentioned previously , that the stm tip does not influence the wire dos , and assume that the wire is coupled to the surface only . if we calculated number of non - equivalent electron paths starting and ending in the surface and passing through a given atom , it turns out that if the number of odd paths ( passing through odd number of atoms ) @xmath76 is larger than the number of even paths ( enclosing even number of atoms ) @xmath77 , then the local dos has a resonance at @xmath71 . thus it behaves as in the case of odd atom wire in two terminal geometry with the end atoms coupled to the leads . for example , for the first atom in @xmath61 atom wire ( thin solid line in the right top panel of fig . [ fig2 ] ) we get @xmath61 non - equivalent odd paths and @xmath60 even paths , thus @xmath78 and the resonance at the @xmath71 is produced . similarly , for the third atom , @xmath79 and @xmath80 , and again @xmath78 , thus we get the resonance at the fermi energy ( see thick solid line in the top left panel of fig . [ fig2 ] ) . opposite is also true , namely , if @xmath77 is larger than @xmath76 , the local dos has similar behavior as in the case of even atom wire , featuring small @xmath81 . for example , @xmath82 and @xmath80 for the second atom in @xmath61 atom wire ( thicker solid line on the same picture as previously ) . this seems to be true for any number of atoms in a wire in the present geometry . of course , this is only intuitive picture , well working in the present case , namely , when the system is in ballistic regime , so the phase coherence length is longer than the wire length . in general , in the presence of interactions ( electron - electron , electron - phonon or scattering on magnetic impurities ) the phase coherence length will be suppressed , and this simple picture can be no longer valid . in this case one has to calculate the contributions from different electron paths . the middle and the bottom panels of fig . [ fig2 ] show the local densities of states @xmath67 at different wire atoms in the case when the impurity with a single atomic level @xmath83 is attached to the first and second wire atom , respectively . the presence of the impurity introduces asymmetry in the dos with respect to the middle of the wire , and the condition @xmath66 does not hold anymore , as a result @xmath67 will be different on each wire atom . in both wires the main modification of the dos features additional resonance and thus splitting by @xmath84 of the resonance around @xmath85 ( @xmath86 ) in the case of @xmath60 ( @xmath61 ) atom wire . the low energy behavior of dos is little affected , in particular for @xmath60 atom wire . accidentally , it may a little bit shift the zero energy resonance , leaving small dos at @xmath71 , as it is seen in the middle right panel of fig . this will be also reflected in the linear conductance , as we will see later . in two terminal geometry , when the end wire atoms are connected to two different leads , the useful quantity is the total density of states , as in this case the conductance of the system is correlated with it @xcite . on the other hand , in the stm geometry we do not expect that the behavior of the conductance will be governed by this quantity . to see this , we plotted the total ( wire plus impurity ) density of states @xmath87 of a chain consisted of @xmath60 atoms ( left panel ) and @xmath61 atoms ( right panel of fig . [ fig3 ] ) . the total dos is defined as a @xmath88 for the wire with no impurity attached , and as a @xmath89 in the case when the impurity is present . we have distinguished two cases , namely , when the impurity has the same value of the atomic energy @xmath90 as for the wire atoms , and when it has different energy @xmath83 . the former case would correspond to the situation when the wire atoms and the impurity are composed of the same material , while in the later one , the impurity is a different atom . first of all , if there is no impurity ( dashed lines ) the total dos shows similar behavior as in the case of the end wire atoms coupled to the leads only . however one can notice different heights of particular resonances . this reflects an effect , known from the studies of two impurities on a surface , and associated with different ( even - odd ) symmetries of electron states in resulting system @xcite . when the impurity is introduced to the system , the total dos is strongly modified due to the parameter @xmath8 , which is responsible for the hoping between those subsystems . in this case @xmath8 is only two times smaller than the wire hoping @xmath5 , thus one should expect modifications of the wire dos . the impurity usually introduces additional resonance to the total dos . in both cases the position of the resonance is around its atomic energy ( compare thin and thick solid lines in in the left panels of fig . [ fig3 ] ) , and slightly modifies the resonances coming from the wire atoms . this behavior only slightly depends on which wire atom is in close connection with the impurity . let us now turn to the transmittance @xmath35 , defined by eq . ( [ transmit ] ) . figure [ fig4 ] shows the transmittance of the system in various stm tip - wire - impurity configurations for @xmath60 atom ( left panels ) and @xmath61 atom wire ( right panels ) . while the transmittance depends on the stm tip position now , it is possible to identify the impurity induced contribution to @xmath35 . the impurity introduces additional resonance to @xmath35 around its atomic energy , i.e. at @xmath91 ( compare the left panels of fig . [ fig4 ] ) . in both cases , i.e. for @xmath60 and @xmath61 atom wires , the transmittance is correlated with the local density of states , and the presence of the impurity leads to similar modifications , especially for @xmath92 ( compare fig . [ fig2 ] ) , although the transmittance is now more asymmetrical . this is in contradiction with the transport along the wire , as in that case the transmittance depends on the total dos . here it depends on the local dos . explanation is simple and intuitive . in the transport along the wire , all the local dos equally contribute to the transmittance @xcite . in the present case , the transport takes place mainly through the wire atom , just below the stm tip , thus @xmath35 depends mainly on the local dos of a particular atom . the presence of the impurity usually leads to higher values of @xmath35 for the resonances of the wire origin . such a behavior is in contradiction with the behavior of @xmath35 when the transport takes place along the wire . in the former geometry this leads to the reduction of the conductance @xcite-@xcite . here however , the situation is different , as the impurity is also connected to the same lead as the wire is , thus the impurity provides additional tunneling channel . nevertheless , the zero energy transmittance is influenced by the impurity , similarly as the local dos ( compare fig . [ fig2 ] ) . at this point we would like to comment on the fano effect , as there are many channels for electron tunneling from stm tip to the surface . even the tip is coupled to a single wire atom , the electron can leave the wire entering the surface electrode through different wire atoms . this is particularly well visible for stm tip placed above second atom in odd atom wire even without impurity attached ( thicker solid line in top right panel of fig . [ fig4 ] ) . in this case odd number atoms in the wire have large density of states at the fermi energy , while even atoms have small dos at @xmath71 ( see fig . [ fig2 ] ) . when the stm tip is placed above first ( thin solid line ) or third ( thickest solid line ) an electron can tunnel through undelaying atom directly to the surface because neighboring atoms have small dos at @xmath71 . the fano effect is negligible in this case . on the countrary , when the tip is placed above second atom , which has small dos at @xmath71 ( thicker solid line ) , the electron has three tunneling channels to the surface , due to large values of dos at neighboring atoms . it can directly tunnel to the surface or go through first or third wire atom . in this case the fano effect is enhanced and is visible as a dip at the fermi energy . the presense of impurity slightly modifies the above picture , usually enhancing the fano effect when the tip is above the wire atom with attached impurity . compare thin solid line in midle right panel of fig . [ fig4 ] and thicker solid line in the bottom right panel . we do not observe fano effect for even atom wire because dos at @xmath71 is always small ( see fig . [ fig2 ] ) . similar effects are reflected in differential conductance @xmath93 , shown in fig . [ fig5 ] . again , the presence of the impurity leads to a increase of @xmath94 around @xmath95 ( note that @xmath96 and @xmath32 ) , thus can help us in studying of the properties of impurities attached to wire in real stm experiments . it also modifies the structures coming from the wire atoms , also leading to larger values of @xmath94 . note however , that the values of @xmath94 are much smaller ( @xmath97 for @xmath60 atom and @xmath98 for @xmath61 atom wire ) than the conductance unit , i.e. @xmath99 per tunneling channel . this is due to small value of the stm tunneling parameter @xmath20 . for @xmath100 , the conductance reaches the unitary limit , i.e. @xmath101 in our case ( two spin channels ) . on the other hand , the linear conductance , i.e. @xmath102 strictly follows the local density of states , similarly as the zero energy transmittance ( compare fig . [ fig5 ] and figs . [ fig2 ] and [ fig4 ] ) . in a special case when the impurity is coupled to the first atom in @xmath61 atom wire , the zero bias maximum is split and the linear transport is strongly reduced for any wire atom ( compare the middle right panel of fig . [ fig2 ] ) . so far we have discussed the various configurations , in which impurity was coupled to a single atom wire . this would correspond to a situation in which the impurity is placed close to one particular wire atom . now , the question arises what will happen if the impurity is side - coupled to two wire atoms , i.e. is placed aside between two wire atoms . figure [ fig6 ] shows the conductance changes due to different couplings of the impurity to one atom ( top panel ) and two atoms in a wire ( bottom panel ) . both panels show the conductance maps vs. bias voltage @xmath103 and corresponding couplings @xmath104 ( @xmath105 ) ( see the insets to the figure ) . in first configuration ( top panel ) the impurity is coupled to one wire atom , and the differential conductance has a maximum in the plane ( @xmath106 ) . the position of the maximum changes linearly with @xmath103 and @xmath104 . on the other hand , in second configuration , when the strength of the impurity coupling changes ( bottom panel ) , i.e. for various positions of the impurity between second and third wire atom , we see that the largest values of @xmath94 are obtained when the impurity is close to the second atom ( @xmath107 , @xmath108 ) and slightly smaller close to the third wire atom ( @xmath109 , @xmath110 ) . for intermediate values of @xmath104 ( @xmath105 ) the conductance is strongly reduced , in particular , when @xmath111 . this is due to the fano effect , which is further enhanced in this case , and leads to similar reduction of @xmath94 in the case of two terminal geometry with the end wire atoms coupled to the leads @xcite-@xcite . the fact , that the maximal values of the differential conductance show asymmetry with respect to @xmath111 point , i.e. they are different for the impurity near the second wire atom and near the third wire atom steams from the fact that stm tip is placed above the second wire atom . finally we would like to comment on the coulomb interactions , as the present model completely neglects them . we expect qualitative modifications of the results ( like even - odd oscillations of density of states ) , especially at low temperatures where the coulomb blockade and the kondo effect take place . the picture can change drastically @xcite . the tunneling can be enhanced when the wire energy levels are placed below the fermi energy and the wire is strongly coupled to the surface ( kondo effect ) or suppressed when it is weakly coupled , leading to the coulomb blockade . the presense of additional tunneling channel ( due to the kondo effect ) can also enhance the fano effect . on the other hand , we do not expect qualitative modifications in the mixed valence and the empty regimes , i.e. when the wire energy levels are above the fermi energy . the present model can be succesfully applied to study monoatomic ( au , ag , pb ) chains grown on various vicinal surfaces @xcite-@xcite . in conclusion we have studied stm tunneling through a quantum wire with a side - coupled impurity in various coupling configurations , i.e. the impurity was coupled to various atoms in a wire , and moreover , it was also connected to more than single wire atom . we have found that the impurity strongly modifies the transport properties . in particular it always produces a resonance around its atomic energy , which can be seen in differential conductance vs. bias voltage . moreover , we have also shown that if the impurity is side - coupled to two wire atoms with equal strength , it leads to the suppression of the conductance , which is a hallmark of the fano effect in such a system . those studies could be potentially useful in stm studying of the impurity induced modifications of the wire properties . this work has been supported by the grant no n n202 1468 33 of the polish ministry of education and science . in this appendix we give analytical solutions / relations for the local density of states ( ldos ) of a wire disturbed by an adatom . ldos is connected with the retarded green s function by the following relation @xmath112 where @xmath113 can be obtained from the equation of motion for green functions . using eq . 2 one can write @xmath114 where @xmath115 is the algebraic complement of the matrix @xmath116 ( cofactor ) . in our calculations we assume the same coupling strengths between atoms , @xmath5 , and the same single particle energies @xmath4 for all atoms in the wire . it is worth noting that the stm is weakly coupled with the wire and thus it does not affect the wire density of states ( in hamiltonian eq . ( [ hamilt ] ) we put @xmath117 ) . first we consider the case of a linear wire without an adatom i.e. @xmath118 . in this case ldos can be obtained from the relation @xmath119 where @xmath120 and @xmath121 , @xmath122 . to obtain ldos one needs to know the determinant of @xmath123 which can be expressed as follows @xmath124 the matrix @xmath125 corresponds to an isolated wire ( non - coupled with the surface ) and is a tridiagonal one , @xmath126 , and can be expressed analytically in terms of chebyshev polynomials of the second kind @xcite . the ldos for a wire coupled with an adatom can be obtained from the relation @xmath127 where @xmath128 and @xmath129 is a vector describing the couplings adatom - qw and adatom - surface , @xmath130 ( @xmath131-th atom of a wire is connected with the adatom . after some algebra the determinant of the above matrix can be written in the form @xmath132 similar equation one can write for @xmath133 and @xmath134 which also can be expressed in terms of @xmath135 but have more complicated structure . for the case @xmath136 ( there is no surface under the wire or the coupling wire - surface is very weak ) one can easily find @xmath137 and the ldos can be obtained fully analytically . it is worth noting that the determinants @xmath138 , @xmath139 and @xmath140 are obtained for arbitrary @xmath131 ( connection adatom - qw atom ) and @xmath46 ( even or odd ) . the minima of @xmath141 determine high value of ldos and the conductance through the system . g. binnig , h. rohrer , ch . gerber , e. weibel , appl . . lett . * 40 * , 178 ( 1982 ) ; phys . lett . * 49 * , 57 ( 1982 ) . g. a. d. briggs , a. j. fisher , surf . rep . * 33 * , 1 ( 1999 ) . w. a. hofer , a. s. foster , a. l. shluger , rev . phys . * 75 * , 1287 ( 2003 ) . m. f. crommie , c. p. lutz , d. m. eigler , science * 262 * , 218 ( 1993 ) ; e. j. heller , m. f. crommie , c. p. lutz , d. m. eigler , nature * 369 * , 464 ( 1994 ) . m. jaochowski , prog . sci . * 74 * , 97 ( 2003 ) . m. krawiec , m. jaochowski , m. kisiel , surf . * 600 * , 1641 ( 2006 ) . f. j. himpsel , k. n. altmann , r. bennewitz , j. n. crain , a. kirakosian , j. -l . lin , j. l. mcchesney , j. phys . : condens . matter * 13 * , 11097 ( 2001 ) . j. n. crain , j. l. mcchesney , fan zheng , m. c. gallagher , p. c. snijders , m. bissen , c. gundelach , s. c. erwin , f. j. himpsel , phys . rev . * b69 * , 125401 ( 2004 ) . m. jaochowski , m. strak , r. zdyb , appl . sci . * 211 * , 209 ( 2003 ) . m. krawiec , t. kwapiski , m. jaochowski , phys . 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the stm tunneling through a quantum wire ( qw ) with a side - attached impurity ( atom , island ) is investigated using a tight - binding model and the nonequilibrium keldysh green function method . the impurity can be coupled to one or more qw atoms . the presence of the impurity strongly modifies the local density of states of the wire atoms , thus influences the stm tunneling through all the wire atoms . the transport properties of the impurity itself are also investigated mainly as a function of the wire length and the way it is coupled to the wire . it is shown that the properties of the impurity itself and the way it is coupled to the wire strongly influence the stm tunneling which is reflected in the density of states and differential conductance . quantum wire , tunneling , stm 68.27.ef , 81.07.vb , 73.40.gk

You are an expert at summarizing long articles. Proceed to summarize the following text: countries were selected by data availability . for each country we require availability of at least one aggregation level where the average population per territorial unit @xmath0 . this limit for @xmath1 was chosen to include a large number of countries , that have a comparable level of data resolution . we use data from recent parliamentary elections in austria , canada , czech republic , finland , russia ( 2011 ) , spain and switzerland , the european parliament elections in poland and presidential elections in the france , romania , russia ( 2012 ) and uganda . here we refer by `` unit '' to any incarnation of an administrative boundary ( such as districts , precincts , wards , municipals , provinces , etc . ) of a country on any aggregation level . if the voting results are available on different levels of aggregation , we refer to them by roman numbers , i.e. poland - i refers to the finest aggregation level for poland , poland - ii to the second finest , and so on . for each unit on each aggregation level for each country we have the data of the number of eligible persons to vote , valid votes and votes for the winning party / candidate . voting results were obtained from official election homepages of the respective countries , for more details see si tab.s1 . units with an electorate smaller than 100 are excluded from the analysis , to prevent extreme turnout and vote rates as artifacts from very small communities . we tested robustness of our findings with respect to the choice of a minimal electorate size and found that the results do not significantly change if the minimal size is set to 500 . the histograms for the 2d - vote - turnout distributions ( vtds ) for the winning parties , also referred to as `` fingerprints '' , are shown in fig.[figure1 ] . of the winning parties as rescaled distributions with zero - mean and unit - variance @xcite . large deviations from other countries can be seen for uganda and russia with the plain eye . for more detailed results see tab.s3.,width=328 ] it has been shown that by using an appropriate re - scaling of election data , the distributions of votes and turnouts approximately follow a gaussian @xcite . let @xmath2 be the number of votes for the winning party and @xmath3 the number of voters in any unit @xmath4 . a re - scaling function is given by the _ logarithmic vote rate _ , @xmath5 @xcite . in units where @xmath6 ( due to errors in counting or fraud ) or @xmath7 @xmath8 is not defined , and the unit is omitted in our analysis . this is a conservative definition , since districts with extreme but feasible vote and turnout rates are neglected ( for instance , in russia 2012 there are 324 units with 100% vote and 100% turnout ) . to motivate our parametric model for the vtd , observe that the vtd for russia and uganda in fig.[figure1 ] are clearly bimodal , both in turnout and votes . one cluster is at intermediate levels of turnout and votes . note that it is smeared towards the upper right parts of the plot . the second peak is situated in the vicinity of the 100% turnout , 100% votes point . this suggests two modes of fraud mechanisms being present , _ incremental _ and _ extreme _ fraud . incremental fraud means that with a given rate ballots for one party are added to the urn and votes for other parties are taken away . this occurs within a fraction @xmath9 of units . in the election fingerprints in fig.[figure1 ] these units are those associated with the smearing to the upper right side . extreme fraud corresponds to reporting a complete turnout and almost all votes for a single party . this happens in a fraction @xmath10 of units . these form the second cluster near 100% turnout and votes for the winning party . ) . for switzerland the fair and fitted model are almost the same . the results for russia and uganda can be explained by the model assuming a large number of fraudulent units.,width=328 ] for simplicity we assume that within each unit turnout and voter preferences can be represented by a gaussian distribution with the mean and standard deviation taken from the actual sample , see si fig.s1 . this assumption of normality is not valid in general . for example the canadian election fingerprint of fig.[figure1 ] is clearly bimodal in vote preferences ( but not in turnout ) . in this case , the deviations from approximate gaussianity are due to a significant heterogeneity within the country . in the particular case of canada this is known to be due to the mix of the anglo- and francophone population . normality of the observed vote and turnout distributions is discussed in the si , see tab.s2 . let @xmath11 be the number of valid votes in unit @xmath4 . the first step in the model is to compute the empirical turnout distribution , @xmath12 , and the empirical vote distribution , @xmath13 , over all units from the election data . to compute the _ model _ vtd the following protocol is then applied to each unit @xmath4 . * for each @xmath4 , take the electorate size @xmath3 from the election data . * model turnout and vote rates for @xmath4 are drawn from normal distributions . the mean of the model turnout ( vote ) distribution is estimated from the election data as the value that maximizes the empirical turnout ( vote ) distribution . the model variances are also estimated from the width of the empirical distributions , see si and fig.s1 for details . * _ incremental fraud_. with probability @xmath9 ballots are taken away from both the non - voters and the opposition and are added to the winning party s ballots . the fraction of ballots which are shifted to the winning party is again estimated from the actual election data . * _ extreme fraud_. with probability @xmath10 almost all ballots from the non - voters and the opposition are added to the winning party s ballots . the first step of the above protocol ensures that the actual electorate size numbers is represented in the model . the second step guarantees that the overall dispersion of vote and turnout preferences of the country s population are correctly represented in the model . given nonzero values for @xmath9 and @xmath10 , incremental and extreme fraud are then applied in the third and fourth step , respectively . for a complete specification of these fraud mechanisms , see the si . values for @xmath9 and @xmath10 are reverse engineered from the election data in the following way . first , model vtds are generated according to the above scheme , for each combination of @xmath14 values , where @xmath9 and @xmath15 . we then compute the point - wise sum of the square difference of model and observed vote distributions for each pair @xmath14 and extract the pair giving the minimal difference . this procedure is repeated for 100 iterations , leading to 100 pairs of fraud parameters @xmath16 . in the following we report the average values of these @xmath9 and @xmath10 values , respectively , and their standard deviations . for more details see si . fig.[figure1 ] shows 2-d histograms ( vtds ) for the number of units for a given fraction of voter turnout ( x - axis ) and for the percentage of votes for the winning party ( y - axis ) . results are shown for austria , canada , czech republic , finland , france , poland , romania , russia , spain , switzerland and uganda . for each of these countries the data is shown on the finest aggregation level , where @xmath0 . these figures can be interpreted as fingerprints of several processes and mechanisms leading to the overall election results . for russia and uganda the shape of these fingerprints differ strongly from the other countries . in particular there is a large number of territorial units ( thousands ) with approximately 100% percent turnout and at the same time about 100 % of votes for the winning party . in fig.[sifigurecoll ] we show the distribution of @xmath8 for each country . roughly , to first order the data from different countries collapse to an approximate gaussian , as previously observed @xcite . clearly , the data for russia falls out out of line . skewness and kurtosis for the distributions of @xmath8 are listed for each data - set and aggregation level in tab.s3 . most strikingly , the kurtosis of the distributions for russia ( 2003 , 2007 , 2011 and 2012 ) exceed the kurtosis of each other country on the coarsest aggregation level by a factor of two to three . values for the skewness of the logarithmic vote rate distributions for russia are also persistently below the values for each other country . note that for the vast majority of the countries skewness and kurtosis for the distribution of @xmath8 are in the vicinity of 0 and 3 , respectively ( which are the values one would expect for normal distributions ) . however , the moments of the distributions do depend on the data aggregation level . fig.[momentaggregate ] shows skewness and kurtosis for the distributions of @xmath8 for each election on each aggregation level . by increasing the data resolution , skewness and kurtosis for russia decrease and approach similar values as observed in the rest of the countries , see also si tab.s3 . these measures depend on the data resolution and thus can not be used as unambiguous signals for statistical anomalies . as will be shown , the fraud parameters @xmath9 and @xmath10 do _ not _ significantly depend on the aggregation level or total sample size . estimation results for @xmath9 and @xmath10 are given in tab.s3 for all countries on each aggregation level . they are zero ( or almost zero ) in all of the cases except for russia and uganda . in the right column of fig.[figure2 ] we show the model results for russia ( 2011 and 2012 ) , uganda and switzerland for @xmath17 . the case where both fraud parameters are zero corresponds to the absence of incremental and extreme fraud mechanisms in the model and can be called the fair election case . in the middle column of fig.[figure2 ] we show results for the estimated values of @xmath9 and @xmath10 . the left column shows the actual vtd of the election . values of @xmath9 and @xmath10 significantly larger than zero indicate that the observed distributions may be affected by fraudulent actions . to describe the smearing from the main peak to the upper right corner which is observed for russia and uganda , an incremental fraud probability around @xmath18 is needed for _ united russia _ in 2011 and @xmath19 in 2012 . this means fraud in about 64% of the units in 2011 and 39% in 2012 . in the second peak close to 100% turnout there are roughly 3,000 units with 100% of votes for united russia in the 2011 data , representing an electorate of more than two million people . best fits yield @xmath20 for 2011 and @xmath21 for 2012 , i.e. two to three percent of all electoral units experience extreme fraud . a more detailed comparison of the model performance for the russian parliamentary elections of 2003 , 2007 , 2011 and 2012 is found in fig.s2 . fraud parameters for the uganda data in fig.[figure2 ] are found to be @xmath22 and @xmath23 . a best fit for the election data from switzerland gives @xmath17 . these results are drastically more robust to variations of the aggregation level of the data than the previously discussed distribution moments skewness and kurtosis , fig.[fraudaggregate ] and tab.s3 . even if we aggregate the russian data up to the coarsest level of federal subjects ( approximately 85 units , depending on the election ) , @xmath10 estimates are still at least two standard deviations above zero , @xmath9 estimates more than ten standard deviations . similar observations hold for uganda . for no other country , on no other aggregation level , such deviations are observed . the parametric model yields similar results for the same data on different levels of aggregation , as long as the values maximizing the empirical vote ( turnout ) distribution and the distribution width remains invariant . in other words , as long as units with similar vote ( turnout ) characteristics are aggregated to larger units , the overall shapes of the empirical distribution functions are preserved and the model estimates do not change significantly . note that more detailed assumptions about possible mechanisms leading to large heterogeneity in the data ( such as the _ qubcois _ in canada or voter mobilization in the helsinki region in finland , see si ) may have an effect on the estimate of @xmath9 . however , these can under no circumstances explain the mechanism of extreme fraud . results for elections in sweden , uk and usa , where voting results are only available on a much coarser resolution ( @xmath24 ) , are given in tab.s4 . another way to visualize the intensity of election irregularities is the cumulative number of votes as a function of the turnout , fig.[figure3 ] . for each turnout level the total number of votes from units with this or lower levels are shown . each curve corresponds to the respective election winner in a different country , with average electorate per unit of comparable order of magnitude . usually these cdfs level off and form a plateau from the party s maximal vote count on . again this is not the case for russia and uganda . both show a boost phase of increased extreme fraud toward the right end of the distribution ( red circles ) . russia never even shows a tendency to form a plateau . as long as the empirical vote distribution functions remain invariant under data aggregation as discussed above , the shape of these cdfs will be preserved too . note that fig.[figure3 ] . demonstrates that these effects are decisive for winning the 50% majority in russia 2011 . we demonstrate that it is not sufficient to discuss the approximate normality of turnout , vote or logarithmic vote rate distributions , to decide if election results may be corrupted or not . we show that these methods can lead to ambiguous signals , since results depend strongly on the aggregation level of the election data . we developed a model to estimate parameters quantifying to which extent the observed election results can be explained by ballot stuffing . the resulting parameter values are shown to be insensitive to the choice of the aggregation level . note that the error margins for @xmath10 values start to increase by decreasing @xmath25 below 100 , see fig.[fraudaggregate]d , whereas @xmath9 estimates stay robust even for very small @xmath25 . it is imperative to emphasize that the shape of the fingerprints in fig.[figure1 ] will deviate from pure 2-d gaussian distributions also as a result of non - fraudulent mechanisms , but due to heterogeneity in the population . the purpose of the parametric model is to quantify to which extent ballot stuffing and the mechanism of extreme fraud may have contributed to these deviations , or if their influence can be ruled out on the basis of the data . for the elections in russia and uganda they can not be ruled out . as shown in fig.s2 , assumptions of their wide - spread occurrences even allow to reproduce the observed vote distributions to a good degree . in conclusion it can be said with almost certainty that an election does not represent the will of the people , if a substantial fraction ( @xmath10 ) of units reports a 100% turnout with almost all votes for a single party , and/or if any significant deviations from the sigmoid form in the cumulative distribution of votes versus turnout are observed . another indicator of systematic fraudulent or irregular voting behavior is an incremental fraud parameter @xmath9 which is significantly greater than zero on each aggregation level . should such signals be detected it is tempting to invoke g.b . shaw who held that `` [ d]emocracy is a form of government that substitutes election by the incompetent many for appointment by the corrupt few . 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making , _ eur . j. b _ * 75 * , 395 - 404 . j. agnew ( 1996 ) , mapping politics : how context counts in electoral geography , _ political geography _ * 15 * ( 2 ) , 129 - 46 . descriptive statistics and official sources of the election results are shown in tab.s1 . the raw data will be made available for download at http://www.complex - systems.meduniwien.ac.at/. they report election results of parliamentary ( austria , canada , czech republic , finland , russia , spain and switzerland ) , european ( poland ) or presidential ( france , romania , russia , uganda ) elections on at least one aggregation level . in the rare circumstances where electoral districts report more valid ballots than registered voters , we work with a turnout of 100% . territorial units with an electorate less than hundred are omitted at each point of the analysis , to avoid extreme vote and turnout rates as spurious results due to small communities . the countries to include in this work have been chosen on the basis of data availability . a country is included , if the voting results are available in electronic form on an aggregation level where a number of @xmath0 vote eligible persons comprises one territorial unit . required data is the number of vote eligible persons @xmath3 , the number of valid votes @xmath11 and the number of votes for the winning party / candidate @xmath2 for each unit . a country is separated into @xmath25 electoral units @xmath4 , each having an electorate of @xmath3 people and in total @xmath11 valid votes . the fraction of valid votes for the winning party in unit @xmath4 is denoted @xmath26 . the average turnout over all units , @xmath27 , is given by @xmath28 with standard deviation @xmath29 , the mean fraction of votes @xmath30 for the winning party is @xmath31 with standard deviation @xmath32 . the mean values @xmath27 and @xmath30 are typically close to but not identical to the values which maximize the empirical distribution function of turnout and votes over all units . let @xmath33 be the number of votes where the empirical distribution function assumes its ( first local ) maximum ( rounded to entire percents ) , see fig . s[sifiguremeth ] . similarly @xmath34 is the turnout where the empirical distribution function of turnouts @xmath35 takes its ( first local ) maximum . the distributions for turnout and votes are extremely skewed to the right for uganda and russia which also inflates the standard deviations in these countries , see tab . s2 . to account for this a left - sided ( right - sided ) mean deviation @xmath36 ( @xmath37 ) from @xmath33 is introduced . @xmath37 can be regarded as the _ incremental fraud width _ , a measurable parameter quantifying how intense the vote stuffing is . this contributes to the smearing out of the main peaks in the election fingerprints , see fig.1 in the main text . the larger @xmath37 , the more inflated the vote results due to urn stuffing , in contrast to @xmath36 which quantifies the scatter of the voters actual preferences . they can be estimated from the data by @xmath38 similarly the _ extreme fraud width _ @xmath39 can be estimated , i.e. the width of the peak around 100% votes . we found that @xmath40 describes all encountered vote distributions reasonably well . for a visualization of @xmath36 , @xmath37 and @xmath39 see fig . s[sifiguremeth ] . while @xmath9 and @xmath10 measure in how many units incremental and extreme fraud occur , @xmath37 and @xmath39 quantify how intense these activities are , if they occur . to get an estimate for the width of the distribution of turnouts over territorial unit which is free of possible fraudulent influences , the _ turnout distribution width _ @xmath41 is calculated from electoral districts @xmath4 which have both @xmath42 and @xmath43 , that is @xmath44 . incremental fraud is a combination of two processes : stuffing ballots for one party into the urn and re - casting or deliberately wrong - counting ballots from other parties ( e.g. erasing the cross ) . which one of these two processes dominates is quantified by the _ deliberate wrong counting parameter _ for @xmath46 the wrong - counting process dominates , for @xmath47 the urn stuffing mechanism is prevalent . in the following @xmath48 denotes a normal distributed random variable with mean @xmath49 and standard deviation @xmath50 . the model is specified by the following protocol , which is applied to each district . * pick a unit @xmath4 with electorate @xmath3 taken from the data . * the model turnout of unit @xmath4 , @xmath51 , is @xmath52 . * a fraction of @xmath53 people vote for the winning party . * with probability @xmath9 incremental fraud takes place . in this case the unit is assigned a fraud intensity @xmath54 . values for @xmath55 are only accepted if they lie in the range @xmath56 . this is the fraction of votes not cast , @xmath57 , which are added to the winning party . votes for the opposition are wrong counted for the winning party with a rate @xmath58 ( where @xmath59 is an exponent ) . to summarize , if incremental fraud takes place the winning party receives @xmath60 votes . * with probability @xmath10 extreme fraud takes place . in this case opposition votes are canceled and added to the winning party with probability @xmath61 ( i.e. the above with @xmath62 replacing @xmath55 ) . acceptable values for @xmath62 are again from the range @xmath63 . , @xmath36 , @xmath37 and @xmath39 are estimated from the election results . @xmath33 is the maximum of the distribution function . @xmath36 measures the distribution width of values to the left of @xmath33 , i.e. smaller than @xmath33 . the incremental fraud with @xmath37 measures the distribution width of values to the right of @xmath33 , i.e. larger than @xmath33 . the extreme fraud width @xmath39 is the width of the peak at 100% votes.,width=328 ] the parameters for incremental and extreme fraud , @xmath9 and @xmath10 , as well as the deliberate wrong counting parameter @xmath59 , are estimated by a goodness - of - fit test . let @xmath64 be the empirical distribution function of votes for the winning party ( the data is binned with one bin corresponding to one percent ) over all territorial units . the distribution function for the model units @xmath65 is calculated for each set of @xmath66 values where @xmath67 . we report values for the fraud parameters where the statistic @xmath68 assumes its minimum , averaged over 100 realizations over the parameter space , see tab.s3 for @xmath9 and @xmath10 . the extreme fraud parameter @xmath10 is zero ( within one standard deviation ) for almost all elections except russia ( 2003 , 2007 , 2011 and 2012 ) and uganda . for very small @xmath25 ( @xmath69 ) estimates for @xmath10 become less robust . these are also the only elections where the incremental fraud parameter @xmath9 is not close to zero . values for @xmath59 for the russian elections are @xmath70 ( 2003 ) , @xmath71 ( 2007 ) , @xmath72 ( 2011 ) , @xmath73 ( 2012 ) , and @xmath74 for uganda . results for @xmath59 from countries where @xmath9 is close to zero can not be detected in a robust way and are superfluous , since there are ( almost ) no deviations from the fair election case . special care is needed in the interpretation of @xmath9 and @xmath10 values in countries where election units contain several polling stations . it may be the case that extreme fraud takes only in a subset of the polling stations within a unit place . in that case extreme fraud would be indistinguishable from the incremental fraud mechanism . it is hard to construct other plausible mechanisms leading to a large number of territorial units having 100% turnout and votes for a single party than urn stuffing . the case is not so clear for the smeared out main cluster . in some cases , namely canada and finland , this cluster also takes on a slightly different form . this effect clearly does not inflate the turnout as much as it is the case in russia and uganda , but it is nevertheless present . in canada the distribution of vote preferences is bimodal , with one peak around 50% and one around 10% ( of the vote eligible population ) , but with similar turnout levels . this is a result of a large - scale heterogeneity in the data : english and french canada . votes are shown for the winning conservatives . looking at their results by province , they tallied 16.5% votes cast in quebec , but more than 40% in eight of the remaining twelve other provinces . as a consequence the logarithmic vote rate kurtosis becomes inflated . however , these statistical deviations are perfectly distinguishable from the traces of ballot stuffing , resulting in vanishing fraud parameters on all aggregation levels . another possible mechanism leading to irregularities in the voting results is successful voter mobilization . this may lead to a correlation between turnout and a party s votes . the finland elections , for example , where marked by radical campaigns by the true finns . they managed to mobilize evenly spread out across the country , with the exception of the helsinki region , where the winning national coalition party performed significantly better than in the rest of the country .

democratic societies are built around the principle of free and fair elections , that each citizen s vote should count equal . national elections can be regarded as large - scale social experiments , where people are grouped into usually large numbers of electoral districts and vote according to their preferences . the large number of samples implies certain statistical consequences for the polling results which can be used to identify election irregularities . using a suitable data collapse , we find that vote distributions of elections with alleged fraud show a kurtosis of hundred times more than normal elections on certain levels of data aggregation . as an example we show that reported irregularities in recent russian elections are indeed well explained by systematic ballot stuffing and develop a parametric model quantifying to which extent fraudulent mechanisms are present . we show that if specific statistical properties are present in an election , the results do not represent the will of the people . we formulate a parametric test detecting these statistical properties in election results . remarkably , this technique produces similar outcomes irrespective of the data resolution and thus allows for cross - country comparisons . free and fair elections are the cornerstone of every democratic society @xcite . a central characteristic of elections being free and fair is that each citizen s vote counts equal . however , already joseph stalin believed that `` it s not the people who vote that count ; it s the people who count the votes . '' how can it be distinguished whether an election outcome represents the will of the people or the will of the counters ? elections can be seen as large - scale social experiments . a country is segmented into a usually large number of electoral units . each unit represents a standardized experiment where each citizen articulates his / her political preference via a ballot . although elections are one of the central pillars of a fully functioning democratic process , relatively little is known about how election fraud impacts and corrupts the results of these standardized experiments @xcite . there is a plethora of ways of tampering with election outcomes , for instance the redrawing of district boundaries known as gerrymandering , or the barring of certain demographics from their right to vote . some practices of manipulating voting results leave traces which may be detected by statistical methods . recently , benford s law @xcite experienced a renaissance as a potential election fraud detection tool @xcite . in its original and naive formulation , benford s `` law '' is the observation that for many real world processes the logarithm of the first significant digit is uniformly distributed . deviations from this law may indicate that other , possibly fraudulent mechanisms are at work . for instance , suppose a significant number of reported vote counts in districts is completely made up and invented by someone preferring to pick numbers which are multiples of ten . the digit `` 0 '' would then occur much more often as the last digit in the vote counts when compared to uncorrupted numbers . voting results from russia @xcite , germany @xcite , argentina @xcite and nigeria @xcite have been tested for the presence of election fraud using variations of this idea of digit - based analysis . however , the validity of benford s law as a fraud detection method is subject to controversy @xcite . the problem is that one needs to firmly establish a baseline of what the _ expected _ distribution of digit occurrences for fair elections should be . only then it can be asserted if _ actual _ numbers are over- or underrepresented and thus suspicious . what is missing in this context is a theory that links specific fraud mechanisms to statistical anomalies @xcite . a different strategy for detecting signals of election fraud is to look at the distribution of vote and turnout numbers as in @xcite . this has been extensively done for the russian presidential and duma elections over the last 20 years @xcite . these works focus on the task of detecting two mechanisms , the stuffing of ballot boxes and the reporting of contrived numbers . it has been noted that these mechanisms are able to produce different features of vote and turnout distributions than those observed in fair elections . while for russian elections between 1996 and 2003 these features were `` only '' observed in a relatively small number of electoral units , they eventually spread and percolated through the entire russian federation from 2003 onwards . according to myagkov and ordeshook @xcite `` [ o]nly kremlin apologists and putin sycophants argue that russian elections meet the standards of good democratic practice '' . this point was further substantiated with election results from the 2011 duma and 2012 presidential elections @xcite . here it was also observed that ballot stuffing not only changes the shape of vote and turnout distributions , but also induces a high correlation between them . unusually high vote counts tend to _ co - occur _ with unusually high turnout numbers . several recent advances in the understanding of statistical regularities of voting results are due to the application of statistical physics concepts to quantitative social dynamics @xcite . in particular several approximate statistical laws of how vote and turnout are distributed have been identified @xcite , some of them are shown to be valid across several countries @xcite . it is tempting to think of deviations from these approximate statistical laws as potential indicators for election irregularities which are valid cross - nationally . however , the magnitude of these deviations may vary from country to country due to different numbers and sizes of electoral districts . any statistical technique quantifying election anomalies across countries should not depend on the size of the underlying sample nor its aggregation level , i.e. the size of the electoral units . as a consequence , a conclusive and robust signal for a fraudulent mechanism , e.g. ballot stuffing , must not disappear if the same dataset is studied on different aggregation levels . in this work we expand earlier work on statistical detection of election anomalies in two directions . first , we test for reported statistical features of voting results ( and deviations thereof ) in a cross - national setting , and discuss their dependence on the level of data aggregation . as the central point of this work we propose a parametric model to statistically quantify to which extent fraudulent processes , such as ballot stuffing , may have influenced the observed election results . remarkably , under the assumption of coherent geographic voting patterns @xcite , the parametric model results do not depend significantly on the aggregation level of the election data or the size of the data sample .

You are an expert at summarizing long articles. Proceed to summarize the following text: the purpose of this paper is to study the uniqueness and non - degeneracy of solutions to a nonlinear schdinger - type equation , arising from the minimization of the following energy functional @xmath2 under the mass constraint @xmath3 . here @xmath4 is the positive part , @xmath5 is a 2-spinor that describes the quantum state of a nucleon ( a proton or a neutron ) , @xmath6 are the pauli matrices and @xmath7 . the equation of interest is @xmath8 with @xmath9 the lagrange multiplier associated with the mass constraint . this equation can as well be written in the form of a system of two coupled dirac - like equations @xmath10 indeed , the above model can formally be deduced from a relativistic model involving one dirac particle coupled with two auxiliary classical fields ( the so - called _ @xmath11 model _ ) , in a specific non - relativistic limit that will be described in detail below . in this limit , the equations for the classical fields can be solved explicitly , leading to the nonlinear system and the corresponding nonlinear energy functional , expressed in terms of @xmath12 only . the term @xmath13 in is the usual nonlinear schrdinger attraction which describes here the confinement of the nucleons . on the other hand , the denominator @xmath14 can be interpreted as a mass depending on the state @xmath12 of the nucleon , and it describes a phenomenon of saturation in the system . a high density @xmath15 generates a lower mass , which itself prevents from having a too high density . mathematically speaking , this term enforces the additional constraint @xmath16 , which is very important for the stability of the energy . without the @xmath12-dependent mass , the model is of course unstable and the energy functional is unbounded from below . the mass term @xmath14 allows us to consider the minimization of the energy in space dimensions @xmath17 without any limitation on @xmath18 and @xmath19 , even if @xmath20 is the interesting physical case . we remark that the upper bound @xmath21 on the particle density @xmath22 arises after an appropriate choice of units . let us emphasize that , in the model presented above , spin is taken into account since @xmath12 takes values in @xmath23 . under the additional assumption that the state of the nucleon is an eigenfunction of the spin operator , the energy must be restricted to functions of the special form @xmath24 leading to the simpler functional @xmath25 it is an open problem to show that minimizers of the original energy are necessarily of the special form . in principle , the spin symmetry could be broken . in this paper we will however restrict ourselves to the simplified functional , which we study in any space dimension @xmath17 . the corresponding euler - lagrange equation simplifies to @xmath26 to our knowledge , the above model was mathematically studied for the first time in @xcite , where esteban and the second author of this paper formally derived the equation from its relativistic counterpart , and then proved the existence of radial square integrable solutions of ( [ eq : nonlinear_nospin ] ) . this result has then been generalized in @xcite , where the existence of infinitely many square - integrable excited states ( solutions with an arbitrary but finite number of sign changes ) was shown . in @xcite , esteban and the second author used a variational approach to prove the existence of minimizers for the spin energy , for a large range of values for the parameter @xmath27 . the model is translation - invariant , hence uniqueness can not hold . usual symmetrization techniques do not obviously apply due to the presence of the pauli matrices @xmath28 s but a natural conjecture is that all minimizers are of the form @xmath29 after a suitable space translation and a choice of spin orientation . the approach of @xcite applies as well to the simplified no - spin model , and the proof works in any dimension . the result in this case is the following . [ thm : minimizers ] let @xmath17 and @xmath30 there exists a universal number @xmath31 such that @xmath32 for @xmath33 , @xmath34 and there is no minimizer ; @xmath32 for @xmath35 , @xmath36 and all the minimizing sequences are precompact in @xmath37 , up to translations . there is at least one minimizer @xmath38 for the minimization problem @xmath39 and it can be chosen such that @xmath40 , after multiplication by an appropriate phase factor . it solves the nonlinear equation for some @xmath41 . the method used in @xcite to prove theorem [ thm : minimizers ] is based on lions concentration - compactness technique @xcite and the main difficulty was to deal with the denominator @xmath42 , for which special localization functions had to be introduced . because the energy depends linearly on the parameter @xmath27 , the function @xmath43 is concave non - increasing , which is another important fact used in the proof of @xcite . the critical strength @xmath44 of the nonlinear attraction is the largest for which @xmath34 and it can simply be defined by @xmath45 it can easily be verified that @xmath46 in dimension @xmath47 , that @xmath48 is related to the gagliardo - nirenberg - sobolev constant in dimension @xmath49 , and that @xmath50 in higher dimensions . estimates on @xmath44 have been provided in dimension @xmath20 in @xcite and similar bounds can be derived in higher dimensions by following the same method . after the two works @xcite , it remained an open problem to show that minimizers are all radial and unique , up to a possible translation and multiplication by a phase factor . the purpose of this paper is to answer this question . our main result is the following . [ thm : main ] the nonlinear equation has no non - trivial solution @xmath51 in @xmath52 when @xmath53 . for @xmath54 , the nonlinear equation admits a unique solution @xmath55 that tends to 0 at infinity , modulo translations and multiplication by a phase factor . it is radial , decreasing , and non - degenerate . this theorem is the equivalent of a celebrated similar result for the nonlinear schrdinger equation ( see , e.g. , ( * ? ? ? b ) and @xcite for references ) . our main contribution is the remark that the equation can be rewritten in terms of @xmath56 as a simpler nonlinear schrdinger equation @xmath57 applying a classical argument of mcleod @xcite ( as explained in ( * ? ? ? b ) and in @xcite ) allows to prove the non - degeneracy and uniqueness in the radial case . that any solution of is necessarily radial decreasing then follows from the moving plane method @xcite . the proof of theorem [ thm : main ] is provided in section [ sec : proof ] below . let us remark that , since equation is invariant under multiplications by a phase factor , we can always suppose that a solution @xmath58 is real - valued . hence , in ( * ? ? ? * appendix a.1 ) it has been proved that any solution @xmath59 is such that @xmath60 a.e . in @xmath61 whenever @xmath62 . as a consequence , the change of variables @xmath63 makes sense whenever @xmath62 . as an application of theorem [ thm : main ] , we are able to construct a branch of solutions of the underlying dirac equation , that converges to the non - relativistic solution @xmath38 in the limit , thereby justifying the formal arguments of @xcite . we explain this now . we restrict ourselves to @xmath20 for simplicity ( but the results are similar in other dimensions ) . we consider one relativistic nucleon in interaction with two scalar fields @xmath64 ( the _ @xmath0field _ ) and @xmath65 ( the _ @xmath1field _ ) . as described for instance in @xcite , the corresponding equation is @xmath66 where @xmath67 are the dirac matrices and @xmath68 is now a 4-spinor . the wavefunction @xmath69 should in principle be normalized in @xmath70 but , here , we think of fixing @xmath71 instead of imposing @xmath72 . any non - trivial solution @xmath69 to also gives a normalized solution after an appropriate change of parameters . in most physics papers , the equation for the @xmath0-field @xmath64 contains a nonlinear term as well ( for instance including vacuum polarization effects @xcite ) , @xmath73 but we restrict ourselves to the simpler linear case for convenience . the fields @xmath64 and @xmath65 are respectively focusing and defocusing , which can be seen from the different signs in the two klein - gordon equations . on the other hand , they have very different effects , since @xmath64 modifies the mass @xmath74 in the same way for the upper and lower spinors , whereas @xmath65 is repulsive for the upper spinor and attractive for the lower spinor . this statement is clarified when the dirac equation is written in terms of @xmath75 as @xmath76 we see that @xmath77 and @xmath78 respectively appear in the two equations . in our units , the non - relativistic limit corresponds to @xmath79 , with all the masses being of the same order . on the contrary to atomic physics , in nuclear physics the coupling constants @xmath80 and @xmath81 are very large , comparable to the masses . it is therefore customary to work in a regime where @xmath82 and @xmath83 are fixed or , even , large . in the two klein gordon equations , the laplacian can then be neglected in such a way that @xmath84 and hence @xmath85 @xmath86 as usual , in the non - relativistic regime , the lower spinor @xmath87 is of order @xmath88 . simple effective equations will then be obtained in the limit . in order to better illustrate the regime of interest for the @xmath0@xmath1 model , let us first discuss the case of the @xmath0 model , in which @xmath89 and @xmath90 . the equation then reduces to @xmath91 the interesting regime is then @xmath83 of order 1 , say @xmath92 fixed . it can be proved that @xmath93 and the usual nls equation is recovered in the limit , after a simple scaling . the precise result is the following . [ thm : sigma ] let @xmath94 be positive constants . then for @xmath74 large enough , the equation admits a branch of solutions of the special form @xmath95 with @xmath96 in the limit @xmath97 , we have @xmath98 strongly in @xmath99 , where @xmath100 is the unique positive radial solution of @xmath101 functions of the form have the lowest possible total angular momentum ( * ? ? ? . the theorem can be shown by following step by step the method of section [ sec : relativistic_limit ] , using the non - degeneracy of the nls ground state . its proof will not be provided in this paper for shortness . theorem [ thm : sigma ] is not satisfactory from a physical point of view . indeed , the limit @xmath100 is considered physically unstable since the corresponding energy functional is unbounded from below in dimension 3 . furthermore , in practice @xmath102 is very large and the corresponding @xmath100 is then very peaked at the origin . in real nuclei , many forces are in action and they tend to compensate in order to avoid this collapse at @xmath103 . it is therefore important to take the @xmath1 field into account . for the @xmath0@xmath1 model , the interesting regime is when the parameters @xmath104 and @xmath105 behave like @xmath74 , whereas @xmath106 stays bounded , which is the cancellation between the two scalar fields mentioned before . even if @xmath104 diverges , the model still has a nice bounded limit @xmath38 , which is precisely the non - relativistic ground state studied in the previous section . [ thm : dirac ] let @xmath107 be positive constants such that @xmath108 . then for @xmath74 large enough , the equation admits a branch of solutions of the special form @xmath109 with @xmath110 in the limit @xmath97 , we have @xmath111 strongly in @xmath99 , where @xmath38 is the unique positive solution of with @xmath112 and @xmath113 . we refer to @xcite , ( * ? ? ? * sec . 3 ) and ( * ? ? 2.3 ) for a discussion of the validity of this regime for standard nucleons . typical physical values for the parameters of the model are provided in ( * ? ? ? * table 3.1 ) . the proof of theorem [ thm : dirac ] is provided in section [ sec : relativistic_limit ] and it is based on the implicit function theorem . in other words , we see as a small perturbation of and we use the non - degeneracy of @xmath38 to construct a solution . remark that , thanks to the non - degeneracy property proved in section [ sec : nondegeneracy_l2 ] , this argument gives also the local uniqueness of the solution to around @xmath58 , modulo translations and multiplication by a phase factor . the exact same reasoning can be used for proving theorem [ thm : sigma ] . a similar argument has for instance been used in @xcite . we hope that our work will stimulate further research on this model . * acknowledgement . * the authors acknowledge financial support from the european research council ( fp7/2007 - 2013 grant agreement mniqs 258023 ) and the anr ( nonap 10 - 0101 ) of the french ministry of research . moreover , the research of the second author was supported by the labex cempi ( anr-11-labx-0007 - 01 ) . this section is devoted to the proof of theorem [ thm : main ] , which is split in several steps . in the next section , we explicit the change of variable @xmath114 and combine it with symmetric rearrangement to deduce that the minimization problem @xmath39 can be restricted to radial decreasing functions . this step is not necessary for our analysis but we mention it for completeness , as it gives a simpler existence proof than in @xcite . then , in section [ sec : moving_planes ] , we use the moving plane method to conclude that positive solutions to the nonlinear equation are radial decreasing . section [ sec : uniqueness_radial ] is devoted to the uniqueness of radial solutions . finally , we prove the non - degeneracy of the linearized operator in the whole of @xmath115 ( modulo the trivial symmetries of the problem ) in section [ sec : nondegeneracy_l2 ] . we recall that the energy functional is @xmath116 which we study on the subset of @xmath38 s in @xmath115 such that @xmath117 and @xmath118 since @xmath119 , it is clear that any such @xmath38 must be in @xmath37 . it was proved in ( * ? ? ? 2.1 ) that it must also satisfy @xmath120 a.e . the nonlinear term @xmath121 is then well defined and , since @xmath120 , we conclude that @xmath122 . by using rearrangement inequalities and the change of variable @xmath114 , we are able to prove that minimizers are always radial - decreasing . this can be used to simplify the proof of theorem [ thm : minimizers ] of @xcite . [ lem : rearrangement ] for every @xmath123 , the minimization problem @xmath39 can be restricted to radial non - increasing functions . furthermore , any minimizer of @xmath39 , when it exists , is positive and radial - decreasing , after a possible translation and multiplication by a phase factor . first we recall that @xmath124 a.e . , see ( * ? ? ? hence @xmath125 and the minimization problem can be restricted to functions satisfying @xmath40 , which we assume from now on . let then @xmath126 be the schwarz rearrangement of @xmath38 . using that @xmath127 we see that @xmath128 next , we have @xmath129 for all @xmath130 and , since @xmath131 is increasing , @xmath132 , by ( * ? ? ? 3 & lem . 7.17 ) . we conclude that @xmath133 and the minimization can be restricted to radial non - decreasing functions . if @xmath38 is a ( possibly non - symmetric ) minimizer with @xmath40 , then @xmath126 is also a minimizer and we have @xmath134 . in general , this does not imply that @xmath38 is itself radial - decreasing , but this will be proved using the nonlinear equation . denoting @xmath114 and @xmath135 , we see that @xmath136 must solve the euler - lagrange equation @xmath137 in particular , @xmath138 must be the first eigenvector of the schrdinger operator @xmath139 and therefore @xmath140 . the real - analyticity of @xmath138 ( see , e.g. , @xcite ) combined with the equality @xmath141 now implies that @xmath142 , hence @xmath143 , after an appropriate space translation , by @xcite . finally , if @xmath38 is an arbitrary minimizer , the equality @xmath144 implies @xmath145 by ( * ? ? ? this concludes the proof of the lemma . in the previous section , we have shown using rearrangement inequalities that minimizers of @xmath146 are necessarily radial - decreasing . here we use the moving plane method to prove that non - negative solutions of the equation are also all radial decreasing , which of course also implies lemma [ lem : rearrangement ] . we recall that the nonlinear equation can be rewritten in terms of @xmath114 as @xmath147 we also remark that @xmath130 when @xmath148 and @xmath149 . for simplicity of notation , we denote @xmath150 let @xmath151 and @xmath152 be a non - trivial solution of with @xmath153 . then , @xmath138 is radial decreasing about some point in @xmath154 . elliptic regularity gives that @xmath155 at infinity . then the result follows immediately from the famous moving plane method . indeed , noticing that @xmath156 and @xmath157 , we may use ( * ? ? ? we have proved that any solution to the equation must be radial - decreasing . the next step consists in studying the uniqueness of radial solutions . in this section , we study radial solutions to the equation , which then solve @xmath158 and we concentrate on showing the uniqueness of positive solutions such that @xmath159 when @xmath160 . in dimensions @xmath161 , the condition @xmath162 is necessary to avoid a singularity at the origin . in dimension @xmath47 , the solution is known to be even about one point and , after a suitable translation we may always assume @xmath162 as well . more precisely , to prove the existence of solutions in dimension @xmath47 , we use the fact that in this case the local energy @xmath163 is conserved along the trajectories . however , in dimension @xmath161 , the energy @xmath164 defined by , decreases : @xmath165 the solutions @xmath166 to are parametrized by @xmath167 . using the same arguments as in the proof of ( 2.6 ) and in particular the fact that the energy @xmath164 is non - increasing , we can easily show that a solution starting at @xmath168 stays bigger than @xmath169 and hence can not tend to @xmath103 at infinity . moreover , note that the equation has the three stationary solutions @xmath170 , @xmath171 and @xmath172 . hence @xmath173 is necessary . the following is a reformulation of the result of @xcite that was expressed in terms of @xmath174 . [ thm : shooting ] for @xmath175 , there is no non - trivial solution @xmath138 to , such that @xmath155 at infinity . for @xmath54 , there exists one positive solution @xmath176 to , such that @xmath177 at infinity . it is decreasing , starts at @xmath178 and has the following behavior at infinity : @xmath179 for some @xmath180 . including the ground state @xmath176 , for @xmath181 and @xmath182 , in dimension @xmath20.[fig : portrait],width=340 ] the proof used in @xcite , which is presented for @xmath20 but can be generalized for all @xmath161 , is based on a shooting method consisting in increasing @xmath183 continuously starting from @xmath103 ( figure [ fig : portrait ] ) . a byproduct of the proof is that all the other solutions @xmath166 with @xmath184 do not tend to @xmath103 at infinity . hence it will remain to prove that the solutions @xmath166 with @xmath185 necessarily vanish at some point @xmath186 . the explicit decay rate was not stated in @xcite , but it is a classical fact whose proof can for instance be read in @xcite . as remarked above , the result of lemma [ lem : rearrangement ] can be use to simplify this proof . for completeness we quickly explain the non - existence part in theorem [ thm : shooting ] which is needed below and is itself taken from ( * ? ? ? * prop 2.1 ) . the idea is to use that the local energy is non - increasing . this implies that any solution satisfying @xmath162 and @xmath187 at infinity must be such that @xmath188 hence @xmath189 is a necessary condition for the existence of @xmath138 . moreover , we see that @xmath190 which is strictly above the stationary solution @xmath191 . the main result of this section is the following [ thm : radial ] for @xmath54 and @xmath17 , the solution @xmath176 of theorem [ thm : shooting ] is the only non - trivial positive solution @xmath138 of such that @xmath187 at infinity . furthermore , @xmath176 is non - degenerate : the unique solution @xmath192 to @xmath193 diverges exponentially fast when @xmath160 . more precisely , if @xmath161 , @xmath192 satisfies @xmath194 and @xmath195 exponentially fast when @xmath160 . in dimension @xmath47 , the result follows immediately from the hamiltonian feature of the problem , based on the energy and the fact that @xmath196 is a non - degenerate critical point of @xmath164 , when @xmath41 . in particular , the divergence of the solution @xmath192 to the linearized equation can be proved by computing the wronskian @xmath197 , using that @xmath198 . we deduce that @xmath199 which can not converge to @xmath103 at infinity . in the following we assume @xmath161 . there are many existing results dealing with the uniqueness ( and , often , the non - degeneracy as well ) of radial solutions to semi - linear equations of the type @xmath200 . after the pioneering works on the nls nonlinearity @xcite , many authors introduced various conditions on the function @xmath201 that ensure uniqueness , see , e.g. @xcite . our particular function @xmath201 as defined in satisfies some of the assumptions required in these works . for instance uniqueness can be directly obtained from ( * ? ? ? 1 ) in dimensions @xmath202 , by means of lemma [ lem : prop_f ] below . on the other hand , the non - degeneracy is sometimes not explicitly stated in those works , although often shown in the middle of the proof . for clarity , we will therefore quickly explain the proof of the theorem , following the approach of mcleod in @xcite and its summary in ( * ? ? ? b ) and @xcite . the main properties of the function @xmath201 that make everything works are summarized in the following [ lem : prop_f ] let @xmath201 be defined as in on @xmath203 , with @xmath54 . then 1 . @xmath201 is negative on @xmath204 and positive on @xmath205 with @xmath206 ; 2 . @xmath207 is decreasing on @xmath205 ; 3 . for every @xmath208 , the function @xmath209 has exactly one root @xmath210 , at which we have @xmath211 . the above properties of @xmath201 are somehow inherited from the nls case , since @xmath212 with @xmath213 . below we will not use the property _ ( 2 ) _ , but rather _ ( 3 ) _ ( which itself follows from _ ( 2 ) _ ) . we however state _ ( 2 ) _ since the monotonicity of @xmath214 appears in many works , including for instance @xcite and @xcite . the proof of lemma [ lem : prop_f ] will be provided at the end of the proof of the theorem . the ` @xmath215 ' function appears as well in @xcite , where an additional assumption on the behavior of @xmath216 was required . now , we assume that @xmath54 and we look at the solutions @xmath166 of with @xmath217 and @xmath162 , and we let @xmath183 vary in @xmath218 . note that the function @xmath219 is smooth ( indeed real - analytic since @xmath201 is analytic ) . following @xcite , we introduce the sets @xmath220 @xmath221 @xmath222 which form a partition of @xmath218 . as we have recalled above , since the energy @xmath164 is decreasing along a solution , we have @xmath223 . this was actually shown in @xcite , where the solution @xmath224 is constructed by looking at the supremum of @xmath225 . in particular , @xmath226 . if @xmath227 we let for convenience @xmath228 . since @xmath229 is smooth , it can easily be proved that @xmath230 is open . the same holds for @xmath225 , but the proof is more difficult . the idea is that the points of @xmath230 are characterized by the fact that the trajectory in phase space crosses first the horizontal axis ( that is , @xmath166 vanishes before @xmath231 ) , whereas for @xmath232 it only crosses the vertical axis ( @xmath231 vanishes and @xmath166 does not ) , see figure [ fig : portrait ] . [ lem : prop_s_- ] let @xmath233 . then @xmath234 on @xmath235 , that is , @xmath166 vanishes before @xmath231 . in particular , @xmath166 is strictly decreasing on @xmath235 . the proof is again based on the monotonicity of the energy @xmath164 and it can for instance be read in ( * ? ? ? the idea is the following . we denote for simplicity @xmath236 and @xmath237 . first , since @xmath238 , then we have from @xmath239 and hence @xmath240 for small @xmath241 . on the other hand @xmath242 ( since @xmath243 is the first root of @xmath236 and the latter can not have double zeroes ) . assuming that @xmath244 changes sign before @xmath243 implies that @xmath138 must have a local strict minimum at some point @xmath245 , at which @xmath246 . then , since @xmath247 , there must be another later point @xmath248 at which @xmath249 . however , we have @xmath250 a contradiction . [ lem : prop_s_+ ] let @xmath232 . then @xmath251 vanishes at least once and , for the first positive root @xmath252 of @xmath251 , we have @xmath253 . the set @xmath225 is open . the proof follows the presentation of @xcite and it goes as follows . we denote for simplicity @xmath236 and @xmath237 . if @xmath254 , then @xmath255 and @xmath256 for all @xmath257 . hence , let @xmath258 . first we claim that @xmath244 must vanish . otherwise @xmath138 is decreasing whenever @xmath259 and increasing if @xmath260 . in both cases , @xmath138 has a positive limit @xmath261 at infinity . using the equation , we see that @xmath262 , hence @xmath263 . next , following @xcite , we look at @xmath264 which solves the equation @xmath265 at infinity we have @xmath266 , hence @xmath267 , which easily leads to a contradiction . we conclude that @xmath244 vanishes and we denote by @xmath252 its first root . next we distinguish two cases . first , if @xmath268 , then @xmath269 and , by , @xmath256 for all @xmath241 and in particular @xmath253 . second , if @xmath270 , then @xmath244 is negative for small @xmath271 ( due to the fact that @xmath272 ) . since @xmath273 ( otherwise @xmath274 and @xmath138 is constant ) , we see that @xmath138 must attain a local minimum at @xmath252 . from the equation , this yields @xmath275 and hence @xmath276 , which implies @xmath253 . finally we prove that @xmath225 is open . we already know that @xmath277 . let then @xmath278 . for @xmath279 in a neighborhood of @xmath183 , @xmath280 possesses a local minimum at @xmath281 at which @xmath282 . since @xmath283 , we get @xmath284 for all @xmath285 and some @xmath286 , and therefore @xmath287 . let now @xmath288 be the unique solution to @xmath289 the main remark is that @xmath290 is the variation of @xmath138 with respect to the initial condition @xmath217 , which implies the following result assume that @xmath227 and that @xmath291 when @xmath292 . then there exists @xmath286 such that @xmath293 and @xmath294 . this is ( * ? ? ? 3(b ) ) and the argument goes as follows . choose first @xmath295 such that @xmath296 on @xmath297 , and then @xmath298 such that @xmath299 for all @xmath300 . finally , choose @xmath301 such that @xmath302 and @xmath303 . for @xmath304 , we then have @xmath305 and @xmath306 . the function @xmath307 is negative at @xmath308 with @xmath309 . if @xmath310 or if @xmath287 , then @xmath311 must tend to @xmath103 or become positive at some point , and therefore it must have a first local ( strict ) minimum at some point @xmath312 , with @xmath313 for all @xmath314 . from the equation we can then write @xmath315 for some @xmath316 . here @xmath317 because of our assumption that @xmath318 and @xmath319 by choice of @xmath320 . now @xmath321 and @xmath322 , which is a contradiction . the argument is the same for @xmath323 . the lemma implies that any @xmath227 for which @xmath324 diverges to @xmath325 must be an isolated point . now , if we can prove that @xmath326 for all @xmath227 then we would clearly be done . indeed , we know that @xmath225 and @xmath230 are open and they can only be separated by points in @xmath327 . but the lemma says that points in @xmath327 can only serve as a transition between @xmath225 below and @xmath230 above . therefore , there can be only one such transition , and we conclude that @xmath327 is reduced to one point . so our goal will be to prove that all the points in @xmath327 have @xmath326 . our argument will be based on the wronskian identity @xmath328 for various functions @xmath329 s . a simple calculation shows that @xmath330 @xmath331 and @xmath332 these three test functions correspond respectively to variations of @xmath166 using multiplication by a constant , dilations and translations . [ lem : v_has_one_root ] for every @xmath227 , the function @xmath288 vanishes exactly once . for simplicity we denote again @xmath236 and @xmath333 . assume on the contrary that @xmath334 for all @xmath335 ( if @xmath192 does not vanish it must be strictly positive since it can not have double zeroes ) . using with @xmath336 , we find @xmath337 and , therefore , @xmath338 is decreasing and vanishes at @xmath339 , hence @xmath340 . since @xmath341 , we conclude that @xmath342 for @xmath343 and thus @xmath344 . as we have said @xmath345 vanishes at @xmath339 and it is decreasing , hence @xmath346 for @xmath343 . however @xmath347 decays exponentially at infinity and hence @xmath348 for @xmath271 large enough . using , this proves that @xmath349 . therefore @xmath350 diverges to @xmath325 exponentially at infinity , which contradicts the assumption that @xmath351 . next , the proof that @xmath192 can only vanish once is the same as in @xcite . indeed , start with @xmath352 at which the solution @xmath353 is stationary . the function @xmath354 vanishes exactly once since @xmath355 decreases from @xmath356 to 0 . taking @xmath357 and using that @xmath358 can not have double zeroes gives that @xmath192 can vanish at most once . we are now able to show that @xmath192 and @xmath350 diverge to @xmath325 . let @xmath227 . then @xmath359 and @xmath360 diverge to @xmath325 as @xmath160 . for simplicity we denote again @xmath236 and @xmath333 . let @xmath361 be the unique root of @xmath192 , at which we must have @xmath362 . let now @xmath363 , which is chosen such that @xmath364 vanishes at the zero @xmath361 of @xmath192 . recall that @xmath240 and @xmath365 for all @xmath241 , by lemma [ lem : prop_s_- ] . then we have from @xmath366 next we remark that the function @xmath367 vanishes both at @xmath339 and at @xmath368 . therefore , its derivative must vanish at least once on @xmath369 , that is , @xmath370 vanishes before @xmath361 . since @xmath138 is strictly decreasing , and since @xmath371 vanishes only once by lemma [ lem : prop_f ] , we conclude that @xmath372 is negative for @xmath373 , hence @xmath367 is strictly decreasing after @xmath361 . in particular , @xmath374 since @xmath375 and @xmath376 go to 0 exponentially at infinity , we conclude that @xmath377 must diverge . more precisely , we have for @xmath271 large enough @xmath378 since @xmath379 and by . hence @xmath380 and after integrating we get @xmath381 . as a consequence , @xmath192 diverge to @xmath325 exponentially . finally using that @xmath382 for large @xmath271 ( since @xmath383 ) , we conclude that @xmath350 diverges to @xmath325 exponentially as well . as we have explained , the fact that all the points @xmath227 are non - degenerate with @xmath326 implies uniqueness , and concludes the proof of theorem [ thm : radial ] . let @xmath213 be the nls polynomial , which is such that @xmath384 . we have @xmath385 which is positive decreasing on @xmath386 . noticing that @xmath387 we find @xmath388 this is negative for @xmath389 . let now @xmath208 , and consider the function @xmath215 in . note that @xmath390 , and hence @xmath391 is positive for small @xmath392 . on the other hand , @xmath393 hence @xmath215 must vanish at least once on the interval @xmath218 . next we remark that @xmath394 . \label{eq : calcul_vanish_once}\end{aligned}\ ] ] note that , for @xmath395 , @xmath396 and @xmath397 from this we conclude that @xmath398 when @xmath399 hence @xmath215 can only vanish on @xmath205 . on this interval @xmath214 is strictly decreasing , as we have shown before , hence @xmath215 can only have one root . the linearized operators at our solution @xmath400 are defined by @xmath401 and @xmath402 more precisely , the linearized operator is @xmath403 . the operator @xmath404 describes variations with respect to @xmath38 for real functions , whereas @xmath405 is related to the invariance of our problem under multiplication by a phase factor . it is easy to verify that both @xmath404 and @xmath405 are self - adjoint operators on @xmath115 , with domain @xmath406 and form domain @xmath37 . the main result of this section is [ thm : non - degenerate ] in @xmath115 , we have @xmath407 and @xmath408 . the operators @xmath404 and @xmath405 both satisfy the perron - frobenius property that their first eigenvalue , when it exists , is necessarily non - degenerate with a positive eigenfunction . this follows for instance from the fact that @xmath409 and from harnack s inequality ( * ? ? ? 6.4 , thm 5 ) which gives the strict positivity of eigenfunctions . since @xmath410 and @xmath38 is positive , we deduce that it must be the first eigenfunction of @xmath405 , and that it is non - degenerate . thus @xmath408 . next , in dimension @xmath47 , we know that @xmath411 and @xmath412 has a constant sign . hence , @xmath103 is the first eigenvalue of @xmath404 and it is non - degenerate which implies @xmath413 . the argument for @xmath404 in dimension @xmath161 is slightly more complicated . a lengthy but straightforward computation shows that @xmath414 since @xmath415 , the multiplier @xmath416 is bounded away from @xmath103 and we deduce that @xmath417 if and only if @xmath418 . hence @xmath419 if and only if @xmath420 . the argument is now classical . the operator @xmath421 commutes with space rotations and it may be written as a direct sum @xmath422 corresponding to the decomposition @xmath423 with @xmath424 the @xmath425th eigenspace of the laplace - beltrami operator on the sphere @xmath426 . in dimension @xmath20 , @xmath427 where @xmath428 are the usual spherical harmonics . the formula for @xmath429 is @xmath430 with an appropriate boundary condition at @xmath339 ( neumann for @xmath431 and dirichlet for @xmath432 ) . each @xmath429 has the perron - frobenius property . since @xmath433 and @xmath434 has a constant sign , we conclude that @xmath103 is the first eigenvalue of @xmath435 and it is non - degenerate , thus @xmath436 . next , for @xmath437 , we simply use that @xmath438 in the sense of quadratic forms , which shows that the first eigenvalue of @xmath429 must be positive and hence @xmath439 . finally , for @xmath431 , the operator @xmath440 was studied in theorem [ thm : radial ] , where we proved that the unique solution to @xmath441 with @xmath442 diverges exponentially at infinity , hence can not be in @xmath443 . we have therefore shown that @xmath444 since @xmath445 , this says that @xmath446 which concludes our proof of theorem [ thm : non - degenerate ] . we want to prove the existence of a branch of solutions to the dirac equation @xmath447 which can be rewritten for @xmath448 as @xmath449 here the parameters are chosen as @xmath450 with @xmath451 fixed such that @xmath452 . it will be convenient to introduce the new fields @xmath453 then , imposing the special form @xmath454 with real - valued functions @xmath455 and @xmath456 , and using and , we obtain the following system @xmath457 which is equivalent to for functions of the above form . next , we consider the following rescaling @xmath458 and we find @xmath459 finally , denoting @xmath460 the perturbative parameter and recalling that @xmath461 we obtain @xmath462 with @xmath463 @xmath464 when @xmath465 , we obtain the system of equations @xmath466 which is equivalent to with @xmath174 and was studied in @xcite . we introduce the map @xmath467 defined by @xmath468 here the spaces @xmath469 are the projections of the usual sobolev spaces @xmath470 to the sector of minimal total angular momentum ( they in particular contain a boundary condition at @xmath339 ) , whereas @xmath471 is the usual projection of @xmath472 to the subspace of radial functions . in what follows , we let @xmath473 , @xmath474 and @xmath475 . solving the system is equivalent to solving @xmath476 . we construct a branch of solutions , by means of an implicit function - type argument . the first step is to prove that @xmath477 is a smooth operator from @xmath478 into @xmath479 . the proof is tedious but elementary . it relies on the fact that @xmath99 is an algebra and that all the functions appearing in the definition of @xmath477 are polynomials in the unknowns @xmath484 . also , it uses that @xmath485 is an isomorphism from @xmath486 to @xmath487 . indeed , this map is related to the restriction of the dirac operator @xmath488 to functions of the form . since @xmath489 , the operator is an isomorphism from @xmath99 to @xmath490 and the same holds in the radial subspaces @xmath486 and @xmath487 . similarly , the operator @xmath491 is the fourier multiplier with the matrix @xmath492 and we claim that @xmath493 for all @xmath494 and all @xmath495 . this estimate shows that the corresponding map is bounded on @xmath99 , as needed . in order to prove , we recall that @xmath496 changing @xmath497 into @xmath498 , it suffices to show that x@xmath499 for all @xmath500 , with @xmath501 the matrix @xmath502 has two real positive eigenvalues @xmath503 and @xmath504 with @xmath505 . hence @xmath506 as a consequence , @xmath507 , for all @xmath494 and for all @xmath495 . finally , the fact that @xmath482 is also continuous can be proved with the same arguments . first we prove that @xmath513 is a one to one operator . let @xmath514 a nontrivial solution to @xmath515 . then , since @xmath516 is bounded , @xmath517 and @xmath518 solves @xmath519 a calculation shows that the radial function @xmath329 solves @xmath520 where @xmath404 is the linearized operator defined in . since the restriction of @xmath404 to radial functions is invertible by theorems [ thm : radial ] and [ thm : non - degenerate ] , we conclude that @xmath521 and @xmath513 is one - to - one . next , we observe that @xmath513 can be written as the sum of two linear operators @xmath522 as we have already said before , the upper part of the operator @xmath523 is a restriction of the dirac operator @xmath524 and it is an isomorphism from @xmath486 to @xmath487 , by definition of these spaces . on the other hand , @xmath525 is compact . therefore @xmath513 is a one - to - one operator that can be written as a sum of an isomorphism and a compact perturbation and it is then an isomorphism . as a conclusion , we can apply the implicit function theorem to find that there exists @xmath526 and a function @xmath527 such that @xmath528 and @xmath529 for @xmath530 . this concludes the proof of theorem [ thm : dirac ] . , _ symmetry of positive solutions of nonlinear elliptic equations in @xmath533 _ , in mathematical analysis and applications , part a , vol . 7 of adv . in math . , academic press , new york - london , 1981 , pp . 369402 . height 2pt depth -1.6pt width 23pt , _ the concentration - compactness principle in the calculus of variations . the locally compact case , part ii _ , ann . h. poincar anal . non linaire , 1 ( 1984 ) , pp . 223283 . , _ nonlinear dispersive equations _ , vol . 106 of cbms regional conference series in mathematics , published for the conference board of the mathematical sciences , washington , dc , 2006 . local and global analysis .

we prove the uniqueness and non - degeneracy of positive solutions to a cubic nonlinear schrdinger ( nls ) type equation that describes nucleons . the main difficulty stems from the fact that the mass depends on the solution itself . as an application , we construct solutions to the @xmath0@xmath1 model , which consists of one dirac equation coupled to two klein - gordon equations ( one focusing and one defocusing ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: the recent discovery at the large hadron collider ( lhc ) of a spin zero resonance @xcite , most certainly the standard model ( sm ) higgs boson , have already given some clues of what kind of new physics we need to proceed beyond . besides , we know that the sm is not the last answer to the questions brought by the huge endeavor carried over the last three decades of researching in particle physics . some of the new challenges involve neutrino s mass and mixing , the dark matter problem , the matter - antimatter asymmetry , among others . this motivates us to search for new broader models and theories for attacking these puzzles in the most simple and easiest way . one such possibility is the enlargement of the gauge group symmetry , implying in new interactions and particle spectrum . here we focus on one of these gauge extensions of the sm , which can be realized at some tev scale , the @xmath2 @xcite gauge group , which we call 341 for short . the 341 model is built as a simple extension of the electroweak symmetry gauge group @xmath3 of sm . it was first suggested in the work of voloshin @xcite , who considered only the leptonic sector . further developments of the model including quarks and electroweak currents are presented in ref . the most attractive feature of this model is the fact that the two chiralities of the lightest leptons are part of a fundamental representation of the gauge group @xmath4 , unifying each lepton family in a single multiplet . also , the explanation of the family replication comes from anomaly cancellation which , along with qcd asymptotic freedom , requires that only three fermion families might be present in the spectrum , as in the @xmath1 ( 331 ) models @xcite . besides , new particles with distinct signature are predicted to appear close to the electroweak scale , like doubly charged vectors and scalars , with some of them carrying two units of lepton number ( the so called bileptons ) , among other features also shared with the 331 models . in the current literature there are 341 models with or without exotic electric charges @xcite . here we deal with the 341 model with exotic electric charges , where we have reduced the scalar sector to only three scalar quadruplets , which are enough to generate the exact number of goldstone bosons needed to break the 341 symmetry to the @xmath5 . in turn , we have disposed of one scalar quartet and the scalar decuplet from the original version . these would be responsible for giving mass for the leptons and some of the quarks , but in our scheme they are not mandatory , since we are going to generate those masses by effective operators , as in ref . our approach poses a considerable reduction in the physical scalar spectrum , which is more suitable for phenomenological studies . moreover , we compute the running coupling of the abelian gauge group , @xmath6 , and show that there exists a landau pole in this model , similar to the minimal 331 model @xcite . nevertheless , in this work the 341 model breaks into a 331 model with right - handed ( @xmath7 ) @xcite , or a 331 model with left - handed heavy neutral fermions ( @xmath8 ) @xcite , and this means that such embedding of the lower symmetry into the 341 model implies a landau pole for these 331 models too . the advantage of such embedding is that such models are known to possess cold dark matter candidates @xcite , which are automatically incorporated into the body of this 341 model . besides , any mechanism that employs non - renormalizable effective operators suppressed by a scale much higher than tev is no longer an option in this scheme , which is the case of neutrino mass generation in some scenarios @xcite . nonetheless , we will show that all fermions , including neutrinos , gain mass from such operators even when the suppression scale is as low as the tev scale gev was needed to implement his scheme , much higher than what we obtain for our model . ] . this work is organized as follows : section [ sec1 ] , we introduce the 341 model with exotic electric charges . in section [ sec2 ] , we discuss the perturbative limit of the model , where we compute the value of its landau pole , with the intent of measuring the energy scale at which the model loses its perturbative character . in section [ sec3 ] , the fermion mass spectrum is obtained , including the neutrinos . finally , in section [ sec5 ] , we draw our conclusions . the 341 model is a gauge extension of the electroweak gauge group of sm , that possesses 12 gauge bosons besides the known 8 gluons of strong interactions and the electroweak , @xmath9 , @xmath10 and @xmath11 of the sm . since the strong gauge group @xmath12 is kept intact under the symmetry breakdown to @xmath5 , we are going to simply omit it throughout this work . in the following we present the field content of the 341 model and write up its scalar potential and the yukawa lagrangian , which are going to be relevant to our purposes in this work . in order to assign the fermionic content of the model , it is useful to recall that the electric charge operator is written as a linear combination of the diagonal generators of the electroweak gauge group , @xmath13 . this allows us to write the electric charge assignment for the fields in the fundamental representation of @xmath14 as , @xmath15 and similarly for the anti - fundamental representation , exchanging the sign of the diagonal generators of @xmath16 , @xmath17 , @xmath18 and @xmath19 . here , @xmath20 and @xmath21 are parameters to be fixed according to the field distribution in the quadruplets , and @xmath22 is the corresponding @xmath23 charge . considering the sm content , which we should recover in the symmetry breakdown process , the first two components of the left handed lepton quartet should recover the known doublet structure , meaning that the first component is a neutrino and the second a negatively charged lepton . moreover , as we are interested in accommodating the associated charge conjugate states for these leptons in the same multiplet , recovering the @xmath7 model in the first symmetry breaking of 341 , the third component of the quartet should be the charge conjugate of the neutrino and the fourth one should be the positive charged partner of the negative lepton , @xmath24 . this association implies that @xmath25 , @xmath26 and @xmath27 , which reduces the operator in eq . ( [ cargas1 ] ) to , @xmath28 then , we can arrange all the leptons ( including right - handed neutrinos ) in quartets of @xmath29 symmetry as follows , @xmath30 where @xmath31 @xmath32 @xmath33 , @xmath34 and the symbol @xmath35 indicates the respective representation for the multiplet under the 341 symmetry . for the first generation of quarks , we can assign the left - handed fields to the fundamental representation of the @xmath29 symmetry , while the right - handed ones transform trivially under this symmetry , @xmath36 where @xmath37 and @xmath38 are the up and down quarks , respectively , @xmath39 and @xmath40 are the new exotic quarks with electric charges , @xmath41 and @xmath42 , respectively . the second and third families of left - handed quarks are arranged in the anti - quartet representation of the @xmath29 symmetry , and right - handed quarks in the singlet representation , @xmath43 where @xmath44 @xmath45 and @xmath46 are exotic quarks with electric charges @xmath47 and @xmath48 , respectively . this arrangement in different representations for the quark families is a consistency requirement such that the model be free of anomalies . next we explore the scalar content of the model . originally @xcite , the 341 model was built with four scalar quartets in order to engender the symmetry breakdown to @xmath5 , give mass to all quarks and avoid mixing among ordinary and exotic quarks . also , one scalar decuplet was introduced so as to generate charged lepton masses . one of the purposes of this work is to show that we can diminish the scalar content and still have all desirable properties concerning the symmetry breakdown and the fermion masses ( including neutrinos ) . here we present the minimal scalar content which will allow us to accomplish this goal , @xmath49 which transform as , @xmath50 , @xmath51 and @xmath52 . the desired pattern of symmetry breakdown can be obtained if we assume that the following neutral components develop nontrivial vacuum expectation value ( vev ) , @xmath53 , @xmath54 and @xmath55 . the vev @xmath56 is responsible for the first step in breaking the 341 symmetry to 331 , while @xmath57 breaks the 331 symmetry to 321 , the sm , and the final breaking to @xmath5 is provided by @xmath58 . in this way we can impose , @xmath59 gev . indeed , as we are going to show later , @xmath60 and @xmath57 should be no more than few tev or we lose perturbativity , which makes this model interesting in the sense it can be promptly tested at lhc or discarded soon . the reason we choose the @xmath61 quadruplet to develop vev only in the third component is related to the fact that we do not want mixing among ordinary and exotic quarks on the yukawa lagrangian , which guarantees the usual ckm mixing in the quark sector . of course this is only possible if we also add a new symmetry to the model , in our case this is going to be a @xmath62 discrete symmetry that we implement later on . with the scalar and fermionic content defined , we can write down the lagrangian of the model , invariant under the gauge symmetry and the additional discrete symmetry mentioned above . next , we add this symmetry and present the yukawa and scalar lagrangian of the model , also defining our gauge boson spectrum . let us impose a discrete @xmath62 symmetry that will be suitable to avoid the mixing among ordinary and exotic quarks but also allow for an appropriate scenario for generating fermion masses through effective operators , as we will see later . the @xmath63 charges are @xmath64 @xmath65 @xmath66 , with @xmath67 , where @xmath68 is the identity element of the group . we then assign @xmath62 charges to the 341 model fields given in table [ table1 ] . .@xmath62 symmetry transformation properties for the 341 fields . those fields not present in this table transform trivially under this discrete group . [ cols="<,<,<,<,<,<,<",options="header " , ] in table [ tab : tab1 ] we have also shown the values of @xmath69 and @xmath70 when the exotic quarks , @xmath71 ( with @xmath72 ) , are decoupled , in which case we used the new value for the rge abelian coefficient in eq . ( [ bi341 ] ) , @xmath73 . we observe that the highest values for the landau pole scale are reached as the 331 symmetry breaking scale approaches that of 341 model , and can be as high as @xmath74 tev when the exotic quarks are decoupled , although perturbativity is lost about half that scale . we do not go beyond @xmath75 tev because our results start becoming inconsistent in the sense that @xmath70 gets smaller than @xmath76 , in other words , we lose perturbativity at lower energies than 341 symmetry breaking scale . this has a deep implication for such class of models , imposing that they must manifest somewhere close to the electroweak scale and may be promptly tested at lhc and/or ilc . finally , the economical 331 model @xcite , along with an extra scalar triplet and three singlet scalars , can be embedded in the model here studied . the same is true for the 331 model when the right - handed neutrino is exchanged by a new neutral fermion , @xmath77 @xcite . this may have interesting implications for these models as well as for the 341 model developed here . for the 331 models , it is automatic that they are not valid up to some arbitrary high energy scale , being limited to the landau pole of our 341 model , @xmath78 tev . such a low cutoff scale is an obstacle for some neutrino mass generation mechanisms in these models @xcite , a problem we address here with an appropriate discrete symmetry and non renormalizable effective operators . in what concerns the advantages for the 341 model , we can mention the fact that some of these models possess natural dark matter candidates @xcite , which can be explored in the context of our 341 model . besides , since the scalar resonance discovered at lhc @xcite can be perfectly adjusted into the 331 models @xcite , its phenomenology can be recovered by our model as well , with some room for extra new gauge bosons and scalars to be tested at lhc . besides , it is possible that the new neutral singlet fields ( under the 331 symmetry ) , may increase the number of dark matter candidates , a question to be investigated elsewhere . next we show that the fermion mass spectrum can be reproduced by means of non renormalizable effective operators , once the @xmath62 symmetry introduced in the last section can be implemented in the model . now that we have established the highest energy scale that works as a cutoff for this 341 model , we can make use of the @xmath62 symmetry imposed through the assignments in table [ table1 ] in order to generate the observed fermion mass spectrum . in the original version of 341 model , this spectrum was obtained by means of four scalar quartets plus a decuplet . the presence of a natural cutoff scale , @xmath74 tev , due to the existence of a landau pole at that scale , turns this scenario suitable for introducing non renormalizable effective operators to get realistic fermion masses and mixing without plaguing the model with too many scalar multiplets . some of the sm quarks get contributions to their masses from the following dimension-6 effective operators : @xmath79 where again , @xmath80 and @xmath72 label the family number ( or generation ) . considering these operators and the yukawa lagrangian , eq . ( [ yukawa ] ) , the up - type quarks mass matrix , in the basis @xmath81 @xmath82 @xmath83 , is written as , @xmath84 while the down - type quarks mass matrix , in the basis ( @xmath85 @xmath86 @xmath87 , is @xmath88 it is remarkable that , by choosing @xmath89 , all entries in these matrices are proportional to the sm symmetry breaking scale , @xmath90 gev , tuned by the dimensionless couplings . in other words , by appropriately tuning these couplings we can recover the sm results concerning quark masses and mixing without much effort . for example , if we take the above matrices as diagonal ( just for the sake of illustration ) we have : @xmath91 where @xmath92 , @xmath93 , @xmath94 , @xmath95 , @xmath96 e @xmath97 are the masses for the up , charm , top , down , strange and bottom quarks , respectively . for @xmath98 tev and @xmath99 tev , and considering the known quark masses , @xmath100 we obtain the following values for the respective yukawa couplings , @xmath101 , @xmath102 , @xmath103 , @xmath104 , @xmath105 , @xmath106 . in what concerns the exotic quarks , their masses are obtained purely from the yukawa lagrangian , eq . ( [ yukawa ] ) , providing the mass for @xmath107 and @xmath108 , @xmath109 it also leads to the exotic quarks mass matrices in the basis @xmath110 @xmath111 , @xmath112 and also in the basis ( @xmath113 @xmath114 , @xmath115 these matrices can be conveniently diagonalized , leaving no massless quarks in the spectrum . instead , just for illustration , we can assume them diagonal and conservatively using , as reference , the lower values obtained by the cms collaboration @xcite for a sequential fourth family ( @xmath116 gev and @xmath117 gev , which we use for the exotically charged quarks as well ) , imply that all yukawa couplings for the exotic quarks have to be approximately bigger than 0.45 . as we saw earlier , charged leptons do not receive mass from yukawa lagrangian , but there is an effective dimension-5 operator that does this job , @xmath118 implying the following mass relation for charged leptons , @xmath119 this term provides an sm - like mass term for charged leptons once we assume @xmath120 , with @xmath121 playing the role of a yukawa coupling . considering the set of values for the mass scale parameter we have been using and plugging in the known masses for the leptons , we obtain the following values for the @xmath121 couplings : @xmath122 , @xmath123 , @xmath124 , which are comparable to those of sm . as for the neutrinos , the lowest order effective operator that engender their masses is dimension-9 , @xmath125 + + h.c.\ , , \label{numassop}\end{aligned}\ ] ] where we are supposing a diagonal basis for simplicity , yielding , @xmath126 with the scales as before , we see that neutrino masses are proportional to @xmath127 gev , which demands tiny values for @xmath128 for a neutrino of @xmath129 ev , not a great improvement if we compare with sm plus right handed neutrinos forming dirac mass terms , but it accounts for sub - ev neutrino masses as well . this procedure has shown to be effective in producing the desired fermion mass spectrum in the 341 model . nevertheless , there is an unwanted side effect that has to be circumvented , it concerns the presence of effective operators that would trigger fast proton decay . the lowest order effective dimension-8 operator that leads to proton decay is , @xmath130 where an anti - symmetric color contraction is implicit . this operator contains the following term , @xmath131 which yields the decay channel , @xmath132 . the simplest way to avoid proton decay is to impose a discrete @xmath133 symmetry over the quark fields , such that , @xmath134 that guarantees the proton stability at any order without compromising any of the results we have obtained . in this work , we have built a 341 gauge model with a lesser scalar content than usual . the original version contains four scalar quartets and a decuplet , while we reduced this sector to only three scalar quartets . that was possible because we have computed the beta function for its abelian gauge coupling , showing that a landau pole exists at a scale not greater than @xmath135 tev . this suggests that some underlying structure could emerge before that scale is reached , allowing us to use non renormalizable effective operators to provide the known mass spectrum for the fermions , which otherwise would need the presence of those extra scalars . it is amazing that such a low scale constrains this class of model at the current energies reachable at colliders such as lhc and/or ilc . besides , a reduced spectrum makes it most attractive for phenomenological studies , competing with supersymmetry , which has not shown any sign of existence till now . it has to be remarked that the existence of a cutoff scale as low as few tev , may circumvent the long standing hierarchy problem . actually , it would appear meaningless to address this problem once the theory enters into a strongly coupled regime at tev scale , where we no longer expect that the sm could be sensitive to . also , an interesting outcome of our scenario is that a 331 models with neutral fermions , when embedded in this 341 model , have to be faced as possessing this same landau pole , a result that is not obvious in this class of 331 models , where the landau pole is believed to appear much beyond the planck scale @xcite . this may have some impact on the phenomenology of such models like the impossibility of implementing neutrino mass under the assumption of existing an arbitrarily high cutoff energy scale for these models . on the other hand , knowing the advantages of these models concerning their dm candidates and higgs phenomenology , we can guarantee the same outcome for our model , besides some new extra particles that can be investigated under the light of current experiments at lhc . this work was supported by conselho nacional de desenvolvimento cientfico e tecnolgico - cnpq ( a.g.d . grant 03094/2013 - 3 , c.a.s.p . grant 306923/2013 - 0 , p.s.r.s . grant 305390/2012 - 0 ) , a.g.d is also supported by the grant 2013/22079 - 8 , so paulo research foundation ( fapesp ) , and coordenao de aperfeioamento de pessoal de nvel superior - capes ( p.r.d.p . phd program scholarship ) . here we derive the mass spectrum for the scalar fields . in order to do that we consider the following shift in the neutral scalars by their vevs , @xmath136 we then obtain the conditions for the minimum of the scalar potential in eq.([pot ] ) , @xmath137 with these constraints we can build the cp - even neutral scalars mass matrix in the basis ( @xmath138 ) , @xmath139 according to the results in section [ sec2 ] , it is natural to assume @xmath140 , which we will use to simplify the computation of eigenvalues and eigenvectors for the cp - even mass matrix above . this diagonalization is performed by employing perturbation theory to the second order , leading to the following mass eigenvalues , @xmath141 where @xmath142 @xmath143 ^{2}}{4c_{1 } % \left[\lambda _ { 5}^{2}-\left ( \lambda _ { 1}-\lambda _ { 3}\right ) \left ( \lambda _ { 3}-\lambda _ { 1}+\sqrt{\left ( \lambda _ { 1}-\lambda _ { 3}\right ) ^{2}+\lambda _ { 5}^{2}}\right ) \right]},\ ] ] @xmath144 @xmath145 ^{2}}{4c_{3 } % \left [ \lambda _ { 5}^{2}+\left ( \lambda _ { 1}-\lambda _ { 3}\right ) \left ( \lambda _ { 1}-\lambda _ { 3}+\sqrt{\left ( \lambda _ { 1}-\lambda _ { 3}\right ) ^{2}+\lambda _ { 5}^{2}}\right ) \right ] } , \ ] ] while the respective eingenstates are obtained by considering first order perturbation theory only , @xmath146 observe that the state @xmath147 is a goldstone boson and @xmath148 is identified with the higgs boson , once it is the only neutral scalar to get mass at the electroweak scale , besides coming from a doublet under @xmath149 symmetry similar to the sm one . the remaining scalars , @xmath150 and @xmath151 , are heavier than the higgs , with masses proportional to the 341 symmetry breaking scale . the neutral cp - odd scalars , @xmath152 @xmath153 @xmath154 and @xmath155 , are all massless . together with @xmath156 , they complete the set of five neutral goldstone modes eaten by the five neutral massive gauge bosons . as for the simply charged scalars , their mass matrices can be divided into two . the mass matrix in the basis ( @xmath157 , @xmath158 ) , is given by , @xmath159 which , upon diagonalization , leads to the mass eigenvalues , @xmath160 and the respective eigenvectors , @xmath161 the second mass matrix , in the basis ( @xmath162 , @xmath163 ) , is @xmath164 yielding the mass eigenvalues , @xmath165 and respective eigenvectors , @xmath166 in the above results @xmath167 and @xmath168 , together with @xmath169 and @xmath170 ( which are already massless eigenstates ) , are all massless and represent the goldstone bosons eaten by the eight simply charged gauge bosons , while the scalar fields , @xmath171 and @xmath172 are massive and remain in the physical spectrum . concerning the doubly charged scalars , we have the mass matrix in the basis @xmath173 , @xmath174 that leads to a null eigenvalue , @xmath175 , and @xmath176 whose eigenvectors are , @xmath177 the scalar @xmath178 is obviously a goldstone boson eaten by the doubly charged gauge boson , while @xmath179 remains in the spectrum , being a characteristic signature in this class of models . below we present the numerical values of couplings chosen to give a higgs mass of 126 gev as well as the masses of all scalars when @xmath180 tev . @xmath181 which yield the cp - even scalar masses , @xmath182 while the simply charged scalars acquired the following masses . @xmath183 finally , the doubly charged scalar gets a mass , @xmath184 we have to keep in mind that the above particles have non - standard couplings to fermions and it is not straightforward to compare them with the existing bounds from colliders , although they are heavy enough to be safe and tested at the next lhc run . a more careful and thorough analysis should be made elsewhere . the effect of 341 to 331 symmetry breaking driven by the vev @xmath60 , causes fermionic quartets to decompose into triplets plus singlets . the 341 model presented here contains as a subgroup the 331 model with neutral fermions in the third component of the lepton triplets . there are two possibilities for this extra neutral fermion ( one for each family ) , it can be the partner right - handed neutrino of the left - handed neutrino in the triplet , or it can be a new neutral fermion not related to the ordinary neutrino . in order to see that , let us decompose the content of the 341 model into multiplets of 331 in the following way : * leptons @xmath185 where , from now on , the transformation properties under parentheses refer to the 331 symmetry . * first quark generation @xmath186 * second and third quark generations @xmath187 * scalars @xmath188 next , we present the breaking of 331 model promoted by @xmath57 that leads to the 321 sm content plus extra singlets . for the leptons we have , @xmath189 where @xmath190 and @xmath191 for the first quark generation we have , @xmath192 where @xmath193 and @xmath194 . for the second and third quark generations , we have @xmath195 where @xmath196 and @xmath197 . the scalar triplets decompose into one doublet and one singlet each , @xmath198 where @xmath199 @xmath200 @xmath201 @xmath202 @xmath203 and @xmath204 . the scalar doublet @xmath205 plays the role of the higgs doublet in the sm . so , we recover all the effective sm doublets and singlets with their respective quantum numbers , plus right - handed neutrinos , extra quarks and scalars . the gauge bosons obtain their masses from the lagrangian , @xmath206 which , after spontaneous symmetry breakdown , leads to the charged ( and non - hermitian ) gauge boson masses , @xmath207 it also leads to the mass matrix for the neutral gauge bosons , in the basis ( @xmath208 @xmath209 @xmath210 @xmath211 ) , @xmath212 where @xmath213 . this mass matrix has determinant equal to zero , which guarantees the existence of a massless gauge boson , that we can associate with the photon . diagonalizing the matrix using the simplifying ( and reasonable ) assumption @xmath214 , we obtain the neutral gauge boson masses , as given below : @xmath215 } { 8h_{w}\left ( 1 - 4s_{w}^{2}\right ) } \approx ( 2.2~{\mbox tev})^2\ , , \label{neutralgbmass}\end{aligned}\ ] ] with the respective eigenstates , @xmath216 , \\ z^{\mu } & = & c_{w}w_{3}^{\mu } -s_{w}\left [ \frac{t_{w}}{\sqrt{3}}\left ( -w_{8}^{\mu } -2\sqrt{2}w_{15}^{\mu } \right ) + \sqrt{1 - 3t_{w}^{2}}w_{x}^{\mu } % \right ] , \\ z^{\prime \mu } & = & \frac{\sqrt{3}}{3}\frac{\sqrt{1 - 3t_{w}^{2}}}{\sqrt{% 1 - 4s_{w}^{2}}}\left ( \sqrt{h_{w}}w_{8}^{\mu } -2\sqrt{2}\frac{s_{w}^{2}}{% \sqrt{h_{w}}}w_{15}^{\mu } \right ) + \frac{s_{w}\sqrt{1 - 3t_{w}^{2}}}{\sqrt{% h_{w}}}w_{x}^{\mu } , \nonumber \\ \\ z^{\prime \prime \mu } & = & \frac{\sqrt{3}\sqrt{1 - 4s_{w}^{2}}}{\sqrt{h_{w}}}% w_{15}^{\mu } + \frac{2\sqrt{2}s_{w}}{\sqrt{h_{w}}}w_{x}^{\mu } , \end{aligned}\ ] ] in the above equations we have used the sine of the electroweak mixing angle written as @xmath217 , and defined @xmath218 , @xmath219 and @xmath220 . from these results we see that @xmath221 , recovering the @xmath222 of the @xmath7 @xcite model , and the @xmath223 boson reproduces the well established massive neutral gauge boson of sm . we then have an extra neutral gauge boson , @xmath224 , which is heavier than @xmath225 , whose phenomenology may easily be probed at lhc and/or the next collider generation . g. aad et al . 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we build a gauge model based on the @xmath0 symmetry where the scalar spectrum needed to generate gauge boson and fermion masses has a smaller scalar content than usually assumed in literature . we compute the running of its abelian gauge coupling and show that a landau pole shows up at the tev scale , a fact that we use to consistently implement those fermion masses that are not generated by yukawa interactions , including neutrino masses . this is appropriately achieved by non renormalizable effective operators , suppressed by the landau pole scale . also , @xmath1 models embedded in this gauge structure are bound to be strongly coupled at this same energy scale , contrary to what is generally believed , and neutrino mass generation is rather explained through the same effective operators used in the larger gauge group . besides , their nice features , as the existence of cold dark matter candidates and the ability to reproduce the observed standard model higgs - like phenomenology , are automatically inherited by our model . finally , our results imply that this model is constrained to be observed or discarded soon , since it must be realized at the currently probed energy scale in lhc . electroweak gauge model , effective theory , fermion mass 12.60.cn , 12.60.fr , 14.60.pq , 14.60.st

You are an expert at summarizing long articles. Proceed to summarize the following text: quantum entanglement plays an important role in various fields of quantum information , such as quantum computation [ 1 ] , quantum cryptography [ 2 ] , quantum teleportation [ 3,4 ] , dense coding [ 5 ] and quantum communication [ 6 ] , etc . it has been recognized [ 7,8 ] that quantum teleportation can be viewed as an achievable experimental technique to quantitatively investigate quantum entanglement . there exists a class of states called maximally correlated states , which have an interesting property , i.e. , the ppt distillable entanglement is exactly the same as the relative entropy of entanglement [ 9 ] . both two - mode squeezed vacuum states and pair cat states [ 10 ] belong to this class . in continuous variable teleportation the entanglement resource is usually the two - mode squeezed state , or the einstein - podolsky - rosen ( epr ) states . continuous variable quantum teleportation of arbitrary coherent states has been realized experimentally by employing a two - mode squeezed vacuum state as an entanglement resource [ 7 ] . theoretical proposals of teleportation scheme based on the other continuous variable entangled states have already been discussed [ 11 ] . update now , little attention has been paid to the entanglement properties of the pair cat state and its possible application of quantum information . gou et al . have proposed a scheme for generating the pair cat state of motion in a two - dimensional ion trap [ 12 ] . in their scheme , the trapped ion is excited bichromatically by five laser beams along different directions in the x - y plane of the ion trap . four of these have the same frequency and can be derived from the same source , reducing the demands on the experimentalist . it is shown that if the initial vibrational state is given by a two - mode fock state , pair cat states are realized when the system reaches its steady state . their work motivates us to investigate the entanglement properties of the pair cat state and its possible application in quantum information processes , such as quantum teleportation . the motivation is two - fold : ( 1 ) the storage of continuous variable entangled states in two - dimensional motional states of trapped ions is feasible in current experimental techniques . ( 2 ) the mapping of steady state entanglement to optical beams is also realizable [ 13 ] . + on the other hand , quantum entanglement is a fragile nature , which can be destroyed by the interaction between the real quantum system and its environment . this effect , called decoherence , is the most serious problem for all entanglement manipulations in quantum information processing . there have several proposals for entanglement distillation and purification in continuous variable systems [ 14 ] . in this paper , we firstly investigate the relative entropy of entanglement of pair cat states in the phase damping channel , and show that the pair cat states can always be distillable in the phase damping channel . then , we explore possible application of pair cat states in quantum information processing , such as quantum teleportation . the fidelity of teleportation protocol in which the mixed pair cat state is used as a entangled resource , is analyzed . + this paper is organized as follows . in section ii , based on the exact solution of the master equation describing phase damping , we give the numerical calculations of relative entropy of entanglement for pair cat states in the phase damping channel and investigate the influence of the initial parameters of these states on the relative entropy of entanglement . in section iii , we analyze the fidelity of teleportation for the pair cat states by using joint measurements of the photon - number sum and phase difference . the influence of phase damping on the fidelity is discussed . a conclusion is given in section iv . the relative entropy of entanglement is a good measure of quantum entanglement , it reduces to the von neumann entropy of the reduced density operator of either subsystems for pure states . for a mixed state @xmath0 , the relative entropy of entanglement [ 15 ] is defined by @xmath1 , where @xmath2 is the set of all disentangled states , and @xmath3 $ ] is the quantum relative entropy . it is usually hard to calculate the relative entropy of entanglement for mixed states . recently , it has been shown [ 27 ] that the relative entropy of entanglement for a class of mixed states characterized by the following density matrix @xmath4 can be written as @xmath5 the separate state @xmath6 that minimizes the quantum relative entropy @xmath7 is @xmath8 where , @xmath9 and @xmath10 are orthogonal states of each subsystem . the states in eq.(1 ) are also called maximally correlated states and are known to have some interesting properties . for example , the ppt distillable entanglement is exactly the same as the relative entropy of entanglement [ 9 ] . + now , we consider the phase damping model . the density matrix satisfies the following master equation in the interaction picture @xmath11 with @xmath12 , \eqno{(5)}\ ] ] where , @xmath13(@xmath14 ) is the ith mode phase damping coefficient , and @xmath15 ( @xmath16 ) is the creation ( annihilation ) operator of the ith mode field . for arbitrary initial states described by the density matrix @xmath17 , the solution of eq.(4 ) can be obtained , @xmath18 where @xmath19 . \eqno{(7)}\ ] ] if we assume the initial density matrix @xmath17 is arbitrary two - mode continuous variable pure states , i.e. , @xmath20 where @xmath21 is two - mode particle number state . then , the time - evolution density matrix with the initial condition is calculated as @xmath22|n , m\rangle\langle{n^{'}},m^{'}| . \eqno{(9)}\ ] ] if the density matrix @xmath23 in eq.(9 ) can be expressed as the similar form of eq.(1 ) , i.e. , @xmath24 where , @xmath25 and @xmath26 are orthogonal states of modes 1 and 2 , the relative entropy of entanglement of @xmath23 in eq.(10 ) can be expressed as , @xmath27 and the separate state @xmath6 that minimizes the quantum relative entropy is @xmath28 in what follows , we investigate the relative entropy of entanglement of pair cat states in phase damping channel . firstly , we will briefly outline the definition of pair cat states and the closely related pair coherent states . for two independent boson annihilation operators @xmath29 , @xmath30 , a pair coherent state @xmath31 is defined as an eigenstate of both the pair annihilation operator @xmath32 and the number difference operator @xmath33 [ 16 ] , i.e. , @xmath34 where @xmath35 is a complex number and @xmath36 is a fixed integer . without loss of generality , we may set @xmath37 and the pair coherent states can be explicitly expanded as a superposition of the two - mode fock states , i.e. , @xmath38 where @xmath39^{-1/2}$ ] is the normalization constant and @xmath40 is the modified bessel function of the first kind of order @xmath36 . it has been suggested by reid and krippner that the non - degenerate parametric oscillator transiently generates pair coherent states , in the limit of very large parametric nonlinearity and high - q cavities[17 ] . recently , munro _ et al . _ have shown that the pair coherent states can be used to improve the detection sensitivity of weak forces[18 ] . pair cat states @xmath41 are proposed by gerry and grobe [ 10 ] , which are defined as superposition of two different pair coherent states , i.e. , @xmath42 , \eqno{(15)}\ ] ] where the normalization constant @xmath43 is given by @xmath44^{-\frac{1}{2}}. \eqno{(16)}\ ] ] it is easy to verify that the states @xmath41 are eigenstates of the operator @xmath45 with eigenvalue @xmath46 . gou et al . have proposed a scheme for generating the pair cat state of motion in a two - dimensional ion trap [ 12 ] . in their scheme , the trapped ion is excited bichromatically by five laser beams along different directions in the x - y plane of the ion trap . four of these have the same frequency and can be derived from the same source , reducing the demands on the experimentalist . it is shown that if the initial vibrational state is given by a two - mode fock state , pair cat states are realized when the system reaches its steady state . our following calculation show that pair cat states hold controllable entanglement . so , it is reasonable to regard the controlled two - dimensional trapped ion as a reliable source of entanglement . recent achievements concerning the transfer of entangled state have provided us a possible way to map the pair cat state of the motional freedom of two dimensional trapped ions into freely propagating optical fields [ 13 ] . when the free photon propagates in the optical fibre , one of the encountered decoherence mechanisms is the phase damping . in the following , we discuss the entanglement of pair cat states in phase damping channel . we assume that the initial state is prepared in pair cat states @xmath41 . by making use of eqs.(6 ) and ( 7 ) , we obtain @xmath47\xi^n\xi^{\ast{m}}(1+(-1)^ne^{i\phi})(1+(-1)^me^{-i\phi } ) } { \sqrt{n!m!(n+q)!(m+q)!}}|n+q , n\rangle\langle{m+q},m| . \eqno{(17)}\ ] ] the relative entropy of entanglement for @xmath23 is calculated as @xmath48 @xmath49 of the pair cat state as a function of the parameter @xmath50 and the degree of damping @xmath51 for @xmath52 with @xmath53 . [ fig.1 ] ] of the pair cat state as a function of the degree of damping @xmath51 and the parameter @xmath54 for @xmath52 with @xmath55 . [ fig.2 ] ] of the pair cat state as a function of the parameter @xmath50 and the parameter @xmath54 for @xmath52 with @xmath56 . [ fig.3 ] ] of the pair cat state as a function of the parameter @xmath54 for three values of @xmath57 and @xmath58 with @xmath56 and @xmath55 . [ fig.4 ] ] in numerical computations throughout this paper , the parameters @xmath59 , @xmath60 are chosen and the truncated photon number has been taken to be @xmath61 , the value of which is sufficiently large for numerical convergence . figures 1,2 and 3 show that the relative entropy of entanglement @xmath62 of the pair cat state increases with @xmath50 and decreases with degree of damping @xmath51 , and can be controlled by adjusting the relative phase @xmath54 . this results can be explained as follows : the entanglement of pair cat states heavily depend on the photon number distribution which can be modified by the relative phase via the interference . similar results have been obtained in ref.[19 ] . in fig.4 , we plot the relative entropy of entanglement @xmath62 of the pair cat state as a function of the relative phase @xmath54 for three values of the parameter @xmath36 . recently , hiroshima has numerically calculated the relative entropy of entanglement of two - mode squeezed vacuum states , defined by @xmath63|vac\rangle$ ] , in phase damping channel [ 20 ] . it has been shown [ 21 ] that the two - mode squeezed vacuum state in phase damping channel is always distillable ( and inseparable ) . in the following , we show that the pair cat states are always distillable ( and inseparable ) in phase damping channel . + for two - mode continuous variable states @xmath64 , @xmath65 where @xmath9 and @xmath10 are orthogonal particle number states of each subsystem and @xmath66 satisfy the normalization condition @xmath67 . the density matrix with the initial condition @xmath68 can be written as @xmath69|\phi_n,\psi_n\rangle\langle\phi_m,\psi_m| . \eqno{(20)}\ ] ] according to ref.[22 ] , if operator @xmath70 is not positive definite , there is always a scheme to distill @xmath23 . here , we find @xmath71 . if there are two nonzero @xmath72 , @xmath73 , it is always possible to choose four vectors @xmath74 , @xmath75 , @xmath76 , @xmath77 . then , we have @xmath78\textrm{re}(f_if^{\ast}_j),\ ] ] @xmath79\textrm{re}(f_if^{\ast}_j),\ ] ] @xmath80\textrm{im}(f_if^{\ast}_j),\ ] ] @xmath81\textrm{im}(f_if^{\ast}_j ) , \eqno{(21)}\ ] ] which satisfy @xmath82 @xmath83(f_if^{\ast}_j)\neq0,\ ] ] @xmath84(f_if^{\ast}_j)\neq0 , \eqno{(22)}\ ] ] eqs.(22 ) show that there is at least one of @xmath85 ( @xmath86 ) , which is negative for any finite @xmath87 . from the above , we obtain the following conclusion : the two - mode continuous variable state @xmath88 , in which @xmath9 and @xmath10 are orthogonal particle number states of each subsystem , is always distillable ( and inseparable ) in phase damping channel , if there are at least two nonzero values of coefficiences @xmath66 . obviously , pair coherent states and pair cat states which belong to the family of states in eq.(20 ) is always distillable in phase damping channel . it should be interesting to consider a slightly modified purification protocol similar to the protocol in ref.[14 ] to distill maximal entangled states from the mixed pair cat states or mixed pair coherent states due to phase damping . recently , cochrane et al . have presented a teleportation protocol by making use of joint measurements of the photon number sum and phase difference on two field modes [ 11 ] . various kinds of two modes entangled states used as the entanglement resource have been discussed and the respective teleportation fidelities have been investigated . in this section , we adopt the protocol of cochrane et al . to investigate the fidelity of teleportation , in which the pair cat state is utilized as the entanglement resource . the influence of phase damping on the fidelity is also discussed . + consider arbitrary target state sent by alice to bob @xmath89 where @xmath90 is the fock state . initially , alice and bob share the two - mode fields in the pair cat state . then , the total state is @xmath91}{\sqrt{n!(n+q ) ! } } of this teleportation protocol can be decomposed as two steps : alice makes a joint measurement of the photon number sum and phase difference of the target state and her component of the pair cat state ; the results of the joint measurement are sent to bob via the classical channel , and bob reproduce the target state after appropriate amplification and phase shift operations according to the results of the joint measurement . the joint measurement of the photon number sum and phase difference has attracted much attention due to its extensive potential applications both in quantum optics and quantum information [ 23,24 ] . in ref.[24 ] , luis et al . introduced the hermitian phase - difference operator @xmath92 with @xmath93 and @xmath94 where @xmath95 is an arbitrary angle . it is obvious that the joint measurement projects the two - mode quantum state onto @xmath96 . in ref.[25 ] , a physical scheme of the joint measurement of the photon number sum and phase difference of two - mode fields was proposed , in which only the linear optical elements and single - photon detector are involved . + if alice measure the number sum @xmath97 of the target and her component of the pair cat state with result @xmath98 , the state of the total system is projected onto @xmath99^{-1/2}\sum^{n}_{l = q } \frac{d_{n - l}\xi^{l - q}[1+(-1)^{l - q}e^{i\phi } ] } { \sqrt{l!(l - q)!}}|l\rangle_a|l - q\rangle_b|n - l\rangle_t , \eqno{(28)}\ ] ] where @xmath100}{l!(l - q ) ! } , \eqno{(29)}\ ] ] is the probability of obtaining the result @xmath98 . further measurement of phase difference with the result @xmath101 performed by alice will project bob s mode onto the pure state @xmath102^{-1/2 } \sum^{n - q}_{n=0}\frac{d_{n - q - n}(e^{-i\theta_{-}}\xi)^n[1+(-1)^ne^{i\phi } ] } { \sqrt{n!(n+q)!}}|n\rangle_b . \eqno{(30)}\ ] ] alice sends the values @xmath98 and @xmath101 to bob , and then bob amplifies his mode so that @xmath103 [ 26 ] and makes a operation @xmath104 for phase shifting his mode . the teleportation protocol is then completed and bob finally has the state in @xmath105^{-1/2 } \sum^{n - q}_{n=0}\frac{d_{n - q - n}\xi^n[1+(-1)^ne^{i\phi}]}{\sqrt{n!(n+q)!}}|n - q - n\rangle_b . \eqno{(31)}\ ] ] the fidelity of this protocol depends on the result @xmath98 and can be obtained as follows @xmath106^{-1}|\sum^{n - q}_{n=0 } \frac{|d_{n - q - n}|^2\xi^n[1+(-1)^ne^{i\phi}]}{\sqrt{n!(n+q)!}}|^2 . \eqno{(32)}\ ] ] the average fidelity defined by @xmath107 is @xmath108}{\sqrt{n!(n+q)!}}|^2 . \eqno{(33)}\ ] ] let the target state be a coherent state @xmath109 . then , the average fidelity can be expressed as @xmath110}{(n - n ) ! \sqrt{n!(n+q)!}}|^2 . \eqno{(34)}\ ] ] with @xmath52 and @xmath111 for different values of @xmath112 , @xmath113 ( solid line ) , @xmath114 ( dash line ) , @xmath115 ( dot line ) . [ fig.5 ] ] in fig.5 , we have plotted the average fidelity as the functions of the parameters @xmath35 for different values of @xmath112 . it is shown that the average fidelity increases with the value of @xmath35 . furthermore , the average fidelity defined above heavily depends on the teleported states . if the teleported state is a coherent state , the smaller the amplitude of coherent states , the higher the average fidelity . its physical reason can be elucidated by two facts : one fact is that this protocol works perfectly if the target is a number state [ 11 ] ; the other fact is that the smaller the amplitude of a coherent state , the closer the state distance between the coherent state and a specific number state , i.e . , the vacuum state . in what follows , we discuss the influence of phase damping on the fidelity of the above teleportation protocol . in this case , the state of the total system can be written as follows @xmath116[1+(-1)^me^{-i\phi } ] e^{-\gamma{t}(n - m)^2}}{\sqrt{n!m!(n+q)!(m+q)!}}|k\rangle_{tt}\langle{l}| \otimes|n+q , n\rangle\langle{m+q},m| . \eqno{(35)}\ ] ] after completing the protocol described above , bob finally achieves the state in his mode expressed by @xmath117^{-1}\sum^{n^{\prime}}_{n , m=0 } \frac{d_{n^{\prime}-n}d^{\ast}_{n^{\prime}-m}\xi^n\xi^{\ast{m}}[1+(-1)^ne^{i\phi}][1+(-1)^me^{-i\phi } ] e^{-\gamma{t}(n - m)^2}}{\sqrt{n!m!(n+q)!(m+q)!}}|n^{\prime}-n\rangle_{bb}\langle{n^{\prime}-m}| , \eqno{(36)}\ ] ] where @xmath118 and @xmath98 is the measured number sum of alice s joint measurement . then , the fidelity of this protocol depends on the result @xmath98 and is @xmath119^{-1}\sum^{n^{\prime}}_{n , m=0 } \frac{|d_{n^{\prime}-n}|^2|d_{n^{\prime}-m}|^2\xi^n\xi^{\ast{m}}[1+(-1)^ne^{i\phi}][1+(-1)^me^{-i\phi } ] e^{-\gamma{t}(n - m)^2}}{\sqrt{n!m!(n+q)!(m+q)!}}. \eqno{(37)}\ ] ] the average fidelity is @xmath120[1+(-1)^me^{-i\phi } ] e^{-\gamma{t}(n - m)^2}}{\sqrt{n!m!(n+q)!(m+q)!}}. \eqno{(38)}\ ] ] for a coherent state @xmath121 teleported by alice , the average fidelity can be rewritten as @xmath122[1+(-1)^me^{-i\phi } ] e^{-\gamma{t}(n - m)^2}}{(n^{\prime}-n)!(n^{\prime}-m)!\sqrt{n!m!(n+q)!(m+q)!}}. \eqno{(39)}\ ] ] in fig.6 , the average fidelity for the coherent states is plotted as a function of @xmath123 for different values of amplitude @xmath112 . it is shown that the phase damping deteriorate the average fidelity of teleportation basing on the pair cat state , which is qualitatively consistent with the behavior of its relative entropy of entanglement in the phase damping channel . as mentioned above , the essential parts of this teleportation protocol are the preparation of the pair cat state and the joint measurement of the number sum and phase difference . the direct preparation of pair cat states in two modes optical fields is still a open question . however , we can map the pair cat state of the motional freedom of two dimensional trapped ions into freely propagating optical fields . the details will be discussed elsewhere . recently , the physical implementation of joint measurement of photon number sum and phase difference of two - mode optical fields is shown to be possible by using only the linear optical elements and the single - photon detector [ 25 ] . so we can conclude that the physical realization of this teleportation protocol is feasible at the present technology . in this paper , we investigate the entanglement of pair cat states in the phase damping channel by employing the relative entropy of entanglement . we give the numerical calculations of the relative entropy of entanglement of this state with @xmath35=30 , @xmath52 and @xmath111 for different values of @xmath112 , @xmath113 ( solid line ) , @xmath114 ( dash line ) , @xmath115 ( dot line ) . 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the entanglement of pair cat states in the phase damping channel is studied by employing the relative entropy of entanglement . it is shown that the pair cat states can always be distillable in the phase damping channel . furthermore , we analyze the fidelity of teleportation for the pair cat states by using joint measurements of the photon - number sum and phase difference . + pacs number : 03.67.-a , 03.65.ud , 42.50.dv

You are an expert at summarizing long articles. Proceed to summarize the following text: as we know the defect structures exist in different branches of physics , such as domain wall , kinks , vertices , monopoles , condensed matter and string theory . in higher space - times dimensions the defect structures are generated by real scalar fields such that the single real scalar field produce just single defect as kink - like and the double sin - gordon model may create two different defects . on the other hand , models containing two or more real scalar fields can give rise to at least two other classes of systems that produce defect that engender internal structure and those that support junctions of defects . also two and three scalar fields describe the regular hexagonal network , higgs model [ 1 - 5 ] and bent brane in five dimensions [ 6 ] . in other hand for field theories involving two and three real scalar fields , the mathematical problem concerning the integrability of equations of motion is much harder , as one deals with a system of two coupled second order nonlinear ordinary differential equations . also the configuration space shows a distribution of minima that allows for a number of topological sectors . one way of simplifying the problem is to consider potentials belonging to large class corresponding to bosonic sector of supersymmetric theories . such systems can be studied by super - potential . this super - potential lead us to consider all second order equations in terms of first order . all bogomolnyi - prasad - sommerfield ( bps ) configuration can be described by this first order equation . for models with three interacting components , the solutions on each topological sector determine orbits in the configuration space , which can be expressed as a constraint equation @xmath2 the equations arising for three fields in deformation procedure more complicated then two and single field , so three deformation functions are required . it is difficult to solve this deformation equation , then we restrict the orbit in field space . therefore , by deforming the first order equations for a three coupled field system we need to impose the orbit constraint . as we know the deformation method for the single and two coupled scalar fields are discussed by bazeia and et al . but we are going to consider three coupled scalar fields in ( 1,1 ) dimensions in flat space - time , where metric is @xmath3 [ 1 - 4 ] . the first - order differential equation and the static solutions such as topological solutions for stable states has been discussed by refs . [ 7,8 ] . + in present paper , we use deformation procedure to obtain defect solutions . this deformation plays an important role to investigate the energy of systems . for example in cosmological model , the energy density , pressure and equation of states can be controlled by the deformation method on the fields . + the above pretext give us motivation to discuss the deformation procedure . so the outline of the paper follows . in the next section three coupled scalar fields with an example is discussed and the analysis of the solution without deformation is shown . the deformation procedure for the three coupled scalar field with diagram is discussed in sec.3 . also we compare two diagrams and show that the variations of the fields with respect to coordinates are changed by scale . lagrangian systems described by coupled scalar fields are gaining renewed attention . in the case of real fields , in particular , the work presented in [ 4 ] has introduced a specific class of systems of two coupled real scalar fields . + let us to start with the lagrangian density with three coupled system , @xmath4 where @xmath5 is in general a non - linear function of the three fields @xmath6 , @xmath7 and @xmath8 here we are using natural units , and so @xmath9 and the metric is such that @xmath10 and @xmath11 + by using euler - lagrange equations , one can obtain the equations of motion as follows ; @xmath12 in order to obtain solution for the equation ( 2 ) , we define super - potential function @xmath13 such that one may write the potential in terms of super potential , @xmath14 where @xmath13 is a smooth function and we have , @xmath15 the energy spectrum associated with these configurations could be written as [ 9 - 10 ] , @xmath16,\end{aligned}\ ] ] we can rewrite it also as following , @xmath17 ^ 2.\end{aligned}\ ] ] here also we set the bps energy , @xmath18 this procedure shows that the energy is minimized to , @xmath19 as we have already learned from [ 4 ] , we impose conditions , @xmath20 in this case we see that the energy gets to its lower bound @xmath21 , and the above first - order equation ( 9 ) solve the corresponding equations of motion ( 2 ) . now we consider special example of three coupled scalar fields which describes the regular hexagonal network . let us consider the system defined by the following super - potential , @xmath22 in this case the first - order equations become , @xmath23 as we know the exact solution for the above equations is not clear , so we shall use the following elliptical orbit procedure [ 6,9,11 ] , @xmath24 finally three - field static solution for the system ( 11 ) is , @xmath25 so we have a three- field model and its general orbit equation depending on two parameters @xmath26 and @xmath27 ( the arbitrary phase ) . + we recall that there are several orbits for solving the equation ( 11 ) . but here we shall consider the condition @xmath28 and the orbit equation ( 12 ) . + by putting equation ( 13 ) in ( 10 ) and ( 3 ) the corresponding super - potential and potential in terms of @xmath29 are given by , @xmath30,\end{aligned}\ ] ] @xmath31.\end{aligned}\ ] ] we are going to apply deformation procedure for three couple scalar fields . as we know the deformation method for two and single field were discussed in ref.s . [ 9 , 12 , 13 ] . + first we transform three initial scalar fields @xmath32 , @xmath33 and @xmath34 into the form of scalar fields @xmath35 , @xmath36 and @xmath37 respectively . + in order to deform three scalar fields we introduce the following deformed function ; @xmath38 where non - deformed function @xmath39 and deformed function @xmath40 are differentiable and invertible . we can write the inverse function as @xmath41 for @xmath42 . for simplicity the deformed functions are just function of single field . the essential condition for the deformation functions are given by [ 12 ] , @xmath43_{orbit}.\end{aligned}\ ] ] which is just the orbit - based deformation procedure . also , we note that in ref . [ 12 ] they applied this procedure in two fields system and we will apply for three fields system where @xmath44 , @xmath45 and @xmath46 . this equation shows that the variation of non - deformed functions are equivalent to the deformed functions . finally the deformed super - potential @xmath47 from eqs . ( 4 ) and ( 9 ) will be as ; @xmath48 with the help of essential condition and eq . ( 3 ) we have , @xmath49 where @xmath50 deformed potential in terms of deformed super - potential is , @xmath51 the energy of deformed defects can be written by following expression , @xmath52 where @xmath53 is deformed bps energy . + + now we apply deformation method for super - potential ( 10 ) . to start , we shall introduce the deformed function @xmath54 , @xmath55 and from eq . ( 9 ) @xmath54 is given by , @xmath56 by using the following expression @xmath57 the orbit equation ( 12 ) will be as , @xmath58 in order to obtain the deformation fields @xmath59 and @xmath60 , we use eqs . ( 16 ) and ( 17 ) , so we shall have ; @xmath61 here we use equations ( 23 ) , ( 24 ) and ( 13 ) the corresponding deformed fields @xmath59 and @xmath60 are respectively ; @xmath62 @xmath63 the deformation functions also will be the following ; @xmath64 @xmath65 the deformed super - potential and deformed potential from equations ( 18 ) and ( 19 ) are given by ; @xmath66 and @xmath67}{(1+tanh^2(2rx))^2}.\end{aligned}\ ] ] finally we can say the topological solutions of non - deformed and deformed equations are compared together by drawing their graphs see figs . ( 1 ) and ( 2 ) . ( 1 ) shows that the plots of non - deformed fields @xmath6 and deformed @xmath54 which are drawn as kink . plots of non - deformed fields @xmath7 and @xmath68 and deformed fields @xmath59 and @xmath60 are lumps . we see that the graphs of three fields for non - deformed and deformed are same . in that case they are just changed by scale . ( 2 ) shows plots of non - deformed and deformed super - potentials . similarly , we see that the variation of two cases are the same , though by scale are different . in this paper , we first have introduced three coupled scalar fields @xmath69 , @xmath70 and @xmath71 [ 11 ] and obtained topological solutions by orbit method . next we have deformed the initial fields as @xmath72 , @xmath73 and @xmath74 also the topological solutions for deformed field are obtained by the orbit method . the solution of deformed and non - deformed three coupled scalar fields lead us to compare these two cases . so , we have shown that the variation of fields for two cases with respect to coordinates in figs.(1 ) and ( 2 ) are same and different just by scale . consequently , different orbit may be lead to different full deformed models and solutions . as we know the deformed fields similar to non - deformed fields are the form of kink and lump solutions , so we have to choose a deformation function such as eq . . therefore we could consider eq . ( 22 ) in different forms , it may be interesting for the future work . r. rajaraman and e. weinberg , phys . rev . d * 11 * ( 1975 ) 2950 . r. rajaraman , phys . lett . * 42*(1979 ) 200 . g. w. gibbons and p. k. townsend , phys . * 83 * ( 1999 ) 1725 . d. bazeia , m. j. dos santos , and r. f. ribeiro , phys . lett . a * 208 * ( 1995 ) 84 . z. surujon , phys . d * 73 * ( 2006 ) 016008 . j. sadeghi and a. mohammadi , eur . phys . j. c * 49 * ( 2007 ) 859 - 864 e. b. bogomolnyi , sov . j. nucl.phys . * 24 * , 449 ( 1976 ) . m. k. prasad and c. m. sommerfield , phys . * 35 * ( 1975 ) 760 . r. rajaraman . solitons and instantons ( north - holland , amsterdam , 1982 ) . d. bazeia , l. losano and j. m. c. malbouisson , phys . d * 66 * ( 2002 ) 101701 ( r ) . d. bazeia , l. losono , c. wotzasek , phys . d * 66 * ( 2002 ) 105025 . v. i. afonso , d. bazeia , m. a. gonzalez leon , l. losano and j. mateos guilarte , phys . d ( 2007 ) * 76 * 025010 . a. de souza dutra , [ arxiv : hep - th/07053237 ] . j. sadeghi and a. mohammadi , mod . lett . a * 22 * , 18 ( 2007 ) 1349 - 1358 .

in this work , we present a deformed solutions starting from systems of three coupled scalar fields with super - potential @xmath0 by orbit method . first , we deform the corresponding super - potential and obtain defect solutions . it is shown that how to construct new models altogether with its defect solutions in terms of the non - deformed model . therefore , we draw the graph of super - potential and different fields in terms of @xmath1 so we observe that the graphs for deformed and non - deformed cases are changed by the scale . + * keywords:*three scalar fields ; deformation method ; orbit solution + + * pacs number : * 11.10.-z , 11.10.lm , 11.27.+d .

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Proceed to summarize the following text: the so(10 ) grand unified theory has several interesting features @xcite . it can accommodate left - right symmetry as one of the intermediate symmetry and hence provides an explanation of parity violation @xcite . @xmath4 is a generator of the group so(10 ) and hence lepton number violation takes place spontaneously . this explains the origin of lepton number violation and neutrino majorana mass naturally . the smallness of neutrino masses is assured by the see - saw mechanism , so that by keeping the scale of @xmath4 violation high the smallness of the neutrino mass is guaranteed . quarks and leptons are treated equally in so(10 ) gut . gauge coupling unification is consistent with all low energy results . on the one hand there are many attractive features of the so(10 ) gut , on the other the predictability becomes low . depending on the symmetry breaking pattern and higgs scalar contents , the model can have widely differing predictions . attempts have been made to construct a minimal model . in one approach the minimal model is constructed with minimum numbers of parameters , while in the other approach minimum numbers of higgs scalars are included in the model . there are also models without any intention of minimality or simplicity , where the main aim is to explain all experiments and have maximum predictability . in the present article we shall study an so(10 ) gut , which has the minimum dimensions of the higgs scalars . in any so(10 ) gut the minimal number of higgs scalar includes an @xmath5 symmetry breaking higgs scalar , which can give masses to the fermions and some scalars that can break the @xmath4 symmetry along with the @xmath6 symmetry and can give majorana masses to the neutrinos . in addition , there is one higgs scalar which breaks the group so(10 ) gut . one conventional model includes a higgs bi - doublet ( a * 10 * -plet of so(10 ) higgs scalar , which is doublet under both the @xmath5 and @xmath7 groups ) and both @xmath5 and @xmath6 higgs triplets . in such models with triplet higgs scalars neutrinos acquire masses at the tree level . the so(10 ) representation that contains this higgs scalar is of * 126 * dimensions . in another version of the model , one breaks the left - right symmetry and the @xmath4 symmetry by doublets of @xmath5 and @xmath6 groups . this higgs field belongs to a * 16 * -plet representation of so(10 ) . for symmetry breaking and giving fermion masses another * 10 * -plet higgs scalar is introduced , which contains the bi - doublet higgs and hence gives tree level masses to the fermions and the neutrinos acquire only dirac masses of the order of other fermion masses . there is no majorana mass term for the neutrinos and hence see - saw mechanism is not possible . however , there exist effective higher - dimensional operators , which can give correct majorana masses to both the left - handed and right - handed neutrinos . recently it has been pointed out that it is possible to consider an so(10 ) gut , which does not have any higgs bi - doublet scalar belonging to a * 10 * -plet of so(10 ) @xcite . the higgs scalar that breaks left - right symmetry and the @xmath4 symmetry belongs to a * 16 * -plet of higgs scalar . since tree level fermion masses are not allowed without the bi - doublet scalar , all fermion masses come from higher - dimensional operators in the see - saw form . in supersymmetric theories the non - renormalizablity theorem does not allow radiative generation of such higher dimensional operators . we shall then restrict ourselves to only non - supersymmetric so(10 ) gut . the source of see - saw suppression for the fermion number violating majorana mass terms are different from the source of see - saw suppression for the fermion number conserving dirac mass terms , which maintain the large hierarchy between the charged fermion masses and the neutrino masses . in this article we shall study some aspects of this model in detail . in the next section we shall discuss the model . in sec . [ hpot ] we shall present details of the scalar potential minimization and the allowed symmetry breaking pattern and in sec . [ fmass ] the generation of fermion masses is discussed . we shall then study the renormalization group equation for this model with the specific choice of the higgs scalars . in sec . [ gcu ] we shall study the gauge coupling unification and in sec . [ yukawa ] the yukawa coupling evolution for the different fermions . since all fermion masses have the same see - saw origin , the perturbative unification becomes an important question in these models . some of the yukawa couplings could become large , although the effective fermion masses still remains small . in sec . [ lepto ] leptogenesis in our model is discussed and in the last section we summarize our results . the starting point for any so(10 ) gut is the choice of the symmetry breaking pattern . there exists many chains of symmetry breaking pattern , which are all consistent with our present knowledge . so the particular choice of a symmetry breaking pattern defines a specific model . we shall consider a symmetry breaking pattern , which requires a minimum number of higgs scalars , given by : @xmath8 \nonumber \\ & { \stackrel{\phi_4 } \longrightarrow } & { \rm su(3)}_c\times{\rm su(2)}_l\times{\rm su(2)}_r\times{\rm u(1)}_{b - l } \quad [ { \rm g}_{3221 } ] \nonumber\\ & { \stackrel{\chi_r}\longrightarrow } & { \rm su(3)}_c\times{\rm su(2)}_l\times{\rm u(1)}_y \quad[{\rm g}_{321 } ] \nonumber \\ & { \stackrel{\chi_l}\longrightarrow}&{\rm su(3)}_c\times { \rm u(1)}_{\rm em}\ , , \label{sbp}\end{aligned}\ ] ] where the higgs fields responsible for the symmetry breakings , @xmath9 , @xmath10 , @xmath11 and @xmath12 are explicitly shown in the above equation . both @xmath9 and @xmath10 are contained in @xmath13 , which transforms as the * 210 * dimensional representation of so(10 ) . the * 210 * -plet decomposes under the pati - salam subgroup @xmath14 of so(10 ) as , @xmath15 in the above decomposition @xmath9 corresponds to @xmath16 and @xmath10 corresponds to ( * 15,1,1 * ) . the higgs fields @xmath12 and @xmath17 belong to the * 16 * dimensional spinor representation @xmath18 of so(10 ) and the other fields @xmath11 and @xmath19 belong to the conjugate representation @xmath20 , called @xmath21 , of so(10 ) . the group transformation properties of the @xmath22 fields under @xmath23 and @xmath14 are as follows : @xmath24 in the above equations @xmath25 , and @xmath26 denote the group transformation properties of the higgs fields under @xmath23 and @xmath14 . at this stage we shall digress to discuss one important feature of the left - right symmetric models , namely the question of parity @xmath27 . the discrete @xmath28 symmetry , that interchanges the two su(2 ) subgroups of the lorentz group o(3,1 ) , is called the parity . this parity can be identified as the discrete @xmath28 symmetry operator that interchanges the groups @xmath5 and @xmath6 of the left - right symmetric model , which implies that under parity @xmath29 . this definition extends to scalars also . that is , an @xmath5 doublet scalar field @xmath12 will transform to an @xmath6 doublet scalar field @xmath11 under the operation of parity @xmath30 . in the conventional left - right symmetric models , the parity is spontaneously broken along with the group @xmath6 . in other words , when the left - right symmetric group @xmath31 is spontaneously broken , parity is also spontaneously broken . there is another possibility of breaking parity spontaneously without breaking the left - right symmetric group . since the scalar fields transform trivially under the lorentz group , the vev of a parity odd field can break parity spontaneously without breaking the left - right symmetry . unlike the conventional case , now the parity acting on the fermions and vector bosons is not spontaneously broken . to distinguish these two cases , this second type of parity is called a @xmath1-parity . thus when @xmath1-parity is broken , the left - handed and right - handed scalars can have different mass and vev and hence the gauge coupling constants of @xmath5 and @xmath6 can also be different . in the present model @xmath1-parity plays a very crucial role , both for symmetry breaking as well as for fermion masses . it also plays some role in gauge coupling unification . the @xmath32 representation of so(10 ) is a totally antisymmetric tensor of rank four @xmath33 and the singlet @xmath9 is the component @xmath34 in the notation , in which , @xmath35 are @xmath36 indices and @xmath37 are @xmath38 indices . thus under @xmath1-parity @xmath9 is odd ( @xmath39 ) and consequently when it gets its vacuum expectation value ( vev ) at the gut scale , @xmath40 , it breaks the left - right parity of the theory . due to this spontaneous breaking of the left - right parity at the @xmath40 scale we will have @xmath41 at a lower energy scale . this @xmath1-parity odd field is also required to give masses to the light neutrinos . next we write down the fermions in our model and their group transformation properties . the left - handed quarks , leptons , anti - quarks and anti - leptons belong to a * 16 * -plet representation of so(10 ) , which transform under @xmath14 as : @xmath42 @xmath43 is the generation index . the right - handed fermions and anti - fermions belong to the conjugate representation , @xmath44 in addition to the above mentioned conventional particles our model consists of an extra so(10 ) gauge singlet fermion per generation : @xmath45 @xmath46 . under @xmath23 the states @xmath47 and @xmath48 transform as : @xmath49 and as a result the fermions can be labelled as : @xmath50 and @xmath51 the generators of the left - right symmetry group @xmath23 are related to the electric charge of the particles by , @xmath52 where @xmath53 in the conventional left - right symmetric models there is one bi - doublet higgs scalar @xmath54 , which gives masses to quarks and charged leptons and a dirac mass to the neutrinos through its couplings of the form @xmath55 . in addition , there are triplet higgs scalars @xmath56 and @xmath57 , which can give majorana masses to the left - handed and right - handed neutrinos through the couplings @xmath58 . in our present model all these higgs scalars @xmath59 , @xmath60 and @xmath61 are absent and hence there are no tree level fermion masses for the quarks and the leptons . after discussing the structure of the higgs vacuum expectation values in this model , we shall come back to the question of fermion masses . in the conventional left - right symmetric models , the combinations of the higgs fields , @xmath59 , @xmath60 and @xmath61 ensures that for certain choices of parameters , @xmath61 can acquire a very large vev compared to other fields breaking left - right symmetry at a large scale . it is clear that in the absence of the field @xmath59 , both the fields @xmath62 would acquire equal vevs . it has also been shown that in the absence of the field @xmath59 in a left - right symmetric model with only the doublet higgs scalars @xmath12 and @xmath11 , the minimization of the potential would result in equal vevs for both @xmath12 and @xmath11 , which would lead to inconsistency and parity will be conserved at low energy . this problem could be solved if parity is broken in these theories either explicitly or spontaneously . in the present model this problem does not occur . it was mentioned in the original version of the model that the @xmath1-parity odd singlet field @xmath63 under g@xmath64 contained in the @xmath32 representation would allow @xmath65 and break left - right symmetry at some high scale . in this section we shall minimize the scalar potential and discuss the various possible solutions , which allows left - right symmetry breaking at some high scale . let us consider the higgs potential @xcite : @xmath66 + m_d \phi ( \gamma^\dagger \gamma ) + \lambda_{\phi \gamma } \phi^2 ( \gamma^\dagger \gamma)\ , . \label{hp}\end{aligned}\ ] ] the coupling @xmath67 is the most important term that is required for the left - right breaking to take place at a higher scale compared to standard model symmetry breaking . @xmath1-parity is broken when @xmath9 acquires a non - vanishing vev , @xmath68 , at the @xmath40 scale , since @xmath9 is odd under @xmath1-parity . @xmath10 will get a non - vanishing vev at the @xmath69 scale . there will be many terms in eq . ( [ hp ] ) including @xmath70 , but as we are analyzing the structure of eq . ( [ hp ] ) in the @xmath23 phase and mainly interested on the vevs of the @xmath22 fields , we do not explicitly write down the terms including @xmath71 . @xmath70 has no important contribution in the expressions of the vevs of the @xmath22 fields . we now discuss the masses of the components of @xmath18 and the vevs . the scalar potential responsible for the masses of the fields @xmath12 and @xmath11 is given by , @xmath72 the masses of these fields are then given by , @xmath73 if @xmath1-parity is conserved , @xmath74 and the masses of both @xmath12 and @xmath11 become equal . since the vev of @xmath9 breaks @xmath1-parity , it will be possible to fine tune parameters to obtain the mass of @xmath12 to be orders of magnitude smaller than the mass of @xmath11 . from phenomenological consideration we also require @xmath75 we shall next check if this widely different vevs for @xmath12 and @xmath11 is possible . @xmath76 breaks the electroweak symmetry , while @xmath77 breaks the left - right symmetry at a very high scale , close to the gut scale . we denote the vevs of the fields @xmath12 and @xmath11 as : @xmath78 instead of minimizing the potential , we shall first write down the potential in terms of the vevs of the fields and then find the conditions satisfied by the vevs . with the above vevs we can write the higgs potential in the @xmath23 phase as : @xmath79 ( v^*_r v_r + v^*_l v_l ) - \frac{\lambda_\gamma}{4}(v^*_r v_r + v^*_l v_l)^2 \nonumber\\ & & -\,\frac{\lambda'_\gamma}{4}(v^4_l + v^4_r + v^{*\,4}_l + v^{*\,4}_r ) + m_d \eta ( v^*_r v_r - v^*_l v_l)\ , . \label{pot1}\end{aligned}\ ] ] the @xmath80 symbols in the above equation stands for terms containing @xmath70 . setting @xmath81 and @xmath82 , which amounts to saying that there is no cp violation and all vevs are considered to be real , the extremum conditions of @xmath83 comes out to be : @xmath84 = 0\,,\nonumber \\ \label{vl}\\ \frac{\partial v}{\partial v_r}&= & 2v_r \left [ m^2_\gamma + ( \lambda_{\phi \gamma } \eta^2 + \cdot \cdot \cdot ) - \frac{\lambda_\gamma}{2 } ( v^{2}_l + v^{2}_r ) -\lambda'_\gamma v^2_r + m_d \eta \right ] = 0\,.\nonumber \\ \label{vr}\end{aligned}\ ] ] the above equations imply , @xmath85=0\ , . \label{vlvr}\end{aligned}\ ] ] neglecting the trivial solution @xmath86 , the other interesting relation between @xmath76 and @xmath77 that comes out from the above equation is , @xmath87 two things can be noted from the above equation . first as it was stated previously , in understanding the relation between @xmath76 and @xmath77 we do not require the vev of @xmath10 . secondly if @xmath88 has some value comparable to @xmath40 and @xmath89 is not too high , then it is apparent from eq . ( [ vlvrsq ] ) that @xmath90 . if the energy scale where @xmath11 and @xmath17 gets a non vanishing vev be @xmath91 then we can say that @xmath92 where @xmath93 gev . thus this model allows left - right symmetry breaking at a much higher scale compared to the standard model symmetry breaking scale . it is clear from the above discussions that this model works only if @xmath1-parity is broken spontaneously . in addition , severe fine tuning is required to obtain and maintain this solution . to make the masses of @xmath12 and @xmath11 different @xmath94 , a fine tuning is required . then the next fine tuning is required to keep the vev @xmath76 to be orders of magnitude smaller than @xmath77 . this is the usual fine tuning required in all non - supersymmetric theories . we can write eq . ( [ vl ] ) as @xmath95 = 0\,.\end{aligned}\ ] ] since the vev @xmath76 will be proportional to @xmath96 , a fine tuning is performed to keep @xmath97 gev . the second fine tuning makes sure that the vev @xmath77 does not destabilize the vev of @xmath76 through radiative corrections . in the left - right symmetric theories the left - handed fermions are doublets under @xmath5 and the right - handed fermions are doublets under @xmath6 . hence the fermion masses would require a bi - doublet higgs scalar . following our discussions at the end of sec . [ sbpat ] , it is clear that in the present model there are no yukawa couplings giving dirac or majorana masses to the quarks and leptons . in this model both the majorana and the dirac masses originate from dimension-5 effective operators , given by : @xmath98 where @xmath99 , @xmath100 and @xmath101 are some heavy mass scales in the theory . in general , the mass scales appearing in the operators which contribute to the dirac masses ( @xmath99 , @xmath100 ) and the mass scales that appear in the operators contributing to the majorana masses ( @xmath101 ) will be different , since in one of them total fermion number is violated by 2 units . when the higgs scalars @xmath12 and @xmath11 acquire vevs , the first two operators @xmath102 and @xmath103 give the quark masses : @xmath104 similarly the third and the fourth operators @xmath105 and @xmath106 contribute to the charged lepton and neutrino dirac masses : @xmath107 while the last two operators @xmath108 and @xmath109 contribute to the majorana masses for the left - handed and right - handed neutrinos respectively : @xmath110 we shall now discuss some of the possible origin of these operators and their consequences . the see - saw masses of the neutrinos in theories with only doublet higgs may arise from various cases as , some higher dimensional effective operators in a non supersymmetric theory , from non - renormalizable gravitational interactions or from supersymmetric extensions of models with doublet higgs @xcite . in the present case the see - saw masses of the neutrinos can be obtained in three different ways . they may be mediated by exchange of scalar fields or fermion fields or may be induced radiatively . as we shall argue now , the first two possibilities are not very attractive and hence we shall study the radiative mechanism in details . when the intermediate field is a scalar , it has to be a field which transforms as @xmath111 and hence the field could be either a @xmath112 or a @xmath113 or a @xmath114 . if the scalar field transform as @xmath113 , the fermion mass matrix will be totally antisymmetric and hence phenomenologically unacceptable . if the scalar field @xmath115 transform as a @xmath116 or a @xmath114 , its components will receive induced vevs through its couplings @xmath117 and @xmath118 . then we can eliminate the @xmath12 and @xmath11 in the resulting theory and revert to the conventional theories with bi - doublet higgs @xmath59 and triplet higgs scalars @xmath62 . so , we shall not discuss this possibility any further in the rest of the article . we shall now consider the possibility of intermediate heavy fermions generating the effective operators for the quark and lepton masses . for each of the operators we require two fermions , one left - handed and the other right - handed , both having same gauge transformation properties . for the majorana mass terms generated by the last two operators a self - conjugate singlet fermion is sufficient . the singlet fermion @xmath119 , we already included in the present model , can give the majorana masses to the left - handed and right - handed neutrinos . to generate the operator @xmath102 , we need two fermions @xmath120 and @xmath121 coupling to @xmath122 and @xmath123 respectively . both these fields should then transform similarly @xmath124 or * 210 * and the lagrangian must contain the couplings @xmath125 to give masses to the up - quarks by the operator @xmath102 . the down quark masses are obtained by an effective operator @xmath126 , which may be generated by adding the field @xmath127 or * 126 * or @xmath128 and introducing the couplings in the lagrangian : @xmath129 the operators @xmath130 and @xmath106 may be obtained by introducing the fields @xmath131 or * 45 * and @xmath132 with the couplings @xmath133 respectively . then we may give masses to the up and the down quarks as well as to the charged leptons and the neutrinos if there are heave fermions transforming as * 120 * and * 45*. the singlet field @xmath119 per generation is required to give majorana masses to the neutrinos with its couplings , which we shall discuss later . we shall now come back to the present model , where the quark and lepton masses are generated radiatively . the fermion content of the model has been discussed in sec . [ sbpat ] . the most general yukawa couplings are then given by , @xmath134 in this expression generation indices have been suppressed . one loop diagram of fig . [ fmass1.f ] then generates effective operators @xmath135 which are of the form @xmath102 and @xmath130 and contributes to the down - quark and charged - lepton masses . on the other hand the one loop diagram of fig . [ fmass2.f ] generates effective terms : @xmath136 which are of the form of the operators @xmath103 and @xmath137 and contributes to the masses of the up - quarks and the dirac masses of the neutrinos . the up - quark , down - quark and charged - lepton masses can now be estimated from fig . [ fmass1.f ] and fig . [ fmass2.f ] to be : @xmath138 here @xmath139 or @xmath140 , depending on whether @xmath141 or @xmath140 is larger and @xmath142 , and @xmath143 we thus obtain different up and down quark masses and on the other hand @xmath144 unification . the other mass relations in the down - quark sector and the charged - lepton mass relations could come from higher order terms , since the remaining matrix elements are of the order of @xmath145 to @xmath146 compared to the 33-element @xcite . for example , operators of the form @xmath147 contribute differently to the down - quark and charged - lepton masses , since the effective vev transform as @xmath148 and @xmath149 , which behaves as the field @xmath150 and hence can solve the fermion mass problem in guts , a la georgi - jarlskog mechanism . the neutrino masses come from the couplings of the neutrinos with the singlet fermions @xmath119 , given by eq . ( [ yukawac ] ) . in the basis @xmath151 the tree level neutrino mass matrix becomes : @xmath152 which gives two heavy states , which are mostly @xmath153 and @xmath154 . in the limit @xmath155 , the two heavy mass eigenvalues are @xmath140 and @xmath156 . on the other hand , when @xmath157 , the two heavy states are almost degenerate with eigenvalues @xmath158 with a mass splitting of about @xmath140 . the latter case may be more interesting for leptogenesis , which we shall discuss at the end . the lightest state @xmath159 remains massless at the tree level . however , if we include the effect of @xmath1-parity violation , this problem could be solved . we thus continue our discussion taking @xmath1-parity violation into consideration . the effective operator : @xmath160 and a similar @xmath1-parity violating effective operator @xmath161 which could come from the fig . [ xyfig3]a and fig . [ xyfig3]b , together give a neutrino mass matrix : @xmath162 where , @xmath163 . this mass matrix is obtained by integrating out the heavy modes @xmath164 . in the absence of @xmath1-parity violation , this mass matrix remains symmetrical and one of the eigenvalues vanishes , leading to a massless left - handed neutrino . when @xmath1-parity violating effect is included , the symmetry between the left and the right handed neutrinos is lost and the left - handed neutrinos become light and massless . in the limit @xmath165 and @xmath166 , diagonalization of this matrix gives a light neutrino with mass @xmath167 this gives the correct order of magnitude for neutrino mass for @xmath168 gev and @xmath169 gev . this tiny neutrino mass is of the see - saw type and in fact all fermion masses are of the see - saw type in this model . in this section we shall study the renormalization group equations for the evolution of the coupling constants in our model . we start with the one - loop renormalization group equation for the gauge coupling constants @xmath170 where @xmath171 where @xmath172 stands for the energy - scale of our theory . @xmath173 is the gauge coupling constant of the group @xmath174 which is a subgroup of the semi - simple gauge group @xmath175 and the beta functions @xmath176 contain contributions from gauge bosons , fermions and scalars as : @xmath177.\end{aligned}\ ] ] to two - loop the @xmath178 functions of any semi - simple gauge group is given as @xcite : @xmath179 where the @xmath180s and @xmath181s are the one - loop and two - loop @xmath178 function coefficients respectively . here @xmath182 is the number of groups whose direct product is the semi - simple gauge group of the theory , @xmath183 takes on values from @xmath184 . first we concentrate on the one - loop effect and later we will see the effects of the two - loop coefficients on the gauge coupling evolutions . the @xmath180s calculated for the various phases are supplied below @xcite . @xmath185 in the above table @xmath186 and the superscripts @xmath187 , @xmath188 , @xmath189 indicates the phase in which the numbers are calculated . the renormalization group ( rg ) equations can now be used to write down the the standard model gauge couplings in terms of the so(10 ) coupling . writing @xmath190 , the gauge coupling constant matching conditions at the @xmath91 scale are : @xmath191_{{\rm g}_{\rm sm}}&= & \left[\frac35\frac{1}{\alpha_{2r}(m_r ) } + \frac23 \frac{1}{\alpha_{b - l}(m_r)}\right]_{{\rm g}_{3221 } } , \\ \left[\frac{1}{\alpha_{2l}(m_r)}\right]_{{\rm g}_{\rm sm}}&= & \left[\frac{1}{\alpha_{2l}(m_r)}\right]_{{\rm g}_{3221 } } , \\ \left[\frac{1}{\alpha_{2l}(m_r)}\right]_{{\rm g}_{\rm sm}}&= & \left[\frac{1}{\alpha_{2r}(m_r)}\right]_{{\rm g}_{3221 } } , \\ \left[\frac{1}{\alpha_{3c}(m_r)}\right]_{{\rm g}_{\rm sm}}&= & \left[\frac{1}{\alpha_{3c}(m_r)}\right]_{{\rm g}_{3221 } } . \label{mmr}\end{aligned}\ ] ] the matching conditions at the @xmath69 scale are : @xmath192_{{\rm g}_{3221}}&= & \left[\frac{1}{\alpha_{4c}(m_r)}\right]_{{\rm g}_{422 } } , \\ \left[\frac{1}{\alpha_{2l}(m_r)}\right]_{{\rm g}_{3221}}&= & \left[\frac{1}{\alpha_{2l}(m_r)}\right]_{{\rm g}_{422 } } , \\ \left[\frac{1}{\alpha_{2r}(m_r)}\right]_{{\rm g}_{3221}}&= & \left[\frac{1}{\alpha_{2r}(m_r)}\right]_{{\rm g}_{422 } } , \\ \left[\frac{1}{\alpha_{3c}(m_r)}\right]_{{\rm g}_{3221}}&= & \left[\frac{1}{\alpha_{4c}(m_r)}\right]_{{\rm g}_{422 } } . \label{mmx}\end{aligned}\ ] ] finally at the @xmath40 scale , @xmath193_{{\rm g}_{422}}&= & \left[\frac{1}{\alpha_{10}(m_u)}\right]_{{\rm so}_{10 } } , \\ \left[\frac{1}{\alpha_{2l}(m_u)}\right]_{{\rm g}_{422}}&= & \left[\frac{1}{\alpha_{10}(m_u)}\right]_{{\rm so}_{10 } } , \\ \left[\frac{1}{\alpha_{2r}(m_u)}\right]_{{\rm g}_{422}}&= & \left[\frac{1}{\alpha_{10}(m_u)}\right]_{{\rm so}_{10 } } . \label{mmu}\end{aligned}\ ] ] with the help of the above matching conditions and the rg equation we can write to one - loop , @xmath194\ , , \label{aly}\\ \frac{1}{\alpha_{2l}(m_z)}&=&\frac{1}{\alpha_{10}(m_u ) } + 8\pi\left [ a^{(sm)}_{2l}\ln\left(\frac{m_r}{m_z}\right ) + a^{(lr)}_{2l } \ln\left(\frac{m_x}{m_r}\right)\right.\nonumber\\ & & + \left.a^{(x)}_{2l}\ln\left(\frac{m_u}{m_x}\right)\right ] , \label{al2l}\\ \frac{1}{\alpha_{3c}(m_z)}&=&\frac{1}{\alpha_{10}(m_u ) } + 8\pi\left [ a^{(sm)}_{3c}\ln\left(\frac{m_r}{m_z}\right ) + a^{(lr)}_{3c } \ln\left(\frac{m_x}{m_r}\right)\right.\nonumber\\ & & + \left.a^{(x)}_{4c}\ln\left(\frac{m_u}{m_x}\right)\right ] . \label{al3}\end{aligned}\ ] ] the linear combinations of the gauge couplings that yields @xmath195 and @xmath196 are the following : @xmath197 where @xmath198 and @xmath196 are related to the electromagnetic and strong interaction coupling constants in the present symmetry broken phase . using the experimental numbers @xcite , @xmath199 eq . ( [ wang ] ) and eq . ( [ as ] ) reduces to the following : @xmath200 the above equations can be utilized for calculating the intermediate scales like @xmath91 and @xmath69 in our theory . here we discuss two cases . in this case from eq . ( [ aly ] ) , eq . ( [ al2l ] ) and eq . ( [ cons1 ] ) and using the @xmath178 function coefficients given in the last table we get , @xmath202=29.66\,.\nonumber\\ \label{mumx1}\end{aligned}\ ] ] similarly from eq . ( [ aly ] ) , eq . ( [ al2l ] ) , eq . ( [ al3 ] ) and eq . ( [ cons2 ] ) and the @xmath178 function coefficients we get , @xmath203=106.444\ , . \label{mumx2}\end{aligned}\ ] ] eliminating @xmath40 from the above two equations we get @xmath204 and if we take @xmath205 gev then @xmath206 gev . the above value of @xmath91 can be taken as the lowest possible value of it in our model and all the predictions in our model will be made assuming @xmath207 gev . in this case the two equations corresponding to eq . ( [ mumx1 ] ) and eq . ( [ mumx2 ] ) are : @xmath209=29.66\ , , \label{munmx1}\end{aligned}\ ] ] and @xmath210=106.444\ , . \label{munmx2}\end{aligned}\ ] ] eliminating @xmath40 from the above two equations gives us a relation between @xmath69 and @xmath91 as , @xmath211\ , . \label{mrmxp}\end{aligned}\ ] ] from the above equation it can be verified that if we take @xmath205 gev and impose @xmath212 then @xmath213 gev . in the next subsection we include the two loop results and the above results are re - derived computationally . from the computational results we see that the above value of @xmath214 gev is two orders of magnitude smaller than the actual one . after the discussion on gauge coupling unification to one - loop we discuss about the two - loop effects of the rg equations . to two - loop the @xmath178 functions are given in eq . ( [ gbeta ] ) . the @xmath180s for the various phases has been supplied in the table appearing in the beginning of this section and the @xmath215s for the various phases are as follows : @xmath216 here @xmath217 , @xmath218 and here @xmath219 , @xmath220 where @xmath221 . s in the sm phase , left - right phase upto @xmath40 . the abscissa is @xmath222 , where @xmath172 initial is @xmath223 gev and @xmath224 gev in our case . the figure in the right shows explicitly the gauge coupling unification at @xmath225 gev.,height=623 ] if we start from @xmath226 gev ( where @xmath227 ) and fix @xmath224 gev then the evolution of the gauge coupling constants are as given in fig . [ gcsm : f ] . in the next phases the coupling constant evolution shows that at @xmath69 both @xmath228 and @xmath229 unite to produce @xmath230 . fig . [ gcsm : f ] shows that from @xmath69 on wards the development of @xmath231 and @xmath232 are identical . at around @xmath225 gev the gauge coupling constants unite . our computational results show that the highest value of @xmath91 is just slightly above @xmath3 gev . if @xmath91 is much above the above mentioned value then @xmath69 comes down and @xmath233 . in models with triplet higgs scalars , it is possible to consider @xmath234 . at the scale @xmath91 , the group @xmath6 is broken into @xmath235 and stays orthogonal to @xmath236 . later when @xmath236 is broken to @xmath237 . subsequently at a much lower scale the symmetry breaking , @xmath238 , takes place . however , this is not possible in scenarios with only doublet higgs scalars , since the right - handed higgs scalar doublet does not have any component with @xmath239 quantum number to be zero and hence it breaks @xmath6 and @xmath240 simultaneously . as a result , the highest value of @xmath91 could be @xmath3 gev . as we discussed earlier @xcite , the gauge coupling unification also requires a lower bound on @xmath241 gev . from this lower ( higher ) bound for @xmath91 , we estimated computationally higher ( lower ) bound for @xmath69 and @xmath40 as shown in figs . [ bound1 ] , [ bound2 ] , [ bound3 ] . the lower and higher bounds for @xmath69 are @xmath242 gev and @xmath243 gev respectively . also lower and higher bounds for @xmath40 are estimated as @xmath244 gev and @xmath245 gev respectively . in this regard it is important to note that the stability of the proton offers further constraints on the parameter space of their model , since the gut scale is lower than most of the conventional models and is smaller than @xmath246gev . the difference between @xmath69 and @xmath91 can go up from 0 ( i.e , @xmath212 ; no intermediate @xmath247 symmetry ) to @xmath248 gev as shown in fig . [ bound4 ] . this range of @xmath91 plays an important role for yukawa coupling unification and leptogenesis in this model . in sec . [ fmass ] , eq . ( [ yukawac ] ) gives the only yukawa couplings of our theory . ( [ yukawac ] ) is valid in the so(10 ) level . in the @xmath14 level the yukawa couplings will become : @xmath249 the gauge transformation properties of the various fields , except @xmath153 which is a gauge singlet , are shown below : @xmath250 in the above equations @xmath251 designates the transformation properties under @xmath14 . the yukawa couplings in the @xmath23 phase is as : @xmath252 the gauge transformation properties of @xmath253 , @xmath254 , @xmath255 , @xmath256 , @xmath19 and @xmath11 are specified in eq . ( [ lql ] ) , eq . ( [ rql ] ) and eq . ( [ higgsconj ] ) . here we specify the gauge transformation properties of the other two fields present in the above equation . @xmath257 in the above equations @xmath258 designates the @xmath259 transformation properties and @xmath260 designates the @xmath14 transformation properties . in the next stage , that is in the @xmath261 phase , the yukawa couplings look like : @xmath262 here the various standard model fermion fields transform under @xmath261 as : @xmath263 similarly the various higgs fields transform as : @xmath264 in the above expressions the first triplet @xmath265 designates the transformation properties under @xmath261 , the next four numbers @xmath258 designates the @xmath23 transformation properties and the triplet @xmath260 stands for the @xmath14 transformation properties . now we give an order of magnitude estimation about the running of the effective top yukawa coupling in our theory . from eq . ( [ mup ] ) it is seen that the effective top quark yukawa coupling looks like @xmath266 . in the @xmath267 phase it looks like @xmath268 in the convention adopted to name the yukawa couplings in eq . ( [ 321 ] ) . calling this effective coupling as @xmath269 it will evolve simply like : @xmath270 as in the standard - model up to the @xmath91 scale . the gauge and quartic coupling contributions will be negligible compared to @xmath269 . starting from the top - quark mass at the electroweak scale , the evolution equation gives the effective top - quark yukawa coupling at the left - right symmetry breaking scale to be of the order of @xmath271 for @xmath272 gev . since the effective coupling constant is a product of three couplings @xmath268 and if @xmath273 then each of these couplings can individually take values large enough as @xmath274 . as a result the yukawa sector becomes non - perturbative in the @xmath259 phase . but on the other hand if we have @xmath275 then the situation changes . in this case the individual coupling becomes of the order of @xmath276 which implies the theory is still perturbative . @xmath69 can be the heavy gauge boson masses or the singlet fermion @xmath153 mass . the previous condition means that the heavy gauge bosons or the singlet fermion can not be heavier than @xmath277 gev in our theory if we take @xmath272 gev for a perturbative scenario in the yukawa sector . above the left - right symmetry breaking scale @xmath91 up to the unification scale @xmath40 , the coupling constants @xmath278 and @xmath279 will evolve separately . the separate yukawa couplings remains finite up to the unification scale . since the neutrino masses now depend on the couplings with the singlets , there is no stringent restriction coming from the up quark masses . as a result , it may be possible to get large neutrino mixing angles . the right - handed neutrinos and the new singlet fermions can now decay into light leptons . the majorana masses of the left - handed and right - handed singlets violate lepton numbers , which in turn can generate enough lepton asymmetry . before the electroweak phase transition this asymmetry can then generate a baryon asymmetry of the universe @xcite . since there is no supersymmetry , the gravitino bounds are not present . the out - of - equilibrium condition can be satisfied near the gut scale since the couplings are large to get the required neutrino mass with large see - saw scale . in this model there is another interesting feature that the singlets combine with the right - handed neutrinos to form pseudo - dirac particles and hence resonant leptogenesis may also be possible @xcite . for leptogenesis , consider the interactions of eq . ( [ yukawac ] ) . unlike usual see - saw models with triplet higgs scalars @xcite , in this model the right - handed neutrinos can not decay into left - handed neutrinos and higgs bi - doublets dominantly . the simplest lepton number violating interactions come from the decays of @xmath164 : @xmath280 the majorana masses of @xmath164 allow the singlet to decay into both leptons and antileptons violating lepton number by two units . in the present model both @xmath254 and @xmath12 are very light and hence these decays are allowed . for cp violation there are two types of one - loop diagrams which interferes with the tree - level diagrams for the decays of @xmath164 . these are the vertex type diagrams of fig . [ vertex ] and fig . [ selfenergy ] . the right - handed neutrinos do not take part in leptogenesis directly , but due to mixing of the right - handed neutrinos with the heavy singlets @xmath164 , the right - handed neutrino decays also enter in the picture of leptogenesis . in the limit of @xmath155 , the right - handed neutrinos and the @xmath164 are both heavy and distinct . in this case the amount lepton asymmetry due to cp violation is given by , @xmath281 } { \sum_\alpha @xmath164 are diagonal and the eigenvalues are hierarchical @xmath282 where @xmath283 are masses of @xmath164 . we can write the effective lepton asymmetry as : @xmath284 where @xmath285 is a suppression factor which depends on the amount of departure from equilibrium . the exact value of @xmath285 can be obtained by solving the boltzmann equation taking all the interactions into consideration . however , it is also possible to make an order of magnitude estimate for the amount of asymmetry , which will be very close to the actual value . since @xmath286 is the lightest of the singlets , the decay of this singlet will be able to generate the lepton asymmetry . the asymmetry generated or washed out by the heavier ones @xmath287 or @xmath288 will be smeared out by the interactions of @xmath286 after @xmath287 and @xmath288 had decayed away . so , for an estimate we shall only consider the decays of @xmath286 . this assumption is well justified when the singlets @xmath164 have a hierarchical mass structure . the out - of - equilibrium condition is parametrized by : @xmath289 when @xmath290 , there is no boltzmann suppression of the generated asymmetry and the out - of - equilibrium condition is satisfied . in this case the generated asymmetry is given by @xmath291 and it is not washed out after it is created and one gets @xmath292 . however , if @xmath293 , then the interaction strength is so slow that the generated asymmetry can never reach the value @xmath291 . although the interactions can not wash out the asymmetry after it is generated , the amount of asymmetry is less than @xmath291 . in the case of @xmath294 , the generated asymmetry is same as the cp asymmetry of @xmath291 , but even after the asymmetry is created , the interaction remains strong enough to deplete the asymmetry . although the depletion is exponentially fast , it can not compete with the expansion of the universe for long and the final amount of asymmetry is not exponentially depleted . it was shown that @xcite the suppression factor @xmath285 is almost linearly proportional to @xmath295 . in the present model we come across this last scenario . in the present model @xmath296 gev . while a lower value of @xmath91 is preferable for out - of - equilibrium condition , since the yukawa couplings grow very fast above the scale @xmath91 we have to consider the highest value of @xmath91 . taking the hierarchical structure of @xmath164 , we consider the mass of the lightest singlet to be around @xmath297 gev . taking @xmath298 gev , @xmath299 and @xmath300 we find that @xmath295 is much lower than 1 , which gives a strong suppression factor of @xmath301 . on the other hand the yukawa couplings in this model comes out to be of the order of 1 and hence we get a large enhancement in the cp asymmetry and @xmath291 in our case can be as large as @xmath302 and so the lepton asymmetry parameter @xmath303 . at this stage , @xmath304 and @xmath305 is given by @xmath291 . thus @xmath306 the final baryon asymmetry after the electroweak phase transition is thus given by , @xmath307 in the case of @xmath157 , the right - handed neutrinos and the @xmath164 singlets of every generation are almost degenerate . the mass splitting between the states @xmath308 and @xmath164 with mass @xmath309 is of the order of @xmath140 . although @xmath164 decays will now generate a lepton asymmetry , both the heavy mass eigenstates contain the states @xmath164 . as a result , when these two almost degenerate states decay , there may be resonant leptogenesis ( which will however require new interactions and fine - tuning ) and hence the scale of leptogenesis could be very low . for the present scenario since @xmath91 and hence @xmath310 can not be much lower , this is not important and hence we shall not discuss it in any further detail . in conclusion , we constructed an so(10 ) gut without any higgs bi - doublets . all the symmetry breaking could be achieved by only two higgs scalars , a @xmath32 and a @xmath311 . by including a massive singlet fermion per generation we break chiral symmetry which can then give masses to all the fermions radiatively without introducing any new scalar fields . all fermion masses have the same see - saw form . the spontaneous parity breaking plays a crucial role in breaking the left - handed and right - handed su(2 ) groups at two widely different scales and also giving masses to the left - handed neutrinos in this scenario . the spontaneous breaking of an ungauged discrete symmetry , the d - parity , which is a special feature of this model may cause formation of very heavy domain walls of gut - 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in this work we study an so(10 ) gut model with minimum higgs representations belonging only to the * 210 * and * 16 * dimensional representations of so(10 ) . we add a singlet fermion @xmath0 in addition to the usual * 16 * dimensional representation containing quarks and leptons . there are no higgs bi - doublets and so charged fermion masses come from one - loop corrections . consequently all the fermion masses , dirac and majorana , are of the see - saw type . we minimize the higgs potential and show how the left - right symmetry is broken in our model where it is assumed that a @xmath1-parity odd higgs field gets a vacuum expectation value at the grand unification scale . from the renormalization group equations we infer that in our model unification happens at @xmath2 gev and left - right symmetry can be extended up to some values just above @xmath3 gev . the yukawa sector of our model is completely different from most of the standard grand unified theories and we explicitly show how the yukawa sector will look like in the different phases and briefly comment on the running of the top quark mass . we end with a brief analysis of lepton number asymmetry generated from the interactions in our model .

You are an expert at summarizing long articles. Proceed to summarize the following text: in the first realizations of bose - einstein condensation ( bec ) in the laboratory @xcite and in many experiments ever since , the bose gas is trapped in a potential that can be considered as parabolic to a very good approximation . in the thermodynamic limit , within the ideal gas approximation , the critical temperature for such a system is given by @xmath0 , where @xmath1 is the geometric mean of the trap frequencies and all the other symbols have their usual meaning ( see e.g. @xcite ) . soon after the first experiments , corrections to this expression , @xmath2 , were found . on the one hand , experiments do not take place in the thermodynamic limit . hence , finite - size corrections are required . on the other hand , the gases are not ideal , having a non - vanishing scattering length . hence , interaction effects must be taken into account . the first order shift @xmath3 due to interactions was determined analytically early on in @xcite within a mean - field approximation , in the form of a linear term in the scattering length . higher order corrections followed in several works @xcite , both numerical and analytical . expansions for @xmath3 in powers of the scattering length down to second order were determined , both within a mean - field approach @xcite and accounting for critical correlations @xcite . the first order finite - size induced shift was given in the isotropic case in @xcite and in the general anisotropic case in @xcite as @xmath4 where @xmath5 . more recently , a higher order result was given in @xcite ( see also @xcite ) . this result relies on the local density approximation , in which the discrete energy levels of the finite system are approximated by a continuum , therefore requiring that the typical thermal energy at the transition be much greater than the typical inter - level spacing ( for example , in the case of an isotropic harmonic trap , @xmath6 ) , i.e. , the thermodynamic limit . moreover , in order to overcome the vagueness ( or non - point - like character ) associated with the critical temperature of the finite system , we believe it would be useful to consider an explicit physical criterion for this critical temperature , related for example to the condensate fraction or the specific heat , when going to the level of detail of higher - order corrections @xcite . strictly speaking , a finite - size correction to @xmath7 is an ill - defined concept when taken on its own because the effect of finite size is to spread out the phase transition from a point to a narrow temperature interval . the first order correction ( [ classic ] ) is typically extracted from a high temperature finite - size expansion of the number of particles , which takes into due account the discreteness of the energy levels and which can be obtained in several ways @xcite . if one attempts to find a second order correction from this expansion , the absence of a true critical temperature makes itself noticed : the next order term in the expansion is divergent at the critical point , ultimately implying the non - existence of bec as a sharp , mathematically defined phase transition in finite systems . it follows that the first order corrected @xmath8 must not be taken too seriously . it merely provides a reference value for signaling the transition . in experimental work where the bec critical temperature is measured @xcite , the expression generally quoted for purposes of comparison with theory , namely for splitting off finite - size effects from interaction effects , is the one in ( [ classic ] ) . now , as mentioned above , this expression should not be taken at face value . thus , there is the possibility that a misinterpretation of the finite - size related shift can lead to a bias in the reported values of the interaction induced shift . it would be of interest to make this matter clearer . what is actually measured in experiments is the number of particles , ground state fraction , trap frequencies and temperature . it is by performing some polynomial fit to a plot involving these quantities that an experimental value for @xmath8 is usually extracted @xcite . in the landmark experiment reported in @xcite the fit is performed in the region where the condensate fraction `` noticeably starts to increase '' . condensate fractions as low as about @xmath9 could be measured in this experiment . if lower condensate fractions could be measured , higher critical temperatures would have been obtained , even rising above @xmath7 for sufficiently small condensate fractions . this is because for finite systems the condensate fraction is not zero for temperatures above the critical region . it is just very small . this fact becomes more conspicuous for low particle numbers . another major experiment in what concerns high precision measurements of @xmath8 is reported in @xcite . here , very much the same comments apply . in this case , condensate fractions as low as @xmath10 could be detected . the authors overcome the problem of isolating interaction from finite - size corrections by performing differential measurements with reference to a standard value of the scattering length . nevertheless , as recently pointed out @xcite , this assumes that finite size and interaction effects are independent . at second order , it might not be the case . our aim in the present work is to obtain higher - order finite - size corrections to the critical temperature of a bose gas in a general harmonic trap . to do this in a meaningful way , which at the same time can connect to experimental procedures , we overcome the non - existence of a true critical temperature by asking instead for the temperature @xmath11 at which the condensate fraction has a given small value @xmath12 , @xmath13 . other criteria could be used , like defining @xmath8 by the maximum of the specific heat or the inflection point of the @xmath14 curve ; but the one we adopt here is probably the most useful because it uses the condensate fraction and it is very simple . from the well known bulk behaviour of the condensate fraction in the bec regime , @xmath15 , we have in the thermodynamic limit @xmath16 . for @xmath17 , this yields @xmath18 . we will provide finite - size corrections to @xmath11 down to third order . stopping at second order is not accurate enough in some circumstances , as detailed below . our approach preserves all the finite - size characteristics of the system , with no approximations involved . the information on the discrete structure of the energy levels is carried in the expansions ( [ condfraction ] ) and ( [ t_k ] ) below . finally , we note that our expressions are also valid ( and highly accurate ) for @xmath19 not small , i.e. , deep into the bec regime . let @xmath20 and @xmath21 . @xmath22 is a rescaled inverse temperature and @xmath23 can be looked at as a rescaled chemical potential . we define the anisotropy vector @xmath24 . using grand - canonical statistics , the number of particles @xmath25 of an ideal bose gas in this trap is given by @xmath26 ^{-1 } = \sum_{\mathbf{n}}\sum_{k=1}^{\infty}e^{-kx(\bm{\lambda}\cdot\mathbf{n}+\epsilon ) } \ ; . \label{nnew}\ ] ] the sum in @xmath27 is over all single particle states , of energy @xmath28 , @xmath29 . let @xmath30 . the usual bulk result for @xmath25 , which is exact in the thermodynamic limit , reads in our variable @xmath31 if @xmath32 and @xmath33 if @xmath34 ( where @xmath35 is the quantity that remains finite in the thermodynamic limit , as opposed to @xmath25 ) . @xmath36 is the polylogarithm of index 3 , with the property @xmath37 . define @xmath38 . as we approach the thermodynamic limit in the usual way ( @xmath39 kept fixed ) we have @xmath40 , or for any fixed temperature , @xmath41 . @xmath22 and eventually @xmath42 will be our expansion parameters . in the bec regime , we have in addition ( still in the thermodynamic limit ) @xmath43 , from where we see that @xmath23 scales as @xmath44 . what we need is an expansion for @xmath25 that contains the finite - size corrections and that is valid _ throughout _ the critical region . this can be achieved by applying a mellin - barnes transform to the exponential inside the @xmath45 summation in ( [ nnew ] ) , as indeed was done before in @xcite . the same procedure was also applied to a bose gas subject to other confinements @xcite . an expansion is obtained by solving a contour integral in the complex plane using the theorem of residues . in this case , the riemann and three - dimensional barnes zeta functions , here denoted @xmath46 and @xmath47 respectively , make their appearance . knowledge of the residues at the poles of these functions is required . we refer the reader to @xcite for details of the procedure . @xmath48 is a multi - dimensional generalization of the hurwitz zeta function , which was studied in depth by barnes in @xcite ( see also @xcite ) . in @xcite the expansion for @xmath25 was calculated to subleading order . however , for our purposes we need also the third and fourth terms . the calculation of the third term , in particular , is more involved due to the existence of a double pole , requiring the knowledge of the finite part at the @xmath49 pole of @xmath47 , not only its residue . specifically , below we need the quantity @xmath50 defined in the following way . let @xmath51 be the finite part at the @xmath52 pole of @xmath47 . then @xmath53 , i.e. , @xmath54 . @xmath55 is a function of @xmath56 only . we obtain the expansion @xmath57 x^{-1 } -\frac{1}{2}{\zeta}_b(0,{\epsilon}|\bm{\lambda } ) + \mathcal{o}(x ) \ ; , \label{nexpansion}\end{gathered}\ ] ] where we have adopted the following notational conventions : @xmath58 and @xmath59 . the first two terms in ( [ nexpansion ] ) were given in @xcite . from @xcite we have that @xmath60 . the full asymptotic expansion for @xmath25 could easily be given , but it is not needed . define the rescaled temperature @xmath61 . in ( [ nexpansion ] ) , change from the variables @xmath25 , @xmath22 and @xmath62 to @xmath42 , @xmath63 and @xmath23 by performing the substitutions @xmath64 and @xmath65 . equation ( [ nexpansion ] ) gives us @xmath23 implicitly as a function of @xmath42 and @xmath63 . since @xmath66 for @xmath67 , we solve for @xmath23 perturbatively by letting @xmath68 and find the coefficients @xmath69 , @xmath70 next we use the expression for the condensate fraction @xmath71 . in this expression , we change again to the variables @xmath42 , @xmath63 and @xmath23 and substitute the newly found expansion for @xmath23 . expanding the resulting expression in powers of @xmath42 yields @xmath72 \\ \times x_0 ^ 2 + \frac{\lambda^3}{{\zeta}(3)}\left [ \frac{{\zeta}(2)}{{\zeta}(3)}\frac{t^3}{1-t^3}+\frac{(\lambda_i^2\lambda_j)}{48\lambda^3}-\frac{7}{16}\right]x_0 ^ 3+\cdots\ ; . \label{condfraction}\end{gathered}\ ] ] this equation gives us the condensate fraction as a function of @xmath63 and @xmath25 ( or @xmath63 and @xmath42 ) . it is valid throughout the bec regime and critical region . note that the first two terms , @xmath73 , are just the bulk result for @xmath74 in the condensate region . we then set the condensate fraction at @xmath75 and solve ( [ condfraction ] ) perturbatively , this time to find @xmath76 as a function of @xmath19 and @xmath42 . this finally yields @xmath77 with the coefficients being given by @xmath78\\ c'({\kappa})&=\frac{9+(\lambda_i\lambda_j)}{(1-{\kappa})^{1/3}36{\zeta}(3)}\\ d({\kappa})&=(1-{\kappa})^{-2/3}\left [ -\frac{{\zeta}(2)^3}{12{\zeta}(3)^3}+\frac{(\lambda_i^2\lambda_j)-21\lambda^3}{144{\zeta}(3 ) } + \frac{{\zeta}(2)}{6{\zeta}(3)^2}\right.\\ & \left . \times \frac{9+(\lambda_i\lambda_j)}{12 } \left ( 1+\frac{1}{3}\ln ( 1-{\kappa } ) \right ) + \frac{{\zeta}(2)\lambda^3}{6{\zeta}(3)^2}\left ( b_0 - 2+\frac{2}{{\kappa}}\right ) \right ] \\ d'({\kappa})&=-\frac{{\zeta}(2)(9+(\lambda_i\lambda_j))}{36{\zeta}(3)^2(1-{\kappa})^{2/3}}\ ; .\end{aligned}\ ] ] note that since @xmath79 this is an expansion in powers of @xmath80 . the leading term is just the bulk result for @xmath81 . the subleading term is the well known first order finite - size correction to the critical temperature given in ( [ classic ] ) : @xmath82 . the higher order terms are new . it is interesting to note that when @xmath83 the coefficient @xmath84 , unlike the other coefficients above , diverges ( due to the @xmath85 in the denominator of the very last term ) . we return to this point in the next section . as mentioned above , this expansion is valid not only in the critical region ( @xmath86 ) , but also throughout the bec regime ( @xmath85 not small ) . in the isotropic case , @xmath87 , @xmath88 , @xmath89 and , by writing @xmath90 in terms of hurwitz zeta - functions , it is easily seen that @xmath91 , where @xmath92 is euler s constant . the @xmath93 and @xmath94 coefficients in ( [ t_k ] ) are then given more simply as @xmath95 where we have used the numerical values of @xmath96 and @xmath97 . the subscript @xmath98 stands for``isotropic '' . in order to use ( [ t_k ] ) ( or for that matter , any of the previous expansions to more than subleading order ) in the case of an anisotropic trap , we must be able to find @xmath55 in the general case . from barnes s work @xcite ( pp.398 and 404 ) , it is easy to arrive at @xmath99 , where the @xmath100 are gamma modular forms , which barnes gives quite generally in terms of contour integrals in the complex plane . application to our case yields @xmath101 \ ; , \label{b0integral}\end{gathered}\ ] ] where we have made use of the fact that @xmath102 , from the definition of @xmath56 . table [ b0list ] presents values of @xmath55 in a few illustrative cases . in the experiment by gerbier _ et al _ @xcite , the trap is cigar shaped with aspect ratio @xmath103 . the widely used trap of ensher _ et al _ @xcite is disc shaped with aspect ratio @xmath104 . we include these two shapes in the table . for a more complete table , see the supplementary material , where we give values of @xmath55 for axially symmetric traps with integer aspect ratios ranging from @xmath105 to @xmath106 . c@d .. 4d .. 4 & & + 2 & 0.1991 & 0.2433 + 3 & 0.9605 & 1.1317 + 5 & 3.0403 & 3.5180 + 10 & 10.8558 & 11.4467 + 30 & 73.7461 & 55.4503 + gerbier _ et al _ & & 102.0169 + ensher _ et al _ & 0.8147 & in fig . [ fig : isotropic ] we plot @xmath107 for @xmath108 and @xmath109 in the isotropic case . we plot both purely numerical results and our analytical results from ( [ t_k ] ) . since ( [ t_k ] ) is an expansion in powers of @xmath80 , its accuracy increases for larger @xmath25 . for small @xmath25 , the value of @xmath85 can not be chosen too small . this is because in this situation we will have @xmath110 due to the spreading out of the phase transition , while we require @xmath111 . this can also be seen by looking at the coefficient of the @xmath94 term in ( [ t_k ] ) , @xmath84 . it contains a term with factor @xmath112 which becomes large if @xmath85 is very small . in fact , if we carry on with the expansion , it is seen that at every third term a new factor @xmath112 will appear . thus , the terms in @xmath94 , @xmath113 and @xmath114 contain the factor @xmath112 , the terms in @xmath115 , @xmath116 and @xmath117 contain the factor @xmath118 and so on . it follows that this expansion is valid only if @xmath119 , or equivalently , @xmath120 . this is a very reasonable condition . it corresponds exactly to temperatures very close to @xmath7 or below it , confirming in this way the original requirement @xmath111 for the validity of the expansion . if @xmath121 just holds but the tighter condition @xmath122 ( equivalently , @xmath123 ) does not , then the expansion is valid but the term in @xmath94 is of the same order as the @xmath42 term ( while the terms in @xmath124 , @xmath125 , will be of smaller order ) . hence , it is very important to include it . this limiting situation happens for example for @xmath126 and @xmath127 or for @xmath128 and @xmath129 . all this is very well corroborated by comparing with numerical results as can be seen in the figure . for medium to large @xmath25 , the analytical and numerical curves are superimposed or hardly distinguishable , due to the high accuracy of ( [ t_k ] ) . for example , for @xmath130 and @xmath108 or @xmath131 and @xmath109 the error in @xmath81 from ( [ t_k ] ) is less than @xmath132 ( in the isotropic case ) . as we lower @xmath25 , the accuracy slowly decreases . as expected , this happens more quickly for @xmath109 . the broadening of the phase transition is observed in the rise of @xmath107 for low @xmath25 . our approximation captures this behaviour . for comparison , we plot the usual first order result , given by eq . ( [ classic ] ) . we see it provides a useful reference value for the transition for @xmath133 , even though it does not have a precise meaning . for lower values of @xmath25 , the new corrections are particularly important . roughly , in most situations , the new corrections should be important for @xmath134 . in such cases , if the interaction strength is small , it is quite possible that these effects become of the same order or even dominant over the interaction effects . we plot @xmath81 for @xmath135 and axially symmetric disc shaped and cigar shaped traps . @xmath81 is plotted as a function of the aspect ratio @xmath136 . for both shapes , the anisotropy causes a decrease in @xmath81 . this effect is more pronounced for the disc shape , which can be understood from the fact that the dependency of @xmath137 ( and hence of @xmath42 ) on the aspect ratio @xmath136 is stronger in the disc case . the effect seen in fig . [ fig : isotropic ] where @xmath81 rises above @xmath105 for low @xmath25 is not seen in fig . [ fig : anisotropic ] because anisotropy lowers the critical temperature without spreading out the phase transition , which remains sharp . it can also be seen that the accuracy of ( [ t_k ] ) slightly decreases with increasing anisotropy . this is expected and is due to the fact that @xmath137 decreases with increasing @xmath136 , which causes @xmath42 to increase . naturally , this is more noticeable for @xmath129 . it is also more noticeable in the cigar shape case . this should be due to the exact nature of the higher order terms ( @xmath113 and higher ) left out of our expansion ( [ t_k ] ) . all in all , it can be seen that ( [ t_k ] ) is still quite accurate in most anisotropic situations ( the exception being the highly anisotropic cigar shaped trap with very small @xmath85 ) . as in the isotropic case , higher values of @xmath25 lead to higher accuracy . we also plot the first order result from ( [ classic ] ) in fig . [ fig : anisotropic ] . again , we see that for this number of particles , eq . ( [ classic ] ) works well for providing a reference value for the transition . let us now take the experimental conditions of smith _ et al _ @xcite : @xmath138 and a nearly isotropic trap . condensate fractions as low as @xmath139 could reliably be measured in this experiment . for @xmath135 and @xmath109 we have @xmath140 from numerical calculations . ( [ t_k ] ) yields @xmath141 whereas the result from ( [ classic ] ) yields @xmath142 , which is lower than @xmath81 by @xmath143 of @xmath7 . in gerbier _ et al _ @xcite , the trap is cigar shaped with aspect ratio @xmath144 and @xmath145 at the transition . condensate fractions as low as @xmath146 could be measured . for this trap shape with @xmath135 and @xmath108 , we have @xmath147 whereas ( [ t_k ] ) yields @xmath148 . the first order result ( [ classic ] ) yields @xmath149 , which is higher than @xmath107 by @xmath9 of @xmath7 . for @xmath150 this difference reduces to @xmath151 of @xmath7 . if the trap had a larger anisotropy , the difference would be larger . in order to obtain the shift @xmath3 due to interactions , the authors in @xcite subtracted the finite - size correction as given in ( [ classic ] ) from their experimental values for @xmath8 . if the authors had used the lowest reliably detected condensate fraction ( @xmath146 ) for defining an experimental critical temperature , the finite - size shift that should be subtracted would be @xmath152 and not the one given in ( [ classic ] ) . however , the procedure for obtaining this temperature was actually more complex than that , involving linear fits to the plots of @xmath153 , @xmath25 and @xmath154 as functions of the trap depth . the main feature of this idea can perhaps be understood by thinking of a linear fit applied to the experimental points near the @xmath108 region of the familiar @xmath155 curve . in this simplified version , the experimental @xmath8 would then be given by the point of intersection of the straight line of the linear fit with the horizontal ( @xmath154 ) axes . due to the complexity of the whole procedure , it is not clear exactly what finite - size correction should be subtracted . nevertheless , we see that an error of the order of 1% of @xmath7 ( not more ) could be involved in the determination of the interaction induced @xmath3 due to the use of eq . ( [ classic ] ) instead of a more precise expression . , an accuracy of about @xmath151 or less in the reported values of @xmath63 should suffice . this requirement seems to be quite realistic and , in particular , it seems to hold in the experiment of @xcite . ] these examples illustrate the relevance of accurate analytical finite - size corrections , while suggesting the usefulness of better defined criteria for measuring the critical temperature , whenever finite - size effects play a role . in particular , in such cases it would seem to us perhaps more relevant to talk about @xmath107 , the temperature at which the condensate fraction is @xmath19 , rather than _ the _ critical temperature . for lower particle numbers , these considerations are even more important , as can be seen in fig . [ fig : isotropic ] . on another note , the use of bec experiments to probe planck - scale physics has been suggested in the last few years ( see @xcite and references therein ) . the idea is that a quantum gravity effect could alter the single particle energy spectrum of the atoms in a harmonic trap , with a consequent shift in @xmath8 . for this effect to show , we would ideally have a very weakly interacting gas and relatively small atom numbers due to finite size is estimated to be about @xmath156 , while the shift due to interactions is less than @xmath157 , but thermal equilibrium can not be assumed . ) ] . finite - size effects would be crucial in such an experiment . in this context , the need has been recognized @xcite for higher order finite - size corrections . we have given these corrections in the present work . this work took as a starting point a certain criterion for the bec critical temperature of a finite system . as mentioned above , other physical criteria could be adopted . in principle , it should be possible to apply the techniques of the present work to these alternative criteria . finally , it would be interesting to study analytically the interplay between finite size and interaction effects . it would not be surprising if there is a second order correction cross term containing a dependency on both the finite size and the interaction strength . 39 natexlab#1#1[1]`#1 ` [ 2]#2 [ 1]#1 [ 1]http://dx.doi.org/#1 [ ] [ 1]pmid:#1 [ ] [ 2]#2 , , , , , ( ) . . , , , , ( ) . . , , , , , , , ( ) . . , , , , ( ) . , , , ( ) . . , , , ( ) . , , ( ) . . , , , ( ) . . , ( ) . , , , ( ) . , , ( ) . , ( ) . , , , , ( ) . , , , , ( ) , , , , ( ) . , , , , ( ) . , , , , , ( ) . , , , ( ) . , ( ) . , , , ( ) . , , ( ) . , , , ( ) . , , , ( ) . , , , , , ( ) . , , , , , , ( ) . , , , , ( ) . , , , , , , ( ) . , , ( ) . , ( ) . , , ( ) . , ( ) . , , , , . , , ( ) . , , ( ) . . , ( ) . , , , , , , , , ( ) .

we obtain second and higher order corrections to the shift of the bose - einstein critical temperature due to finite - size effects . the confinement is that of a harmonic trap with general anisotropy . numerical work shows the high accuracy of our expressions . we draw attention to a subtlety involved in the consideration of experimental values of the critical temperature in connection with analytical expressions for the finite - size corrections . bose - einstein condensation , bose gas , finite - size effects 03.75.hh , 05.30.jp

You are an expert at summarizing long articles. Proceed to summarize the following text: first observed in the @xmath5 meson system at argus , the neutral b meson transition from the particle to anti - particle state , and vice versa , occurs through a second order weak transition or `` box diagram '' . the frequency of the oscillation is proportional to the small difference in mass between the two eigenstates , @xmath6 , and for the @xmath5 system can be translated into a measurement of the ckm element @xmath7 . @xmath7 can be used to constrain the unitarity triangle and thereby yield information on the _ cp _ violating phase @xcite . @xmath2 has been precisely measured ( the world average is @xmath8 @xcite ) but large theoretical uncertainties dominate the extraction of @xmath7 from @xmath2 . this problem can be reduced if the @xmath9 mass difference , @xmath10 , is also measured . @xmath7 can then be extracted with better precision from the ratio : @xmath11 where @xmath12 is estimated from lattice qcd calculations to be @xmath13 @xcite . the above has motivated many experiments to search for @xmath9 oscillations and though a statistically significant signal hasnt been observed yet , a lower limit ( @xmath14 ps@xmath15 at 95% c.l . ) has been set . since this current limit indicates that the @xmath9 oscillations are at least 30 times faster than the @xmath16 oscillations , a @xmath9 mixing measurement is experimentally very challenging . the d experiment at the fermilab tevatron , a @xmath17 collider at 1.96 tev center of mass energy , is well equipped to search for @xmath9 oscillations . the large muon acceptance and forward tracking coverage of the d detector ( pseudorapidity coverage of @xmath18 for the muon , @xmath19 for the tracking and @xmath20 for the silicon sub - detectors ) , along with a robust muon trigger are highly effective in exploiting the large @xmath21 cross - section resulting in some of the largest semileptonic @xmath9 yields . [ dsphipi ] shows the yield for the decay @xmath22 in @xmath3250 pb@xmath15 of data . the decay @xmath23 has also been reconstructed and is expected to contribute significantly to the total @xmath9 yield . other decays including hadronic @xmath9 decays are being studied as well . other elements essential to a mixing analysis are given by the expression for the average statistical significance : @xmath24 where @xmath25 is the number of signal events , @xmath26 is a measure of the effectiveness of a flavor tagger ( @xmath27 is the tagging efficiency while @xmath28 is the `` dilution '' ) , @xmath29 is the number of background events and @xmath30 is the proper time resolution . as indicated above , tagging the meson flavor ( @xmath9 or @xmath31 ) at decay time ( final - state tagging ) and at production time ( initial - state tagging ) are crucial . for the semileptonic modes used for mixing studies at d the final state particles provide the decay - time tag . for initial - state tagging , different techniques are being studied and optimized by doing measurements of @xmath2 . three tagging algorithms have been developed so far : opposite - side muon tagging , opposite - side jet charge tagging , and same - side ( soft - pion ) tagging . the muon tagger relies on identifying the flavor of the other b meson in the event using the sign of the muon it decayed to - a negative muon corresponds to a @xmath32 quark , and vice - versa . for the decay used for @xmath16 mixing studies - @xmath33 where @xmath34 and @xmath35 - both muons having the same sign would indicate that one b hadron had oscillated while opposite signs would indicate that neither ( or both ) had oscillated . [ mixing_slt ] shows the measured asymmetry between the non - oscillated and oscillated mesons as a function of the visible proper decay length ( vpdl ) . the fit to the asymmetry gives @xmath36 ps@xmath15 with an efficiency of @xmath37 % and @xmath38 % . the opposite - side jet charge tagging algorithm calculates a @xmath39-weighted average charge of tracks : @xmath40 . a @xmath41 value corresponds to a @xmath32 quark for the b meson on the opposite side , and vice - versa . the same - side pion tagging algorithm makes use of the fact that the charge of the fragmentation pion correlates with the flavor of the reconstructed b meson . a positive pion corresponds to a @xmath42 quark ( i.e. @xmath16 ) if the reconstructed @xmath29 meson is neutral , but to a @xmath32 quark ( i.e. @xmath43 ) if the reconstructed @xmath29 meson is charged , and vice versa . oscillations have been seen using both the opposite - side jet - charge and the same - side pion taggers , and work is ongoing to compute systematics errors and optimize performance . lastly , since the significance scales as @xmath44 , for large values of @xmath10 , a precise measurement of the proper decay time is crucial . efforts are ongoing in this direction and we hope to have preliminary @xmath10 results soon . a preliminary measurement of @xmath2 has been made with the upgraded d detector . the opposite - side soft muon tagger was used to obtain @xmath36 ps@xmath15 . this measurement , along with the development and optimization of other tagging algorithms , and the reconstruction of different @xmath45 decay modes , is an important step towards a @xmath0 mixing measurement . d.abbaneo , `` review of experimental results on neutral b meson oscillations , '' arxiv : hep - ex/0012010 . et al . _ , `` the review of particle physics by the particle data group , '' phys . lett . * b592 * , 1 ( 2004 ) . o.schneider , `` @xmath46 mixing , '' arxiv : hep - ex/0012010 .

measurement of the @xmath0 oscillation frequency via @xmath0 mixing analyses provides a powerful constraint on the ckm matrix elements . the study of @xmath1 oscillations is an important step towards @xmath0 mixing and a preliminary measurement of @xmath2 has been made with @xmath3 250 pb@xmath4 of data collected with the upgraded run ii d detector . different flavor tagging algorithms have been developed and are being optimized for use on a large set of @xmath0 mesons that have been reconstructed in different semileptonic decay modes . = 14.5pt

You are an expert at summarizing long articles. Proceed to summarize the following text: our approach to understanding the evolution of genetic architectures combines standard models from quantitative genetics @xcite with the wright - fisher model from population genetics @xcite . in its simplest version , our model considers a continuous trait whose value , @xmath4 , is influenced by @xmath5 loci . each locus @xmath6 contributes additively an amount @xmath7 , so that the trait value is defined as the mean of the @xmath7 values across contributing loci . this trait definition means that a gene s contribution to a trait is diluted when @xmath5 is large , which prevents direct selection on gene copy numbers when genes have similar contributions @xcite . we discuss this definition below , along with alternatives such as the sum . the fitness of an individual with trait value @xmath4 is assumed gaussian with mean @xmath8 and standard deviation @xmath9 , so that smaller values of @xmath9 correspond to stronger stabilizing selection on the trait @xcite . individuals in a population of size @xmath10 replicate according to their relative fitnesses . upon replication , an offspring may acquire a point mutation that alters the direct effect of one locus , @xmath6 , perturbing the value of @xmath7 for the offspring by a normal deviate ; or the offspring may experience a duplication or a deletion in a contributing locus , which changes the number of loci @xmath5 that control the trait value in that individual ( see methods ) . point mutations , duplications , and deletions occur at rates @xmath11 , @xmath12 , @xmath13 , which have comparable magnitudes in nature ( table s1 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? finally , an offspring may also increase the number of loci that contribute to its trait value by recruitment that is , by acquiring a recruitment mutation , with probability @xmath14 , in some gene that did not previously contribute to the trait value ( see methods ) . over successive generations in our model , the genetic architecture underlying the trait that is , how many loci contribute to the trait s value , and the extent of their contributions varies among the individuals in the population , and evolves . the genetic architectures that evolve in our model represent the complete genetic determinants of a trait , which may include but do not correspond precisely to the genetic loci that would be detected based on polymorphisms segregating in a sample of individuals in a qtl study . we discuss this important distinction below , when we compare the predictions of our model to empirical qtl data . we studied the evolution of genetic architectures in sets of @xmath15 replicate populations , simulated by monte carlo , with different amounts of selection on the trait . we ran each of these simulations for @xmath16 million generations , in order to model the extensive evolutionary divergence over which genetic architectures are assembled in nature . the form of the genetic architecture that evolves in our model depends critically on the strength of selection on the trait . in particular , we found a striking non - monotonic pattern : the equilibrium number of loci that influence a trait is greatest when the strength of selection on the trait is intermediate ( fig . 1 ) . moreover , the variability in the contributions of loci to the trait value ( fig . s1 ) and the effects of deleting or duplicating genes ( fig . s2 ) are also greatest for a trait under intermediate selection . in other words , our model predicts that traits under moderate selection will be encoded by many loci with highly divergent effects ; whereas traits under strong or weak selection will be encoded by relatively few loci . we also studied how epistatic interactions among loci influence the evolution of genetic architecture . to incorporate the influence of locus @xmath17 on the contribution of locus @xmath6 we introduced epistasis parameters @xmath18 so that the trait value is now given by @xmath19 where @xmath20 is a standard sigmoidal filter function ( * ? ? ? * see methods and fig . s4 ) . as with the direct effects of loci , the epistatic effects were allowed to mutate and vary within the population , and evolve . although significant epistatic interactions emerge in the evolved populations ( fig . s3b ) , the presence of epistasis does not strongly affect the average number of loci that control a trait ( figs . s3a and s4 ) . epistasis is not required for the evolution of large @xmath5 , nor does it change the shape of its dependence on the strength of selection . there is an intuitive explanation for the non - monotonic relationship between the selection pressure on a trait and the number of loci that control it . for a trait under weak selection ( high @xmath9 ) , changes in the trait value have little effect on fitness . thus , even if deletions , recruitments and duplications change the trait value , these changes are nearly neutral ( fig . 2 ) . as a result , the number of loci controlling the trait evolves to its neutral equilibrium , which is small because deletions are more frequent than duplications and recruitments ( see methods , figs . 1 and s3 ) . on the other hand , when selection on a trait is very strong ( low @xmath9 ) , few point mutations , and only those with small effects on the trait , will fix in the population . as a result , all loci have similar contributions to the trait value ( fig . 2 row 1 ) , and so duplications or deletions again have little effect on the trait or on fitness ( fig . 2 rows 2 and 3 ) . in this case , the equilibrium number of loci is given by the value expected when deletions and duplications , but not recruitments , are neutral ( figs . 1 and s3 ) . only when selection on a trait is moderate can variation in the contributions across loci accrue and impact the fixation of deletions and duplications ( fig . 2 row 4 ) , by a process called compensation : a slightly deleterious point mutation at one locus , which perturbs the trait value , segregates long enough to be compensated by point mutations at other loci @xcite . compensation increases the variance in the contributions among loci ( fig . 2 , row 1 ) , as has been observed for many phenotypes in plants and animals @xcite . finally , even though duplications and deletions are mildly deleterious in this regime , there is a bias favoring duplications over deletions ( fig . 2 row 3 ) . this bias arises because duplications increase the number of loci in the architecture , which attenuates the effect of each locus on the trait ( fig . 2 row 2 ) . thus when selection is moderate , duplications and recruitments fix more often than deletions and drive the number of contributing loci above its neutral expectation ( fig . 2 rows 4 and 5 ) . as the number of loci increases the bias is reduced ( fig . 2 rows 4 and 5 ) , and so @xmath5 equilibrates at a predictable value ( figs . 1 and s3 ) . duplications and recruitments might also be slightly favored over deletions under intermediate selection , because architectures with more loci also have reduced genetic variation @xcite . this effect which would positively select for an increase in gene copy numbers is likely weak in our model , as duplications and recruitments are deleterious on average under intermediate selection , only less so than deletions ( fig . 2 rows 4 and 5 ) . ( line 1 ) , and we then measured the effects of recruitments , deletions and duplications on the trait value ( line 2 ) and on fitness ( line 3 ) . from the latter , we calculated the rate at which deletions , recruitment and duplications enter and fix in the population ( line 4 ) , and the resulting rate of change in the number of loci contributing to the trait ( line 5 ) . * line 1 : * for @xmath21 , the variation in direct effects ( @xmath7 ) and indirect effects among controlling loci ( @xmath22 ) increases as selection on the trait is relaxed . * line 2 : * as a consequence of this variation among loci , the average change in the trait value following a duplication or a deletion also increases as selection on the trait is relaxed . * line 3 : * changes in the trait value are not directly proportional to fitness costs , because the same change in @xmath4 has milder fitness consequences when selection is weaker ( larger @xmath9 ) . as a result , the average fitness detriment of duplications and deletions is highest for traits under intermediate selection . * line 4 : * consequently , the fixation rates of duplications and deletions are smallest under intermediate selection . * line 5 : * the equilibrium number of loci controlling a trait under a given strength of selection is determined by that value of @xmath5 for which duplications and recruitments on one side , and deletions on the other , enter and fix in the population at the same rate . for example , when @xmath23 these rates are equal when @xmath5 is close to @xmath24 ( black arrow ) , so that the equilibrium genetic architecture contains @xmath25 loci on average ( compare fig . s3 black arrow).,width=680 ] the predictions of our model notably , that the number of loci in a genetic architecture is greatest for traits under intermediate selection are robust to choices of population - genetic parameters . the non - monotonic relation between selection pressure on a trait and the size of its genetic architecture , @xmath5 , holds regardless of population size ; but the location of maximum @xmath5 is shifted towards weaker selection in larger populations ( fig . this result is compatible with our explanation involving compensatory evolution : selection is more efficient in large populations , and so compensatory evolution occurs at smaller selection coefficients . likewise , when the mutation rate is smaller the resulting equilibrium number of controlling loci is reduced ( fig . this result is again compatible with the explanation of compensatory evolution , which requires frequent mutations . increasing the rate of deletions relative to duplications also reduces the equilibrium number of loci in the genetic architecture , but our qualitative results are not affected even when @xmath13 is twice as large as @xmath12 ( fig . s7 ) . finally , increasing the rate of recruitment @xmath26 ( or the genome size ) increases the number of loci contributing to all traits except those under very strong selection , as expected from fig . our prediction that traits under intermediate selection are encoded by the richest genetic architectures is insensitive to changes in this parameter , and it holds even in the absence of recruitment ( fig . our analysis has relied on several quantitative - genetic assumptions , which can be relaxed . first , we assumed that all effects of locus @xmath6 ( _ i.e. _ @xmath7 and all @xmath27 and @xmath18 ) are simultaneously perturbed by a point mutation . relaxing this assumption , so that a subset of the effects are perturbed , does not change our results qualitatively ( fig . second , we assumed that point mutations have unbounded effects so that variation across loci can increase indefinitely . to relax this assumption we made mutations less perturbative to loci with large effects ( see methods ) . even a strong mutation bias of this type led to very small changes in the equilibrium behavior ( fig . third , we assumed no metabolic cost of additional loci , even though additional genes in _ saccharomyces cerevisiae _ are known to decrease fitness slightly @xcite . nonetheless , including a metabolic cost proportional to @xmath5 does not alter our qualitative predictions ( fig . s11 ) . finally , we defined the trait value as the average of the contributions @xmath7 across loci , as opposed to their sum . this definition reflects the intuitive notion that a gene product s contribution to a trait will generally depend on its abundance relative to all other contributing gene products . moreover , this assumption that increasing the number of loci influencing a trait attenuates the effect of each one is supported by empirical data : changing a gene s copy number is known to have milder phenotypic effects when the gene has many duplicates @xcite . nonetheless , alternative definitions of the trait value , which span from the sum to the average of contributions across loci , generically exhibit the same qualitative results ( text s1 and fig . although robust to model formulation and parameter values , our results do depend in part on initial conditions . when selection is strong , the initial genetic architecture can affect the evolutionary dynamics of the number of loci ( fig . this occurs because the initial architecture may set dependencies among loci that prevent a reduction of their number . this result indicates that only those architectures of traits under very strong selection should depend on historical contingencies . we have also studied a multitrait version of our model , where genes participating in other traits can be recruited or lost through mutation . even though this model features pleiotropy , and the effects of recruitments evolve neutrally , our qualitative results remained unaffected ( text s3 and fig . s15 ) . previous models related to genetic architecture have been used to study the evolutionary fate of gene duplicates . these models typically assume that a gene has several sub - functions , which can be gained ( neo - functionalization ; * ? ? ? * ) or lost ( sub - functionalization ; * ? ? ? * ; * ? ? ? * ) in one of two copies of a gene . such `` fate - determining mutations '' @xcite stabilize the two copies , as they make subsequent deletions deleterious . such models complement our approach , by providing insight into the evolution of discrete , as opposed to continuous or quantitative , phenotypes . yet there are several qualitative differences between our analysis and previous studies of gene duplication . most important , our model considers the dynamics of both duplications and deletions , in the presence of point mutations that perturb the contributions of loci to a trait . this co - incidence of timescales is important in the light of empirical data @xcite showing that changes in copy numbers occur at similar rates as point mutations ( table s1 ) . under these circumstances , a gene may be deleted or acquire a loss - of - function mutation before a new function is gained or lost . our model includes these realistic rates , and accordingly we find that duplicates are very rarely stabilized by subsequent point mutations . instead , the number of loci in a genetic architecture may increase , in our model , because compensatory point mutations introduce a bias towards the fixation of duplications as opposed to deletions . like most evolutionary models , our analysis greatly simplifies the mechanistic details of how specific traits influence fitness in specific organisms . as a result , our analysis explains only the broadest , qualitative features of how genetic architectures vary among phenotypic traits , leaving a large amount of variation unexplained . this remaining variation may be partly random ( as predicted by the distributions of the number of evolving loci , see _ e.g. _ fig . 1 ) , and partly due to ecological and developmental details that our model neglects . due to this variation , a quantitative comparison between our model and empirical data would require information about the genetic architectures for at least hundreds of traits ( see below , for our analysis of expression qtls ) . nevertheless , the qualitative , non - monotonic predictions of our model ( fig . 1 ) may help to explain some well - known trends in the genetics of human traits . for instance , in accordance with our predictions , human traits under moderate selection , such as stature or susceptibility to mid - life diseases like diabetes , cancer , or heart - disease , are typically complex and highly polygenic ; whereas traits under very strong selection , such as those ( _ e.g. _ mucus composition or blood clotting ) affected by childhood - lethal disease like cystic fibrosis or haemophilias are often mendelian ; and so too traits under very weak selection ( such as handedness , bitter taste , or hitchhiker s thumb ) are often mendelian . our analysis provides an evolutionary explanation for these differences , and it delineates the selective conditions under which we may expect a mendelian , as opposed to fisherian , architecture . we tested our evolutionary model of genetic architectures by comparison with empirical data on a large number of traits . such a comparison must , of course , account for the fact that our model describes the true genetic architecture underlying a trait , whereas any qtl study has limited power and describes only the associations detected from polymorphisms segregrating in a particular sample of individuals . accounting for this discrepancy ( see below ) , we compared our model to data from the study of @xcite , who measured mrna expression levels and genetic markers in @xmath28 recombinant strains produced from two divergent lines of _ s. cerevisiae_. for each yeast transcript we computed the number of non - contiguous markers associated with transcript level , at a given false discovery rate ( see methods ) . we also calculated the codon adaptation index ( cai ) of each transcript an index that correlates with the gene s wildtype expression level and with its overall importance to cellular fitness @xcite . we found a striking , non - monotonic relationship between the cai of a transcript and the number of loci linked to variation in its abundance ( fig . thus , assuming that cai correlates with the strength of selection on a transcript , @xcite detected more loci regulating yeast transcripts under intermediate selection than transcripts under either strong or weak selection . we compared the empirical data on yeast eqtls ( fig . 3a ) to the predictions of our evolutionary model . in order to make this comparison , we first evolved genetic architectures for traits under various amounts of selection ( fig . s3 ) , and for each architecture we then simulated a qtl study of the exact same type and power as the yeast eqtl study : that is , we generated 112 crosses from two divergent lines using the yeast genetic map ( text s2 ) . as expected , the simulated qtl studies based these 112 segregants detected many fewer loci linked to a trait than in fact contribute to the trait in the true , underlying genetic architecture ( fig . 3b versus fig . 1 ) . this result is consistent with previous interpretations of empirical eqtl studies @xcite . the simulated qtl studies revealed another important bias : a locus that contributes to a trait under weak selection is more likely to be correctly identified in a qtl study than a locus that contributes to a trait under strong selection ( fig . furthermore , our simulations demonstrate that the number of associations detected in such a qtl study depends on the divergence time between the parental strains used to generate recombinant lines ( fig . finally , traits under weaker selection may be more prone to measurement noise , which we also simulated ( fig . s18 ) . despite these detection biases , which we have quantified , the relationship between the selection pressure on a trait and the number of _ detected _ qtls in our model ( fig . 3b and figs . s18 and s19 ) agrees with the relationship observed in the yeast eqtl data ( fig . importantly , both of these relationships exhibit the same qualitative trend : traits under intermediate selection are encoded by the richest genetic architectures . populations of genetic architectures , using the parameters corresponding to fig . s3 . from each such population , we then evolved two lines independently for @xmath29 generations in the absence of deletions , duplications and recruitment , to mimic the divergent strains used in the yeast cross of @xcite . from these two divergent genotypes we then created @xmath28 recombinant lines following the genetic map from @xcite . we then analyzed the resulting simulated data with _ r / qtl _ in the same way as we had analyzed the yeast data ( text s2 ) . the distribution of qtls detected and their means are represented as in fig . 1 , for each value of selection strength @xmath9.,width=680 ] many interesting developments lie ahead . our model is far too simple to account for tissue- and time - specific gene expression , dominance , context - dependent effects , etc @xcite . how these complexities will change predictions for the evolution of genetic architectures remains an open question . nonetheless , our analysis shows that it is possible to study the evolution of genetic architecture from first principles , to form _ a priori _ expectations for the architectures underlying different traits , and to reconcile these theories with the expanding body of qtl studies on molecular , cellular , and organismal phenotypes . we described the evolution of genetic architectures using the wright - fisher model of a replicating population of size @xmath10 , in which haploid individuals are chosen to reproduce each generation according to their relative fitnesses . the fitness of an individual with @xmath5 loci encoding trait value @xmath4 is @xmath30 where @xmath31 denotes the density at @xmath4 of a gaussian distribution with mean @xmath8 and standard deviation @xmath32 , and the second term denotes the metabolic cost of harboring @xmath5 loci , which depends on a parameter @xmath33 . the trait value of such an individual , given the direct contributions @xmath7 and epistatic terms @xmath18 is described by eq . ( 1 ) where @xmath34 is a sigmoidal curve , so that the epistatic interactions either diminish or augment the direct contribution of locus @xmath6 depending on whether @xmath35 is positive or negative ( fig . s4 ) . in general , loci do not influence themselves ( @xmath36 ) and , in the model without epistasis , all @xmath37 and @xmath38 . if an individual chosen to reproduce experiences a duplication at locus @xmath6 then the new duplicate , labelled @xmath39 , inherits its direct effect ( @xmath40 ) and all interaction terms ( @xmath41 and @xmath42 for all @xmath43 ) , with the interaction terms @xmath44 and @xmath45 initially set to zero . recruitment occurs with probability @xmath26 per mutation of one of the @xmath46 genes not contributing to the trait . the initial direct contribution @xmath7 of recruited locus @xmath6 is drawn from a normal distribution with mean zero and standard deviation @xmath47 ; its interaction terms with other loci ( @xmath39 ) , @xmath44 and @xmath45 , are initially set to zero . note that this assumption is relaxed in the multilocus version of our model , where the direct and indirect effects of recruitments evolve neutrally ( text s3 and fig . s15 ) . in general a point mutation at locus @xmath6 changes its contribution to the trait , @xmath7 , and all its epistatic interactions , @xmath27 and @xmath18 , each by an independent amount drawn from a normal distribution with mean zero and standard deviation @xmath47 . the normal distribution satisfies the assumptions that small mutations are more frequent than large ones @xcite , and that there is no mutation pressure on the trait @xcite . we relaxed the former assumption by drawing mutational effects from a uniform distribution without qualitative changes to our results ( fig . s13 ) . in order to relax the latter assumption we included a bias towards smaller mutations in loci with large effects , so that the mean effect of a mutation at locus @xmath6 now equals @xmath48 and @xmath49 , respectively for @xmath7 and @xmath27 @xcite . we also considered a model in which a mutation at locus @xmath6 affects only a proportion @xmath50 of the values @xmath7 , @xmath27 , and @xmath18 . by default , simulations were initialized with @xmath51 and @xmath52 ; alternative initial conditions were also studied , as shown in fig . s14 . when deletions and duplications are neutral , and recruitments strongly deleterious , the evolution of the number of loci @xmath5 in the genetic architecture is described by a markov - chain on the positive integers . the probability of a transition from @xmath53 to @xmath54 equals @xmath55 , and that of a transition from @xmath6 to @xmath56 is @xmath57 . we disallow transitions to @xmath58 , assuming that some regulation of the trait is required . we obtained the stationary distribution of @xmath5 by setting the density of @xmath59 of individuals in stage @xmath60 to @xmath60 and calculating the density @xmath61 of individuals in the following stages as @xmath62 the equilibrium probability of being in state @xmath6 was calculated as @xmath63 and the expected value of @xmath5 was calculated as @xmath64 . with @xmath65 and @xmath66 , we found an equilibrium expected @xmath5 of @xmath67 . when deletions , duplications and recruitments are all neutral , equation ( 4 ) can be replaced by : @xmath68 this equation illustrates the fact that the rates of deletions ( which include loss of function mutations ) and duplication depend on the number of loci in the architecture , whereas the rate of recruitments does not . with @xmath69 and @xmath70 , we found an equilibrium expected @xmath5 of @xmath71 . we first evolved populations to equilibrium with a fixed number of controlling loci @xmath5 , and we then measured the effects of deletions , duplications or recruitments introduced randomly into the population . we simulated the evolution of the genetic architecture with @xmath5 fixed in @xmath15 replicate populations , over @xmath74 generations for deletions and @xmath75 generations for duplications , reflecting the unequal waiting time before the two kinds of events . we used @xmath75 generations for recruitment as well , although different durations did not affect our results . for each genotype @xmath39 in each evolved population , we calculated the fitness @xmath76 of mutants with locus @xmath6 deleted or duplicated . we calculated the corresponding selection coefficients as : @xmath77 where @xmath78 denotes mean fitness in the population . we calculated @xmath72 as the mean across loci and genotypes of @xmath79 , weighted by the number of individuals with each genotype . we calculated the probability of fixation of a duplication , deletion or recruitment as @xmath80 and obtained the mean @xmath81 using the same method as for @xmath72 . rates of deletions and duplications fixing were calculated per locus ( fig . 2 ) as @xmath13 or @xmath12 times @xmath82 . the total probability of a duplication or a deletion entering the population and fixing is , of course , also multiplied by @xmath5 . however , recruitment rates remain constant as @xmath5 changes . therefore , we divided the rate of recruitments by @xmath5 in fig . 2 , for comparison to the per - 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[ fig : epsilon ] ) . interestingly , the mean number of loci also shows a non - monotonic trend in this situation , with a small peak at @xmath93 . this trend is likely driven by the fixation rate of recruitment mutations ( as seen in fig . this explains why it is much less pronounced than those in figs . 1 and s3-a , which involve a higher fixation rate of duplications over deletions . for any other value of @xmath94 , we found a non - monotonic relationship similar to the one reported in the main text . thus , our qualitative results hold for all models provided the trait is not defined strictly as the sum of contributions across loci . when the trait value equals the sum of the contributions of all locus ( @xmath92 ) , the effect of a gene deletion , knock - out or knock - down is independent of the number of copies of the gene . conversely when the trait value is the mean of the contributions ( @xmath95 ) , or is some function between the mean and the sum ( @xmath96 ) , the effect of a deletion decreases with the number of loci in the genetic architecture . as shown by @xcite in @xmath97 , the number of detectable knock - down phenotypes decreases with the number of copies of genes in a gene family , suggesting that @xmath90 does indeed exceed @xmath8 in this species . a similar stronger effect of the deletion of a singleton compared to that of a duplicate has also been observed in _ s. cerevisiae _ @xcite . we analyzed the genetic architectures that evolved under our population - genetic model using a simulated qtl study of the exact same type and power as the yeast eqtl study @xcite . specifically , @xmath98 evolved populations were taken from simulations with parameters corresponding to fig . s2 for the model with epistasis . from each population , we evolved two lines independently for @xmath99 generations in the absence of deletions , duplications and recruitment . we then used the most abundant genotype from each line to create parental strains , mimicking the diverged by and rm parental strains in @xcite . a few populations were polymorphic for the number of loci initially , sometimes resulting in two lines with different values of @xmath5 , which we discarded . in each parent , we assigned the @xmath5 contributing loci randomly among @xmath83 simulated marker sites , and also assigned their associated @xmath7 values and the interactions @xmath27 between loci . we constructed @xmath28 recombinant haploid offspring by mating these two parents according to the genetic map inferred from brem et al . each offspring inherited each @xmath7 value , and the set of interactions towards other loci ( @xmath27 @xmath100 ) , from either one or the other parent . the trait value in each offspring was calculated as in eq . ( 1 ) and then was perturbed by adding a small amount of noise ( normally distributed with mean @xmath8 and standard deviation @xmath101 ) , to simulate measurement noise . we then analyzed these artificial genotype and phenotype data following the same protocol we used for the real yeast eqtls data ( i.e. using rqtl ) . we repeated this entire process with @xmath98 different pairs of parents for each value of @xmath9 . s18 shows the relationship between the selection pressure @xmath9 and the number of linked loci detected in this simulated qtl study , for different divergence times between the two lines and different values of @xmath101 . in fig . s19 , we increased @xmath101 proportionally to @xmath102 , from @xmath103 to @xmath104 . we also calculated the probability that a locus known to influence the trait in the true architecture ( fig . s3 ) is in fact detected in the qtl study . this probability is plotted as a function of @xmath9 for different values of the noise @xmath101 ( fig . s16 ) and of the time of divergence ( fig . we simulated the evolution of the genetic architecture underlying multiple traits with a model slightly modified from the single - trait version . in this model , the phenotype consists of @xmath105 traits , each trait @xmath39 under a different selection pressure @xmath106 ( the values of @xmath106 are those used in independent simulations of the single - locus model ; see the x - axis of fig . s15 ) . in the multiple traits version , @xmath5 denotes the total number of loci forming the architecture of the @xmath105 traits . @xmath5 can change when loci are duplicated at rate @xmath12 and deleted at rate @xmath13 . each locus participates to a set of traits . the direct effect of locus @xmath6 on trait @xmath107 is now denoted @xmath108 and the indirect effect of locus @xmath6 on the part of locus @xmath17 that contributes to @xmath107 is denoted @xmath109 . to allow for partial gains and losses of function , we define two new matrices @xmath110 and @xmath111 , which have the same dimensions as @xmath112 and @xmath113 . the functions corresponding to @xmath108 and @xmath109 are ` on ' when @xmath114 or @xmath115 , respectively , and are ` off ' otherwise . similarly to eq . ( 1 ) in our single - trait model , we calculate the trait value @xmath107 as : @xmath116 where @xmath20 is the sigmoidal function defined in eq ( 3 ) . point mutations of locus @xmath6 alter all @xmath108 and @xmath109 by a normal deviate . moreover , a mutation can change @xmath117 and @xmath118 to @xmath8 with probability @xmath119 and to @xmath60 with probability @xmath120 . over successive generations , the genetic architecture underlying each trait evolves through gene deletions and duplications , and through recruitments and losses of new functions . in this model , only the @xmath5 genes in the simulated architecture can be recruited _ i.e. _ we do not assume a fixed number of genes that can be recruited at any time . therefore , the phenotypic effects of recruitment evolve during our simulation , instead of being sampled from a given distribution . if @xmath121 for any trait @xmath107 , the individual is considered non - viable and fitness @xmath122 equals @xmath8 . otherwise , fitness is the product of gaussian functions for each trait times the cost associated to the number of loci , as follows : @xmath123 we simulated the evolution of the genetic architecture through a wright fisher process , with population genetics parameters identical to the default values in table s2 , except @xmath124 @xcite ) . the results of @xmath125 simulations are represented in fig . s15 . .estimates of rates of mutations @xmath11 , gene duplications @xmath12 and deletions @xmath13 . all rates are per gene per generation . @xmath11 is the rate of non - silent mutations @xcite ( @xmath126 the per - nucleotide mutation rate ) . when the mutation rate was given per nucleotide , we multiplied it by the average gene length in eukaryotes @xcite ( @xmath127 bp ) . for _ d. melanogaster _ @xcite , the rate of detectable mutations was used , after correcting for the length of the @xmath88 loci in the study . the scale of analysis can be the whole genome ( wg ) , or a specific set of loci , in which case the number of loci is denoted in the table . [ cols="<,^,^,^,^,^,>",options="header " , ] , depends on the strength of selection on the trait , in the model without epistasis . a : traits under intermediate selection ( intermediate values of @xmath9 ) have more variable effects . under strong selection , the variance across loci is low because mutations changing the trait value are eliminated shortly and can not be compensated by other mutations . under weak selection , variance can increase through compensatory evolution when the architecture includes multiple loci , but this variance goes to @xmath8 when the number of loci reaches @xmath60 . this occurs often enough ( see the distributions in fig . 1 ) to strongly reduce the mean variance of the phenotypic effects across loci . parameters are set to their default values ( table s2 ) . b : this difference across traits under various strengths of selection is also apparent in the distribution of @xmath7 . the genetic architectures of traits under strong selection , and to a lesser extent of traits under weak selection are dominated by loci with small individual effects . traits under intermediate selection rely on loci with more diverse contributions . ] ) evolve genetic architectures with the greatest number of controlling loci . the rectangle areas are proportional to the number of wright - fisher simulations ( among @xmath15 per value of @xmath9 ) in which the number of loci on the y - axis evolved . dots denote the ensemble mean of each distribution . the neutral expectations for the equilibrium number of loci ( see methods ) are represented as grey lines , when recruitment events are neutral ( top line ) or not ( bottom line ; deletions and duplications are neutral in both cases ) . the black arrow represents the number of loci for which the number of deletions fixing approximately equals that of duplications or recruitments for @xmath23 ( fig . 2 ) , where the mode of the distribution is expected . b : standard deviations of @xmath7 ( direct effects ) and @xmath22 ( indirect ) are maximum under intermediate selection . parameters are set to their default values ( table s2 ) . ] at which the expected number of loci @xmath5 is maximum . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . ] increases with the mutation rate @xmath11 . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . a mutation rate of @xmath131 was used to sample recruitment events , so the overall probability of recruitment remains constant . ] decreases as deletions become more frequent ( _ i.e. _ @xmath13 increases ) . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . ] increases as the probability of recruitment , @xmath26 , increases . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . ] , has similar effects as decreasing @xmath11 . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . ] is not strongly affected by mutation biases in @xmath112 or @xmath113 . a strong bias ( @xmath132 ) reduces the maximum variation across loci and therefore reduces @xmath5 when @xmath133 . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . one data point was omitted : @xmath134 at @xmath135 and @xmath136 . ] decreases with the metabolic cost @xmath33 . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . ] whenever @xmath90 is higher than @xmath8 ( text s1 ) . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . ] is not affected by the form of the distribution of mutation effects . all values represent the ensemble average of @xmath15 replicate simulations run for @xmath130 generations . all other parameters are set to their default values ( table s2 ) . ] loci and variable effects ( see description in the methods section ) . compared to fig . s3-a , only the architectures of traits under strong selection have changed . the mean number of loci under strong selection would be expected at the bottom grey line if deletions and duplications had neutral effects . instead , this initial variation across loci prevents deletion or duplication , so the mean number of loci remains close to its initial value . architectures of traits under intermediate and weak selection are not affected . ]

in the classic view introduced by r.a . fisher , a quantitative trait is encoded by many loci with small , additive effects . recent advances in qtl mapping have begun to elucidate the genetic architectures underlying vast numbers of phenotypes across diverse taxa , producing observations that sometimes contrast with fisher s blueprint . despite these considerable empirical efforts to map the genetic determinants of traits , it remains poorly understood how the genetic architecture of a trait should evolve , or how it depends on the selection pressures on the trait . here we develop a simple , population - genetic model for the evolution of genetic architectures . our model predicts that traits under moderate selection should be encoded by many loci with highly variable effects , whereas traits under either weak or strong selection should be encoded by relatively few loci . we compare these theoretical predictions to qualitative trends in the genetics of human traits , and to systematic data on the genetics of gene expression levels in yeast . our analysis provides an evolutionary explanation for broad empirical patterns in the genetic basis of traits , and it introduces a single framework that unifies the diversity of observed genetic architectures , ranging from mendelian to fisherian . * the evolution of genetic architectures underlying quantitative traits * + etienne rajon@xmath0 , joshua b. plotkin@xmath1 + @xmath2 department of biology , university of pennsylvania , philadelphia , pa 19104 , usa + @xmath3 e - mail : [email protected] + a quantitative trait is encoded by a set of genetic loci whose alleles contribute directly the trait value , interact epistatically to modulate each others contributions , and possibly contribute to other traits . the resulting genetic architecture of a trait @xcite influences its variational properties @xcite and therefore affects a population s capacity to adapt to new environmental conditions @xcite . over longer timescales , genetic architectures of traits have important consequences for the evolution of recombination @xcite , of sex @xcite and even reproductive isolation and speciation @xcite . although scientists have studied the genetic basis of phenotypic variation for more than a century , recent technologies , as well as the promise of agricultural and medical applications , have stimulated tremendous efforts to map quantitative trait loci ( qtl ) in diverse taxa @xcite . these studies have revealed many traits that seem to rely on fisherian architectures , with contributions from many loci @xcite , whose additive effects are often so small that qtl studies lack power to detect them individually @xcite . other traits , however , are encoded by a relatively small number of loci including the large number of human phenotypes with known mendelian inheritance . the subtle statistical issues of designing and interpreting qtl studies in order to accurately infer the molecular determinants of a trait are already actively studied @xcite . nevertheless , distinct from these statistical issues of inferences from empirical data , we lack a theoretical framework for forming _ a priori _ expectations about the genetic architecture underlying a trait @xcite . for instance , what types of traits should we expect to be monogenic , and what traits should be highly polygenic ? more generally , how does the genetic architecture underlying a trait evolve , and what features of a trait shape the evolution of its architecture ? to address these questions we developed a mathematical model for the evolution of genetic architectures , and we compared its predictions to a large body of empirical data on quantitative traits .

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Proceed to summarize the following text: it is now widely accepted that the horizontal structure of the sunspot penumbra is composed of two magnetic components ( solanki 2003 ; bellot rubio 2003 ) . one of them possesses a somewhat inclined ( @xmath3 with respect to the vertical direction to the solar surface ) and strong ( @xmath4 g ) magnetic field , whereas the other is characterized by a weaker and more horizontal one ( lites et al . 1993 ; redi et al . 1998 ; bellot rubio et al . 2004 ; borrero et al . 2004 , 2005 ) . traditionally , these two magnetic components have been identified with a horizontal flux tube , that carries the evershed flow , and is embedded in a more vertical background magnetic field : _ uncombed _ model ( solanki & montavon 1993 ; schlichenmaier et al . 1998 ; borrero 2007 ) . recently , this view has been challenged by spruit & scharmer ( 2006 ) and scharmer & spruit ( 2006 ) , who propose instead that the penumbra is formed by magnetic field - free plumes ( connected to the underlying convection zone ) that pierce the penumbral magnetic field from beneath . this is the so - called _ gappy _ penumbral model . so long as these two different magnetic structures ( weak / horizontal and strong / vertical ) have remained spatially ( horizontally ) unresolved , distinguishing between the uncombed and gappy penumbral scenarios has not been possible . however , with the new spectropolarimeter on board of the japanese spacecraft hinode ( kosugi et al . 2007 ; shimizu et al . 2007 ) it is now possible to obtain high spatial resolution ( @xmath5 " ) observations of the sunspot penumbra . this could be sufficient to distinguish between the uncombed and gappy models , since they both postulate the existence of flux tubes or field - free gaps that are about 200 - 300 km in diameter ( martnez pillet 2000 ; spruit & scharmer 2006 ) . this feature is particularly interesting , because the _ uncombed _ and _ gappy _ models predict very different vertical stratifications in the magnetic field strength across the weak / horizontal magnetic field component : which is identified with an embedded flux tube in the uncombed model , but with a field - free gap in the gappy model . in the latter , the magnetic field decreases monotonically with depth , whereas the former possesses a magnetic field that decreases with depth only initially , since once the boundary of the flux tube is reached , the magnetic field can either decrease or increase depending upon the strength of the magnetic field inside the tube . in this paper we will focus on obtaining the vertical stratification of the magnetic field for penumbral filaments ( where the magnetic field is more horizontal and weaker ) using high spatial resolution spectropolarimetric observations from hinode , in order to establish which penumbral model is more realistic . the observations are described in section 2 . section 3 describes our data analysis and results from our inversion technique . section 4 compares our findings with the predictions made by the uncombed and gappy penumbral models . in section 5 we make a thourough investigation of the effects of the scattered light . finally , section 6 summarizes our findings . on may 3rd 2007 , between 10:15 and 11:40 am ut , the active region ar 10953 was mapped using the spectropolarimeter of the solar optical telescope on - board of the hinode spacecraft ( lites et al . 2001 ) . the active region was located at a heliocentric angle of @xmath6 . it was scanned in a thousand steps , with a step width of 0.148 `` and a slit width of 0.158 '' . the spectropolarimeter recorded the full stokes vector ( @xmath7 , @xmath8 , @xmath9 and @xmath10 ) of the pair of neutral iron lines at 630 nm with a spectral sampling of 21.53 m . the integration time was 4.8 seconds , resulting in an approximate noise level of @xmath11 ( in units of the normalized continuum intensity ) . in the absence of the telluric oxygen lines we proceeded with two different wavelength calibration methods that were cross - checked for consistency . the first method was obtained by matching the average quiet sun profile with the fts spectrum , whereas the second calibration assumes that the average umbral profile exhibits no velocities . a map of the continuum intensity at 630 nm of the scanned region is shown in figure 1 . the white arrow indicates the direction of the center of the solar disk . the penumbra on the center side is heavily distorted and therefore left out from our analysis . on the limb side the penumbra is more uniform , with radially aligned filaments . the region enclosed by the white rectangle has been chosen for our study . this sunspot has negative polarity ( magnetic field in the umbra points towards the solar interior ) , however the results presented hereafter are shown , in order to facilitate the interpretation , as if the sunspot had positive polarity . we have applied the sir inversion code ( ruiz cobo & del toro iniesta 1992 ) to our spectropolarimetric observations to retrieve the physical properties of the solar atmosphere . this code allows all relevant physical parameters to be a generic function of the optical depth : @xmath12 , @xmath13 , @xmath14 , @xmath15 , etc . in addition , a depth - dependent temperature stratification @xmath16 models the atmospheric thermodynamics under local thermodynamic equilibrium ( lte ) conditions . sir retrieves the values of the parameters at a number of optical depth points called _ nodes_. the final stratification is obtained by interpolating splines across those nodes . note however , that sir employs equivalent response functions ( del toro iniesta 2003 ) , which ensures sensitivity to the atmospheric layers located between nodes . each node represents a free parameter in the inversion . in our investigation we will employ increasingly complex models ( i.e. : more free parameters ) according to the amount of information we hope to extract from the profiles . given the high spatial resolution of hinode s observations , we will consider only one magnetic component . a non - magnetic component is also considered to account for the scattered light . in this section , the scattered light profile is obtained by averaging the intensity profiles of those pixels with polarization signals below the noise level ( quiet sun granulation around the sunspot ) . the same scattered light profile is used in the inversion of all pixels . in sect . 5 we make a thourough analysis of the effects that different treatments for the scattered light have on our results . note that using one single magnetic component is equivalent to assuming that the penumbral structure is horizontally resolved . this is clearly not the case if we look into continuum images at even higher spatial resolution ( scharmer et al . however , our assumption would still be valid if the ( unresolved ) variations of the magnetic field inside the weak / horizontal magnetic component are much smaller than the differences between the weak / horizontal and strong / vertical components . since the former remain unresolved , we can not assess the validity of this assumption . this question should be addressed as better spectropolarimetric observations become available . in order to locate the intraspinal pixels we have carried out a first inversion where all physical parameters , with the exception of the temperature , are constant with optical depth . we therefore have one single node for @xmath12 , @xmath13 , @xmath14 , @xmath15 . to account for unresolved velocity fields , we also consider depth - independent micro and macroturbulent velocities : @xmath17 and @xmath18 . another free parameter , @xmath19 , represents the fraction of the observed intensity , stokes @xmath7 , that corresponds to scattered light . finally , three nodes are given to the temperature @xmath16 . in total , this first inversion has 10 free parameters . since @xmath20 , @xmath21 , @xmath22 and @xmath23 are constant with optical depth , the retrieved values indicate some kind of average over the region where the spectral lines are formed . westendorp plaza et al . ( 1998,2001 ) studied this issue in detail and found that the largest contribution for this pair of fe i lines ( sect . 2 ) comes from @xmath24 . figure 2 displays the resulting values for the line - of - sight velocity and magnetic field vector in the selected box in fig . 1 . regions of weak , @xmath25 g , and highly inclined , @xmath26 , magnetic field can be clearly distinguished in this figure . they are also characterized by the presence of large red - shifted velocities ( evershed flow ) . these are the so - called penumbral intraspines , and therefore the most likely locations where field - free gaps or horizontal flux - tubes can be found . also visible are structures characterized by a stronger and more vertical magnetic field , as well as by a strongly reduced evershed flow . these are usually referred to as spines . spines and intraspines are also seen at moderate ( @xmath27 1 `` ) spatial resolution ( lites et al . 1993 ; stanchfield et al . 1997 ; mathew et al . 2003 ) but the associated changes in their properties ( field strength , inclination , etc ) are larger if observed at high spatial resolution ( bello gonzlez et al . 2005 ; langhans et al . 2005 ) . bellot rubio et al . ( 2004 ) interprets this result as a consequence of these structures not being spatially resolved at 1 '' resolution . as demonstrated by borrero et al . ( 2008 ) they are indeed horizontally resolved in hinode observations ( 0.32 " ) . + + in figure 2 we also indicate with black and white dots a large number ( total of 7520 ) of intraspinal pixels . they have been found as those satisfying the following conditions : @xmath28 g , and @xmath29 $ ] km s@xmath30 . since the main difference between spines and intraspines is the presence of a strong evershed flow , we use @xmath23 to distinguish among them . however , we do not consider the few pixels where @xmath31 km s@xmath30 , since they usually present extremely abnormal stokes v profiles , usually a sign of the existence of horizontally unresolved structure . we do not constrain the values of the magnetic field inclination and strength ( here we use only a lower limit to avoid taking pixels outside the visible boundary of the sunspot ) because the magnetic properties of the spines in the outer penumbra are very similar to those of the intraspines in the inner penumbra . the final selected pixels represent about 39 % of all penumbral pixels in figure 2 . note that they are mostly located in the middle and outer penumbra : @xmath32 ( @xmath33 being the sunspot radius ; umbral - penumbral boundary is located at @xmath34 ) . note also that , even though we have not constrained the values of the magnetic field strength an inclination , all intraspinal pixels are located in regions where the magnetic field is highly inclined and weak . in order to investigate the depth variation of the physical parameters in intraspines , we performed a renewed inversion of the pixels selected in fig . 2 , where we now allow for two nodes in @xmath12 , @xmath13 , @xmath14 , @xmath15 ( linear variations with optical depth ) . the total number of free parameters is now 14 . results from this new inversion are presented in figure 3 : @xmath16 ( upper - left ) , @xmath15 ( upper - right ) , @xmath12 ( lower - left ) , and @xmath13 ( lower - right ) . all inverted pixels display similar stratifications of @xmath15 and @xmath13 : both increase monotonically with optical depth : @xmath35 , @xmath36 . the magnetic field strength @xmath12 , however , can either increase ( in 66 % of inverted pixels : 4971 ) or decrease ( 34 % ; 2249 pixels ) with optical depth . in either case , the retrieved gradient is relatively small : @xmath37 gauss km@xmath30 . an important feature to note is that pixels displaying a decreasing magnetic field towards deeper layers , @xmath38 , are mostly located in the inner penumbra , whereas pixels showing @xmath39 are mostly found in the outer penumbra ( see figure 4 ) . we now perform a more complex inversion of the same pixels as in sect . 3.2 . in this case we allow for 4 nodes in @xmath12 , @xmath13 , @xmath14 , and @xmath15 . these nodes are located at optical depth positions : @xmath40 $ ] . in total , this new inversion has 22 free parameters . figure 5 shows the results from the 4-node inversion of the 7250 intraspinal pixels selected in sect . the stratifications are very similar to those already obtained through the 2-node inversion ( see fig . 3 ) . the larger scatter ( pixel - to - pixel variations ) in the 4-node inversion is due to the larger amount of free parameters , which are more weakly constrained by the observations . since now we allow for 4 nodes to the stratification of the magnetic field strength it is not easy to classify our results between those where the magnetic field increases or decreases with optical depth . to showcase the differences between the possible stratifications we have taken separately those pixels where , in the 2-node inversion , showed @xmath38 ( family 1 ) or @xmath39 ( family 2 ) and obtained the averaged stratification for the 2 and 4-node inversion . results for family 1 and 2 are presented in figures 6 and 7 respectively . + interestingly , the magnetic field strength , that in the 2-node inversion showed different gradients , now shows an initial decrease up to @xmath41 ( approximately 100 km above the continuum level ) , where it starts to increase again towards deep layers . this happens for both families of magnetic structures , and thus could indicate that they are indeed closely related . a closer look reveals that both families possess a similar magnetic field strength in deep layers : @xmath42 g , but slightly different higher up : @xmath43 g ( family 1 ; fig . 6 ) and @xmath44 g ( family 2 ; fig . 7 ) . this effect explains why the 2-node inversion ( sect . 3.2 ) retrieves different overall gradients for the magnetic field strength : it is due to a large variation in the magnetic field at around @xmath45 since the magnetic field deeper down is basically the same in both cases . to further confirm these results we have repeated our 4-node inversion with the nodes located at slightly different positions . sir always places 2 nodes at the uppermost and deepest @xmath46-locations of the discretized atmosphere , while spreading the rest equidistantly in between . therefore , to keep the same number of nodes and , at the same time , change their @xmath46-positions we must change the initial and last @xmath46-points of the atmosphere . in our first set of inversions the atmosphere is discretized between @xmath47 $ ] . changing this to @xmath48 $ ] and @xmath49 $ ] would position the 4 nodes at @xmath50 $ ] and @xmath51 $ ] , respectively . we have inverted all pixels again in these two cases and confirmed that our results ( fig . 5,6 and 7 ) do not change . this is due to the use that sir makes of equivalent response functions ( see section 3 ) . if we consider a ray passing through the center of an intraspine , the _ gappy _ and _ uncombed _ penumbral models predict a very similar stratification of the magnetic field strength above the field - free gap or flux tube , but very different ones inside them . figure 8 illustrates some possible stratifications predicted by these two models , where the upper boundary of the field - free gap and flux tube is located at @xmath52 . above @xmath52 they both share the same stratification for the surrounding magnetic atmosphere . here we present two examples , one where the surrounding magnetic field is weak : @xmath53 ( dashed line ; meant to represent the outer penumbra , @xmath54 ) , and another case where the surrounding field is stronger ( solid line ) : @xmath55 ( meant to represent to inner penumbra , @xmath56 ) . note that the magnetic field strength in the surrounding atmosphere decreases towards deeper layers . this is due to the fact that the vertical component of the surrounding field must vanish ( or nearly vanish in the case of a cusp - shaped boundary ) at the flux tube s or gap s boundary . these two examples are actual solutions of analytical models ( fig . 5 in spruit & scharmer 2006 ; fig . 3 in scharmer & spruit 2006 ; see also eqs . 33 - 34 in borrero 2007 ) . below the boundary of the flux tube or field - free gap , @xmath52 , both models predict a very different situation . in the case of the _ gappy _ penumbra this region is void of magnetic fields : @xmath57 ( hollow circles ) . in contrast , the _ uncombed _ model assumes the existence of a flux tube where the magnetic field is strong @xmath58 g ( filled circles ) . if we compare fig . 8 with our 2-node inversion ( fig . 3 ) of intraspinal pixels we find that , on the one hand , the _ gappy _ penumbral model can only explain the slowly decreasing magnetic field , observed for 34 % of intraspinal pixels ( family 1 ) , if the @xmath2 level is formed above the gap s boundary , otherwise a much more sudden drop would be observed ( hollow circles in fig . 8) . on the other hand , this model does not offer any explanation for the 66 % of the intraspinal pixels that present an increasing magnetic field strength towards deeper layers ( family 2 ) . however , the _ uncombed _ penumbral model can explain both observed situations . it all depends on the strength of the flux tube s magnetic field as compared to the magnetic field high above it : @xmath59 versus @xmath60 . a magnetic field that decreases smoothly towards the interior of the photosphere can be explained by a flux tube ( of any field strength ) whose upper boundary layer lies below @xmath2 . if the upper boundary is above @xmath2 , it can also be explained with a magnetic field inside the flux tube that is weaker than the magnetic field a few hundred kilometers above ( solid lines plus filled circles in fig . in addition , a magnetic field that increases towards the interior of the photosphere is compatible with a flux tube with an upper boundary layer above @xmath2 , and with a stronger magnetic field than the one above ( dashed line and filled circles in fig . 8) . a more complex ( 4-nodes ) inversion of intraspinal profiles indicates that , what appeared as two different families of structures using a 2-node inversion , are likely to correspond to one single kind of magnetic structure , where the magnetic field exhibits an initial decrease between @xmath61 $ ] , but increases between @xmath62 $ ] ( see figs . 6 - 7 ) . while the _ gappy _ model offers no explanation for this effect , it can indeed be explained by the _ uncombed _ penumbral model , by means of a magnetic field whose strength decreases initially but increases once the line - of - sight crosses the flux tube s boundary ( see fig . furthermore , although intraspinal families 1 and 2 appear to be the equivalent in the 4-node inversion , they still present a subtle yet important difference : family 1 ( more commonly found in the inner penumbra ; see fig . 4 ) displays a much stronger initial decrease as compared to family 2 , which is usually found in the outer penumbra ( compare lower - left panels in figs . 6 and 7 ) . this can be explained , in terms of the _ uncombed _ model if , at small - intermediate radial distances , the horizontal flux tube possesses a weaker magnetic field than the field in the atmosphere in which it is embedded : @xmath63 at @xmath64 small ( compare solid line plus filled circles in fig . 8 with green solid in fig . 6 ) . as we move towards larger radial distances , and assuming that the magnetic field inside the flux tube remains constant , the surrounding magnetic field weakens and falls below the flux tube s field strength : @xmath65 at @xmath64 large ( compare dashed line plus filled circles in fig . 8 with solid green in fig . note that the assumption that the magnetic field in the flux tube remains constant is in agreement with a surrounding magnetic field whose strength decays much more rapidly towards the outer penumbra than inside the flux tube ( see fig . 4 in borrero et al . 2004 ; fig . 6 in borrero et al . 2005 and fig . 4 in borrero et al . 2006 ) . here we find that this known feature of the penumbral intraspines helps to explain , within the frame of the uncombed model , differences in the stratification in the magnetic field strength across instrapines at different radial distances , as deduced from high resolution spectropolarimetric observations . one of the most critical issues in the inversion of spectropolarimetric data is the treatment of the scattered light . in order to properly model its contribution , detailed measurements of the telescope s psf are needed . since these are not usually available , the scattered light is often treated as a non - polarized contribution to the total observed light ( see sect . 3 ) . in our study this is particularly important because one of the models under study ( _ gappy _ model ) postulates the existence of field - free regions around the @xmath2 level in the penumbra . these regions will naturally produce a non - polarized contribution to the total observed stokes vector . therefore , there is a potential risk of not detecting the field - free gaps due to an incorrect treatment of the scattered light . + if our inversions are affected by this degeneracy between scattered light and field - free gaps , it is expected that those pixels where the intraspines are located show larger values for the amount of scattered light retrieved by the inversion ( @xmath19 ) . to study this possibility we have plotted in figure 9 the variations of the magnetic field strength ( top panel ) , line - of - sight velocity ( middle panel ) and inclination angle ( bottom ) along an azimuthal cut at @xmath66 . other azimuthal cuts at different radial distances show very similar behaviors . the values are taken at an optical depth of @xmath2 from the 2-node inversion in sect . this plot includes , not only those pixels selected in fig . 2 as intraspines , but all of them . therefore regions where the magnitude of evershed flow is reduced and the magnetic field is more vertical and strong ( penumbral spines ) are also visible . all three panels also show the amount of scattered light @xmath19 ( dashed lines ) . there is no particular correlation between the location of penumbral intraspines ( high velocities , weak and very inclined fields ) and the regions where @xmath19 is largest . similar variations are observed if we plot the values of the magnetic field strength and inclination , and line - of - sight velocity , at an optical depth of @xmath67 . this rules out the possibility that our inversions do not show field - free regions , in the deep photospheric layers , where intraspines are located at the expense of an enhanced scattered light contribution . recently , orozco surez et al . ( 2007a ; 2007b ) have presented inversions of stokes spectra measured using hinode s spectropolarimeter in the quiet sun . these authors claim that for this instrument it is more appropriate to consider a local ( unpolarized ) scattered light profile . this is obtained by averaging the observed stokes @xmath7 profiles over a small region ( about 1 arcsec ) around the pixel that is being studied . in this case , a different scattered light profile is used in the inversion of each pixel . this approach can be justified by the fact that the focus of the narrow - band filter ( nbi ) on hinode is favored when simultaneous observations are carried out with both instruments . in our inversions , we have however used a global scattered light profile , where we average the stokes @xmath7 signal emerging from the quiet sun region far away from the sunspot . in order to test whether our results depend on the use of a different scattered light profile , we have repeated our 2-node inversion using the same approach as orozco surez et al . the results are presented in fig . fig . 3 ) . the percentage of intraspinal pixels with @xmath68 is even larger than before ( 90 % ) . alternatively , orozco surez et al . ( 2007a ; 2007b ) point out that the most realistic way to account for the scattered light would be to consider a local * and * polarized scattered light profile , where not only stokes @xmath7 is averaged , but also stokes @xmath8 , @xmath9 and @xmath10 . we have also tested this possibility . unfortunately , this yields unrealistically high values for @xmath19 during the inversion : @xmath69 . this indicates that the inversion code tries to dominantly reproduce the observed stokes profiles using the scattered light contribution . this is not surprising since the neighboring stokes profiles often look very similar to those in the pixel under study . therefore we conclude that this is not a reasonable approach when inverting sunspot data . we can not rule out however , that this treatment will work in quiet sun regions . as a final test , we have repeated our 2-node inversion but neglecting any scattered light : @xmath70 . this test is very appropriate because , according to spruit & scharmer ( 2006 ) , inversions of spectropolarimetric data fail to detect field - free regions in the penumbra as a consequence of these being already included in the scattered light profile . if their hypothesis is correct , not accounting for the scattered light contribution should uncover these regions with @xmath71 near @xmath2 . results ignoring the effects of the scattered light ( figure 11 ) are essentially unchanged if compared to those where we used a global ( fig . 3 ) or local ( fig . 10 ) scattered light profile , with 68 % of the pixels showing @xmath68 . the only difference is the increased temperatures obtained when we impose @xmath70 . in particular we do not see any pixel where the magnetic field reaches very small values in the deepest photospheric layers . taking into account all tests carried out in this section , it seems unlikely that the scattered light can significantly bias our magnetic field stratifications , consequently making highly unlikely that we are missing the detection of field - free regions near @xmath2 . the _ uncombed _ model postulates that penumbral intraspines are characterized by the presence of horizontal flux tubes embedded in a surrounding atmosphere that possesses an inclined magnetic field . according to this model , looking along these regions should reveal a magnetic field that smoothly decreases at first , but once the flux tube contribution starts , the field strength could either increase or decrease . alternatively , the _ gappy _ penumbral model postulates that instraspines correspond instead to regions where convective field - free gaps penetrate the penumbral field . in this case the magnetic field strength should also decrease with optical depth at first , but suffer a much larger drop once the line - of - sight crosses the field - free gap . in order to differentiate between these two models , we have used polarimetric data at very high spatial resolution , recorded with the spectropolarimeter on - board of the japanese spacecraft hinode , to investigate the depth variation of the magnetic field strength in the penumbra . we have selected a large number ( @xmath277500 ) of pixels that are representative of weak and horizontal magnetic fields ( i.e. : penumbral intraspines ) carrying strong evershed flows . from the inversion of the stokes profiles at these locations we find that the magnetic field strength can either increase or decrease with optical depth . a more detailed inversion of the average stokes vector over the selected pixels , shows that the magnetic field initially decreases , between @xmath72 $ ] , but increases thereafter until @xmath73 . the _ gappy _ penumbral model can explain a smoothly decreasing magnetic field strength only if the @xmath2 level is formed above the field free gap , otherwise a much more sudden decrease would be observed as the line - of - sight penetrates the field - free plasma . a partial solution to this problem can be found if we assume that the gap is not fully evacuated of magnetic field . however , it offers no explanation for about 66 % of the selected pixels , where an increasing magnetic field strength with optical depth is observed . the absence of field - free gaps , as indicated by the inversion , does not in itself imply that there is no form of convection present in the penumbra , but rather suggests that the convective energy transport takes places in the presence of a magnetic field ( see zakharov et al . 2008 ; rempel & schssler 2008 ) . an example is the roll convection proposed by danielson ( 1961 ) . all inferred stratifications are compatible with the scenario proposed by the _ uncombed _ model . a magnetic field that decreases smoothly towards the interior of the photosphere can be explained by either a flux tube ( of any field strength ) whose upper boundary layer lies below @xmath2 or , if the upper boundary is above @xmath2 , with a magnetic field inside the flux tube that is weaker than the magnetic field a few hundred kilometers above . in addition , a magnetic field that increases towards the interior of the photosphere is compatible with a flux tube with an upper boundary layer above @xmath2 , but with a stronger magnetic field than the one above . this very same configuration can explain a magnetic field that first decreases and then increases with optical depth , as inferred from the inversion of averaged intraspinal profiles . we have also studied the effects of the scattered light in our inversions . we have seen that any inaccuracies in its treatment are unlikely to be a source of error in the stratification of the magnetic field strength . it would be very desirable to make a robust confirmation of our findings , namely @xmath74 in the outer penumbra , for a larger number of sunspots at different heliocentric angles and including also the disk - ward side of the penumbra . a natural extension of this work would be to use the fe i lines at 1.56 @xmath75 m ( which are formed deeper in the photosphere ) to confirm the absence of field - free regions around @xmath2 . unfortunately , no such observations exist at the spatial resolution needed to resolve penumbral intraspines ( @xmath76 0.4 `` ) . indeed , some studies at slightly lower resolution ( 0.6 - 0.7 '' ) have been presented by cabrera solana et al . ( 2008 ) , who used simultaneous observations of fe i 630 nm and 1.56 @xmath75 m recorded with the tip ( martnez pillet et al . 1999 ) and polis ( schmidt et al . 2003 ) instruments . they found that in the outer penumbra , the horizontal magnetic field component ( carrying the evershed flow ) was no longer weaker than the more vertical one . this can be used as an independent confirmation of our work , where we routinely find @xmath74 at large radial distances from the center of the sunspot . in addition , flux tubes with stronger magnetic field than the one of the environment in which they are embedded are also necessary to explain certain aspects of the net circular polarization observed in the outer penumbra of sunspots ( tritschler et al . 2007 ; ichimoto et al . 2008 ) . bello gonzlez , n. , okunev , o.v . , domguez cerdea , i. , kneer , f. , puschmann , k.g . 2005 , a&a , 434 , 317 bellot rubio , l.r . , baltasar , h. & collados , m. 2004 , a&a , 427 , 319 bellot rubio , l.r . 2003 , in proceedings of the solar polarization conference . eds : javier trujillo and jorge snchez almeida . series , vol . 307 , p 301 . borrero , j.m . , solanki , s.k . , bellot rubio , l.r . , lagg , a. & mathew , s.k . 2004 , a&a , 422 , 1093 borrero , j.m . , lagg , a. , solanki , s.k . & collados , m. 2005 , a&a , 436 , 333 borrero , j.m . , solanki , s.k . , lagg , a. , socas - navarro , h. & lites , b.w . 2006 , a&a , 450 , 383 borrero , j.m . 2007 , a&a , 471 , 967 borrero , j.m . , lites , b.w . & solanki , s.k . 2008 , a&a , 481 , l13 cabrera solana , d. , bellot rubio , l.r . , borrero , j.m . & del toro iniesta , j.c . 2008 , a&a , 477 , 273 danielson , r.e . 1961 apj , 134 , 289 del toro iniesta , j.c . introduction to spectropolarimetry . cambridge university press ( cambridge , uk ) ichimoto , k. , tsuneta , s. , suematsu , y. , katsukawa , y. , shimizu , t. , lites , b.w . , kubo , m. , tarbell , t.d . , shine , r.a . , title , a.m. , nagata , s. , a&a , 481 , l9 kosugi , t. , matsuzaki , k. , sakao , t. et al . 2007 , , 243 , 3 langhans , k. , scharmer , g. , kiselman , d. , lfdahl , m.g . & berger , t.e . 2005 , a&a , 436 , 1087 lites , b.w . , elmore , d.f . , seagraves , p. & skumanich , a.p 1993 , apj , 418 , 928 lites , q.w . , elmore , d.f . & streander , k.v . 2001 , in asp conf . 236 , advanced solar polarimetry , ed . m. sigwarth ( san francisco : asp ) , 33 martnez pillet et al . 1999 , in asp conf . 183 , high resolution solar physics : theory , observations and techniques , ed . rimmele , k.s . balasubramanian & r.r . radick ( san francisco : asp ) , 264 martnez pillet , v. 2000 , a&a , 361 , 734 mathew , s.k . , lagg , a. , solanki , s.k . 2003 , a&a , 410 , 695 orozco surez , d. , bellot rubio , l.r . & del toro iniesta , j.c . 2007a , apj , 662 , l31 orozco surez , d. , bellot rubio , l.r . , del toro iniesta , j.c . et al . 2007b , apj , 670 , l61 rempel , m. & schssler , m. 2008 , _ in preparation _ redi , i. , solanki , s.k . , keller , c.u . & frutiger , c. 1998 , a&a , 338 , 1089 ruiz cobo , b. & del toro iniesta 1992 , apj , 398 , 375 scharmer , g.b . , gudiksen , b.v . , kiselman , d. , lfdahl , m. g. ; rouppe van der voort , l.h . m. 2002 , nature , 420 , 151 scharmer , g. & spruit , h.c . 2006 , a&a , 460 , 605 schlichenmaier , r. , jahn , k. & schmidt , h.u . 1998 , a&a , 337 , 897 schmidt , w. , beck , c. , kentischer , t. , elmore , d. & lites , b.w . 2003 , astron . nach . , 324 , 300 shimizu , t. , nagata , s. , tsuneta , s. et al . 2007 , , _ in press _ solanki , s.k . , & montavon , c.a.p . 1993 , a&a , 275 , 283 solanki , s.k . 2003 , a&arv , 11 , 153 spruit , h.c . & scharmer , g.b . 2006 , a&a , 447 , 343 stanchfield , d.c.h . , thomas , j.h . & lites , b.w . 1997 , apj , 447 , 485 tritschler , a. , mller , d.a.n , schlichenmaier , r. & hagenaar , h.j . 2007 , apj , 671 , l85 westendorp plaza , c. , del toro iniesta , j.c . , ruiz cobo , b. , martnez pillet , v. , lites , b.w & skumanich , a. 1998 , apj , 494 , 453 westendorp plaza , c. , del toro iniesta , j.c . , ruiz cobo , b. , martnez pillet , v. , lites , b.w & skumanich , a. 2001 , apj , 547 , 1130 zakharov , v. , hirzberger , j. , riethmueller , t. , solanki , s.k . & kobel , p. 2008 , a&a , _ submitted _

the vertical stratification of the magnetic field strength in sunspot penumbrae is investigated by means of spectropolarimetric observations at high spatial resolution from the hinode spacecraft . assuming that the magnetic field changes linearly with optical depth we find that , in those regions where the magnetic field is more inclined and the evershed flow is strongest ( penumbral intraspines ) , the magnetic field can either increase or decrease with depth . allowing more degrees of freedom to the magnetic field stratification reveals that the magnetic field initially decreases from @xmath0 until @xmath1 , but increases again below that . the presence of strong magnetic fields near the continuum is at odds with the existence of regions void of magnetic fields at , or right below , the @xmath2 level in the penumbra . however , they are compatible with the presence of a horizontal flux - tube - like field embedded in a magnetic atmosphere . *l *

You are an expert at summarizing long articles. Proceed to summarize the following text: for 32 edge - on spiral galaxies , spanning hubble type and mass , we fit the edge - on infinite disk model by @xcite on the irac mocaics : @xmath0 . we applied stellar dominated 3.6 and 4.5 @xmath1 m channels and the pah emission at 8.0 @xmath1 m , with the stellar contribution subtracted ( using the prescription from * ? ? ? * ) . the data is from our dedicated go program ( go 20268 ) and the _ spitzer _ archive . our program aims to populate the lower range of disk masses . from the fit we obtain the scale - length ( @xmath2 ) , the scale - height ( @xmath3 ) , and the face - on central surface brightness ( @xmath4 ) . the edge - on disk model fits the stellar channels very well but the pah channel less so because of star - formation structures , such as hii regions , in the pah maps . the disk s total luminosity is inferred from the fitted model : @xmath5 , with @xmath2 the scale - length and @xmath4 the face - on central surface brightness ( see * ? ? ? combined with the rotational velocities from hyperleda , we can construct the tully - fisher relation for our galaxy disks and a plot of disk oblateness as a function of rotational velocity ( and hence dynamical mass ) . figure [ f : tf ] shows the inferred tully - fischer relation for these disks in one of the stellar channels , 4.5 @xmath1 m . notably , the slope ( @xmath6 ) is 3.5 , similar to what @xcite found but contrary for the increasing trend of slope with redder filters . the effects of age and metallicity of the stellar population become independent and opposite effects on the color - m / l relation in nir ( see * ? ? ? * figure 2d ) . hence , the shallower slope in the irac stellar channels , could well be this metallicity effect starting to dominate . the tully - fisher relation for disks in our sample . notably , the slope is very similar to the one found by @xcite for the sings sample . open circles are the poorer disk fits ( @xmath7).,scaledwidth=80.0% ] figure [ f : hz ] shows the relation between the disk oblateness the ratio of scale - length and -height , @xmath8 relative to the oblateness in 4.5 @xmath1 m . the 3.6 and 4.5 @xmath1 m channels both trace the stellar population and their oblateness measurements agree well . the pah mosaic ( 8 @xmath1 m ) , shows flattening compared to stellar disk : @xmath9 . the pah disk s flattening is substantial in the case of the more massive disks ( @xmath10 ) . this appears to corroborate the reasoning of @xcite that massive disks are vertically unstable and hence form the characteristic dust lanes . the oblateness ( @xmath8 ) of the 3.6 @xmath1 m fit ( open triangles ) and 8.0 @xmath1 m pah map ( filled squares ) in terms of the 4.5 @xmath1 m . a similar oblateness as the 4.5 @xmath1 m . the oblateness of the stellar filters , 3.6 and 4.5 @xmath1 m is very similar . the pah disk is flatter than the stellar , especially in massive disks ( @xmath10 , vertical dotted line).,scaledwidth=80.0% ] hence , from our disk models of irac data , we conclude : * the tully - fisher relation has a shallower slope ( @xmath11 ) than naively expected from trends with filter ( figure [ f : tf ] ) , which may be a metallicity effect of the stellar population s m / l . * the pah disk is flatter than the stellar one , especially in more massive spirals ( figure [ f : hz ] ) . the effect is not as pronounced as found by @xcite and @xcite based on dust lanes . the unique perspective of _ spitzer _ on edge - on spirals allows us to quantify the vertical structure in spiral disks . the pah maps are the product of both the ism density and star - formation . the stellar channels are effectively unaffected by extinction and can therefore serve as a reference to quantify dust lanes and smaller dust extinction structures . the phenomenon of truncation the sudden change in exponential behavior of disks can also be characterized for all irac bands .

edge - on spiral galaxies offer a unique perspective on disks . one can accurately determine the height distribution of stars and ism and the line - of - sight integration allows for the study of faint structures . the spitzer irac camera is an ideal instrument to study both the ism and stellar structure in nearby galaxies ; two of its channels trace the old stellar disk with little extinction and the 8 micron channel is dominated by the smallest dust grains ( polycyclic aromatic hydrocarbons , pahs ) . @xcite probed the link between the appearance of dust lanes and the disk stability . in a sample of bulge - less disks they show how in massive disks the ism collapses into the characteristic thin dust lane . less massive disks are gravitationally stable and their dust morphology is fractured . the transition occurs at 120 km / s for bulgeless disks . here we report on our results of our spitzer / irac survey of nearby edge - on spirals and its first results on the nir tully - fischer relation , and ism disk stability .

You are an expert at summarizing long articles. Proceed to summarize the following text: the @xmath5 galaxy fsc 10214 + 4724 was one of the most remarkable objects detected by the survey . originally proposed to be the most luminous galaxy known ( rowan - robinson 1991 ) , multi - wavelength observations subsequently showed that it is lensed by an intervening @xmath6 galaxy , boosting its intrinsic emission by a factor of @xmath7 10100 ( depending on the location and extent of the unlensed emission with respect to the caustic ; e.g.,broadhurst & lehar 1995 ; downes 1995 ; trentham 1995 ; eisenhardt 1996 ; evans 1999 ) . optical and near - ir spectroscopic / polarimetric observations have unambiguously shown that fsc 10214 + 4724 hosts an obscured active galactic nucleus ( agn ; e.g.,elston 1994 ; soifer 1995 ; goodrich 1996 ) . multi - wavelength analyses have suggested that the agn is powerful ( e.g. , goodrich 1996 ; granato 1996 ; green & rowan - robinson 1996 ) , although it is generally accepted that star - formation activity dominates the bolometric output ( e.g.,rowan - robinson 1993 ; rowan - robinson 2000 ) . the lensing - corrected properties of fsc 10214 + 4724 are similar to those of galaxies ( e.g. , blain 2002 ; ivison 2002 ; smail 2002 ; chapman 2003 ; neri 2003 ) : it lies at @xmath8 , is optically faint with @xmath9 25 , has an 850@xmath10 m flux density of a few mjy and a 1.4 ghz flux density of @xmath0 25 @xmath10jy , is massive ( a molecular gas mass of @xmath11@xmath12 @xmath13 ) , and has a bolometric luminosity of @xmath14 l@xmath15 ( e.g. , rowan - robinson 1993 ; downes 1995 ; eisenhardt 1996 ) . galaxies appear to be massive galaxies undergoing intense star - formation activity and are likely to be the progenitors of the local @xmath7 @xmath16 early - type galaxy population ( e.g. , blain 2004 ; chapman 2004 ) . deep x - ray observations have shown that a large fraction ( at least @xmath0 40% ; alexander 2003b , 2004 ) of the galaxy population is actively fuelling its black holes during this period of intense star formation , which can account for a significant fraction of the black - hole growth in massive galaxies ( alexander 2004 ) . the agn in fsc 10214 + 4724 was identified via optical / near - ir observations , and the large lensing boost of the agn emission provides interesting insight into agn activity in the galaxy population . in this letter we present the results from a @xmath0 20 ks observation of fsc 10214 + 4724 with the _ chandra x - ray observatory _ ( hereafter _ chandra _ ; weisskopf et al . 2000 ) . due to the large lensing boost of fsc 10214 + 4724 , these observations provide the equivalent sensitivity of an up - to @xmath0 4 ms exposure . the galactic column density toward fsc 10214 + 4724 is ( stark et al . @xmath17 km s@xmath3 mpc@xmath3 , @xmath18 , and @xmath19 are adopted . fsc 10214 + 4724 was observed with _ chandra _ on 2004 march 4 ( observation i d 4807 ) . the advanced ccd imaging spectrometer ( acis ; garmire et al . 2003 ) with the ccd s3 at the aim point was used for the observation ( the ccds s1s4 , and i2i3 were also turned on ) ; the optical position of fsc 10214 + 4724 is @xmath20 10@xmath21 24@xmath22 3454 , @xmath23 @xmath2409@xmath25 . faint mode was used for the event telemetry format , and the data were initially processed by the _ chandra _ x - ray center ( cxc ) using version 7.1.1 of the pipeline software . [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ] notes : @xmath26offset between the _ chandra _ full - band source position and the _ hst _ source position ( component 1 ) of eisenhardt et al . @xmath27source counts and errors . `` fb '' indicates full band , `` sb '' indicates soft band , and `` hb '' indicates hard band . the source counts are determined with wavdetect . the errors correspond to 1 @xmath28 and are taken from gehrels ( 1986 ) . @xmath29ratio of the count rates in the 2.08.0 kev and 0.52.0 kev bands . the errors were calculated following the numerical method " described in 1.7.3 of lyons ( 1991 ) . @xmath30effective photon index for the 0.58.0 kev band , calculated from the band ratio . the photon index is related to the energy index by @xmath31 = @xmath32 where @xmath33 . @xmath34fluxes in units of @xmath35 erg @xmath36 s@xmath3 . these fluxes have been calculated from the count rate in each band using the cxc s portable , interactive , multi - mission simulator ( pimms ) assuming @xmath37 ; they have been corrected for galactic absorption . + the reduction and analysis of the data used _ chandra _ interactive analysis of observations ( ciao ) version 3.0.2 tools . the ciao tool acis_process_events was used to remove the standard pixel randomisation . the data were then corrected for the radiation damage sustained by the ccds during the first few months of _ chandra _ operations using the charge transfer inefficiency ( cti ) correction procedure of townsley et al . all bad columns , bad pixels , and cosmic ray afterglows were removed using the `` status '' information in the event files , and we only used data taken during times within the cxc - generated good - time intervals . the background light curve was analysed to search for periods of heightened background activity using the contributed ciao tool analyze_ltcrv with the data binned into 200 s intervals ; there were no periods of high background ( i.e. , a factor of @xmath7 2 above the median level ) . the net exposure time for the observation is 21.26 ks . the _ asca _ grade 0 , 2 , 3 , 4 , and 6 events were used in all subsequent analyses . the pointing accuracy of _ chandra _ is excellent ( the 90% uncertainty is @xmath38 ) . however , for our observation we wanted to improve the source positions to provide an unambiguous distinction between fsc 10214 + 4724 and nearby objects ( e.g. , the @xmath39 lensing galaxy lies @xmath0 @xmath40 from fsc 10214 + 4724 ; eisenhardt et al . 1996 ) . to achieve this we matched sources detected in the observation to sources found in the sloan digital sky survey ( sdss ) , which has a positional accuracy of @xmath41 ( rms ; pier et al . source searching was performed using wavdetect ( freeman 2002 ) with a false - positive threshold of @xmath42 in the full ( fb ; 0.58.0 kev ) , soft ( sb ; 0.52.0 kev ) , and hard ( hb ; 28 kev ) bands ; we used wavelet scale sizes of 1 , 1.44 , 2 , 2.88 , 4 , 5.66 , and 8 pixels . the resulting source lists were then merged with a @xmath43 matching radius , producing a catalog of 37 sources . the three brightest x - ray sources in this catalog ( those with @xmath4420 counts in the full band ) that lay within @xmath45 of the aim point were matched to sdss sources in the data release 2 catalog ( dr2 ; abazajian 2004 ) with a @xmath43 search radius . the mean sdss-_chandra _ positional offset of the matched sources ( excluding fsc 10214 + 4724 ) was ( right ascension ; ra ) and ( declination ; dec ) . these corrections were applied to the x - ray source positions . fsc 10214 + 4724 is detected in all three bands ( see table 1 ) . the x - ray source position ( taken from the full band ) lies @xmath46 from the position measured by the _ hubble space telescope _ ( hereafter ) in the f814w band ( eisenhardt et al . 1996 ; see table 1 & figure 1 ) ; the x - ray position is also @xmath47 offset from both the co(32 ) position ( downes 1995 ) and the 1.49 ghz radio position ( lawrence 1993 ) . the x - ray source is clearly identified with fsc 10214 + 4724 rather than the @xmath39 lensing galaxy ( see figure 1 ) . the x - ray properties of fsc 10214 + 4724 are shown in table 1 . although this observation achieves the equivalent sensitivity of an up - to @xmath0 4 ms exposure ( e.g. , a 10-count source has a full - band flux of @xmath7 @xmath48 erg @xmath36 s@xmath3 for a lensing boost of @xmath49 ; compare to table 9 in alexander 2003a ) , only a few x - ray counts are detected . the @xmath0 10 counts in the soft band correspond to a significant detection in the rest - frame band while the @xmath0 4 counts in the hard band correspond to a weak detection in the rest - frame 6.626.3 kev band . the soft - band flux is @xmath0 10 times below the 2 @xmath28 constraint reported in lawrence ( 1994 ) and the hard - band flux is @xmath0 20 times below the upper limit reported in iwasawa ( 2001 ) . we re - examined the pspc image and could not find unambiguous evidence of x - ray emission at the location of fsc 10214 + 4724 . from our analyses of the pspc image we determine a 3 @xmath28 0.52.0 kev upper limit of @xmath50 erg @xmath36 s@xmath3 . the observed ( uncorrected for gravitational lensing ) full - band luminosity is @xmath51 2.4 @xmath52 erg s@xmath3 . the band ratio ( i.e. , the ratio of the hard to soft - band count rate ) of fsc 10214 + 4724 implies an effective photon index of @xmath53 ( see table 1 ) . this is generally consistent with that of an unobscured agn ( i.e. , @xmath54 2.0 ; e.g.,nandra & pounds 1994 ; george 2000 ) ; however , due to the large uncertainties and comparatively high redshift of fsc 10214 + 4724 , this could also be consistent with a column density of @xmath55 @xmath36 at @xmath5 ( for an intrinsic x - ray spectral slope of @xmath56 ) . the latter would be more consistent with the obscured agn classification of fsc 10214 + 4724 than the former ( e.g. , elston 1994 ; soifer 1995 ) . see 3.1 for further obscuration constraints . the gravitational lensing boost of fsc 10214 + 4724 is unknown in the x - ray band . since the lensing boost is a function of the source size , basic constraints can be placed from the extent of the x - ray emission ( e.g. , see 2 of broadhurst & lehar 1995 ) . in figure 2 we show the full - band profiles ( s - n and e - w orientations ) of fsc 10214 + 4724 and compare them to the on - axis acis - s point spread function ( psf ) . while this analysis is limited by small - number statistics , the extent of fsc 10214 + 4724 is consistent with that of an unresolved x - ray source ( @xmath0 1@xmath57 ) . the half - power radius of fsc 10214 + 4724 ( i.e. , the radius over which the central seven counts are distributed ; ) is also consistent with that of an unresolved x - ray source . although somewhat uncertain , this suggests that the magnification in the x - ray band is @xmath7 25 ( e.g. , compare to the extent and magnification found in the 2.05 @xmath10 m nicmos observations of evans 1999 ) . we can compare this estimate to the expected lensing boost from other observations . the _ hst _ observations and source model of nguyen ( 1999 ) suggest that the central source ( i.e. , the emitting agn ) is @xmath0 100 pc from the caustic , indicating that the magnification of the agn emission is likely to be @xmath0 100 . the strong optical polarisation and prominent high - excitation emission lines also indicate that the caustic lies close to the central source ( e.g. , broadhurst & lehar 1995 ; lacy 1998 ; simpson 2004 ) . -axis corresponds to the offset from the wavdetect - determined position , and the @xmath58-axis error bars correspond to 1 @xmath28 uncertainties ( gehrels 1986 ) . the psf was simulated using the ciao tool mkpsf and has been normalised to the peak of the s - n orientation profile . although the signal - to - noise ratio of the data is low , the profiles and the half - power radius ( @xmath0 05075 ) are consistent with those of an unresolved source , suggesting a lensing boost in the x - ray band of @xmath7 25 ; see with a @xmath0 20 ks acis - s observation we have shown that fsc 10214 + 4724 is comparatively weak at x - ray energies ( @xmath59 erg s@xmath3 , @xmath60 erg s@xmath3 , and @xmath61 erg s@xmath3 for a lensing boost of @xmath62 ) . previous studies have suggested that fsc 10214 + 4724 hosts both a powerful starburst and a powerful agn ( goodrich 1996 ; granato 1996 ; green & rowan - robinson 1996 ) . in this final section we predict the expected x - ray emission from both star formation and agn activity in fsc 10214 + 4724 and compare it to the observed x - ray emission . we also compare the x - ray properties of fsc 10214 + 4724 to those of high - redshift galaxies and discuss the x - ray identification of compton - thick agns at high redshift . @xmath635007 luminosity of fsc 10214 + 4724 for a lensing boost of 100 ( simpson 2004 ; see 3.1 ) ; the light shaded region shows the variance in the @xmath64}$]/@xmath65 relationship ( mulchaey 1994 ) . the filled circles indicate the x - ray detected galaxies from alexander ( 2003b ) ; the crosses indicate sources classified as agns , and the `` u ' 's indicate sources with unknown classifications but with x - ray properties consistent with those of starburst galaxies . the dark shaded region denotes the 1 @xmath28 dispersion in the locally determined x - ray - radio correlation for star - forming galaxies ( see figure 6 of shapley , fabbiano , & eskridge 2001 ; bauer 2002 ; ranalli 2003 ) . the rest - frame @xmath66 8 kev emission from fsc 10214 + 4724 is consistent with that expected from star - formation activity ; see 3.1.,width=321 ] many studies have shown a correlation between the 1.4 ghz radio luminosity density and the x - ray luminosity of star - forming galaxies ( e.g. , shapley 2001 ; bauer 2002 ; ranalli 2003 ) . since the radio emission from fsc 10214 + 4724 is consistent with star - formation activity ( e.g. , lawrence 1993 ; rowan - robinson 1993 ; eisenhardt 1996 ) , we can use this correlation to predict the expected x - ray emission from star formation . the radio extent and morphology of fsc 10214 + 4724 are similar to those found in the rest - frame ultra - violet , suggesting that the radio emission is lensed by a factor of ( eisenhardt 1996 ) . in figure 3 we show the rest - frame 1.4 ghz radio luminosity density versus the rest - frame 0.58.0 kev luminosity for fsc 10214 + 4724 ; the rest - frame 0.58.0 kev luminosity was calculated from the rest - frame 1.66.6 kev luminosity ( observed soft band ) assuming @xmath37 and is shown for a range of lensing boosts . the rest - frame emission from fsc 10214 + 4724 is entirely consistent with that expected from star formation for the range of probable lensing boosts at radio wavelengths . the x - ray - to - optical flux ratio is also concordant with that expected from star formation [ @xmath67 ; see 4.1.1 of bauer 2004 ] when appropriate @xmath68 corrections and lensing boosts are applied [ @xmath37 with lensing boosts of 25100 in the x - ray band , and the host galaxy templates of mannucci ( 2001 ) with a lensing boost of 100 in the @xmath69-band ] . these results imply that the agn in fsc 10214 + 4724 is either heavily obscured or intrinsically weak . we can estimate the instrinsic luminosity of the agn in fsc 10214 + 4724 using the [ oiii]@xmath635007 luminosity ( e.g. , mulchaey 1994 ; bassani 1999 ) . taking the [ oiii]@xmath635007 luminosity from serjeant ( 1998 ) , the [ oiii]@xmath635007 to x - ray correlation of mulchaey ( 1994 ) , and assuming the lensing boost to the [ oiii]@xmath635007 emission - line region is @xmath70 ( simpson 2004 ) , the predicted rest - frame 0.58.0 kev luminosity is @xmath71 erg s@xmath3 ( with a variance of @xmath52 erg s@xmath3 ) ; see figure 3 . , a typical intrinsic x - ray spectral slope for agns . ] these predicted x - ray luminosities are within the range expected for quasars and would be even higher if the [ oiii]@xmath635007 emission - line region suffers from reddening ( e.g. , elston 1994 ; soifer 1995 ; cf serjeant 1998 ) . since the lensing boost to the [ oiii]@xmath635007 emission - line region is a lower limit , these constraints should be considered upper limits . however , given that the [ oiii]@xmath635007 emission - line region is likely to be more extended than the central source , the [ oiii]@xmath635007 emission is unlikely to be much more magnified than the x - ray emission . these results suggest that the agn in fsc 10214 + 4724 is powerful . however , the rest - frame 1.626.3 kev luminosity is approximately 12 orders of magnitude below the constraint estimated from the [ oiii]@xmath635007 luminosity . this is significant since rest - frame @xmath72 kev emission is not easily attenuated [ e.g. , one order of magnitude of extinction at @xmath72 kev requires compton - thick obscuration ( @xmath73 @xmath36 ) ; see appendix b in deluit & courvoisier 2003 ] . hence , if fsc 10214 + 4724 hosts a quasar , as previously suggested , then it must be obscured by compton - thick material ; these general conclusions are consistent with those found for other _ iras _ galaxies of similar luminosity ( e.g. , iwasawa 2001 ; wilman 2003 ) . under this assumption , the observed x - ray emission from the agn would be due to reflection and scattering and a strong fe k@xmath31 emission line should be detected . with only @xmath0 14 x - ray counts , the current x - ray observations can not provide good constraints on the presence of fe k@xmath31 ; however , a scheduled @xmath0 50 ks observation ( pi : k. iwasawa ) may be able to place some constraints . the lensing - corrected properties of fsc 10214 + 4724 are similar to those of galaxies ( see 1 ) , sources that are probably the progenitors of massive galaxies ( and hence massive black holes ) in the local universe . the steep x - ray spectral slope and moderately luminous x - ray emission of fsc 10214 + 4724 contrasts with the x - ray properties of the five x - ray detected galaxies classified as agns in alexander ( 2003b ) ; however , the x - ray properties of fsc 10214 + 4724 are similar to those of the two x - ray detected galaxies classified as unknown ( see figure 3 ) . since we know that an agn is present in fsc 10214 + 4724 ( possibly powerful and compton thick ) , this suggests that further unidentified agns may be present in the galaxy population and the @xmath0 40% agn fraction ( alexander 2003b , 2004 ) should be considered a lower limit . many compton - thick agns may be present in ultra - deep x - ray surveys ( e.g. , fabian 2002 ) . however , the presence of vigorous star formation may make them difficult to identify on the basis of their x - ray properties alone . we acknowledge support from the royal society ( dma ) , pparc ( feb ; cs ) , cxc grant go4 - 5105x ( wnb ) , nsf career award ast-9983783 ( wnb ) , and miur ( cofin grant 03 - 02 - 23 ; cv ) . we thank p. eisenhardt , d. hogg , k. iwasawa , and n. trentham for useful discussions .

we present a @xmath0 20 ks _ chandra _ acis - s observation of the strongly lensed ultra - luminous infrared galaxy fsc 10214 + 4724 . although this observation achieves the equivalent sensitivity of an up - to @xmath0 4 ms _ chandra _ exposure ( when corrected for gravitational lensing ) , the rest - frame 1.626.3 kev emission from fsc 10214 + 4724 is weak ( @xmath1 2@xmath2 erg s@xmath3 for a lensing boost of @xmath4 ) ; a significant fraction of this x - ray emission appears to be due to vigorous star - formation activity . if fsc 10214 + 4724 hosts a quasar , as previously suggested , then it must be obscured by compton - thick material . we compare fsc 10214 + 4724 to high - redshift galaxies and discuss the x - ray identification of compton - thick agns at high redshift . [ firstpage ] x - rays : individual : fsc 10214 + 4724 galaxies : active gravitational lensing

You are an expert at summarizing long articles. Proceed to summarize the following text: effects of ferromagnetism have been known since antiquity . but a mathematical understanding of the microscopic origin of ferromagnetism has remained somewhat elusive , until today ! pauli paramagnetism , ferro- , ferri- and antiferromagnetism are quantum phenomena connected to the spin of electrons and to pauli s exclusion principle . the theory of _ paramagnetism _ in ( free ) electron gases is quite straightforward , @xcite . _ antiferromagnetism _ is relatively well understood : a mechanism for the generation of antiferromagnetic exchange interactions has been proposed by anderson @xcite , who discovered a close relationship between the half - filled hubbard model and the heisenberg antiferromagnet using perturbative methods ; ( see also @xcite for mathematically more compelling and more general variants of anderson s key observation ) . it has been proven rigorously by dyson , lieb and simon @xcite , using the method of infrared bounds previously discovered in @xcite , that the _ quantum heisenberg antiferromagnet _ with nearest - neighbour exchange couplings exhibits a phase transition accompanied by spontaneous symmetry breaking and the emergence of gapless spin waves , as the temperature is lowered , in _ three _ or _ more _ dimensions . ( the mermin - wagner theorem says that , in ( one and ) two dimensions , continuous symmetries can not be broken spontaneously in models with short - range interactions , @xcite . ) our mathematical understanding of _ ferromagnetism _ is far less advanced . some kind of heuristic theory of ferromagnetism emerged , long ago , in the classic works of heisenberg , bloch , stoner , dyson , landau and lifshitz , and others @xcite . various insights have been gained on the basis of some form of mean - field theory , with small fluctuations around mean - field theory taken into account within a linear approximation . this approximation , however , is known to break down in the vicinity of the critical point of a ferromagnetic material , where nonlinear fluctuations play a crucial role , @xcite . in a variety of tight - binding models of itinerant electrons , ferromagnetic order has been exhibited in the _ ground state _ ( i.e. , at zero temperature ) ; see @xcite,@xcite,@xcite . one of these models is a fairly natural two - band model in which ferromagnetism arises from a competition between electron hopping , coulomb repulsion and an on - site hund s rule , @xcite . ( hund s rule says that the spin - tripled state of two electrons occupying the same site is energetically favoured over the spin - singlet state . it should be emphasized , however , that a mathematically rigorous derivation of hund s rule in atomic physics from first principles has _ not _ been accomplished , so far . ) none of the results in @xcite,@xcite,@xcite comes close to providing some understanding of ferromagnetic order and of an order - disorder phase transition at _ positive _ temperature . it is not known how to derive , with mathematical precision , an effective hamiltonian with explicit ferromagnetic exchange couplings from the microscopic schrdinger equation , or a tight - binding approximation thereof , of ferromagnetic materials . but even if we resort to a phenomenological description of such materials in terms of models where ferromagnetic exchange couplings have been put in _ by hand _ we face the problem that we are unable to exhibit ferromagnetic order at low enough temperature and to establish an order - disorder phase transition in three or more dimensions . _ no mathematically rigorous proof _ of the phase transition in the _ quantum heisenberg ferromagnet _ is known , to date ! ( such a result has , however , been established for _ classical _ heisenberg models in @xcite . ) ab - initio quantum monte carlo simulations of models of quantum ferromagnets are plagued by the well known `` sign- ( or complex - phase ) problem '' . thus , until now , there are neither substantial mathematically rigorous results on , nor are there reliable ab - initio numerical simulations of , realistic models of ferromagnetic metals , such as ni , co or fe ! given that ferromagnetism is among the most striking _ macroscopic _ manifestations - apparent , e.g. , in the needle of a compass - of the _ quantum - mechanical _ nature of matter , this is clearly a desolate state of affairs . in the present paper we shall not remedy this unsatisfactory situation . however , first , we attempt to draw renewed attention to it , and , second , we outline a formalism and some fairly elementary analytical observations of which we hope that they will ultimately lead to a better , mathematically rather precise understanding of ferromagnetism . were it not known already , our analysis and the one in @xcite would make clear that ferromagnetism is a non - perturbative phenomenon involving strong correlations and gapless modes . to understand it mathematically will most probably necessitate a full - fledged _ multi - scale _ ( renormalization group ) _ analysis_. the formalism presented in this paper and our calculations are intended to provide a convenient starting point for such an analysis . analytical work on ferromagnetism may seem to be rather unfashionable . however , there are recent developments , such as spintronics , fast magnetic devices , etc . that may make work like ours appear worthwhile . our own motivation for the work that led to this paper actually originated in studying recent experiments with beams of spin - polarized , hot electrons shot through ferromagnetically ordered films consisting of ni , co or fe that were carried out in the group of h.c . siegmann at eth ; see @xcite,@xcite . back in 1998 , it became clear to one of us that the concept of the `` _ weiss exchange field _ '' ( see @xcite,@xcite ) would play a useful role in a theoretical interpretation of the experimental results reported in @xcite,@xcite . more generally , the weiss exchange field appears to offer a key to a systematic study of phase transitions in magnetic materials , magnetic order , spin precession and magnon dynamics . in this paper , we focus on elucidating the microscopic origin of the weiss exchange field , the role it plays in the theory of magnetism , and its dynamics . our paper is organized as follows . in section 2 , we describe the experiments reported in @xcite,@xcite and sketch a phenomenological interpretation , based on scattering theory , of the results found in these experiments , merely adding some conceptual remarks to the discussion of our experimental colleagues and describing the role played by the weiss exchange field . we also propose some further experiments , in particular a stern - gerlach experiment for electrons traversing an inhomogeneous exchange field . a mathematically precise analysis of the scattering of electrons ( or neutrons , photons , ... ) at dynamical targets , such as magnetic films , metallic solids , liquid droplets , ... , will appear elsewhere ; ( some first results appear in @xcite ) . in section 3 , we reformulate one - band @xmath1 and hubbard models in terms of a dynamical weiss exchange field . for this purpose , the standard imaginary - time functional integral formalism for the analysis of thermal equilibrium states of quantum many - body systems is recalled . the _ weiss exchange field _ is seen to be a lagrange - multiplier field in a hubbard - stratonovich transformation of the original functional integral that renders the action functional quadratic in the grassmann variables describing the electronic degrees of freedom ; see e.g. @xcite,@xcite . the effective field theory of the weiss exchange field is obtained after integrating over those grassmann variables . the effective ( imaginary - time ) action functional of the exchange field and identities for green functions of spin operators are derived . in section 4 , we determine the leading terms of the effective action of the weiss exchange field , @xmath2 , in the approximation where fluctuations of the _ length _ of the exchange field are neglected . for this purpose , we derive the ferro- and antiferromagnetic mean - field equations from the exact effective action of the exchange field . by solving these equations we determine the most likely length , @xmath3 , of the exchange field . from that point on , the length of the exchange field is frozen to be @xmath4 . we then consider a one - band hubbard model with a half - filled band and find that , in this situation , the effective action of @xmath5 is the one of a nonlinear @xmath6-model with a minimum that favours nel order . this result is found on the basis of controlled perturbative calculations and goes beyond linear stability analysis of the antiferromagnetic mean - field solution , ( which has been presented , e.g. , in @xcite ) . it represents a functional - integral version of anderson s basic observations @xcite . we then turn to ferromagnetically ordered mean field solutions and show that @xmath7-independent fluctuations are not a source of instability of such a solution . then we consider a one - band hubbard model with a weakly filled , fairly flat band . in this situation one expects that ferromagnetism prevails . indeed , we find that , at low temperatures , the ferromagnetic mean - field equation has a non - trivial solution , and that this solution belongs to a _ quadratically stable _ critical point of the effective action of w. this conclusion is the result of somewhat subtle calculations involving processes close to the fermi surface , which make the dominant contribution ( but would lead to small - energy denominators in a purely perturbative analysis ) . details will appear in @xcite . our calculations support the idea that the one - band hubbard model with a weakly filled , fairly flat band describes coexistence of metallic behaviour with ferromagnetic order , at sufficiently low temperatures . a similar conclusion was reached , tentatively , in @xcite for some two - band hund - hubbard models . the methods of the present paper also apply to the model discussed in @xcite ; see @xcite . in the next to last subsection of section 4 , we exhibit a universal wess - zumino term in the effective action of the weiss exchange field @xmath5 and calculate its coefficient , which is purely imaginary . the wess - zumino term is `` irrelevant '' for antiferromagnets , but plays a _ crucial _ role in the dynamics of magnons in _ ferromagnetically _ ordered systems . repeating arguments in @xcite , we derive the landau - lifshitz equations for magnons in a ferromagnet . finally , we draw attention to two well known arguments explaining why there is no magnetic ordering at positive temperature , in one and two dimensions ; ( but see @xcite ) . in section 5 , we sketch novel rigorous proofs , based on analyzing the effective field theory of the exchange field , @xmath5 , of the existence of phase transitions and magnetic order at low temperatures in a class of @xmath0-models , heisenberg antiferromagnets and ferromagnets of localized @xmath8- spins , for @xmath9 . see @xcite for the original results . our proof is based on establishing reflection positivity of the effective field theory of the exchange field @xmath5 and then using the original techniques developed in @xcite ; ( see also @xcite , @xcite ) . it should be emphasized that the concept of the weiss exchange field has a number of further , quite exciting applications . we hope to return to these matters in future papers . some of the material in this paper has a review character ; but some of it is new . we hope it is fairly easy to read . if it draws renewed attention to some of the deep technical problems in the quantum theory of magnetism it has fulfilled its purpose . we gratefully dedicate this paper , belatedly , to two great colleagues and friends of the senior author ( j.f . ) : g. jona - lasinio , on the occasion of his seventieth birthday , and h .- c . siegmann , on the occasion of his retirement from eth . _ acknowledgements . _ j. frhlich thanks s. riesen and h .- c . siegmann for very stimulating discussions of the experiments in @xcite , p. wiegmann for some crucial advice with the calculations in section 4 , and the ihs for hospitality during much of the work on this paper . we all thank m. azam for very useful and pleasant discussions on the uses of the exchange field . we start this section with a brief description of recent experiments carried out by oberli , burgermeister , riesen , weber and siegmann at eth - zrich @xcite , @xcite . in these experiments , a beam of hot , spin polarized electrons is shot through a thin ferromagnetic film ( ni , co , or fe ) and the polarization of the outgoing beam is observed . their experimental setup is as described in fig . [ fig : exp1 ] . * the thickness , @xmath10 , of the film ; @xmath10 is a few nanometers . * the average energy , @xmath11 , of an incident electron ; if @xmath12 denotes the fermi energy of the magnetic film then @xmath13 varies between 4@xmath14 and 16@xmath14 . the group velocity of the electrons inside the film is denoted by @xmath15 ; it is not directly measurable , but it is comparable to @xmath16 , where @xmath14 is the average potential energy of an electron and @xmath17 its effective mass inside the film . * the degree , @xmath18 , and the direction , @xmath19 , of the spin polarization of the incident electron beam ; ( in fig . [ fig : exp1 ] , @xmath19 is parallel to the @xmath7-direction , @xmath20 to the @xmath21-direction ) ; the same quantities , @xmath22 and @xmath23 , for the outgoing beam . * the direction of the magnetization , @xmath24 , of the film ( in fig . [ fig : exp1 ] chosen to be parallel to the @xmath25-axis ) ; the angles , @xmath26 and @xmath27 , between @xmath28 and @xmath24 and between @xmath23 and @xmath29 , respectively ( @xmath30 , in fig . [ fig : exp1 ] ) . experimentally , the angle @xmath27 is found to be considerably smaller than @xmath26 , i.e. , the spins of the transmitted electrons rotate into the direction of the spontaneous magnetization @xmath24 of the film . this is interpreted as being mainly due to an enhanced absorption of minority - spin electrons , as compared to majority - spin electrons ; see ( vi ) . ( it appears that the contribution of spin flip processes accompanied by magnon emission into the film `` stoner excitations '' to the total spin rotation is only around 5% , @xcite ) . * the spin precession angle , @xmath31 , between the projections of @xmath32 and of @xmath33 onto the plane perpendicular to @xmath24 ( the @xmath34-plane of fig . [ fig : exp1 ] ) ; @xmath31 is found to be `` large '' ( tens of degrees ) . * let @xmath35 be the intensity of the incident beam , and let @xmath36 and @xmath37 be the intensities of the outgoing beam of electrons with spin parallel or antiparallel to @xmath24 , respectively , assuming the incident beam has intensity @xmath35 and the spins of its electrons are parallel to @xmath24 ( @xmath28 parallel to @xmath24 , @xmath38 ) , or antiparallel to @xmath24 ( @xmath28 anti - parallel to @xmath24 , @xmath39 ) , respectively . then @xmath40 , or @xmath41 , respectively , and @xmath42 . @xmath43 and @xmath44 can be measured and yield the spin - transmission asymmetry @xmath45 @xmath46 is found to be positive and large . this is interpreted in terms of rates of transitions of electrons into unoccupied 3d states ( holes ) : there are more unoccupied 3d states in the film with spin antiparallel to @xmath24 ( minority spin ) than with spin parallel to @xmath24 ( majority spin ) . this explains qualitatively the experimental results found for @xmath46 and for @xmath47 ( see ( iv ) ) ; @xcite . * the orbital deflection angle , @xmath48 , between the directions of the incident and the transmitted beam ( not indicated in fig . [ fig : exp1 ] ) . experimentally @xmath48 is found to be negligibly small . this tells us that the integrated lorentz force on the electrons transmitted through the film is tiny . the precession of the spins of the electrons when they traverse the film can therefore _ not _ be explained by zeeman coupling of the spins to the magnetic field inside the layer . it is mainly due to zeeman coupling of the spins to what will be called the _ weiss exchange field_. in iron , the weiss exchange field causing the observed spin precession would correspond to a magnetic field of roughly 8000 tesla ( which is gigantic ) . a theoretical interpretation of the experimental results reported in @xcite can be attempted within the formalism of _ scattering theory_. if the luminosity of the incident beam is low we can consider a single incoming electron . the incoming state is described as a tensor product of a pauli spinor , @xmath49 , describing the incident electron and a state , @xmath50 , of the film . typically , @xmath50 is the ground state ( temperature @xmath51 ) or a thermal equilibrium state ( @xmath52 ) of the film . the outcoming state , long after the interactions between an outgoing electron and the film have taken place , is more complicated and will , in general , exhibit _ entanglement _ between the electron and the degrees of freedom of the film . if only measurements far away from the film are performed , as in @xcite , the outgoing state can be described as a density matrix @xmath53 where @xmath54 is a non - negative , trace - class operator on the hilbert space of @xmath55 _ outgoing electrons _ ( the incident electron has knocked @xmath56 electrons out of the film ) , @xmath57 ; @xmath58 is a non - negative number , the _ absorption probability _ , @xmath59 is a non - negative trace - class operator on the hilbert space @xmath60 of square - integrable pauli spinors and describes the ( generally _ mixed _ ) state of _ one _ outgoing electron . `` conservation of probability '' implies that @xmath61 the state @xmath62 is obtained by taking a partial trace of the outgoing state of the _ total system , including _ the film , over the degrees of freedom of the film . this is justified , because the degrees of freedom of the film are _ not _ observed in the experiment . if the energy , @xmath11 , of the incident electron is below ( or comparable to ) the threshold , @xmath63 , for emission of two or more electrons from the film then @xmath64 in the interpretation of the experimental data provided in @xcite , this is tacitly assumed . experimentally , the absorption probability @xmath58 and the spin polarizations @xmath65 and @xmath66 of the incoming and the outgoing electron , respectively , are measured . the vectors @xmath67 and @xmath66 are given by @xmath68 and @xmath69 where @xmath49 is the wave function of the incident electron , @xmath70 is the vector of pauli matrices , and @xmath71^{-1 } \rho_1 $ ] is the _ conditional _ state of the outgoing electron , given that it has _ not _ been absorbed in the film . if ( [ eq : rhon ] ) is assumed to hold then @xmath72 it would be highly interesting to estimate , experimentally , the amount of entanglement with the film ( or _ decoherence _ ) in the state of the outgoing electron by measuring the quotient @xmath76 . if @xmath77 then @xmath78^{-1 } \rho_1 $ ] is _ not _ a pure state , anymore , meaning there is entanglement with the film . apparently , @xmath76 has not been measured accurately , yet . a moment s reflection shows that spin flip processes accompanied by magnon emission in the film ( `` stoner excitations '' ) lead to entanglement ; while absorption of electrons into unoccupied 3d states need not be correlated with entanglement of the states of _ those _ electrons that _ do _ traverse the film . in fact , it is implicitly assumed in @xcite that if stoner excitations are neglected then @xmath74 is close to a pure state ( at least in spin - space ) . the experimental techniques of @xcite could be used to test this hypothesis . next , we express the spin transmission asymmetry @xmath46 ( see ( vi ) ) in terms of outgoing states . let @xmath79 and @xmath80 be incoming states with spin polarization @xmath81 parallel to @xmath24 ( majority spin ) and @xmath82 antiparallel to @xmath83 ( minority spin ) . let @xmath84 and @xmath85 be outgoing states corresponding to @xmath86 , @xmath87 , respectively . then @xmath88 as follows from ( [ eq : cons ] ) . more interesting would be measurements of @xmath89 clearly @xmath90 are parallel or anti - parallel to @xmath83 ; but their lengths @xmath91 ought to be measured . in @xcite , it is tacitly assumed that @xmath92 , and that the states @xmath93 are close to _ pure _ states ; but serious experimental data backing up this hypothesis appear to be lacking . it is clear that it would be invalidated if `` stoner excitations '' played an important role . in the following , we outline a phenomenological description of the experiments in @xcite , assuming that ( [ eq : rhon ] ) and the hypothesis just discussed ( purity of @xmath93 ) are valid . ( a more detailed discussion of the scattering approach to electron transmission and reflection experiments will be presented elsewhere . ) when an incoming electron enters the film it occupies an empty state of the film . if the film is crystalline this state belongs to a band of states ; let @xmath48 be the corresponding band index . the state of an electron in band @xmath48 is described by a pauli spinor @xmath94 where @xmath95 and @xmath96 are the components of @xmath97 with spin parallel or anti - parallel to the magnetization @xmath24 , respectively . adopting the approximation of the peierls substitution , the pauli equation for @xmath97 in configuration space has the form @xmath98 where @xmath99 is the band function of band @xmath48 , @xmath100 here @xmath101 denotes a ( complex ) electrostatic potential , @xmath102 is a ( complex ) _ weiss exchange field _ , @xmath103 is the electromagnetic vector potential , and @xmath104 is an @xmath105-vector potential responsible for _ spin - orbit interactions_. as discussed in @xcite , ( [ eq : cd0 ] ) displays electromagnetic @xmath106-_gauge invariance and _ @xmath105-_gauge invariance _ , i.e. , covariance with respect to local @xmath105-rotations in spin space . a number of important consequences of these gauge symmetries have been pointed out in @xcite . according to ( vii ) above , effects of the electromagnetic vector potential @xmath107 are apparently negligible ; so @xmath107 is set to 0 . the electrostatic vector potential @xmath101 is given , approximately , by @xmath108 where @xmath14 is the surface exit work , and the imaginary part , @xmath109 , of @xmath101 provides a phenomenological description of spin - independent inelastic absorption processes inside the film . velocity - dependent spin - flip processes due to _ spin - orbit interactions _ appear to play a very minor role in the experiments reported in @xcite ; so we may set @xmath110 to 0 , @xmath111 . the weiss exchange field @xmath112 is given by @xmath113 where the real part @xmath114 describes _ exchange interactions _ between the incoming electron and the electron density of the film , and the imaginary part , @xmath115 , yields a phenomenological description of spin - dependent absorption processes . let @xmath116 denote the pauli spinor describing the state of an electron when it enters the film at some time @xmath117 . ( [ eq : cd0 ] ) can be solved for @xmath118 , @xmath119 , with @xmath120 . the solution is explicit if @xmath121 , @xmath122 , @xmath111 . let us suppose that the real part @xmath114 and the imaginary part @xmath115 of the exchange field @xmath112 are both anti - parallel to the magnetization @xmath24 of the film . then ( [ eq : cd0 ] ) leads us to consider two simple , quasi - one - dimensional scattering problems in a complex potential well of depth @xmath123 with @xmath124 and @xmath125 , for electrons with spin parallel to @xmath126 , or anti - parallel to @xmath127 , respectively . the solution to these scattering problems can be found in every book on elementary quantum mechanics . for the purposes of interpreting the results in @xcite , a semi - classical treatment appears to be adequate . the group velocity , @xmath128 , of an incoming electron wave with energy peaked at @xmath11 and spin up @xmath129 , or down @xmath130 , inside the film can be found by solving the equation @xmath131 for @xmath132 and then setting @xmath133 where @xmath134 is a solution of ( [ eq : xip ] ) chosen such that @xmath135 points in the positive @xmath21-direction . the sojourn time @xmath136 of the wave inside the film is then given by latexmath:[\[\tau_{\pm } = \frac{d}{v_{\pm } } , \quad \text{with } \quad v_{\pm } = state of the electron with spin up @xmath129 , or spin down @xmath130 , respectively , when it enters the film its state @xmath138 upon leaving the film is then given , approximately , by . ] @xmath139\right ) \phi_{\text{in}}^{\pm } .\ ] ] the presence of inelastic absorption processes implies that @xmath140 . setting @xmath141\right ) = c \sqrt{1 \pm a},\ ] ] we find that @xmath142 where @xmath46 is the spin transmission asymmetry ( see point ( iv ) and ( [ eq : asy ] ) ) , @xmath31 is the spin precession angle , and @xmath27 is an ( unimportant ) spin - independent phase . if @xmath143 then @xmath144 , and ( [ eq : phiout ] ) yields @xmath145 thus , measuring @xmath31 and @xmath10 and estimating @xmath146 yields an approximate value for the size of the spin precession angular velocity @xmath147 and hence of the size of the weiss exchange field . ( [ eq : phiout ] ) and ( [ eq : phiout2 ] ) could be mistaken for equations describing _ kaon_- or _ neutrino oscillations _ and are analogous to the equations describing the _ faraday rotation _ of light traversing a magnetized medium . the origin of the weiss exchange field @xmath114 is hardly a mystery : the spin of an electron traversing the film in a band @xmath48 apparently experiences ( exchange ) interactions with the spins of the occupied states of the film . since states with spin up and with spin down are occupied asymmetrically ( corresponding to the fact that @xmath148 ) , the net spin density , @xmath149 , at a point @xmath7 in the film is different from zero . the weiss exchange field @xmath150 is given by @xmath151 where @xmath152 is the strength of the _ exchange coupling _ between spins in the @xmath153 band and those in the occupied band . the theoretical discussion above is based on a _ mean field ansatz _ : the exchange field @xmath114 in ( [ eq : cd-0 ] ) , ( [ eq : w0c ] ) is chosen to be @xmath154 ( our conventions are such that @xmath155 is parallel to the magnetization @xmath24 . ) one of the surprising implications of the experimental results of @xcite is that , apparently , @xmath152 is quite large , even for rather high - lying bands ( @xmath48 ) , implying that the orbitals of states in such bands must have substantial overlap with those of states in the partially occupied , spin - polarized band . ( [ eq : wx ] ) makes it clear that @xmath156 is a _ dynamical _ field . one of the main purposes of this paper is to derive the effective quantum dynamics of @xmath156 within a lagrangian functional integral formalism and to sketch what can be accomplished with this formalism . * by applying an external magnetic field to the film rotating in the @xmath34-plane with angular velocity @xmath157 , the exchange field @xmath114 could be made to rotate around the @xmath25-axis : @xmath158 a polarized beam of electrons shot through this film must exhibit _ bloch spin resonance _ ; but , in this experiment , it would be due to the rotation of the exchange field . * one may envisage a _ stern gerlach experiment _ for electrons . one would start by constructing a sandwich of two ferromagnetic metals , i ( e.g. fe ) and ii ( e.g. ni ) , with exchange fields @xmath159 and @xmath160 of different strength , joined by a transition region , a mixture of i and ii , of width @xmath161 . the transition region would be parallel to the @xmath34-plane ( see fig . [ fig : exp2 ] ) . one shoots an unpolarized beam of ( not very hot ) electrons through the sandwich along the transition region between i and ii , as shown in fig . [ fig : exp2 ] . one would expect to detect two beams emerging on the other side of the film in slightly different directions that are spin - polarized in opposite directions . + the force , @xmath162 , in the @xmath25-direction on an electron with spin up / down inside the film is given , approximately , by @xmath163 it yields a change in the @xmath25-component of the momentum of the electron during its passage through the film given by @xmath164 where @xmath146 is the average group velocity . the deflection angle @xmath48 is found from @xmath165 of course , as discussed above , the intensity , @xmath166 of the upper beam can be expected to be much larger than the intensity , @xmath167 , of the lower beam , due to spin - asymmetric absorption inside the film . in order to achieve adequate focussing , the incident beam could come from the tip of a scanning tunnelling microscope . * in an experimental set - up similar to the one above , one would force an electron current , spin - polarized in the @xmath25-direction , through the film . then a hall tension in the @xmath25-direction should be observed ( hall effect for spin currents ; see @xcite ) . e. brezin , j. c. le guillou and j. zinn - justin , _ field theoretical approach to critical phenomena , _ in _ phase transitions and critical phenomena , _ c. domb and m.s . green ( eds . ) volume 6 , 125 - 247 , london : academic press 1976 . d. oberli , _ the ferromagnetic spin filter , _ phd thesis eth zrich , nr . 12933 , ( 1998 ) . j. frhlich , u. m. studer and e. thiran , _ quantum theory of large systems of nonrelativistic matter _ , in _ fluctuating geometries in statistical mechanics and field ( les houches 1994 ) , f. david , p. ginsparg , j. zinn - justin ( eds . ) ; elsevier 1996 , [ arxiv : cond - mat/9508062 ] . t. chen , j. frhlich and m. seifert , _ renormalization group methods : landau - fermi liquid and bcs superconductor , _ in _ fluctuating geometries in statistical mechanics and field theory _ , f. david , p. ginsparg , j. zinn - justin ( eds . ) ; elsevier 1996 , [ arxiv : cond - mat/9508063 ] . j. frhlich and t. spencer , _ phase transitions in statistical mechanics and quantum field theory , _ in _ new developments in quantum field theory and statistical mechanics _ ; m. lvy , p. mitter ( eds . ) , new york , london : plenum 1977 .

the huge spin precession frequency observed in recent experiments with spin - polarized beams of hot electrons shot through magnetized films is interpreted as being caused by zeeman coupling of the electron spins to the so - called _ weiss exchange field _ in the film . a `` stern - gerlach experiment '' for electrons moving through an inhomogeneous exchange field is proposed . the microscopic origin of exchange interactions and of large mean exchange fields , leading to different types of magnetic order , is elucidated . a microscopic derivation of the equations of motion of the weiss exchange field is presented . novel proofs of the existence of phase transitions in quantum @xmath0-models and antiferromagnets , based on an analysis of the statistical distribution of the exchange field , are outlined .

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Proceed to summarize the following text: the study of _ diffusion _ in nearly - integrable hamiltonian dynamical systems of the form @xmath5 where @xmath6 are n - dimensional action - angle variables and @xmath0 is a small parameter , constitutes a central problem in hamiltonian dynamical systems theory , in view , in particular , of its multiple applications in physics and astronomy ( see @xcite@xcite for an introduction , the basic review paper @xcite , or @xcite@xcite @xcite for recent advanced reviews emphasizing various aspects of this subject ) . it is a well established result that , if @xmath7 , and @xmath8 satisfies appropriate convexity and analyticity conditions ( see section 2 below ) , two distinct regimes characterize the laws of diffusion as a function of @xmath0 : for @xmath9 , where @xmath10 is a threshold value , the onset of the so - called ` nekhoroshev regime ' takes place @xcite@xcite@xcite @xcite@xcite@xcite . in this case , the nekhoroshev theorem provides an @xmath11 $ ] upper bound for the speed of diffusion . the exponent @xmath12 depends on the number of degrees of freedom @xmath13 , while its precise value in local domains of the action space depends also on the multiplicity of the resonance conditions holding in such domains ( see e.g. @xcite@xcite@xcite ) . furthermore , the mechanism of diffusion caused by _ transition chains _ , as demonstrated in one special example by arnold @xcite ( see also @xcite ) , is conjectured to hold in more general systems of the form ( [ hamgen ] ) ( e.g. @xcite ; note , however , that no formal proof of this fact has been given to date ) . on the other hand , for @xmath14 , the diffusion is driven mainly by the mechanism of _ resonance overlap _ @xcite@xcite @xcite . in this case , one expects a power - law dependence of the speed of diffusion on @xmath0 ( see e.g. @xcite ; a power law is also found in the case of the so - called ` fast arnold diffusion ' @xcite ) . the diffusion in weakly chaotic systems has been a subject also of extensive numerical studies over the last three decades ( some indicative references are @xcite@xcite@xcite@xcite @xcite@xcite@xcite@xcite ) . a detailed study , however , of the very slow diffusion characterizing the ` nekhoroshev regime ' has become possible only in recent years . in this respect , we note in particular the series of instructive works @xcite @xcite@xcite@xcite@xcite , where , using the so - called fast lyapunov indicator ( fli ; see @xcite ) , a method was found to depict the resonant structure of the action space in models of three degrees of freedom , or 4d and 6d symplectic mappings being in the nekhoroshev regime @xcite . in @xcite , the mean - square spread in action space @xmath15 was measured as a function of the time @xmath16 for orbits along the chaotic border of a _ simply - resonant _ domain ( see section 2 for a precise definition ) . it was found that i ) the local character of diffusion is normal , i.e. @xmath17 , and ii ) the diffusion coefficient @xmath18 decreases with @xmath0 faster than a power law . the exponential fit @xmath19 was given in a subsequent study @xcite . the estimate obtained in @xcite , through interpolation over five orders of magnitude of the perturbation parameter , yields with with certainty the first digit of the exponent @xmath20 , but the errors in the interpolation make uncertain the second digit in both the above estimates . in @xcite , @xmath1 was measured as a function of the separatrix splitting @xmath21 of the asymptotic manifolds of simply unstable two - dimensional tori lying at the borders of simple resonances ( see also @xcite@xcite ) . the measurement of @xmath21 itself was based on employing the fli . it was found that @xmath22 , with @xmath23 and @xmath24 in two resonances of increasing order respectively . finally , the laws of diffusion in systems violating one or more necessary conditions of the nekhoroshev theorem were investigated in @xcite@xcite , leading to a number of interesting results regarding the dynamical consequences of such violations . the motivation for the present study stems primarily from the results reported in refs @xcite@xcite @xcite @xcite , and it can be described as follows . the results obtained so far are very satisfactory from the numerical point of view . they require , however , computations involving large ensembles of orbits and integration times of the order of billions , or even trillions of periods . on the other hand , we can remark that , in principle , the analytical methods involved in the main theories of chaotic diffusion lend themselves also conveniently to getting quantitative predictions regarding the value of the diffusion coefficient , or the scaling laws of diffusion , in general , in the weakly chaotic regime . for such a goal , however , to be accomplished , it is required that one should be able to carry on expansions of certain quantities up to a very high order in the small parameter @xmath0 ( usually with the aid of a computer ) . this fact is explicit in nekhoroshev theory , where one needs to reach an expansion order high enough for the asymptotic behavior of the perturbation series to show up . this has been realized in studies seeking to determine the _ range _ ( in the small parameter value ) and/or the _ conditions of applicability _ of nekhoroshev theory , or , finally , the _ domain of practical stability _ for motions in simple physical systems or models inspired mainly from solar system dynamics ( see e.g. @xcite@xcite @xcite@xcite@xcite@xcite @xcite@xcite@xcite ) . these studies notwithstanding , the question of central interest in the present paper , namely how to obtain relevant quantitative estimates of the _ local value of the diffusion coefficient d _ in resonant domains ( of various multiplicities ) of the action space via _ high order expansions _ of perturbation theory , remains , to our knowledge , largely unexplored . regarding now this last question , it should be noted that the formal analytical apparatus of nekhoroshev theory , entailing the construction of a _ normal form _ in local domains covering the action space of systems of the form ( [ hamgen ] ) , aims to transform the original hamiltonian into one in new canonical variables resuming the form @xmath25 , where @xmath26 , the normal form , corresponds to a simple dynamics , while @xmath27 , the remainder , induces a perturbation to this dynamics . the so - called ` geometric part ' of nekhoroshev the theorem ensures that , despite allowing in general for chaotic motions , the flow under a multiply - resonant normal form _ alone _ would imply perpetual confinement of all chaotic orbits in balls of radius @xmath28 in the action space . nevertheless , this picture is altered due to the effects of the remainder which eventually causes the orbits to diffuse away of their initial @xmath28 domain . now , via a sequence of hamiltonian normalization steps we find that there is an optimal order at which the size of the remainder becomes exponentially small in a power of @xmath29 . this , in turn , implies an exponentially small _ semi - analytic upper bound _ of the value of the diffusion coefficient @xmath1 . unfortunately , such a bound turns usually to be very unrealistic , as it overestimates by a large factor the true value of @xmath1 ( or , equivalently , it underestimates the time of practical stability ) . we are thus led to conclude that , whereas the remainder @xmath27 constitutes a quantity of primary interest in quantitative applications of nekhoroshev theory , the precise relation between @xmath27 and @xmath1 is apparently very different from what upper bound estimates would suggest . instead , a detailed analysis of the _ effects of the remainder on dynamics _ appears to be necessary in order to formulate a more precise theory of the relation between @xmath27 and @xmath1 . in the sequel , we present such an analysis in systems of three degrees of freedom . in this analysis , we still have to rely on an assumption for which numerical indications are available , namely that the local character of diffusion in sufficiently small domains of the action space is ` normal ' , that is , the mean square spread of the actions of the chaotic orbits grows linearly with time ( there are indications that _ global diffusion _ , which concerns ensembles or orbits diffusing in a substantial part of the arnold web over much longer timescales , could also be described as ` normal ' ( see @xcite ) ; however , the issue of the laws of global diffusion can only be hoped to tackle after the laws of local diffusion have been adequately understood ) . in the rest of our analysis , we proceed by expressing all quantities of interest in terms of the remainder function , which , in turn , is calculated in concrete examples by a well - defined algebraic procedure . finally , we estimate via this analysis how @xmath1 depends on the size @xmath2 of the remainder at the optimal normalization order . it should be noted that the idea that the stability properties of the orbits in nearly - integrable systems depend on the size of the optimal remainder is not new , but it is one permeating nearly all forms of canonical perturbation theory . the novel feature here , instead , is to use @xmath2 not as an upper bound for @xmath1 , but as a way to estimate @xmath1 via examining the relation between the two quantities as determined by independent numerical experiments . one main prediction is that this relation is altered according to the _ multiplicity of resonance conditions _ holding in the action domain of interest . more concretely , we predict that the diffusion coefficient @xmath1 scales with @xmath2 as a power - law @xmath30 in simply resonant domains , for some constant @xmath31 . a combination of theoretical arguments found in @xcite@xcite , together with quantitative estimates on the relation between the size of the so - called separatrix splitting ( see subsection 2.3.2 ) and the normal form remainder given in @xcite , suggest @xmath32 , i.e. @xmath33 in simply resonant domains . this agrees with the numerical results obtained in a previous study @xcite . in @xcite , a computer - algebraic program was written in order to calculate the optimal normal form as well as the remainder function @xmath34 at the optimal mormalization order in a case of simple resonance , employing the same hamiltonian model as in @xcite . this operation involved computing about @xmath35 fourier coefficients , at a truncation order in fourier space as high as @xmath36 . comparing the computed size of @xmath2 versus available numerical data on @xmath1 from @xcite , the scaling @xmath37 was found by numerical fitting . in the present paper , after presenting some theoretical results , we make a similar numerical calculation as in @xcite but in the case of a double resonance . in order to reach the optimal normalization , we had to extend all normal form calculations up to the fourier order @xmath38 ( @xmath39 coefficients ) . we thus determined the size of the optimal remainder @xmath2 for many different values of the small parameter @xmath0 . in the same time , we computed the diffusion coefficient @xmath1 for the same values of @xmath0 by a purely numerical procedure involving runs of ensembles of chaotic orbits ( see section 3 ) . finally , we made two independent numerical comparisons of the relation between @xmath1 and @xmath2 . the latter yield the power laws @xmath40 is close to 2 in doubly resonant domains , albeit with a small noticeable difference even in this case , which probably requires a more precise theory to interpret . besides the above computation , our analysis using high order normal forms resulted in a relevant result regarding the possibility to visualize how the phenomenon of arnold diffusion proceeds locally , within a doubly - resonant domain , by materializing the computation of a convenient set of variables helping to this purpose , that were proposed in the work @xcite . we note that numerical evidence for arnold diffusion of orbits entering from simple to double resonances was presented in @xcite . here , we provide a detailed topological description of this phenomenon . the whole computation consists of : i ) computing a set of resonant canonical action - angle variables via a sequence of lie canonical transformations , ii ) taking a 2d poincar surface of section of the _ doubly - resonant normal form dynamics _ ( which represents a system of two degrees of freedom ) , and ( more importantly ) iii ) using the _ energy @xmath41 of the normal form _ as the third variable , showing the effect of arnold diffusion . according to theory , the value of @xmath41 changes exponentially slowly in time due to the effect of the remainder . in the sequel we refer to this phenomenon as ` drift ' , although in reality it means that a number of quantities can be characterized as undergoing random walk during the whole diffusion process . besides setting the timescale of diffusion , the drift can be viewed also as the source of a dynamical phenomenon , namely the communication between chaotic domains that would be otherwise isolated under the doubly - resonant normal form hamiltonian flow . we show in a true example the excursion of a chaotic orbit within the doubly - resonant domain as it appears in the above proposed set of variables . we thus identify a sequence of chaotic transitions of such an orbit from one resonant domain to another . in fact , in each transition the orbit bypasses the barriers imposed by normal form dynamics via a ` third dimension ' , i.e. the slowly drifting value of @xmath41 . we finally argue that , besides their practical utility , such illustrations are also suggestive of the geometric structure underlying the asymptotic manifolds of lower - dimensional tori filling the phase space in the domain of a double resonance . these manifolds are important , because , following the spirit of arnold s original work @xcite , it has been widely conjectured that their heteroclinic intersections constitute a primary cause of arnold diffusion . of course , proving this fact represents a well known important open problem of dynamical systems theory . the structure of the paper is as follows : section 2 presents the theory , focusing on the normal form algorithm , multiply - resonant dynamics , effect of the remainder , and , finally , on the relation between @xmath1 and @xmath2 . we describe in some length all necessary theoretical steps in order to render the paper as self - contained as possible . section 3 then passes to the numerical results . we present i ) the results from the normal form computer - algebraic construction , ii ) the visualization of arnold diffusion using appropriate variables based on the normal form computation , iii ) the numerical calculation of the diffusion coefficient @xmath1 , and , finally iv ) the comparison of @xmath1 with @xmath2 . section 4 summarizes the main conclusions of the present study . most statements made in subsections 2.1 and 2.2 below , regarding the properties of the hamiltonian models considered as well as the algorithm by which we perform hamiltonian normalization , are applicable to systems of an arbitrary number of degrees of freedom . in order , however , to be consistent with the rest of the paper , we use everywhere a notation referring to systems of three degrees of freedom . on the other hand , the analysis of subsection 2.3 applies to the study of the diffusion in doubly or simply resonant domains . in systems of three degrees of freedom , the latter represent the only possible multiplicities of a resonance condition , while in systems of more than three degrees of freedom there are also cases of intermediate resonance multiplicities between one and the maximal . the latter s study , nevertheless , is well beyond our present computational capacity , and thus it is left as an open problem . we consider three degrees of freedom systems of the form ( [ hamgen ] ) , where @xmath8 satisfies the following analyticity and convexity conditions : \i ) _ analyticity : _ @xmath8 is assumed to be an analytic function in a complexified domain of its arguments . namely , we assume that there is an open domain @xmath42 and a positive number @xmath43 such that for all points @xmath44 and all complex quantities @xmath45 satisfying the inequalities @xmath46 , the function @xmath47 admits a convergent taylor expansion @xmath48 where @xmath49 , and @xmath50 are the entries of the hessian matrix of @xmath47 at @xmath51 . furthermore , we assume that there is a positive constant @xmath52 such that for all @xmath53 , @xmath54 admits an absolutely convergent fourier expansion @xmath55 in a domain where all three angles satisfy @xmath56 , @xmath57 . by the fourier theorem ( see e.g. @xcite ) , this condition implies that the coefficients @xmath58 decay exponentially with the @xmath59modulus @xmath60 , that is , there is a positive constant @xmath61 such that the bound @xmath62 . we finally assume that all coefficients @xmath63 admit taylor expansions with respect to @xmath51 @xmath64 ( where @xmath65 are the entries of the hessian matrix of @xmath58 at @xmath51 ) , which are convergent in the same union of domains as for @xmath47 . \ii ) _ convexity : _ for the hessian matrix @xmath66 , which is real symmetric , we assume a simple quasi - convexity condition , namely that for all @xmath67 either two of the ( real ) eigenvalues of @xmath66 have the same sign and one is equal to zero , or all three eigenvalues have the same sign . furthermore , we define two constants : @xmath68 where @xmath69 is a label of only non - zero eigenvalues @xmath70 of @xmath66 , i.e. @xmath71 or @xmath72 if there are two or three non - zero eigenvalues respectively . as will be discussed in detail in subsection 2.3 , the quasi - convexity condition is essential , since it introduces a confinement of the orbits for exponentially long times on a surface arising from the condition of preservation of the energy ( see @xcite ) . we now give some definitions allowing to characterize resonant dynamics . a _ resonant manifold _ @xmath73 associated with a non - zero wavevector @xmath74 with co - prime integer components @xmath75 is the two - dimensional locus defined by @xmath76 where @xmath77 . let @xmath67 be such that all three frequencies @xmath78 , @xmath79 are different from zero . we now distinguish the following three cases : + i ) _ non - resonance : _ no resonant manifold @xmath73 contains @xmath51 . \ii ) _ simple resonance : _ one resonant manifold @xmath73 contains @xmath51 . \iii ) _ double resonance : _ more than one resonant manifolds contain @xmath51 . in the latter case , it is possible to choose two linearly independent vectors @xmath80 such that all resonant manifolds @xmath81 containing @xmath51 are labeled by vectors @xmath74 which are linear combinations of the chosen vectors @xmath80 with rational coefficients . the intersection of these manifolds forms a one - dimensional _ resonant junction_. a doubly - resonant point @xmath51 always corresponds to the intersection of a resonant junction with a constant energy surface @xmath82 . + in the above definitions , resonant manifolds @xmath73 of all possible wavevectors @xmath74 have been considered . it is well known , however , that in normal form theory a natural truncation limit @xmath83 arises in fourier space ( see below ) . accounting for this possibility , we call a point @xmath67 i ) non - resonant , ii ) simply resonant , or iii ) doubly resonant _ with respect to a k truncation _ , if the number of resonant manifolds @xmath73 with @xmath83 passing through @xmath51 are i ) zero , ii ) one and iii ) more than one respectively . finally , it will be convenient to introduce a definition concerning _ open domains _ in @xmath84 . let @xmath85 be a ball of radius @xmath86 around one point @xmath51 in @xmath84 . if @xmath47 satisfies convexity conditions as assumed above , for @xmath86 small whatsoever the domain @xmath85 is crossed by a dense set of resonant manifolds @xmath73 . however , for any fixed value of the positive integer @xmath87 , only a finite subset of the manifolds @xmath73 satisfy @xmath83 . the domain @xmath85 is then called : i ) non - resonant , ii ) simply - resonant , and iii ) doubly - resonant with respect to the @xmath87truncation if @xmath51 is , respectively , non - resonant , simply - resonant or doubly - resonant , and no other resonant manifolds @xmath73 with @xmath83 cross @xmath85 except for the ones passing through @xmath51 . all our estimates on the speed of diffusion are based on an appropriate normal form construction . in this , we adopt the method exposed in detail in @xcite , which lends itself conveniently to i ) developing a computer - algebraic program , and ii ) deriving analytical estimates on the size of various quantities appearing in the course of hamiltonian normalization . the main elements of this method are : + _ expansion centers . _ the action space can be covered by domains @xmath85 , centered around points @xmath51 which serve as expansion centers of both the original hamiltonian and the normal form . we choose the points @xmath51 to belong to the set of all doubly - resonant points of @xmath84 , denoted by @xmath88 , and by setting @xmath86 as of order @xmath28 . the covering is possible because @xmath88 is dense in @xmath84 . a normal form construction as done below is valid within one domain @xmath85 ( this is essentially the same starting point as in lochak s @xcite analytic construction leading to a proof of the nekhoroshev theorem ) . a crucial remark is that the characterization of dynamics within @xmath85 as non resonant , simply resonant , or doubly resonant depends on @xmath0 . this is because , as shown below , the optimal normal form truncation order @xmath89 in fourier space depends on the value of @xmath0 . furthermore , for a given value of @xmath87 , the set @xmath88 can be decomposed in three disjoint sets @xmath90 , containing all non - resonant , simply resonant and doubly resonant points respectively with respect to the @xmath87truncation . thus , the characterization of resonant dynamics within @xmath85 depends on whether , according to the value of @xmath87 , @xmath51 belongs to @xmath91 , @xmath92 , or @xmath93 . + _ resonant module : _ let @xmath51 be a point of @xmath88 and @xmath94 , @xmath95 two linearly independent vectors such that @xmath96 for @xmath97 . more than one choices of @xmath98 and @xmath99 are possible . in the sequel we choose @xmath98 and @xmath99 so that @xmath100 is minimal . the vector @xmath101 defined by @xmath102 is parallel to the vector @xmath103 since @xmath104 for all @xmath74 satisfying @xmath105 . if @xmath106 , @xmath107 , @xmath108 are not co - prime integers , we re - define @xmath109 by dividing the @xmath110 by their greatest common divisor . the set @xmath111 is hereafter called the resonant module associated with the point @xmath112 . the resonant module includes wavevectors @xmath74 whose respective trigonometric terms @xmath113 are to be retained in the normal form . + _ action re - scaling : _ from now on we focus on the construction of the normal form in one specific domain @xmath85 . it has been mentioned already that it is convenient to choose @xmath86 as a quantity scaling proportionally to @xmath114 . the simplest way to accommodate such a choice is by introducing the following re - scaling of all action variables within @xmath85 : @xmath115 this re - scaling greatly simplifies the normal form algorithm , because it formally removes all terms besides linear in the actions from the kernel of the so - called homological equation ( see below , or @xcite for details ) by which the normalizing generating functions are determined . eq.([resc ] ) does not define a canonical transformation . however , the correct equations of motion in the variables @xmath116 are produced by the hamiltonian function @xmath117 , i.e. ( neglecting a constant ) @xmath118 where the first line in the above equation comes from the integrable part @xmath47 of the original hamiltonian ( eq.([h0exp ] ) ) , while the second line comes from the perturbation @xmath54 ( eq.([h1four ] ) ) given the series expansion of the fourier coefficients as in eq.([hkexp ] ) . + _ book - keeping : _ we now split the hamiltonian ( [ hamexp2 ] ) in parts of different order of smallness , which are to be normalized step by step . the function ( [ hamexp2 ] ) contains terms of various orders in the small parameter @xmath114 . however , the presence of a second ` small parameter ' @xmath119 is implied in ( [ hamexp2 ] ) by the exponential decay of all fourier coefficients @xmath120 , @xmath65 , etc . , due to eq.([anal ] ) ( see @xcite pp.90 - 91 for a thorough exposition of the role of this small parameter in nekhoroshev theory ) . we take both parameters into account by introducing an integer @xmath121 such that @xmath122 , i.e. by setting : @xmath123~~.\ ] ] using @xmath121 , the hamiltonian ( [ hamexp2 ] ) can be split in groups of practically the same order of smallness . this is realized by artificially introducing a ` book - keeping ' coefficient @xmath124 in front of each term in ( [ hamexp2 ] ) , whose numerical value is set equal to unity at the end of the calculation . furthermore , for a term of the form @xmath125 we set @xmath126+\mu$ ] . regarding the above ` book - keeping ' process it is worth noting the following : i ) this way of splitting the hamiltonian in different orders of smallness results in a finite number of terms appearing in every power of @xmath127 . ii ) this technique is suggested already by poincar @xcite and arnold @xcite . in fact , the dependence of @xmath121 on @xmath0 is weak , since it is logarithmic , so that an alternative choice to the ` ansatz ' ( [ kprime ] ) is to set @xmath128 . in fact , according to giorgilli @xcite this is an optimal choice . iii ) since , at every normalization order , we have a reduction of the analyticity domain , one could consider re - defining @xmath121 at every normalization step . however , this is hardly tractable from an algorithmic point of view . instead , keeping @xmath121 constant at all normalization orders should be viewed as a rule indicating the sequence by which the various terms in the hamiltonian are normalized , i.e. , the terms or order @xmath129 are normalized in the r - th step . albeit not necessarily optimal regarding the grouping of the terms according to their size , this rule proves simple to implement and sufficient in practice . returning to the form of the hamiltonian after introducing the book - keeping factor @xmath127 , the hamiltonian reads : @xmath130}\epsilon^{1/2}h_{k*}\\ & + & \lambda^{2+[|k|/k']}\epsilon\nabla_{i_*}h_k\cdot j + \lambda^{3+[|k|/k']}{\epsilon^{3/2}\over 2 } \sum_{i=1}^3\sum_{j=1}^3h_{k , ij*}j_ij_j + \ldots\bigg ) \exp(ik\cdot\phi)\nonumber~~.\end{aligned}\ ] ] + setting @xmath131 , the hamiltonian ( [ hamexp3 ] ) resumes the form @xmath132 where i ) the superscript @xmath133 denotes zeroth - step of the normalization procedure (= original hamiltonian ) , ii ) the exponent of @xmath127 in different terms keeps track of their true order of smallness , and iii ) the functions @xmath134 are of the form @xmath135 where @xmath136 are polynomials containing terms of degree @xmath137 or @xmath138 in the action variables @xmath139 . precisely , we have : @xmath140 if @xmath141 , or @xmath142 @xmath143 if @xmath144 . + _ hamiltonian normalization : _ we use the algorithm of composition of lie series in order to perform the hamiltonian normalization . let us recall that the purpose of the normalization is to introduce a sequence of canonical transformations @xmath145 @xmath146 @xmath147 so that the hamiltonian expressed as a function of the new variables allows one to more easily identify the main features of dynamics . after @xmath148 normalization steps , the old variables @xmath149 are expressed in terms of the new variables @xmath150 , and the hamiltonian @xmath151 takes the form @xmath152 the terms @xmath153 and @xmath154 are called the _ normal form _ and the _ remainder _ respectively . the normal form is a finite expression which contains terms up to order @xmath148 in the book - keeping constant @xmath127 , while the remainder is a series containing terms of order @xmath155 and beyond . the mathematical structure of the normal form term @xmath156 is such as to imply an easily identifiable dynamics in the variables @xmath157 ( e.g. an oscillator or pendulum dynamics ) . on the other hand , the remainder is a _ convergent _ series in a restriction of the domain of analyticity of the original hamiltonian , which represents a perturbation with respect to the hamiltonian flow of @xmath156 . an optimal normalization order @xmath158 exists ( see below ) where the process must be stopped . the hamiltonian normalization is implemented step - by - step by the recursive equation : @xmath159 where @xmath160 is the r - th step lie generating function and @xmath161 is the poisson bracket operator . both @xmath162 and @xmath160 are functions of the variables @xmath163 . the generating function is defined by the solution of the homological equation @xmath164 where @xmath165 denotes all terms of @xmath166 which i ) have a book - keeping coefficient @xmath129 in front , and ii ) belong to the range of the operator @xmath167 . given the definition of the resonant module @xmath168 in eq.([resmod ] ) , one has the relation @xmath169 where @xmath170 are all the terms of @xmath166 having a factor @xmath129 , and @xmath171 are the _ normal form _ terms of @xmath170 , that is all the trigonometric terms whose wavevectors @xmath74 belong to @xmath168 . it follows immediately that @xmath162 has the form @xmath172 where all terms in the functions @xmath173 have a factor @xmath174 , while @xmath175 is a series in powers of @xmath127 starting with terms of order @xmath155 . + _ optimal truncation : _ in the analytical part of the nekhoroshev theory it is demonstrated that the whole normalization process has an asymptotic character . namely , i ) the domain of convergence of the remainder series @xmath175 shrinks as the normalization order @xmath148 increases , and ii ) the size @xmath176 of @xmath175 , where @xmath177 is a properly defined norm in the space of trigonometric polynomials ( see below ) , initially decreases , as @xmath148 increases , up to an optimal order @xmath158 beyond which @xmath176 increases with @xmath148 . in the nekhoroshev regime , one has @xmath178 . thus , stopping at @xmath158 best unravels the dynamics , which is given essentially by the hamiltonian flow of @xmath179 slightly perturbed by @xmath180 . the long term consequences of this perturbation , which determine the speed of diffusion , will be analyzed in subsection 2.3 . the normal form @xmath181 contains trigonometric terms @xmath113 of order not greater than @xmath182 . let @xmath158 be the optimal normalization order . it is well known that the dependence of @xmath158 on @xmath0 is given by an inverse power - law , namely @xmath183 the exponents @xmath184 , @xmath185 and @xmath186 , referring to the non - resonant , simply resonant , and doubly resonant normal form constructions respectively , are found in @xcite . we emphasize that , while , due to the introduction of the book - keeping process , the algorithm of hamiltonian normalization analyzed above is not technically identical with the usual normalization procedure used in the proof of the nekhoroshev theorem ( e.g. as in @xcite ) , in practice we recover the estimate ( [ ropt ] ) , and the resulting exponents , both in the simply resonant case ( see @xcite ) and in the doubly resonant case , as confirmed by numerical experiments in section 3 below . in particular , we find that since the leading terms in the remainder are @xmath187 , the size of the remainder is of order @xmath188 , implying ( viz.eq.([kprime ] ) ) : latexmath:[\[\label{remropt } i.e. the remainder is exponentially small in @xmath29 in accordance with the nekhoroshev theorem . the fourier order @xmath190 is hereafter called the optimal k truncation order . all the normal form terms of @xmath191 have fourier orders satisfying @xmath192 . + we are now in a position to discuss the essence of all the previous definitions . the key point is to observe that , depending on the value of @xmath0 , _ the same _ expansion point @xmath112 of the normal form construction turns to be either non - resonant , or simply or doubly resonant with respect to the optimal k truncation . in particular , let @xmath98 and @xmath99 be two linearly independent vectors of @xmath168 such that for all @xmath193 one has @xmath194 . we then distinguish the following three regimes : i ) @xmath195 . then , the point @xmath51 is doubly - resonant with respect to the optimal k truncation . this is the case we mainly focus on in the sequel . the main theoretical results are given in subsection 2.3.1 , while the main numerical results are given in section 3 . ii ) @xmath196 . then , @xmath51 is simply - resonant with respect to the optimal k truncation . one such example was dealt with in the numerical study @xcite . further theoretical analysis of this case is made in subsection 2.3.2 . iii ) @xmath197 . then , @xmath51 is non - resonant with respect to the optimal k truncation . since @xmath198 decreases as @xmath0 increases , for fixed @xmath100 this inequality always occurs if @xmath199 , where @xmath200 is a threshold depending on @xmath98 , @xmath99 . the case @xmath201 , where @xmath202 is the critical threshold for the onset of the nekhoroshev regime , presents no practical interest . if , however , @xmath203 , then , for all values of @xmath0 in the interval @xmath204 the optimal normal form describes a true non - resonant dynamics . note that in order to describe the dynamics close to a point @xmath205 of the action space corresponding to diophantine frequencies @xmath206 , it suffices to choose @xmath51 such that @xmath207 corresponds to a very high order rational approximation of @xmath207 , i.e. the numbers @xmath208 are high order finite digit approximants of the numbers @xmath209 . then , @xmath100 becomes very large , and @xmath200 approaches very close to zero . in this case , for @xmath0 sufficiently small , we expect the existence of a set of points of large measure within @xmath85 , corresponding to kolmogorov - arnold - moser tori in the neighborhood of the point @xmath51 . however , these tori can not fill an open domain . thus , the diffusion in action space is topologically possible for ( very weakly ) chaotic orbits wandering through the set of kam tori . however , in the absence of significant resonant chaotic layers ( since no important resonances cross @xmath85 ) , the question of whether or not the diffusion can be observed is of no practical interest , since its rate would be extremely slow to be of any relevance in applications . thus , the non - resonant case is no further considered below . as long as @xmath210 , the point @xmath51 is doubly - resonant with respect to the optimal k truncation . in this case , the normal form contains either terms independent of the angles , or trigonometric terms of the form @xmath211 , @xmath212 and their multiples in the exponents . writing explicitly only the most important terms , the normalized hamiltonian takes the form : @xmath213 the main feature of the hamiltonian ( [ hamres ] ) is that , since in @xmath214 there are coupling terms between more than one resonant angles , the normal form @xmath26 _ alone _ is _ non - integrable_. in fact , @xmath26 can be decomposed into an integrable system of one degree of freedom and a non - integrable system of two degrees of freedom ( see @xcite ) . the decomposition is done by the linear canonical transformation @xmath215 @xmath216 @xmath217 defined by @xmath218 where @xmath101 has been defined in eq.([mvec ] ) . the hamiltonian in the new variables reads ( apart from a constant ) @xmath219 where @xmath220 and the remainder @xmath221 is exponentially small in @xmath29 . since @xmath222 is ignorable in @xmath26 , @xmath223 is an integral under the flow of the normal form . on the other hand , the remaining degrees of freedom @xmath224 and @xmath225 are coupled under the flow of @xmath26 due to the trigonometric terms @xmath226 . the main characteristics of motion can be understood by the following remarks : refer to non - scaled values , i.e. before the re - scaling of eq.([resc ] ) is implemented , while all action variables defined by a symbol starting with the letter @xmath139 have re - scaled values , according to eq.([resc ] ) . thus , in the domains considered below , all quantities of the form @xmath227 , where @xmath51 is the selected central doubly - resonant point of interest , scale proportionally to @xmath114 , while all actions denoted by a letter @xmath139 exhibit no scaling with @xmath0 . furthermore , all hamiltonian - type functions denoted by @xmath228 , @xmath162 , @xmath26 , or @xmath27 , are expressed in re - scaled variables ; only the original hamiltonian ( eq.([hamgen ] ) ) is expressed in non - scaled action variables @xmath229 . finally , the quantities @xmath230 ( eq.([eneprime ] ) ) and @xmath41 ( eq.([ez ] ) ) scale proportionally to @xmath114 . ] \i ) the constant - valued action @xmath223 can be viewed as a parameter in the two degrees of freedom hamiltonian @xmath26 . furthermore , except for the case of some very low resonances satisfying @xmath231 , all coefficients @xmath232 in ( [ nfdble ] ) are of order @xmath114 or higher . thus , the terms @xmath233 define an ` integrable part ' of @xmath26 , while the remaining terms depending on the resonant angles can be considered as a perturbation . the terms quadratic in @xmath234 in the r.h.s . of ( [ nfdble0 ] ) define the quadratic form @xmath235 in appendix a it is demonstrated that , due to the quasi - convexity condition assumed for the hessian matrix @xmath50 , the quadratic form ( [ z02 ] ) is positive definite . thus , the constant level curves of the quantity @xmath236 on the plane @xmath237 , given by @xmath238 are ellipses centered at @xmath239 ( the role of the elliptic structures formed around double resonances in the nekhoroshev theorem is discussed extensively in @xcite ) . if the higher order terms in the action variables of the development of eq.([nfdble0 ] ) are taken into account , the constant energy condition of eq.([eneprime ] ) yields deformed ellipses on the plane @xmath237 . if @xmath240 or @xmath241 , the slow frequencies @xmath242 , @xmath243 are non - zero , and they are given by @xmath244 on the other hand , due to the definition ( [ restra ] ) one has @xmath245 which is valid for any value of @xmath246 in the domain of convergence of the series ( [ nfdble0 ] ) . it follows that all the resonant manifolds defined by relations of the form @xmath247 intersect any of the planes @xmath237 corresponding to a fixed value of @xmath223 . using the notation @xmath248 the intersection of one resonant manifold with the plane @xmath237 is a curve . in the linear approximation , we have @xmath249 the above equation defines a ` resonant line ' , which is the local linear approximation to a ` resonant curve ' . all resonant lines ( or curves ) pass through the point @xmath250 , which , therefore , belongs to the resonant junction defined by the wavevectors @xmath80 . to each resonant curve we can associate a resonant strip in action space whose width is proportional to the separatrix width for that resonance . if , for a single pair of integers @xmath251 , we only isolate the resonant terms @xmath252 in the normal form @xmath26 ( eq.([hamres ] ) ) , we obtain a simplified resonant normal form @xmath253 corresponding to the limiting case of a single resonance . in a strict sense , @xmath254 describes well the dynamics far from the resonant junction . however , it can also be used in order to obtain estimates of the resonance width along the whole resonant curve defined by the integer pair @xmath251 . to this end , the leading terms of @xmath253 are ( apart from constants ) : @xmath255 + \ldots\nonumber\end{aligned}\ ] ] where the coefficients @xmath256 satisfy the estimate @xmath257 ) . after still another transformation @xmath258 , @xmath259 , @xmath260 , @xmath223 becomes a second integral of motion of @xmath253 , which takes the form @xmath261\end{aligned}\ ] ] where @xmath262 and @xmath263 are constants of the hamiltonian flow of ( [ hamresn12n ] ) . combining ( [ gn12 ] ) and ( [ hamresn12n ] ) , the separatrix width can be estimated as @xmath264 eq.([sepwidth ] ) allows to estimate the width of a resonant strip in the direction normal to a resonant curve on the plane @xmath237 . using the relations @xmath265 ( for @xmath266 ) , this estimate takes the form @xmath267 by the overlapping of various resonant strips whose limits ( pairs of parallel red lines ) correspond to separatrix - like thin chaotic domains around each resonance . two constant normal form energy ellipses @xmath268 and @xmath269 are also shown . right : the front and back panels show the phase portraits corresponding to a surface of section ( in one of the pairs @xmath270 or @xmath271 ) under the normal form dynamics alone , for the energies @xmath268 ( front panel ) and @xmath269 ( back panel ) . the blue curly arrows in both panels indicate the directions of a possible ` drift ' motion ( = slow change of the value of @xmath230 ) due to the influence of the remainder on dynamics . ] the outcome of the analysis so far can be visualized with the help of figure [ fgdbleresmodel ] ( schematic ) . the left panel shows the structure of a doubly - resonant domain in the plane of the resonant action variables @xmath237 . the two bold ellipses correspond to the constant energy condition for two different values of @xmath230 , namely @xmath268 and @xmath269 with @xmath272 . their common center is the point @xmath273 defined in eq.([jr0 ] ) . the three pairs of parallel red lines depict the borders of the separatrix - like thin chaotic layers of three resonances passing through the center . infinitely many such resonances exist , corresponding to different choices of integer vectors @xmath274 ; however , their width decreases as @xmath275 increases , according to eq.([reswidth ] ) . we thus show schematically only three resonances with a relatively low value of @xmath275 , named by the letters ` a ' , ` b ' and ` c ' . the blue curly curves indicate a slow drift undergone by the chaotic orbits along the resonance layers , allowing for a transition from one resonance to another . this phenomenon , which will be addressed in detail below , is due to the influence of the _ remainder terms _ of the normalized hamiltonian on dynamics . here , however , we discuss first the ( non - trivial ) influence of the _ normal form terms _ on dynamics , by considering the hamiltonian flow under the approximation @xmath276 . then , the following facts hold : + - for any fixed value of @xmath230 , and a fixed section in the angles , the motion is confined on one ellipse . + - for @xmath230 large enough ( @xmath268 , outermost ellipse in the left panel of fig.[fgdbleresmodel ] ) , the various resonant strips intersect the ellipse @xmath268 at well distinct arcs , i.e. there is no resonance overlap . the right front panel in fig.[fgdbleresmodel ] shows schematically the expected phase portrait , which can be obtained by evaluating an appropriate poincar surface of section , e.g. in the variables @xmath224 or @xmath225 . the dashed lines show the correspondence between the limits of various resonant domains depicted in the left and right panels . in particular , the intersection of each resonant strip in the left panel with the ellipse @xmath268 corresponds to the appearance of an associated _ island chain _ in the right panel . the size of islands is given essentially by the separatrix width estimate of eq.([reswidth ] ) . hence , the size of the islands decreases exponentially with the order of the resonance @xmath277 . however , the main effect to note is that , since all resonant strips are well separated on the ellipse , the thin separatrix - like chaotic layers marking the borders of each of their respective island chains do not overlap . as a result the local chaos around one resonance is isolated from the local chaos around the other resonances . in fact , the normal form dynamics induces the presence of rotational kam tori which , in this approximation ( @xmath276 ) , completely obstruct the communication among the resonances . note that a detailed study of the dynamics of the above type , induced by the doubly - resonant normal form , was recently presented in @xcite . + - far from the domain of resonance overlap , the size of the islands corresponding to each resonance is nearly independent of the energy @xmath230 , as it depends essentially only on the size of the fourier coefficient of the corresponding harmonics in the hamiltonian . however , the separation of the islands is reduced as the energy _ decreases _ , since this separation is given essentially by the separation between the distinct arcs in fig.[fgdbleresmodel ] at which the various resonances intersect the ellipse corresponding to a fixed energy @xmath230 . as a result , below a critical energy @xmath278 , significant resonance overlap takes place , leading to the communication of the chaotic layers of the various resonances and an overall increase of chaos . this is shown in the left panel of fig.[fgdbleresmodel ] for an ellipse @xmath279 , with the corresponding phase portrait shown in the right back panel . we note in particular the ` merging ' of all three resonant domains one into the other , which produces a large connected chaotic domain surrounding all three island chains ( and many other smaller chains , not visible in this scale ) . the value of the critical energy @xmath280 marking the onset of large scale resonance overlap can be estimated as follows : each resonant strip intersects one fixed energy ellipse on one arc segment . also , eq.([reswidth ] ) can be replaced by the estimate @xmath281 where @xmath277 , @xmath282 , and @xmath283 , with the constants @xmath284 defined as in eq.([mh ] ) . the total length @xmath285 of all segments can be now estimated by summing , for all @xmath13 , the estimate ( [ reswdk ] ) , namely @xmath286 on the other hand , the total circumference of the ellipse for the energy @xmath230 is estimated as @xmath287 where @xmath288 is the geometric mean of the ellipse s major and minor semi - axes . for @xmath288 one has the obvious estimate @xmath289 , whence @xmath290 the critical energy @xmath291 can now be estimated as the value where @xmath292 , implying that the associated ellipse is fully covered by segments of resonant strips . thus @xmath293 eq.([epcrit ] ) implies that @xmath280 is a @xmath294 quantity . when @xmath309 , @xmath51 is simply - resonant with respect to the optimal k truncation . in this case , the normal form contains terms either independent of the angles , or depending on them via trigonometric terms of the form @xmath310 , @xmath311 . using the same notations as in the previous subsection , the transformed hamiltonian reads : @xmath312 , but for a simple resonance . in this case , any other resonance crossing the main ( guiding ) resonance has an exponentially small width and acts as a ` driving ' resonance for diffusion . ] repeating all steps as in the case of double resonance leads to the normal form @xmath313 the main difference with respect to the doubly - resonant normal form ( [ hamresn12 ] ) is that , the angle @xmath314 being ignorable , the action @xmath315 ( or @xmath316 ) is an integral of the flow of @xmath254 , in addition to @xmath223 . thus , @xmath254 defines an integrable hamiltonian . a pair of constant values @xmath317 , @xmath318 defines a straight line @xmath319 which corresponds to the unique resonance @xmath320 . this will be called ` main resonance ' (= the ` guiding resonance ' in @xcite ) . in figure [ fgspleresmodel ] ( schematic ) , the domain of the main resonance is delimited by two vertical thick red lines corresponding to the separatrix - like thin chaotic layers at the boundary of the resonance similarly to fig.[fgdbleresmodel ] . using similar arguments as in the derivation of eq.([reswdk ] ) , the separatrix width can be estimated as @xmath321 under the normal form dynamics , motions are allowed only across the resonance , i.e. in the direction @xmath322 . in fig.[fgspleresmodel ] this is the horizontal direction . the thin strip delimited by two horizontal red lines corresponds to the resonance with resonant wavevector @xmath99 , which , since @xmath323 , is now of width exponentially small ( @xmath324 ) . thus , it will be called a ` secondary ' resonance . in order to estimate the speed of diffusion as a function of the optimal remainder in this case , let us note first that the influence of the remainder on dynamics is to slowly change the value of the two approximate integrals @xmath223 and @xmath316 , that would be exactly preserved under the normal form dynamics . in view of eq.([hamresn1 ] ) , the hamiltonian ( [ hamressple0 ] ) can be approximated by @xmath325+ ... \bigg]\end{aligned}\ ] ] where i ) the ( non - integer ) vectors @xmath326 , @xmath79 come from the solution of the right eqs.([restra ] ) for the angles @xmath327 in terms of the angles @xmath328 , @xmath314 , and @xmath329 , and ii ) we approximate all the fourier coefficients in the remainder series by their constant values @xmath330 at the points @xmath331 ( we set @xmath332 for @xmath333 ) . the latter approximation is sufficient for estimates regarding the speed of diffusion . the key remark is that for all the coefficients @xmath330 the bound @xmath334 holds , while , for the leading fourier term @xmath335 in the remainder we have @xmath336 . in fact , we typically find that the size of the leading term is larger from the size of the remaining terms by several orders of magnitude , since this term contains a repeated product of small divisors of the form @xmath337 ( see appendix a ) . furthermore , using an analysis as in @xcite , we readily find @xmath338 , where @xmath339 is a so - called ( in @xcite ) ` delay ' constant . we note in passing that the fourier terms of the form @xmath335 are called ` resonant ' in @xcite . the value of the diffusion coefficient can now be estimated by applying the heuristic theory of chirikov ( @xcite , see also @xcite and @xcite ) in the hamiltonian model ( [ hamressple ] ) . the estimate @xmath340 holds , where @xmath341 , @xmath342 is an average period of motion within the main resonance separatrix - like thin chaotic layer , of width @xmath343 , @xmath61 is the melnikov function with argument @xmath344 ( see appendix b of @xcite ) , the vector @xmath345 being defined by the relation @xmath346 . the estimate @xmath347 holds . in view of eq.([restra ] ) however , we have that @xmath348 . since @xmath349 ( see appendix b ) , and @xmath350 , it follows that @xmath351 , for an exponent @xmath31 . putting these estimates together , we finally arrive at a steeper dependence of the diffusion coefficient @xmath1 on the optimal remainder @xmath2 in the case of simple resonance than in the case of double resonance , namely : @xmath352 regarding now the precise value of @xmath4 , it is hardly tractable to determine this on the basis exclusively of the behavior of the melnikov integrals discussed above . we note , however , that the quantity @xmath353 yields the size of the ` splitting'@xmath21 of the separatrix of the main ( guiding ) resonance due to the effects of the leading term in the remainder function . the relation between the separatrix splitting and the size of the optimal remainder has been examined in @xcite and later in @xcite . in the latter work , the estimate @xmath354 was predicted and probed by numerical experiments , where @xmath138 ( in the notation of @xcite ) is the effective size of the perturbation to the normal form pendulum dynamics caused by the remainder . setting thus @xmath355 suggests the scaling @xmath356 , whereby the constant @xmath4 can be estimated as @xmath357 . hence ( in view of [ difcfsple ] ) @xmath358 in simply resonant domains . despite the heuristic character of the above derivation , it seems that the value @xmath357 is supported by the results of numerical experiments . in particular , in @xcite the diffusion coefficient @xmath1 along a simple resonance was compared directly to the size of the optimal normal form remainder . it was found that @xmath37 , essentially confirming that @xmath359 . we point out , however , that in @xcite a different exponent was found @xmath360 regarding the same resonance as in @xcite , while it was found that @xmath23 in the case of a very low order simple resonance ( with @xmath231 ) , which is not discussed in our present work . these exponents , on the other hand , depend on the chosen definition of the numerical measure used to estimate both @xmath21 and @xmath2 . thus , a detailed quantitative comparison of the works cited above is left as on open problem for future study . in our numerical work we employ the same hamiltonian model of three degrees of freedom as in @xcite . the hamiltonian reads : @xmath361 this model has a particularly simple , yet sufficient for our purpose , structure , allowing to probe numerically all steps of the previous section . in particular : the function ( [ hamfr ] ) is polynomial in the action variables , thus it is analytic in any complex extension of @xmath362 . on the other hand , the domain of analyticity in the angle variables was examined in @xcite . it was found that analyticity can be established in a set @xmath363 , @xmath364 , @xmath79 , for a positive constant @xmath52 estimated semi - analytically as @xmath365 . accordingly , the coefficients @xmath63 of the fourier development @xmath366 where @xmath75 , @xmath367 , decay exponentially . the distance of the nearest singularity , with respect to each of the angles @xmath368 , from the real axis is given by the solution of @xmath369 , or @xmath370 . thus , the following bound holds : latexmath:[\[\label{expdecay } as regards convexity , for all @xmath67 the matrix @xmath66 has a particularly simple structure , since we have @xmath372 , and @xmath373 for all other @xmath374 . thus there are two positive eigenvalues equal to unity and one equal to zero , while @xmath375 . the constant energy condition @xmath376 defines a paraboloid in the action space . the resonant manifolds are planes , since @xmath377 , @xmath378 , @xmath379 , whereby the resonance conditions @xmath380 for all @xmath75 define planes normal to the @xmath381 plane . it follows that , when projected to the @xmath381 plane , the intersections of all resonant manifolds with a surface of constant energy of the unperturbed problem yield a set of straight lines . this greatly facilitates the numerical study , since all diffusing orbits in the perturbed problem follow piecewise straight paths nearly parallel to one or more resonant lines of the unperturbed problem , while the orbits can only change direction by approaching close to resonance junctions . examples of diffusion of this type along a simple resonance where studied in @xcite , while the case of consecutive encounters with doubly - resonant domains was examined in a mapping model @xcite variant of the hamiltonian model ( [ hamfr ] ) . the connection between the size of the optimal remainder @xmath2 and the diffusion coefficient @xmath1 in a case of simple resonance was the main subject of a previous study @xcite . following the same terminology and notations as in section 2 above , the point @xmath51 in the normal form construction in @xcite was chosen as @xmath382 . for this point we have ( viz . eq.([mvec ] ) ) @xmath383 , @xmath384 , @xmath385 . the optimal truncation order in all calculations of @xcite varied from @xmath386 to @xmath387 ( depending on the value of @xmath0 in the range considered ) . thus , in all cases we have @xmath388 , that is the so - chosen point @xmath51 was found to be simply resonant with respect to the optimal k truncation . following fig.5 of @xcite it was then found by numerical fitting that the diffusion coefficient @xmath1 scales with the optimal remainder as @xmath37 . a theoretical justification for this ` steepening ' of the power - law with respect to the exponent @xmath389 holding in double resonances was given in subsection 2.3.2 . in order to probe now the dependence of @xmath1 on @xmath2 in the case of a double resonance , in the sequel we focus our numerical study on a different point of @xmath88 , namely @xmath390 . the basic resonant wavevectors are @xmath391 the hamiltonian normalization is carried out as exposed in subsection 2.2 . the interval of values of @xmath0 considered is @xmath392 which , according to @xcite is below the critical value for the onset of the ` nekhoroshev regime ' ( @xmath393 ) . furthermore , it will be shown below that for all values of @xmath0 in the above interval the optimal fourier - truncation order @xmath198 turns to be much larger than @xmath394 . on the other hand , for the basic wavevectors we have @xmath395 , @xmath396 . thus , for all considered values of @xmath0 one has @xmath397 , that is , the point @xmath51 is doubly resonant with respect to any of the optimal k truncations considered in the sequel . due to eq.([kprime ] ) , the constant @xmath121 in terms of which book - keeping is implemented changes with @xmath0 . however , one notices that , because of the logarithmic dependence of @xmath121 on @xmath0 , in the largest part of the interval @xmath392 , where we focus , one has a constant value @xmath398 , while one has @xmath399 only close to the upper limit @xmath400 and @xmath401 close to the lower limit @xmath402 . for simplicity , we thus fixed the value of @xmath121 as @xmath398 in all normal form computations . doing so , computer memory limitations restrict all computed expansions to a maximum order @xmath403 in the book - keeping parameter @xmath127 , or maximum order @xmath404 in fourier space . in fact , for @xmath405 we perform at most 14 normalization steps , so that the remainder contains terms of at least three consecutive orders in @xmath127 , namely @xmath406 , @xmath407 and @xmath408 . as explained below , this allows us to perform some numerical tests regarding the convergence of the remainder series when the optimal normalization order is as high as @xmath158=14 ( or @xmath409 ) . on the other hand , for @xmath410 we allow for one more normalization step ( @xmath406 ) in order to get as close as possible to the optimal order , which , as shown below for @xmath411 is larger than 14 . thus , for the calculation of the corresponding remainder value at this order ( @xmath406 ) we necessarily have to rely on the sum of only two rather than three or more consecutive terms . writing the truncated ( at order 17 ) remainder function as : @xmath412 where @xmath413 are the terms of order @xmath414 in the book - keeping parameter @xmath127 allows us to probe numerically the convergence of the remainder function within any chosen domain @xmath85 in action space . to this end , at any normalization order @xmath148 , let us consider a disk @xmath415 in the space of the transformed action variables ( we neglect the action @xmath416 which , in the particular case of the hamiltonian ( [ hamfr ] ) , is dummy , i.e. it does not appear in any higher order term of either the normal form or the remainder ) . this is a deformed disk also in the old canonical variables @xmath417 , limited by a boundary given approximately by @xmath418 ( cf . the action re - scaling given by eq.([resc ] ) ) . for the hamiltonian ( [ hamfr ] ) one can readily check that all the terms in @xmath413 are trigonometric polynomials of maximum degree @xmath419 whose coefficients are polynomial of maximum degree @xmath420 in the actions , namely @xmath421 furthermore , in the disk @xmath422 the obvious bound @xmath423 holds . we thus define the norm @xmath424 can be obtained . in fact , by calculating the truncated sums @xmath425 for any fixed choice of @xmath43 , where @xmath426 takes all values @xmath427 , we can have a clear numerical indication of whether the remainder function was calculated up to a sufficiently high order for convergence to have been practically reached . the maximum value of @xmath43 for which the series @xmath428 converges absolutely sets the size of the doubly - resonant domain @xmath429 ( in non - scaled variables ) where the normal form calculations are valid . in practice , we are interested in the diffusion of orbits with initial conditions inside this domain . in particular , in subsection 3.2 we will consider orbits starting on the circle @xmath430 . all our numerical orbits are studied up to a time in which their distance from the center of the double resonance changes significantly less than @xmath431 ( see below ) . variations of this order at maximum are found when we measure @xmath43 either in the original canonical action variables or in the variables after the optimal canonical transformation . thus , for all the orbits we can set a safe outer boundary @xmath432 within which they are well confined . we then verify numerically that this domain belongs to the analyticity domain of the various transformations employed in the form of series ( of the new variables in terms of the old variables or vice versa ) . this check is made by finding whether the fourier coefficients of the series exhibit an exponential decay . an example is given in fig.[fggksigma ] . we consider the fourier series yielding the new transformed canonical action @xmath433 as a function of the old canonical variables , for @xmath434 , at the normalization orders @xmath435 and @xmath436 . writing this as a series @xmath437 we define the coefficients @xmath438 ( see text ) as a function of the fourier order @xmath439 , for @xmath434 , at the normalization orders @xmath440 , @xmath441 , and @xmath442 ( lower , middle and upper set of points respectively ) . all three curves exhibit an exponential decay for large @xmath439 , with nearly the same asymptotic law . the straight line has inclination @xmath443 . ] figure [ fggksigma ] shows the coefficients @xmath444 for @xmath434 , @xmath445 , and @xmath435 and @xmath436 . we observe that all three curves exhibit a tail showing exponential decay of the fourier coefficients . however , it is remarkable that the asymptotic exponential slope seems to change only marginally . instead , the main change , as @xmath148 increases , regards that formation of a ` plateau ' of fourier coefficients of nearly constant size formed for small @xmath439 . namely , the width of the plateau increases as @xmath148 increases . it is remarkable that the asymptotic tail laws for all @xmath148 appear to follow an exponential decay with the same constant @xmath446 , i.e. with nearly the same value as the constant appearing in the analyticity condition of the original hamiltonian ( cf . eq.([expdecay ] ) ) . this effect shows that , while in the usual proofs of the nekhoroshev theorem one requires a reduction of the analyticity domain at every normalization step , i.e. one considers bounds of the form @xmath447 with @xmath448 , in practice the dependence of the coefficients @xmath449 on @xmath439 is more complicated than a simple exponential decay law . in fact , the constants @xmath450 reflect an average exponential slope that compensates between the plateau , for small @xmath439 , and the exponential tail , for large @xmath439 . namely , as the width of the plateau increases with @xmath148 , one obtains smaller and smaller values of the average exponential decay constant @xmath450 . as a function of the truncation order @xmath426 when @xmath434 , @xmath430 , and the normalization orders are @xmath451 ( upper curve ) , @xmath442 ( lower curve ) and @xmath452 ( middle curve ) . ( b ) the value of @xmath453 as a function of @xmath148 for different values of @xmath0 . for @xmath454 and @xmath455 , the dashed curves after the order @xmath406 are found by quadratic extrapolation . no attempt to extrapolate was made for @xmath456 and @xmath402 . ( c ) the optimal normalization order @xmath158 as a function of @xmath0 together with a power - law best fitting curve . ] fig.[fgremopt]a shows now an example of the behavior of the truncated remainder function for @xmath457 and @xmath434 . the upper curve shows the value of @xmath458 at the normalization order @xmath451 as a function of @xmath426 for @xmath459 . clearly , after @xmath460 the cumulative sum ( [ sumrem ] ) shows no further substantial variation , which indicates that the remainder series converges after three consecutive terms @xmath461 and @xmath462 ( this is verified also by computing numerically a convergence criterion like dalembert s criterion ) . the lower and middle curves show now the same effect for the normalization orders @xmath442 and @xmath452 respectively . note that the three consecutive truncation orders @xmath463 and @xmath408 allowed for the computation of the remainder at the normalization order @xmath452 are essentially sufficient to demonstrate the convergence of the remainder . hence , @xmath464 represents a good numerical estimator of the value of the remainder series for normalization orders up to @xmath452 . however , the main effect to note is that the estimated remainder value @xmath465 found for @xmath452 is larger than the one for @xmath442 , implying that the _ optimal _ normalization order @xmath158 is below @xmath452 . fig.[fgremopt]b shows , precisely , the asymptotic character of the above normalization , showing @xmath465 against the normalization order @xmath148 for various values of @xmath0 as indicated in the figure . for all values down to @xmath466 we now observe the asymptotic behavior , namely the size of the remainder initially decreases as @xmath148 increases , giving the impression that the normalization might be a convergent procedure . however , this trend is reversed after an optimal order @xmath158 , where the remainder reaches its minimum value , while , for @xmath467 the remainder increases with @xmath148 and eventually goes to infinity . we also observe that for @xmath468 the optimal order is beyond @xmath406 . however , for @xmath454 and @xmath455 , the computed remainder values are close to the minimum . the dashed extensions of the numerical curves shown in fig.[fgremopt]b correspond to an extrapolation obtained by quadratic fitting of the available numerical points near the corresponding minima . using this extrapolation , we obtain an estimate of the optimal remainder size for the values @xmath454 and @xmath455 , that will be used in some calculations below . on the other hand , for @xmath456 and @xmath402 , even using the extrapolation we find that the optimal normalization is beyond any reliable possibility to estimate given our computing limitations . as discussed above , the estimate @xmath469 holds @xcite , i.e. @xmath158 is expected to be a decreasing function of @xmath0 . fig.[fgremopt]c shows the numerical estimate for @xmath158 as a function of @xmath0 from the points of minima of fig.[fgremopt]b . the blue curve is a power - law fitting , yielding the exponent 0.52 , i.e. very close to the one predicted by theory . since the value @xmath470 depends on @xmath0 , from the above procedure we obtain numerically pairs of values @xmath471 . in subsection 3.4 below , we will numerically calculate the value of the diffusion coefficient @xmath1 for each one of the selected values of @xmath0 , thus allowing for a probe of the dependence of @xmath1 on @xmath2 and a comparison with the results of subsection 2.3.1 . the resonant structure in the action space ( around @xmath51 ) can be visualized by employing the method of the fli map as in @xcite . we recall that the fast lyapunov indicator ( fli ) is a numerical indicator of chaos , defined for one orbit by @xmath472 where @xmath473 is a deviation vector , i.e. in our case @xmath474 found after solving the variational equations of motion up to the time @xmath16 from some initial conditions @xmath475 . by properly choosing a threshold value @xmath476 , orbits with @xmath477 are characterized as regular , and those with @xmath478 as chaotic . furthermore , a convenient use of the fli in the visualization of the arnold web is found by producing fli color maps @xcite . considering a grid of initial conditions in the action space , we assign to each initial condition a color corresponding to the fli value found for the resulting orbit integrated up to a sufficiently long time ( of the order 100 1000 periods ) . this allows for illustrating the resonant structure in action space , as shown in fig.[fgdbleres ] , which is an fli map in an action domain including our chosen doubly - resonant point @xmath479 for three different values of @xmath0 . in all three panels , there are resonances projecting on @xmath381 as single yellow or orange thick lines , while other resonances project as strips with a green or blue interior zone delimited by pairs of nearly parallel yellow or red lines . as explained in @xcite , this difference is only due to the particular choice of surface of section ( @xmath480 , @xmath481 , similar to @xcite ) . namely , the yellow lines marking all resonances represent the intersection of the thin separatrix - like chaotic layers formed around each resonance with the chosen surface of section . this produces a pair of nearly parallel yellow or orange lines for any resonance ( of the form @xmath482 ) whose leading fourier coefficient @xmath63 of the resonant term @xmath113 in the original hamiltonian expansion has a negative real part , while it produces a single yellow or orange thick line if @xmath483 is positive . in the latter case , the domain of regular orbits inside the resonance has no projection on the chosen surface of section , while in the former case it projects as a strip of green or blue color . of the hamiltonian [ hamfr ] for @xmath480 , @xmath484 , around the doubly - resonant point @xmath479 for ( a ) @xmath402 , ( b ) @xmath466 , ( c ) @xmath485 . the color scale represents the computed value of the fli ( see text ) in the intervals @xmath486 ( magenta , most ordered ) , @xmath487 ( blue ) , @xmath488 ( green ) , @xmath489 ( orange ) , @xmath490 ( yellow , most chaotic ) . ] when @xmath402 ( fig.[fgdbleres]a ) , we easily distinguish four main resonances passing through @xmath479 . the biggest resonant domain ( green , from bottom left to top right ) corresponds to the resonance @xmath491 , whose corresponding wave - vector is the basic resonant wavevector @xmath98 . similarly , the single yellow - red thick line going from bottom right to top left is the resonance @xmath492 , whose corresponding ( also basic ) wavevector is @xmath99 . we also clearly distinguish two resonances of order @xmath493 , namely @xmath494 ( blue ) , and @xmath495 ( green ) . many other higher order resonances cross the central doubly - resonant point @xmath479 , denoted hereafter by o , but they are not so visible in the scale of fig.[fgdbleres]a . the resonant strips of all previous resonances join each other forming a domain of double resonance around o. the extent of this domain can be determined roughly by drawing concentric circles around the point o. such circles correspond to nearly constant normal form energy values , as can be seen by noting that , for the particular hamiltonian function ( [ hamfr ] ) , the coefficients @xmath496 of eq.([hamresn12 ] ) have the values @xmath497 @xmath498 , and @xmath499 . applying eqs.([z02],[ene0],[jr0 ] ) for the particular resonant wavevectors given by ( [ resvectors ] ) , the doubly - resonant normal form of the hamiltonian ( [ hamfr ] ) expressed in resonant variables takes the form @xmath500 where the @xmath501 terms are trigonometric polynomials of the resonant angles @xmath502 , @xmath503 , while @xmath504 is a constant which appears only in the numerical values of the quantity @xmath505 called , hereafter , the normal form energy ( @xmath41 differs from the quantity @xmath230 defined in eq.([eneprime ] ) only by the constant @xmath506 ) . we note that the estimate @xmath501 for the trigonometric terms in @xmath26 follows from the estimate ( [ gn12 ] ) for the size of the corresponding fourier coefficients , taking into account that @xmath507 , according to eq.([kprime ] ) . since the angle @xmath222 is ignorable in the hamiltonian ( [ ene02fr ] ) , @xmath223 is an integral of the flow of @xmath26 . furthermore , since for the particular choice of hamiltonian model ( [ hamfr ] ) the action @xmath416 is dummy , implying that @xmath416 can be assigned any arbitrary value without affecting the dynamical evolution of any other canonical variable , we can always choose the value of @xmath416 so that @xmath508 . then , the normal form energy condition @xmath509 implies @xmath510 where @xmath43 is a @xmath511 quantity . transforming to the original non - scaled action variables we also find @xmath512 whereby @xmath513 is interpreted as the radius of a circle , around o , corresponding to a constant normal form energy condition . it follows that the set of all possible normal form energy values are represented on the @xmath381 plane as a set of concentric circles around o. three such circles are drawn in fig.[fgdbleres]a , corresponding to @xmath402 and @xmath514 ( outer circle ) , @xmath515 ( middle circle ) , and @xmath516 ( inner circle ) . their main difference concerns the degree of resonance overlapping in each case . namely , for a value of @xmath517 , corresponding to the outer circle @xmath514 , the main visible resonances of fig.([fgdbleres]a ) intersect the circle in some arcs only , while the remaining parts of the circle lie in the regular ( non - resonant ) domain . in the latter parts , the normal form dynamics alone would imply the existence of a set of kolmogorov - arnold - moser invariant tori of large measure . on the contrary , in the inner circle , corresponding to @xmath518 , all resonances essentially overlap , producing a strongly chaotic domain . the middle circle corresponds to @xmath519 , which is close to the critical energy below which resonance overlapping dominates the dynamics . the remaining panels of fig.[fgdbleres ] show what happens when @xmath0 is increased by a factor 5 ( @xmath466 , fig . [ fgdbleres]b ) , or 15 ( @xmath485 , fig . [ fgdbleres]c ) with respect to fig.[fgdbleres]a . a main feature to notice is that , by increasing @xmath0 , many more resonances ` show up ' in the fli map . furthermore , the size of all resonant domains grows proportionally to @xmath114 , as verified in fig.[fgdbleres ] , where by augmenting the scale in panels ( b ) and ( c ) by a factor @xmath520 and @xmath521 respectively with respect to panel ( a ) , the widths of all resonant strips passing through @xmath522 remain essentially unaltered in all three panels . thus , the only essential change is the increase of chaos as @xmath0 increases . namely , we see that the chaotic layers delimiting the borders of each resonance become thicker as @xmath0 increases . this also increases the resonance overlapping locally , close to the points of resonance crossings . defined as in eqs.([restra ] ) for the hamiltonian ( [ hamfr ] ) , for the same surface of section and using the same color scale as in fig.([fgdbleres ] ) . the circle in ( a ) corresponds to the constant normal form energy value @xmath523 in ( a ) and @xmath524 in ( b ) . the phase portraits of the normal form dynamics for the values ( c ) @xmath523 and ( d ) @xmath525 . the plotted surfaces of sections are @xmath270 whenever the quantity @xmath526 ( where all symbols denote the new canonical coordinates and momenta after the optimal lie normalization ) crosses a multiple value of @xmath527 . the main resonances are identified as : @xmath491 ( vertical ) , @xmath492 ( horizontal ) , @xmath494 ( bottom left to top right diagonal ) , @xmath495 ( top left to bottom right diagonal ) . ] focusing , now , on one value @xmath528 , figure [ fgresportrait ] shows in detail the implications of normal form dynamics in the two regimes when there is no resonance overlap ( @xmath529 , figs . [ fgresportrait]a , c ) , or when there is substantial resonance overlap ( @xmath524 , figs.[fgresportrait]b , d ) . the upper panels correspond to fli maps as in fig.([fgdbleres ] ) . here , however , instead of the action variables ( @xmath530 ) we use the resonant re - scaled actions @xmath237 , defined as in eq.([restra ] ) , where , for each point in the action space of the original variables we compute the values of the transformed actions @xmath531 , @xmath79 by the composition of the lie canonical transformations resulting from the computer - algebraic program calculating the optimal normal form @xmath26 . since the same program renders also the algebraic form of @xmath26 , we use this expression to derive the hamiltonian equations of motion of the normal form alone , namely @xmath532 , @xmath533 , @xmath534 , @xmath535 , @xmath536 , while we set @xmath537 . for each value of @xmath41 , we then compute _ numerical orbits under the normal form dynamics _ alone via the previous equations . finally we plot a convenient surface of section of the normal form flow , taken by the condition @xmath538 . these sections are shown in figs.[fgresportrait]c , d , for the normal form energy values @xmath529 and @xmath539 respectively . the corresponding circles , through eq.([eres ] ) , are shown in panels ( a ) and ( b ) , superposed to the color background yielding the fli map in the resonant action variables for @xmath528 . the main feature of this plot is the exact correspondence between the values of @xmath540 where each resonance intersects the circle corresponding to @xmath509 in panels ( a ) and ( c ) , and the projection of these values to thin chaotic layers delimiting the same resonance in the corresponding surface of section . in fact , inside each resonance we have regular orbits corresponding to islands of stability on the surface of section . furthermore , while at the normal form energy value @xmath529 there are many rotational kam tori separating these resonances , at the value @xmath524 these tori are destroyed and substantial resonance overlap takes place . this fact leads to the creation of a connected chaotic domain surrounding all main resonances in the surface of section of fig.[fgresportrait]d . this , in turn , implies that under the normal form dynamics alone no communication is allowed from one resonance to the other for the normal form energy value @xmath529 ( which in this approximation remains constant in time ) , while such communication is possible throughout the whole connected chaotic domain for @xmath524 . in fact , the phase portrait of fig.[fgresportrait]d renders visually clear that chaos is rather strong in this case . however , as emphasized in section 2 , this fact has no consequences regarding the possibility of long excursions in the action space , since all motions in this approximation would be bounded on circles like those of figs.[fgresportrait]a , b . on the contrary , such excursions are only possible due to the effect of the remainder , which causes the chaotic orbits to slowly ` drift ' from circle to circle as the value of @xmath41 changes slowly in time . to this we now turn our attention . the main effect , of local diffusion within the doubly - resonant domain , can now be demonstrated with the help of figure [ fgarnolddif ] . the time evolution of one chaotic orbit is shown in this figure , as the orbit moves within the doubly - resonant domain along some of the main intersecting resonances . in this example as well we take @xmath528 ( as in fig.[fgresportrait ] ) . ) for @xmath528 . after computing the optimal normal form , we find , via the lie canonical transformations , the values of all transformed variables @xmath541 and @xmath542 corresponding to particular values of the old variables @xmath543 and @xmath544 stored at many different times @xmath16 within an interval @xmath545 along the numerical run . using the numerical values of the computed transformed variables , ( a ) shows the variation of the normal form energy @xmath546 as a function of @xmath16 in the intervals @xmath547 ( blue ) , @xmath548 ( red ) , and @xmath549 ( green ) . the initial and final values are equal to @xmath550 , while the minimum value , occurring around @xmath551 is @xmath524 . ( b ) the evolution of the orbit in the action space @xmath237 , using the same colors as in ( a ) for the corresponding time intervals . in the first time interval ( blue ) , the orbit wanders in the thin chaotic layer of the resonance @xmath494 . in the second time interval ( red ) it jumps first to the resonance @xmath495 , and then to the resonance @xmath491 . in the third time interval ( green ) the orbit recedes from the doubly - resonant domain along the resonance @xmath491 . ( c ) 3d plot in the variables @xmath552 , visualizing arnold diffusion for the same orbit . taking 20 equidistant values of @xmath553 , @xmath554 in the interval @xmath555 , we first find the times @xmath556 in the interval @xmath557 when the normal form energy value @xmath546 of the numerical orbit approaches closest to the values @xmath553 . for each @xmath558 , starting with the momentary values of all resonant variables at @xmath556 , we then compute 1000 poincar consequents of the normal form flow on the same section as in figs.[fgresportrait]c , d . the same procedure is repeated in a second interval @xmath559 . as a net result , the orbit at the beginning and end of the calculation is found on the same section ( corresponding to @xmath529 ) , but in a different resonant layer , having by - passed the barriers ( invariant tori of the normal form dynamics ) via a third dimension ( here parameterized by the time - varying value of @xmath41 ) . ] in fig.[fgarnolddif ] , the evolution of the orbit is shown for a total time @xmath560 . the optimal normal form for @xmath528 has also been computed , whose optimal normalization order is @xmath561 , corresponding to an optimal fourier order @xmath562 . since the corresponding lie generating functions are known , we compute , via the composition of lie canonical transformations , the values of all transformed variables @xmath541 and @xmath542 corresponding to particular values of the old variables @xmath543 and @xmath563 stored at many different times during the numerical run , i.e. as @xmath16 varies within the interval @xmath545 . finally , since the exact algebraic expression for the normal form @xmath26 is known , we compute the precise numerical value of the normal form energy @xmath546 at the same times . fig.[fgarnolddif]a shows the variation of the normal form energy @xmath546 as a function of the time @xmath16 in the intervals @xmath564 ( blue ) , @xmath548 ( red ) , and @xmath549 ( green ) . the final time is such that the initial and final values of @xmath41 are equal , namely @xmath550 . on the other hand , as @xmath41 slowly changes during the run , it acquires a minimum value around @xmath551 , which is @xmath565 . such evolution corresponds to the process described schematically in fig.[fgdbleresmodel ] ( section 2 ) . namely , from the previous figure ( fig.[fgresportrait ] ) we conclude that the two extreme values of @xmath41 acquired during the numerical run are such that @xmath566 while @xmath567 , where @xmath568 is the critical energy corresponding to a large scale overlapping of resonances ( subsection 2.3.1 ) . furthermore , as we will see in the next subsection , the chaotic excursions of the orbits , and , consequently , time evolution of @xmath41 , can be approximated by a normal diffusion process . furthermore , the fastest evolution takes place in the intervals @xmath569 , and @xmath570 , in both of which the total variation of @xmath41 is of the order of @xmath571 , or a ` per step ' variation of the order of @xmath572 . it should be stressed that these extremely small variations are possible to unravel numerically only because we use the new canonical variables deduced by the normalizing sequence of lie canonical transformations . when the old variables are used , instead , we find that the there are large variations ( of order @xmath114 ) of all quantities depending on the actions . these variations are , in fact , dominated by the so - called ( in the nekhoroshev theory ) ` deformation ' effects ( which are also of order @xmath114 ) , hence completely covering the drift effects which are much smaller in size . this feature of the optimal canonical transformations will be exploited in the measurement of the diffusion coefficient @xmath1 as described in the next subsection . fig.[fgarnolddif]b shows the diffusion of the orbit in the action space @xmath237 , using the same colors as in fig.[fgarnolddif]a for the corresponding time intervals ( the background produced by the fli map is shown here in gray scale ) . in the first time interval ( blue ) , the orbit wanders chaotically within the thin chaotic layer of the resonance @xmath494 . it should be stressed that this wandering has a random walk character , i.e. the orbit makes several reversals of its drift direction , sometimes approaching and other times receding from the center of the double - resonance . on average , however , the drift is in the inward direction ( this is a statistical effect ; for other initial conditions the average drift turns to be outwards ) . in the second time interval ( red ) , the orbit jumps first to the domain of the resonance @xmath495 . now , however , the chaotic motion takes place with a relatively high speed ( of order @xmath114 ) in the direction across resonances . as a result , the orbit fills nearly ergodically the whole connected chaotic domain surrounding the main overlapping resonances , while , at the end of this time interval , the orbit is closer to the resonance @xmath491 . finally , in the third time interval ( green ) the orbit recedes from the doubly - resonant domain ( this is also a statistical effect ) being trapped along the domain of the resonance @xmath491 . in this way , at the time @xmath560 , the orbit is found at about the same distance from the center as initially ( at @xmath573 ) , but on a different resonance . fig.[fgarnolddif]c , now , shows a 3d plot in the variables @xmath552 , visualizing the ` third dimension ' along which the arnold diffusion progresses for the same orbit . from this plot we can clearly see the effect of the remainder , which can be considered as a very slow modification of the normal form dynamics acting on a timescale of the order of @xmath574 periods . the normal form dynamics , on the other hand , describes well the motion over shorter timescales , of the order of @xmath575@xmath576 periods . in order to show the dynamical effects happening on both timescales , we adopt the following numerical procedure : taking 20 equidistant values of @xmath553 , @xmath554 in the interval @xmath555 , we first find the times @xmath556 within the interval @xmath557 ( where the motion is , in general , in the inward direction ) when the normal form energy value @xmath546 of the numerical orbit approaches the closest possible to the values @xmath553 . then , for each @xmath558 , we set the momentary values of all canonical variables of the numerical orbit at the time @xmath556 as initial conditions via which we compute the corresponding values of all the new resonant canonical variables following the composition of the corresponding lie canonical transformations . with these values as initial conditions , we compute 1000 poincar consequents of the normal form flow alone on the same surface of section as defined in figs.[fgresportrait]c , d . the same procedure is repeated in the second interval @xmath559 , where the motion is in general in the outward direction . the whole set of poincar consequents ( points @xmath270 gathered in this way are plotted in the 2d sections of the parallelepiped of fig.[fgarnolddif]c , along with the variations of the value of the normal form energy @xmath546 ( sampled more frequently ) which are shown in the third dimension . the details of the filling process of the various resonant chaotic layers located in the doubly - resonant domain are now clearly seen . in particular , we note that the chaotic orbit fills the whole separatrix layer of the initial resonance @xmath494 in a timescale much shorter than the one required for substantial drift in the @xmath41 direction . after a transient ` back and forth ' motion around @xmath577 , the orbit then moves slowly towards the value @xmath524 , where all important resonances overlap . in the intermediate time interval ( red ) , we clearly see the filling of the stochastic layers of both resonances @xmath495 and @xmath491 , while global transport is allowed by the normal form dynamics from one resonance to the other . as , however , the remainder effect causes a new motion of the orbit outwards ( i.e. towards higher values of @xmath41 ( green ) ) , the orbit is eventually captured at the resonance @xmath491 , and stays there until the end of the simulation at @xmath560 . it should be emphasized that the fact that the orbit moves in the outward direction at @xmath560 does not guarantee that there will be no further return inwards . in fact , we find that most orbits undergo several ` in - out ' cycles like the one described in fig.[fgarnolddif ] , before eventually abandoning the doubly - resonant domain . as an estimate , for @xmath528 we find that the number of cycles before a final exit from the doubly - resonant domain is of the order of 10 , while the total time required for this effect is of the order of @xmath578 to @xmath579 periods . furthermore , the probability of exit along one particular resonance decreases as the order of the resonance increases . this is expected , since the width of resonances scales with their order @xmath439 as @xmath580 , while the fast filling of the innermost chaotic domains where all the resonances overlap is nearly ergodic . finally , we point out that a visualization of the diffusion process like in fig.[fgarnolddif]c clearly suggests that the diffusion is driven by the intersections of the asymptotic manifolds of lower - dimensional objects ( like hyperbolic 2d tori ) all along the path in which the diffusion takes place . however , locating such tori , and studying their manifolds is a task that can not be accomplished by the use of the birkhoff normal form as above . on the other hand , the latter provides good initial conditions for a numerical search of such tori . this subject is proposed for future study . our final goal is to obtain numerical estimates of the value of the diffusion coefficient @xmath1 as well as its relation to the size @xmath2 of the optimal normal form remainder as @xmath0 is varied in the interval @xmath581 . to this end we implement the following numerical procedure : for any fixed value of @xmath0 , using the information from the fli maps , we first select 100 initial conditions corresponding to on a circle defined as in eq.([eres12 ] ) , where the radius is chosen equal to @xmath582 for such a choice of @xmath43 , the corresponding circle lies inside the resonance overlap domain , ensuring that the short time dynamics is dominated by the doubly - resonant normal form . however , in longer times all these orbits exhibit weakly chaotic diffusion . the complete set of initial conditions for one orbit on the circle @xmath583 are found by solving simultaneously for @xmath584 and @xmath585 the equation of the circle ( eq.([eres ] ) ) as well as an equation for the initial angle @xmath586 $ ] , where , for each initial condition , @xmath587 is chosen by visual inspection so as to correspond to an initial condition in the domain of each one of the main overlapping resonances . with @xmath430 , for ( a ) @xmath454 , ( b ) ) @xmath588 and ( c ) @xmath434 . the black points show the orbits consequents on the surface of section up to a time @xmath589 . all three orbits are diffusing outwards . the circle with radius @xmath590 is shown in pink . ] we then follow numerically these orbits for a time long enough so that the mean change of their radial distance from the center is large enough to allow for a reliable computation of the diffusion coefficient . let @xmath591 be the instantaneous value of the distance from the center for any such orbit . the quantity @xmath592 ^ 2 $ ] changes as an orbit slowly drifts from one circle to another . figure [ fgdifinout ] shows this effect for three orbits corresponding to the same initial angle @xmath587 but for three different values of @xmath0 , namely @xmath454 ( fig.[fgdifinout]a ) , @xmath588 ( fig.[fgdifinout]b ) , and @xmath434 ( fig.[fgdifinout]c ) . the orbits are shown by the black points on the section @xmath593 , @xmath480 , superposed as usually to the colored background of the fli map . the pink circles in each panel are the circles @xmath594 , where the orbits initial conditions lie . apart from an overall change of the size of the circle of initial conditions with @xmath0 , a simple visual comparison of the three panels suffices to conclude that they imply quite different diffusion rates of their depicted orbits . in all three panels , the orbits ( black points ) are shown up to a time @xmath589 , which is quite long compared to the time needed to fill the chaotic domain along the circle @xmath457 . however , when @xmath454 ( fig.[fgdifinout]a ) , the orbit s plot shows that the orbit exhibits no discernible transverse motion with respect to this circle , despite the fact that the orbit lies entirely within a rather strong chaotic domain ( yellow in the fli scale ) . on the other hand , when @xmath0 is raised to @xmath588 ( fig.[fgdifinout]b ) , the orbit is observed to create a small ring around its initial circle , implying that the diffusion is visible in this timescale . increasing @xmath0 still further ( @xmath434 , fig.[fgdifinout]c ) , causes now a rather fast diffusion , which leads to the orbit following clearly a preferential ` exit resonance ' , where the diffusion continues essentially as in the simple resonance case ( subsection 2.3.2 ) . and @xmath223 ( see text ) , using the new transformed canonical variables ( first and second panel ) , and @xmath595 and @xmath223 using the original canonical variables ( third and fourth panel ) , for two chaotic orbits of our chosen ensemble for @xmath528 ( upper and lower row).,title="fig : " ] and @xmath223 ( see text ) , using the new transformed canonical variables ( first and second panel ) , and @xmath595 and @xmath223 using the original canonical variables ( third and fourth panel ) , for two chaotic orbits of our chosen ensemble for @xmath528 ( upper and lower row).,title="fig : " ] a key remark , now , is the following : similarly to the case of the orbit of fig.[fgarnolddif ] , whose dynamical features were possible to unravel using the _ new , i.e. , transformed _ canonical variables after an optimal normalizing transformation , exploiting the same variables , instead of the original ones , allows to observe the random walk - like drift of one orbit in the action space _ in a much shorter integration time than by the use of the original variables . _ an example is given in fig.[fgdrift ] , for @xmath528 . we compute , via the optimal normalizing canonical transformation , a time sequence of the values of all the transformed canonical variables @xmath596 from the available sequences of values of the original variables @xmath597 along the numerical orbits . the four panels in each row show the time evolution , for one chaotic orbit on the circle @xmath430 , of the quantities i ) @xmath41 computed in the transformed canonical variables , ii ) @xmath598 computed in the transformed variables , iii ) @xmath595 computed in the original canonical variables , and iv ) @xmath223 computed in the original variables . we note immediately the gain by passing the data through the optimal normalizing transformation , namely the fact that this transformation absorbs all ` deformation ' effects , allowing to see the very slow drift due to the weakly chaotic diffusion in a timescale @xmath599 . in fact , the quantity @xmath41 can only be computed in the transformed canonical variables , in which , for both orbits , it undergoes variations of the order @xmath600 . in comparison , the analog of @xmath41 in the original variables , i.e. , @xmath595 , undergoes variations in the second digit , and the corresponding time evolution is dominated by @xmath28 oscillations , which completely hide the slow drift process in the radial direction with respect to the central doubly resonant point . the comparison is even more straightforward in the variables @xmath223 computed by the transformed and by the original action variables . in the former , we can clearly see the drift phenomenon for both orbits , which results in a slow change of the value of @xmath223 ( which is an approximate integral ) at the fifth digit . in contrast , this phenomenon is completely hidden when @xmath223 is computed in the original variables , since the corresponding plot is dominated by oscillations of at least one order of magnitude larger amplitude than the drift effect . ( see text ) , in our ensemble of numerical data for @xmath528 , when computed by use of the new transformed canonical variables ( left ) , or the original canonical variables ( right ) . the evolution is shown up to the time @xmath601 . ] in order , now , to measure the value of the diffusion coefficient , using the data from all 100 orbits , we define the mean square deviation : @xmath602 where @xmath603 , and @xmath604 stands for any of the four quantities shown in fig.[fgdrift ] . plotting @xmath605 against the time @xmath16 allows to estimate the diffusion coefficient . figure [ fgdif008 ] shows an example of this calculation , setting @xmath606 equal to @xmath223 in the transformed variables ( left panel ) , or the original variables ( right panel ) . we note again that it becomes possible to observe the diffusion in a timescale @xmath607 using the ensemble of data in the transformed variables , while this time is quite short to reveal any linear trend of @xmath608 with the time @xmath16 in the original variables . in fact , in the original variables it was possible to measure reliably the diffusion coefficient only after an integration time @xmath609 . furthermore , this time increases even more for smaller values of @xmath0 . ( upper set of points ) and @xmath610 ( lower set of points ) on the optimal normal form remainder @xmath611 , using numerical data from the integration of orbits ( see text ) . the points correspond to the values of @xmath0 ( from left to right ) 0.003 , 0.004 , 0.005 , 0.007,0.008 , 0.01 , 0.012 , 0.013 , 0.015 , 0.018 , 0.020 . the straight lines represent the power - law fits @xmath612 ( upper ) and @xmath613 ( lower ) . ] figure [ fgdifcfrem ] shows the final result . computing , as indicated above , the diffusion coefficients @xmath614 and @xmath610 in the transformed canonical variables , for eleven different values of @xmath0 as noted in the caption , we also use the data from fig.[fgremopt ] , whereby we obtain the optimal remainder value @xmath2 for the same values of @xmath0 ( from the minima of the curves of fig.[fgremopt ] ) . we then plot @xmath614 and @xmath610 against @xmath2 in a log - log scale . despite some scatter , the correlation of both independent estimates of the diffusion coefficient with @xmath611 can be described as a power - law . the power - law exponents found by best - fitting are @xmath615 for the data of @xmath614 and @xmath616 for the data of @xmath610 . in these best fittings we excluded the two points for @xmath455 and @xmath454 , since the value of the optimal remainder found by extrapolation is uncertain for these values of @xmath0 . however , we note that the corresponding points in fig.[fgdifcfrem ] are still very close to the fitting law found by the remaining data . the exponents found in fig.[fgremopt ] are not far from the theoretical estimate @xmath617 derived in section 2 ( eq.([difcfest ] ) ) . however , we have made various trials to determine @xmath426 via alternative definitions of the diffusion coefficient , and we always find estimations of @xmath426 somewhat larger than 2 . we thus conjecture that this difference from @xmath617 is a real effect ( not due to numerical uncertainties ) , which , however , requires a more detailed theory to interpret . on the other hand , the corresponding analysis for simple resonances ( subsection 2.3.2 ) as well as the numerical results of @xcite indicate that the steepening of the power law in simple resonances of order not smaller than @xmath121 is quite substantial , leading closer to @xmath33 . in the latter case , another independent example @xcite yields @xmath618 . the issue of how exactly to quantify the steepening of the power - law remains open . we examined in detail the phenomenon of weak chaotic diffusion in doubly or simply resonant domains of hamiltonian systems of three degrees of freedom satisfying the necessary conditions for the holding of the nekhoroshev theorem . the aim was to determine a quantitative relation between the diffusion coefficient @xmath1 and the size of the optimal remainder @xmath2 of a resonant normal form constructed according to the requirements of the analytical part of the nekhoroshev theorem . our main results are the following : \1 ) we propose an efficient algorithm for hamiltonian normalization , which is implemented as a computer algebraic program performing expansions up to a high order . we explain the practical aspects of this algorithm , and show how it can be used in order to compute i ) the optimal normalization order @xmath158 as a function of the small parameter @xmath0 , and ii ) an estimate of the size of the remainder @xmath2 at the order @xmath158 . the dependence of @xmath158 on @xmath0 is found to be an inverse power - law with an exponent in agreement with theory . \2 ) we construct estimates on the speed of diffusion in doubly resonant domains . to this end , we examine first the dynamics under the hamiltonian flow induced by the normal form alone ( i.e. neglecting the remainder ) . the role of the convexity conditions assumed for the original hamiltonian is analyzed in the context of the normal form dynamics . we then discuss the influence of the remainder on dynamics . estimates on the value of the diffusion coefficient @xmath1 are quantified by considering a ` random walk ' model for the slow drift of the value of the normal form energy due to the remainder . the final prediction is a power - law estimate @xmath619 with @xmath389 in doubly resonant domains . \3 ) we perform detailed numerical experiments aiming to test the above predictions , employing the same hamiltonian model as in @xcite as well as the ` fli map ' method . using the information from the computed normalizing canonical transformations , we propose a convenient set of variables in which the arnold diffusion in the doubly resonant domains is clearly visualized . furthermore , using ensembles of chaotic orbits , we make two independent numerical calculations of the diffusion coefficient @xmath1 for various values of @xmath0 . the relation between @xmath1 and @xmath2 found by the two calculations is @xmath620 and @xmath621 respectively . \4 ) finally , we make some theoretical estimates on the relation between @xmath1 and @xmath2 in simply resonant domains . in this case , we combine the basic theory developed in @xcite together with estimates given in @xcite regarding the dependence of the size of the separatrix splitting on the optimal normal form remainder in simply resonant domains . we are thus led to the prediction @xmath622 , where @xmath357 , or @xmath359 , holding for all simple resonances of order higher than @xmath121 , where @xmath121 is defined in eq.([kprime ] ) . the latter result interprets the results obtained in an earlier study @xcite by purely numerical means . + + * acknowledgements : * we thank two anonymous referees for a thorough revision of our manuscript , with many constructive suggestions , as well as prof . g. contopoulos for careful reading of the manuscript . c.e . acknowledges fruitful discussions with the group of c. froeschl , m. guzzo and e. lega . arnold , v.i . , 1963 : _ russ . math . surveys _ * 18 * , 9 . arnold , v.i . , 1964 : _ sov . dokl . _ * 6 * , 581 . benettin , g. , galgani , l. , and giorgilli , a. : 1985 , _ cel . _ * 37 * , 1 . benettin , g. , and gallavotti , g. : 1986 , * j. stat . phys . * * 44 * , 293 . benettin g. , fass f. , guzzo m. : 1998 , _ regular chaot . _ , * 3 * , 56 . benettin , g. : 1999 , in a. giorgilli ( ed . ) ` hamiltonian dynamics . theory and applications ' . notes math . _ * 1861 * , 1 . cachucho , f. , cincotta , p.m. , and ferraz - 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( eds ) _ discrete dynamical systems _ , gordon and breach science publishers , pp.91 - 106 . efthymiopoulos , c. , giorgilli , a. , and contopoulos , g : 2004 , _ j. phys . a math . * 37 * , 10831 . efthymiopoulos , c. : 2005 , _ cel . astron . _ * 92 * , 29 . efthymiopoulos , c. , and sandor , z. : 2005 , _ mon . not . r. astron . soc . _ * 364 * , 253 . efthymiopoulos , c. : 2008 , _ cel . astron . _ * 102 * , 49 . ferraz - mello , s. : 2007 , _ canonical perturbation theories . degenerate systems and resonance_. springer ( new york ) . froeschl , c. , guzzo , m. , and lega , e. : 2000 , _ science _ * 289 * ( 5487 ) , 2108 . froeschl , c. , guzzo , m. , and lega , e. : 2005 , _ cel . astron . _ * 92 * , 243 . gelfreich , v. , sim , c. , and vieiro , a. : 2013 , _ physica d _ * 243 * , 92 . giordano , c.m . , and cincotta , p.m. : 2004 , _ astron . astrophys . _ * 423 * , 745 . giorgilli a. , delshams a. , fontich e. , galgani l. , sim c. : 1989 , _ j. differ . _ , * 77 * , 167 . giorgilli a. , and locatelli , u. : 1997 , _ zamp _ , * 48 * , 220 . giorgilli a. , and skokos , c. : 1997 , _ astron . _ , * 317 * , 254 . giorgilli , a. : 1999 , in : c. simo ( ed . ) , _ hamiltonian systems with three or more degrees of freedom _ , kluwer , dordrecht . giorgilli , a : 2002 , _ notes on exponential stability of hamiltonian systems _ , in dynamical systems . part i : hamiltonian systems and celestial mechanics , pubblicazioni della classe di scienze , scuola normale superiore , pisa . giorgilli , a. , locatelli , u. , and sansoterra , m. : 2009 , _ cel . * 104 * , 159 . guzzo , m. , lega , e. , and froeschl , c. : 2002 , _ physica d _ * 163 * , 1 . guzzo , m. , lega , e. , and froeschl , c. : 2005 , _ dis . b _ * 5 * , 687 . guzzo , m. , lega , e. , and froeschl , c. : 2006 , _ nonlinearity _ * 19 * , 1049 . guzzo , m. , lega , e. , and froeschl , c. : 2011 , _ chaos _ * 21 * , 033101 . kaneko , k. , and konishi , t. : 1989 , _ phys . rev . a _ * 40 * , 6130 . laskar , j. : 1993 , _ physica _ * d67 * , 257 . lega , e. , guzzo , m. , and froeschl , c. : 2003 , _ physica d _ * 182 * , 179 . lega , e. , froeschl , c. , and guzzo , m. : 2007 , _ lect . notes phys . _ * 729 * , 29 . lega , e. , guzzo , m. , and froeschl , c. : 2009 , _ cel . astron . _ * 104 * , 191 . lega , e. , guzzo , m. , and froeschl , c. : 2010a , _ cel . * 107 * , 129 . lega , e. , guzzo , m. , and froeschl , c. : 2010b , _ cel . astron . _ * 107 * , 115 . lhotka , ch . , efthymiopoulos , c. , and dvorak , r. : 2008 , _ mon . not . r. astron . soc . _ * 384 * , 1165 . liechtenberg , a.j . , and lieberman , m.a . : 1992 , _ regular and chaotic dynamics _ , springer , berlin . lochak , p. : 1992 surv . _ * 47 * , 57 . mather , j. : 2004 , _ j. math . _ * 124 * , 5275 . morbidelli , a. : 2002 , _ modern celestial mechanics . aspects of solar system dynamics _ , taylor and francis , london . morbidelli , a. , and guzzo , m. : 1997 , _ cel . astron . _ * 65 * , 107 . morbidelli , a. , and giorgilli , a. : 1997 , _ physica d _ * 102 * , 195 . neishtadt , a.i . : 1984 , _ mech . _ * 48 * , 133 . nekhoroshev , n.n . : 1977 , _ russ . surv . _ * 32*(6 ) , 1 . pavlovic , r. , and guzzo , m. : 2008 , _ mon . not . r. astron . soc . _ * 384 * , 1575 . poincar , h : 1892 , _ mthodes nouvelles de la mcanique cleste _ , gautier - vilard , paris . pshel , j. : 1993 , _ math . z. _ * 213 * , 187 . rosenbluth , m. , sagdeev , r. , taylor , j. , and zaslavskii , m. : 1966 , _ nucl . fusion _ * 6 * , 217 . sim , c. , and valls , c. : 2001 , * nonlinearity * * 14 * , 1707 . tennyson , j. : 1982 , _ physica d _ * 5 * , 123 . skokos , c. , contopoulos , g. , and polymilis , c. : 1997 , _ cel . astr . _ * 65 * , 223 . wood , b.p . , lichtenberg , a.j . , and lieberman , m.a . : 1990,_phys . a _ * 42 * , 5885 . the quadratic form @xmath623 given by eq.([z02 ] ) can be written as : @xmath624 where @xmath625 is a @xmath626 matrix whose first and second line are given by @xmath627 and @xmath628 respectively . since the matrix @xmath66 is real symmetric , it can be writen in the form @xmath629 , where @xmath630 , with @xmath631 the eigenvalues of @xmath66 , while @xmath632 is an orthogonal matrix with columns equal to the normalized eigenvectors of @xmath66 . using the above expression for @xmath66 , eq.([z02matr ] ) resumes the form @xmath633 where @xmath634 is a @xmath626 matrix . writing @xmath623 as @xmath635 , and denoting by @xmath636 the elements of @xmath606 , the discriminant @xmath637 is given by : @xmath638\ ] ] since we have assumed ( subsection 2.1 ) that either all three eigenvalues @xmath639 have the same sign , or two of them have the same sign and one is zero , by eq.([z02anal ] ) we have that @xmath640 . that is , the quadratic form @xmath623 is positive definite .

a detailed numerical study is presented of the slow diffusion ( arnold diffusion ) taking place around resonance crossings in nearly integrable hamiltonian systems of three degrees of freedom in the so - called ` nekhoroshev regime ' . the aim is to construct estimates regarding the speed of diffusion based on the numerical values of a truncated form of the so - called remainder of a normalized hamiltonian function , and to compare them with the outcomes of direct numerical experiments using ensembles of orbits . in this comparison we examine , one by one , the main steps of the so - called analytic and geometric parts of the nekhoroshev theorem . thus : i ) we review and implement an algorithm @xcite for hamiltonian normalization in multiply resonant domains which is implemented as a computer program making calculations up to a high normalization order . ii ) we compute the dependence of the optimal normalization order on the small parameter @xmath0 in a specific model and compare the result with theoretical estimates on this dependence . iii ) we examine in detail the consequences of assuming simple convexity conditions for the unperturbed hamiltonian on the geometry of the resonances and on the phase space structure around resonance crossings . iv ) we discuss the dynamical mechanisms by which the remainder of the optimal hamiltonian normal form drives the diffusion process . through these steps , we are led to two main results : i ) we construct in our concrete example a convenient set of variables , proposed first by benettin and gallavotti @xcite , in which the phenomenon of arnold diffusion in doubly resonant domains can be clearly visualized . ii ) we determine , by numerical fitting of our data the dependence of the local diffusion coefficient @xmath1 on the size @xmath2 of the optimal remainder function , and we compare this with a heuristic argument based on the assumption of normal diffusion . we find a power law @xmath3 , where the constant @xmath4 has a small positive value depending also on the multiplicity of the resonance considered . hamiltonian systems ; arnold diffusion ; normal forms ; nekhoroshev theorem .

You are an expert at summarizing long articles. Proceed to summarize the following text: enormous effort has gone into understanding the nature of sub - mm selected galaxies since they were first discovered with the submillimetre common user bolometer array ( scuba , holland et al . 1999 ) just under a decade ago ( e.g. smail et al . 1997 ; barger et al . 1998 ; hughes et al . 1998 ) . a general understanding of the sub - mm population and its role in galaxy evolution is severely limited by the poor spatial resolution of current sub - mm telescopes and the faintness of these sources at other wavelengths ( e.g. pope et al . the sub - mm population appears to consist of massive objects ( borys et al . 2005 ; greve et al . 2005 ) at @xmath1 ( chapman et al . 2005 ; pope et al . 2005 ) . these galaxies are rare and have very high star formation rates ( sfrs , lilly et al . 1999 ; scott et al . 2002 ) , and hence may represent an early phase in the evolution of massive elliptical galaxies . an outstanding question remains : are the incredible infrared luminosities of sub - mm galaxies powered by huge bursts of star formation or agn activity ? to assess in detail what powers the intense ir luminosity of sub - mm galaxies it is important to obtain a complete multi - wavelength picture . the sample of sub - mm galaxies discussed in this paper is taken from the goods - n scuba super - map ( borys et al . 2003 ; pope et al . 2005 ) . this is a purely sub - mm selected sample and it is almost completely identified at other wavelengths thanks to the deep multi - wavelength data available in the goods - n field ( see pope et al . 2006 for more details ) . the addition of the mips photometry to the multi - wavelength dataset for sub - mm galaxies provides powerful constraints on the shape of the sed , the nature of the power source , and the source redshift . in the left panel of fig . 1 , we plot the @xmath2 flux ratio as a function of redshift as compared to several models . in general , this flux ratio as a function of redshift is lower than that of other ultra luminous infrared galaxies ( ulirgs ) of the same luminosity at low and high redshift . this suggests that sub - mm galaxies either have higher levels of extinction or cooler dust temperatures . this can be tested further by looking in detail at the mid - ir spectrum . using deep multi - wavelength follow - up observations of sub - mm galaxies , it is possible to study their seds to derive fundamental properties such as dust properties and sfrs . in the right panel of fig . 1 , we plot a composite rest - frame sed for the sub - mm sources with spectroscopic redshift as compared to several models . we find that sub - mm galaxies have cooler average dust temperatures than those of local ulirgs of the same luminosity ( see pope et al . 2006 for more details ) . this result is confirmed with the addition of deep @xmath3 m imaging of these sub - mm galaxies ( huynh et al . this may indicate that their far - ir emission is more extended than that of local ulirgs , in which the majority of the ir emission comes from within the central kpc ( charmandaris et al . the seds of sub - mm galaxies are also different from those of their high redshift neighbours selected at near - ir wavelengths ( e.g. bzk galaxies , daddi et al . 2005 ) , whose mid - ir to radio seds are more like those of local ulirgs . we may understand this difference in terms of the evolutionary scenario advocated by vega et al . ( 2005 ) , in which enshrouded star forming galaxies undergo four distinct phases characterized by different mid- and far - ir colours . under their scheme , the sub - mm galaxies , with cooler temperatures , may be an earlier phase in the star formation than bzk galaxies . many sub - mm galaxies show evidence of an agn , as determined either through x - ray observations ( alexander et al . 2005 ) , or with optical spectra ( swinbank et al . 2004 ; chapman et al . 2005 ) , however it is unclear if the agn is a significant contributor to the bolometric luminosity of the galaxy . mid - ir spectroscopy can help determine what powers sub - mm galaxies since these wavelengths contain a number of features which are sensitive to the nature of the energy source ( clavel et al . 2000 ; hudgins & allamandola 2004 ) . we obtained _ spitzer _ irs observations of a sample of the sub - mm galaxies and agn in goods - n ( go2 pi : r. chary ) . 2 shows the rest - frame raw irs spectra for 10 sub - mm sources from this sample , offset for clarity . all of these spectra show at least weak polycyclic aromatic hydrocarbon ( pah ) emission . we have obtained several new redshifts from these irs spectra ( pope et al . in preparation ) . 3 shows the mean irs spectrum for these sub - mm galaxies compared to those of several local galaxies . this composite spectrum was calculated by taking the mean of the individual spectra . note that using the median instead of the mean does not make a significant difference to the shape of the composite spectrum . we have excluded the top source in fig . 2 since its steeply rising mid - ir continuum is not seen in the other 9 spectra in our sample . this source is not typical of our sample and is likely to have a stronger agn component . in the composite spectrum , we see the 6.2 , 7.7 , 8.6 and 11.3@xmath4 m pah features much more clearly . while arp220 is often considered the typical local analog to high redshift sub - mm galaxies , it appears to have more silicate absorption and less 6.2@xmath4 m pah emission than the average sub - mm galaxy . interestingly , the mid - ir spectra of the sub - mm galaxies seems to resemble that of m82 with the addition of a shallow power - law component ( @xmath5@xmath6 ) . the total contribution of the power - law component to the total luminosity in this wavelength range is @xmath7 . m82 is a typical local starburst galaxy but it is at least 2 orders of magnitude less luminous in the ir than the average sub - mm galaxy . this composite suggests that the mid - ir emission in sub - mm galaxies is dominated by star formation activity and there is little contribution from an agn . the average 7.7@xmath4 m line to continuum ratio for this sample is @xmath83.5 . comparing this with figure 4 from genzel et al . ( 1998 ) , we find that this ratio for sub - mm galaxies is consistent with the ratios for sbs and ulirgs and not as low as those found for agn . our preliminary analysis suggests that sub - mm galaxies are powered primarily by sbs and not agn activity . we are currently working on fitting templates to quantify the level of agn contributing to the ir luminosity and investigating various emission and absorption line strengths as diagnostics for the star formation activity . combined with the other multi - wavelength data that are already tabulated on these sub - mm sources , we will create a complete picture of the energetics of sub - mm galaxies ( pope et al . in preparation ) . these data will help determine if sub - mm galaxies and agn populations are connected via an evolutionary sequence . thanks to the organizers of this conference for a very fruitful meeting . we also thank our collaborators involved in the goods - n irs project . this work was supported by the natural sciences and engineering research council of canada and the canadian space agency . this work is based in part on observations made with the _ spitzer space telescope _ , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . support for goods , part of the _ spitzer space telescope _ legacy science program , was provided by nasa through contract number 1224666 issued by jpl , caltech , under nasa contract 1407 . alexander d.m . , et al . , 2005 , apj , 632 , 736 barger a.j . , et al . , 1998 , nat , 394 , 248 borys c. , et al . , 2003 , mnras , 344 , 385 borys c. , et al . , 2005 , apj , 635 , 853 chapman s.c . , et al . , 2005 , apj , 622 , 772 charmandaris , v. , et al . , 1999 , , 266 , 99 charmandaris v. , stacey g.j . , gull g. , 2002 , apj , 571 , 282 chary r. , elbaz d. , 2001 , apj , 556 , 562 clavel j. , et al . , 2000 , a&a , 357 , 839 daddi e. , et al . , 2005 , apj , 631 , l13 draine b.t . , 2003 , ara&a , 41 , 241 fischer , j. , et al . 1999 , , 266 , 91 frster schreiber , n. m. , et al . , 2003 , , 399 , 833 genzel , r. , et al . 1998 , , 498 , 579 greve t.r . , et al . , 2005 , mnras , 359 , 1165 holland w.s . , et al . , 1999 , mnras , 303 , 659 hudgins d. m. , allamandola l. j. , 2004 , aspc , 309 , 665 hughes d.h . , 1998 , nature , 394 , 241 huynh m. et al . , 2006 , apj submitted lilly s.j . , et al . , 1999 , apj , 518 , 641 pope a. , et al . , 2005 , mnras , 358 , 149 pope a. , et al . , 2006 , mnras , 370 , 1185 scott s.e . , et al . , 2002 , mnras , 331 , 817 smail i. , ivison r.j . , blain a.w . , 1997 , apj , 490 , l5 swinbank a.m. , et al . , 2004 , apj , 617 , 64 vega o. , et al . , 2005 , mnras , 364 , 1286

submillimetre ( sub - mm ) galaxies have very high infrared ( ir ) luminosities and are thousands of times more numerous at @xmath0 than local ultra - luminous ir galaxies . they therefore represent a key phase in galaxy evolution which can be missed in optical surveys . determining their contribution to the global star formation rate requires dissecting their ir emission into contributions from starbursts ( sb ) and active galactic nuclei ( agn ) . there are several examples of agn systems which masquerade as sbs in either the ir or x - ray , and sbs can often look like agn in some wavebands . a combination of sb and agn emission is not unreasonable , given models of merger - driven evolution . to assess in detail what powers the intense ir luminosity of sub - mm galaxies it is important to obtain a complete multi - wavelength picture . mid - ir spectroscopy is a particularly good probe of where the intense ir luminosity is coming from . we present the first results from a program to obtain _ spitzer _ irs spectroscopy of a sample of high redshift galaxies in the goods - n field , a large fraction of which are sub - mm galaxies . this field is already home to the deepest x - ray , optical , ir and radio data . we piece together the sub - mm data with the _ spitzer _ photometry and irs spectra to provide a well sampled ir spectral energy distribution ( sed ) of sub - mm galaxies and determine the contribution to the bolometric luminosity from the agn and sb components .

You are an expert at summarizing long articles. Proceed to summarize the following text: in recent years , great progress has been made in quantifying the evolution of the galaxy population at the end of cosmic reionization around @xmath15 . deep hubble space telescope ( hst ) legacy fields , such as the hubble ultra - deep field ( hudf ; * ? ? ? * ) or goods @xcite , and wide area ground - based imaging , have made it possible to study the evolution of the uv luminosity function ( lf ) across @xmath16 to great accuracy ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . over the last two years , with the installation of the wide field camera 3 ( wfc3 ) onboard the hst , the observational frontier of galaxies has now been pushed into the reionization epoch , as deep wfc3/ir data led to the identification of more than 100 galaxy candidates at @xmath17 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this is essential for estimating the contribution of galaxies to cosmic reionization . one of the most important conclusions from these studies is thus the realization that the uv luminosity density ( ld ) emitted by the galaxy population gradually falls towards higher redshifts . for example , the ld of the @xmath18 galaxy population is about an order of magnitude larger than that of the @xmath12 population , about 1.5 gyr earlier . how this evolves to even higher redshifts is still very unclear . a sizable galaxy population at @xmath19 is expected based on the first estimates of stellar population ages of @xmath20 galaxies , indicating that these sources very likely started forming stars already at @xmath21 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this is still somewhat uncertain due to possible nebular line emission contaminating the spitzer photometry ( e.g. * ? ? ? nonetheless , an early epoch of star - formation is also required by the mean redshift of reionization as measured by wmap ( @xmath22 ; * ? ? ? * ) , if galaxies are assumed to be the main drivers for this process . however , previous searches for @xmath19 sources in the pre - wfc3 era only resulted in very small samples of relatively low reliability candidates , none of which have been confirmed ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this is mainly due to the extreme faintness of the @xmath0 galaxy population . not only are these galaxies fainter due to their increased distance , but also they are expected at intrinsically lower luminosities . additionally , the detection of such high - redshift sources is further complicated by the fact that they are invisible in optical data . due to the highly neutral inter - galactic medium before the end of cosmic reionization , their uv photons are absorbed shortward of the redshifted ly@xmath23 line , which shifts to @xmath24 at @xmath25 . thus , these sources can only be seen in the nir , where previous detectors were significantly lagging behind optical technology . with 40@xmath26 higher efficiency relative to nicmos to detect high redshift galaxies in the nir , wfc3/ir has the potential to change this and to push galaxy studies to beyond @xmath27 . several deep and wide area wfc3/ir data sets have been taken already and several more are upcoming . the challenge for identifying genuine @xmath28 sources in these data sets , is that these galaxies will only be visible in one band ( @xmath4 ) . this has already led to some controversy in the first searches for @xmath0 sources in the first - epoch wfc3/ir data over the hubble ultra deep field ( see e.g. * ? ? ? * ; * ? ? ? in our recent analysis , which includes the full two - year wfc3/ir data over the hudf as well as the shallower , but wider early release science ( ers ) data , @xcite found only one single galaxy candidate detected at @xmath29 with an estimated redshift at @xmath30 . given that about three should have been detected , if the evolution of the lf continued as extrapolated from the trends established across @xmath31 to @xmath15 , this provided first tentative evidence for an accelerated evolution in the galaxy population from @xmath12 to @xmath0 . in this paper we significantly expand on our first @xmath0 analysis from wfc3/ir data presented in @xcite by extending the search to all the deep wfc3/ir fields in the chandra deep - field south area that have since become available . the inclusion of the two deep hudf09 parallel fields is especially useful , since both reach just @xmath32 mag shallower than the ultra - deep hudf field but triple the search area for @xmath33 ab mag sources . additionally , we use different analysis tools developed by the first author that provide an independent analysis of the hudf and ers data . while the lyman - break approach is similar in principle to that of @xcite the use of independently tested software and procedures for the source detection and its analysis provides confirmation and validation . the expanded data set also covers @xmath34 the area at moderately deep @xmath35 ab mag , thanks to the inclusion of the first epochs of candels data over these fields . this will be used to constrain the evolution of the galaxy population over the @xmath36 myr from @xmath0 when the universe was @xmath37 myr old to @xmath12 at @xmath38 myr . we start by describing the full data set in [ sec : data ] and present the @xmath39 candidate selection and its efficiency in [ sec : selection ] . in [ sec : lf ] we present our new constraints on the lf at @xmath0 . we will refer to the hst filters f435w , f606w , f775w , f850lp , f098 m , f105w , f125w , f160w as @xmath40 , @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath45 , @xmath7 , @xmath4 , respectively . throughout this paper , we adopt @xmath46 kms@xmath47mpc@xmath47 , i.e. @xmath48 . magnitudes are given in the ab system @xcite . and table [ tab : data ] ) . the hudf / hudf09 - 1/hudf09 - 2 fields comprise the deepest nir data available to date , reaching to @xmath49 ab mag . these three wfc3/ir pointings are covered in a total of 192 hst orbits , which obtained photometry in @xmath45 , @xmath7 , and @xmath4 . the ers field consists of 60 orbits of wfc3/ir imaging in @xmath44 , @xmath7 , and @xmath4 spread over a 2@xmath265 tile . the candels program is an on - going multi - cycle treasury survey consisting of two parts : candels - deep ( @xmath50 arcmin@xmath2 , limited at @xmath51 ) and candels - wide ( @xmath52 arcmin@xmath2 , limited at @xmath53 ) . only @xmath7 and @xmath4 data obtained before august 6 , 2011 is used in this analysis ( together with the shallow @xmath45 imaging of candels wide ) . ] lcccccccccc hudf09 & 4.7 & 29.2 & 29.6 & 29.4 & 28.8 & 29.1 & 29.3 & 29.4 + hudf09 - 1 & 4.7 & & 28.9 & 28.7 & 28.6 & 28.6 & 28.8 & 28.6 + hudf09 - 2 & 4.7 & 28.8 & 29.3 & 28.9 & 28.7 & 28.6 & 28.9 & 28.9 + ers & 41.3 & 27.8 & 28.0 & 27.5 & 27.2 & 27.4 & 27.8 & 27.6 + candels - deep & 63.1 & 27.8 & 28.0 & 27.5 & 27.2 & & 27.7 & 27.5 + candels - wide & 41.9 & 27.8 & 28.0 & 27.5 & 27.2 & 27.1 & 27.2 & 26.9 our analysis is based on the public wfc3/ir data sets that are available over the goods south fields @xcite as a result of three different programs ( hudf09 , ers , and candels ) , which we describe below . the outline of all fields is shown in figure [ fig : fields ] . the key feature of these fields is that they have wfc3/ir coverage in @xmath7 and @xmath4 , which will be used to select @xmath0 sources . additionally , they have deep optical coverage as well as deep irac imaging , which is essential to exclude low redshift contamination ( see section [ sec : selection ] ) . all the wfc3/ir data has been reduced following standard procedures outlined , e.g. , in @xcite . in particular , our reduction pipeline includes the subtraction of a super median image , careful image registration to the acs frames and an automatic elimination of pixels affected by persistance . the final pixel scale of the images in our analysis is set to @xmath54 . a summary of the hst data used in this analysis can be found in table [ tab : data ] . the final resolution of the wfc3/ir data is @xmath55 ( fwhm ) , and @xmath56 in the optical acs data . the acs data on the goods fields we used are the v2 reductions that are publicly available from mast ( m. giavalisco and the goods team , in preparation ) . the spitzer data are the goods irac images @xcite as made publicly available by the simple team ( see e.g. * ? ? ? additionally , we include the newly acquired spitzer irac [ 3.6 ] and [ 4.5 ] data over the hudf field from the iudf10 survey ( proposal 70145 , pi : labb ) . this so far adds @xmath57h of observations , which increases the depth in both filters by an additional @xmath10.4 mag . where available , we also matched our wfc3/ir sources with the publicly available goods - music multi - band photometry catalog of @xcite . the hudf09 program ( pi : illingworth ; * ? ? ? * ) consists of 192 hst orbits to provide ultra - deep wfc3/ir imaging over three pointings centered on the hudf @xcite and its two parallel fields from the udf05 program ( pi : stiavelli ; * ? ? ? the program has been completed , providing the deepest ir images ever taken . it comprises @xmath58 arcmin@xmath2 imaging in the three filters @xmath45 , @xmath7 , and @xmath4 , reaching down to @xmath59 , and 28.9 ( @xmath60 in 05 diameter apertures ) for the hudf , hudf09 - 1 , and hudf09 - 2 , respectively ( see fig . [ fig : fields ] ) . for a more detailed description of this data set and the data reduction see @xcite . in addition to the ultra - deep hudf09 data , we also analyzed shallower , wider area wfc3/ir imaging from the ers and candels programs , in order to constrain the volume density of more luminous star - forming galaxies at @xmath0 . the early release science data ( ers ) provide wfc3/ir imaging of @xmath61 arcmin@xmath2 of the northern part of the goods south field . two orbits of wfc3/ir imaging were obtained in each of the filters @xmath44 , @xmath7 , and @xmath4 , over a @xmath62 grid of pointings ( 60 orbits in total ) . these data are reduced in an analogous way to our hudf09 data , and are aligned and drizzled to the goods acs mosaics after rebinning to a @xmath54 pixelscale . these data reach to @xmath63 ( see also * ? ? ? for a more detailed description of this data set see @xcite . the last two fields included in our analysis are obtained as part of the multi - cycle treasury program candels ( pi : faber / ferguson ; * ? ? ? * ; * ? ? ? in particular , we include the first six visits of the candels - deep program ( obtained until august 6 , 2011 ) , which covers the central part of goods south in @xmath64 tiles with @xmath65 s exposures in both @xmath7 and @xmath4 in a total of 92 orbits . this data covers @xmath66 arcmin@xmath2 and reaches to @xmath51 mag . additionally , we also included the imaging data of the supernova follow - up program of candels ( pi : riess ) , which adds imaging over two pointings over candels deep ( one of which is essentially centered on the hudf ) . finally , we made use of the 29 orbits of wfc3/ir data of the candels - wide survey ( @xmath45 , @xmath7 and @xmath4 , obtained until march 29 , 2011 ) . these comprise 9 wfc3/ir pointings ( @xmath67 arcmin@xmath2 ) , completing the coverage of the goods south field , and reach to @xmath68 ( @xmath69 s exposures ) . as for the ers , the wfc3/ir data of the candels program has been aligned to the goods acs mosaics with a pixel scale of @xmath54 . the part of the candels field overlapping with the wfc3/ir hudf has been omitted when analyzing this data set in order not to duplicate the analysis of that area . we will subsequently refer to the combination of the ers and the two candels fields as ` wide fields ' . source catalogs are derived with the sextractor program @xcite , which is used to detect galaxies in the @xmath4 images and perform matched aperture photometry on psf - matched images . the colors used here are based on isophotal apertures derived from the @xmath4 images , and total magnitudes are measured in standard 2.5 kron apertures , corrected by 0.2 mag in order to account for flux loss in the psf wings . the detection significance of sources was established in @xmath70 radius apertures . the rms maps were scaled based on the detected variance in 1000 random apertures for each wfc3/ir frame on empty sky regions after @xmath71 clipping . this procedure ensures that the sextractor weight maps correctly reproduce the actual noise in the images . subsequently only sources with signal - to - noise ratios larger than 5 ( in @xmath70 radius apertures ) in @xmath4 are considered . galaxies ( [ sec : candsel ] ) . shown are the @xmath72 colors of different types of galaxies as a function of their redshift . star - forming galaxies are shown as solid blue lines . the lighter blue spectral energy distribution ( sed ) is reddened by e(b - v)@xmath73 mag using a @xcite dust law . as the ly@xmath23 absorption due to inter - galactic neutral hydrogen shifts into the @xmath7 band , galaxies start to exhibit progressively redder colors beyond @xmath12 . a selection with @xmath6 thus identifies galaxies at @xmath5 . strong balmer and 4000 breaks in evolved @xmath18 galaxies , combined with some dust obscuration can also result in very red @xmath72 colors . the dashed lines correspond to more evolved galaxies dominated by progressively older stellar populations from the library of @xcite as well as a 1 gyr old single stellar population ( ssp ) from the library of @xcite . spitzer irac data provides a way to separate these different populations at intermediate and very high redshifts ( see text and fig . [ fig : maglim ] ) . ] ) of some of the data used in this analysis . the red arrows indicate the limits of the acs , wfc3/ir and irac data over hudf , while blue arrows correspond to the ers data set . the sed of a @xmath74 star - forming galaxy is plotted in purple , showing the distinguishing feature of complete absorption shortward of the redshifted ly@xmath23 line . evolved and dusty galaxy seds can mimic the same features as a @xmath0 source , i.e. red @xmath72 and undetected in the optical data , unless this is extremely deep . such sources are typically detected in the irac data . alternatively , the solid and dashed green lines show possible seds of @xmath75 galaxies which could escape the irac detection at @xmath76 . these correspond to evolved galaxies ( 500 myr ) , with moderate amounts of dust extinction , and are normalized to the 5 @xmath3 detection limits of the @xmath4 bands . note that much larger amounts of dust can be hidden for interloper galaxies at the detection limit of the hudf ( green solid , @xmath77 mag ) than for the ers ( green dashed , @xmath78 mag ) . thus even with irac it can be a challenge to separate real @xmath79 sources from lower redshift contaminants . ] galaxies at @xmath80 are expected to exhibit very red @xmath72 colors since the redshifted , strong ly@xmath23 absorption ( by the predominantly neutral inter - galactic hydrogen ) cuts into the flux in the @xmath7 filter ( see figure [ fig : colsel ] ) . this makes such high redshift galaxies completely invisible blueward of @xmath7 , and we use this fact for their identification . as can be seen from figure [ fig : colsel ] , however , also passively evolving or dusty galaxies at intermediate redshifts ( @xmath81 ) can exhibit similarly red colors in @xmath72 . while the requirement of optical non - detections removes the bulk of lower redshift contamination , certain intermediate redshift galaxies with evolved or dusty stellar populations can still be included due to the fact that the optical data does not reach deep enough , if at similar depth as the ir ( see fig . [ fig : maglim ] ) . deep spitzer irac data provides a way to identify contaminating galaxies . these are expected to exhibit very red @xmath82 $ ] colors , which discriminates them from genuine @xmath0 candidates . to exclude possible low - redshift contamination , we thus use two steps to select @xmath5 galaxy candidates . for the first step , the primary criteria are based on hst data only : @xmath83 @xmath84 additional to excluding objects that are detected in any band blueward of @xmath7 at more than @xmath76 , we include a cut in the optical @xmath85 value of a galaxy ( see e.g. * ? ? ? * ; * ? ? ? * ) . this is computed from the @xmath70 radius aperture fluxes as @xmath86 , where the sum runs over all the bands available in the given data set blueward of @xmath7 , i.e. it includes all the available optical data as well as the nir band @xmath45 for the hudf09 data , and @xmath44 in the ers . the relatively large apertures were chosen in order sample @xmath87 of the light of point - like sources . the limiting @xmath88 are derived from photometric scatter simulations . they are set to exclude the majority of interlopers which remain undetected at @xmath76 purely due to photometric noise , but not to cut a substantial fraction of galaxies with real zero flux in the optical bands . the scatter simulations utilize all galaxies in our catalogs that are @xmath89 mag above the completeness limit , applying photometric gaussian noise from 1 mag fainter sources . from these simulations it is clear that contamination is mainly an issue at 0.75 mag above the completeness limits , but that @xmath90% of contaminants can be eliminated by using a @xmath85 limit of @xmath91 or 2.4 , for 5 filters or 4 filters , respectively . in the hudf09 data , the resulting number of expected contaminants due to photometric scatter is thus reduced from @xmath32 source per wfc3/ir field to @xmath92 source . on the other hand , the adopted @xmath88 limits do remove an additional @xmath93% of sources with real zero flux , simply due to gaussian statistics . this reduction of the real galaxy sample is reflected in our subsequent analysis in the reduction of the selection volume . lcccccccccc hudfj-39546284 & 03:32:39.54 & -27:46:28.4 & @xmath94 & @xmath95 & 6.3 all galaxies passing the above selection criteria , using both the acs and wfc3/ir data , are retained and analyzed individually . these total to 17 sources with @xmath96 in the range @xmath97 mag ; one source in the hudf , none in the parallel hudf09 fields , three in the ers and 8 and 5 in the candels deep and wide , respectively ( see tables [ tab : phot ] and [ tab : photcontamin ] ) . interestingly , only one source ( the previously reported galaxy with @xmath98 mag from bouwens et al . 2011 ) did pass our selection in the three deep hudf09 fields , while the shallower candels and ers fields contribute a total of 16 sources ( all with @xmath99 mag ) . upon inspection of their images , it turns out that all these brighter sources are very well detected in the irac data , even in the shallow 8.0@xmath100 band . their measured @xmath101 $ ] colors are in the range @xmath102 , which , for a @xmath0 source , would correspond to a uv continuum slope @xmath103 , or a dust reddening with @xmath104 mag . given that galaxies at the bright end of the @xmath105 population are measured to have very low extinction values and continuum slopes of @xmath106 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , the extremely red colors of these galaxies rules out @xmath19 solutions with any sensible sed . all sources with irac detections are thus removed from our sample of potential @xmath0 galaxies , reducing the sample to one single candidate in the hudf , previously reported in ( * ? ? ? * see figure [ fig : stampz10 ] and table [ tab : phot ] ) . the 16 removed sources are shown in the appendix in figure [ fig : stampscontamin ] and listed in table [ tab : photcontamin ] . images of the only possible @xmath0 galaxy candidate are shown in figure [ fig : stampz10 ] . the source is detected at 6.3 @xmath3 in @xmath4 ( measured in circular apertures of @xmath70 radius ) . this is higher but completely consistent with the @xcite significance estimates , which are based on smaller apertures . as can be seen , the source is not significantly detected in any other band . its value of @xmath107 is very close , but just below the limit of @xmath91 . this is mainly due to a 1.5@xmath3 flux excess in @xmath42 , which appears to be due to an extended structure in the background of that image . when adopting smaller apertures , the @xmath85 value is found to be reduced , indicating also that this excess of flux is not associated with the source itself . the spitzer irac 3.6@xmath108 m data shows some flux from a nearby source . however , after subtraction of all the neighboring sources in irac , the candidate is undetected ( @xmath109 ) , with a @xmath76 upper limit on its irac [ 3.6 ] magnitude of @xmath110 mag ab ( gonzalez et al . , in prep . ) . this thus corresponds to @xmath101<1.6 $ ] at @xmath76 , which is much bluer ( by @xmath111 mag ) than the typical low - redshift contaminants that we culled from our sample ( see also next section ) . after adding the newly acquired irac data from the iudf10 program ( spitzer proposal 70145 , pi : labbe ) to the goods irac data and removing neighboring sources , the @xmath0 candidate is also undetected at [ 4.5 ] ( @xmath112 ) providing added weight to the likelihood of it being at high rather than low reshift . we derive the photometric redshift of the candidate using the code zebra @xcite with synthetic stellar population models from @xcite to which we added nebular continuum and line emission following , e.g. , @xcite . using the full 11 band fluxes and flux errors , we derive a photometric redshift for this source of @xmath113 , with a likelihood of a low redshift solution at @xmath114 of @xmath115 . the full spectral energy distribution ( sed ) of the source and its redshift likelihood function are shown in figure [ fig : jdropsed ] . the best - fit sed corresponds to a very young , dust - free star - burst ( see also gonzalez et al . in prep . ) . the best low redshift solution is found at @xmath116 , for an evolved , very low mass galaxy sed ( @xmath117 @xmath118 ) with moderate extinction . interestingly , this sed is expected to be detected only at @xmath119 in @xmath7 , but it nevertheless has a significantly higher @xmath120 value ( 13.7 compared to @xmath121 ) . deeper hst data shortward of the break would be extremely useful in order to constrain the possible non - detection of the source shortward of 1.4 . as with other high - redshift catalogs , the added shorter - wavelength optical / near - ir data would play a key role in helping to further tighten its photometric redshift measurement . candidate , previously reported in @xcite . the measured photometry is shown with the red circle with errorbars and with 2@xmath3 upper limits in case of non - detections . all fluxes and flux errors were used in the sed fit , however , even if they were negative . the best fit sed is found at @xmath122 and is shown as a solid line ( @xmath123 ) , corresponding to a dust - free , young galaxy . the best template for a low redshift solution is also shown as the dashed green line . this sed has a redshift of @xmath116 , and is passive with an age of 650 myr and of rather low mass ( only @xmath124 @xmath118 ) . additionally , the low - z sed is reddened with @xmath125 mag . based on its larger @xmath120 value ( @xmath126 ) , this sed is formally excluded at @xmath127 probability , however . the redshift probability function is shown in the inset in the upper left . the sharp decrease above @xmath128 of @xmath129 is due to the use of a lf prior at @xmath130 and due to the fact that ly@xmath23 absorption starts affecting the @xmath4 band ( which would require the source to be brighter intrinsically ) . ] the properties of the 16 intermediate brightness sources that did not pass the irac non - detection criteria are discussed in the appendix . from fitting their seds , these sources are found to be mostly massive galaxies ( @xmath131 @xmath118 ) with obscured but evolved stellar populations at @xmath8 . interestingly , all these galaxies are essentially limited to @xmath9 mag . this is @xmath11 mag brighter than the detection limits of our bright survey fields ( see figure [ fig : contaminmz ] in the appendix ) , and thus suggests that such very red galaxies have a somewhat peaked luminosity function . this will have to be confirmed and quantified with future wide area data . however , at face value , it would appear that contamination from such sources with @xmath6 , and with @xmath101\gtrsim2 $ ] is less problematic at fainter magnitudes . we note , however , that contamination from similar galaxies with less extreme colors ( which may be more abundant also at @xmath132 mag ) may still be non - negligible due to photometric scatter ( see also [ sec : additionalcontamin ] ) . furthermore , it is interesting to note that it is very difficult to construct clean @xmath0 galaxy samples purely based on hst data alone . even if we were to increase the color criteria to @xmath133 , there would still be two contaminating galaxies in the sample together with our only viable @xmath0 candidate . thus , also for future constraints on the bright end of the @xmath0 lf , it will be important to perform additional follow - up studies to validate the candidates , e.g. with spitzer . this will be less of a concern for future @xmath27 galaxy samples , which can be obtained , e.g. , based on new f140w filter data . in such a data set intrinsically red galaxies can be sorted out by requiring a blue continuum across f140w and @xmath4 . here , we only give a brief summary of the possible sample contamination . for a thorough discussion we refer the reader to @xcite . essentially , the only probable chance for contamination is due to photometric scatter of a red lower redshift source . we estimate this to be a 10% chance based on our photometric scatter experiments described in section [ sec : candsel ] , including the @xmath85 cuts . other typical contaminants to lbg selections such as very cool dwarf stars and supernovae can essentially be excluded based on the relatively blue @xmath72 ( @xmath134 ) colors of stellar seds , on the fact that the source is detected in both the first and the second year of the hudf09 wfc3/ir data , and due to the fact that the candidate shows clear signs of an extended morphology . additionally , it is very unlikely that this source is spurious , since the flux distribution in circular apertures randomly distributed over empty regions of the hudf @xmath4 image are nearly exactly gaussian , and the source is well detected at @xmath135 . in this section we compute the expected abundance of @xmath0 galaxies in our dataset , and derive constraints on the @xmath0 lf based on our data . in order to estimate the number of sources we expect in our data from a given lf , we have to estimate the completeness as a function of magnitude @xmath136 and selection function as a function of redshift and magnitude @xmath137 . following @xcite , this is done by inserting artificial galaxies in the observational data and rerunning the source detection with the exact same setup as for the original catalogs . this is done for each of our fields individually . two sets of simulations were run . in the first set , we follow @xcite , where the artificial galaxies are ` cloned ' from the @xmath31 dropout sample of the goods and hudf fields . the images of these sources are adjusted for surface brightness dimming , the difference in angular diameter distance , as well as a size scaling of @xmath138 as observed for the lyman break galaxy population across @xmath139 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? these are then inserted in the observed images with galaxy colors as expected for star - forming galaxies between @xmath140 and @xmath141 . when computing the galaxy colors we assume a distribution of uv continuum slopes with @xmath142 ( see e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . from these simulations , we compute the completeness as a function of observed @xmath96 magnitude for each field , taking into account the scatter and offsets between input and output magnitudes . additionally , we compute the selection probabilities as a function of redshift and magnitude by measuring the fraction of sources that meet our selection criteria . by construction , galaxies are selected at @xmath143 at redshifts @xmath5 . in the second set of simulations , we repeat the above procedure . however , instead of using observed galaxy images , we use theoretical galaxy profiles from a mix of exponential ( sersic @xmath144 ) and de vaucouleur ( sersic @xmath145 ) profiles . the size distribution is chosen to be log - normal , again with the same size scaling as a function of redshift . the completeness and selection functions of our two procedures are in excellent agreement . this demonstrates the reliability of our approach , which appears to be essentially independent of the adopted galaxy profiles ( unlike what has been claimed elsewhere , e.g. * ? ? ? * but see also bouwens et al . 2010b ) . candidates in the different fields assuming the lf to be as measured at @xmath146 ( top ; * ? ? ? * ) or as expected from extrapolating the @xmath11 trends to @xmath0 ( bottom ) . the one detected source is in stark contrast to the @xmath147 sources expected if the lf was constant across the 170 myr from @xmath0 to @xmath12 . the lf appears to evolve at a significantly accelerated pace with respect to the empirical evolution observed across @xmath11 . 6 sources at @xmath0 should be detected when extrapolating the lower redshift trends , while only one probable candidate is found.,title="fig : " ] candidates in the different fields assuming the lf to be as measured at @xmath146 ( top ; * ? ? ? * ) or as expected from extrapolating the @xmath11 trends to @xmath0 ( bottom ) . the one detected source is in stark contrast to the @xmath147 sources expected if the lf was constant across the 170 myr from @xmath0 to @xmath12 . the lf appears to evolve at a significantly accelerated pace with respect to the empirical evolution observed across @xmath11 . 6 sources at @xmath0 should be detected when extrapolating the lower redshift trends , while only one probable candidate is found.,title="fig : " ] the expected number of sources in a given magnitude bin @xmath148 can be estimated for any given lf , @xmath149 , through : @xmath150)\ ] ] the result of this calculation is shown in figure [ fig : nexp ] for each individual field using the observed lfs at @xmath12 @xcite , as well as that expected at @xmath0 . the latter is based on extrapolating the evolution of the schechter function parameters presented in @xcite , which is based on fitting the likelihood contours of lbg lfs across @xmath11 . in particular , significant evolution is only seen in the characteristic luminosity @xmath151 , which is found to dim by 0.33 mag per unit redshift . the other two schechter function parameters were kept constant , in agreement with the small evolutionary trends that were found at low significance in @xcite . for reference , the final parameter evolution we use in this work is : @xmath152 @xmath153 @xmath154 as can be seen in figure [ fig : nexp ] , the wide - area , shallow data of candels and ers are very useful for constraining the evolution in the bright end of the lf . in particular , if there was no evolution in the lf from @xmath0 to @xmath12 , the wide area data should contain 8.5 @xmath0 sources . in total , we would expect to see @xmath147 sources if the lf was unchanged over the 170 myr from @xmath12 to @xmath0 . this implies that the wide area data provides essentially 30% of the total search power in case of an lf evolution in @xmath155 only . given that we only detect one probable candidate , the lf appears to drop faster than expected from the empirical lower redshift extrapolation . in particular , we do not detect the @xmath156 galaxies that we would have expected to find at @xmath0 if the lower redshift trends remained valid . from these extrapolations we predicted to find three sources in the hudf ( consistent with the expectations from * ? ? ? * ) and about one in each of the two hudf09 parallel fields . the poissonian probability to find @xmath157 source , given that 6 are expected is @xmath158 . this remains significant , even after including cosmic variance , which adds an additional uncertainty on these low number counts of about @xmath159 for an individual wfc3/ir pointing ( see e.g. , * ? ? ? * ; * ? ? ? we derive an upper limit of 6% to the probability of finding @xmath157 source in our search area , based on the cosmic variance calculator of @xcite and combining the number counts uncertainty in the different fields assuming the final distribution is gaussian ( justified by the central limit theorem ) . therefore , the detection of an accelerated evolution relative to the low - redshift extrapolation is significant at @xmath160 . the accelerated evolution in the uv lf can also be seen from our constraints on the @xmath0 lf . the stepwise lf is computed using an approximation of the effective selection volume as a function of observed magnitude @xmath161 . the lf in bins of absolute magnitude is then given by @xmath162 . this is shown in figure [ fig : lfevol ] , where the lf was evaluated in bins of 0.5 mag , and non - detections correspond to @xmath163 upper limits including the effects of 50% cosmic variance per pointing . as can be seen in figure [ fig : lfevol ] , the current constraints on the @xmath0 lf are significant even at bright magnitudes where the wide area data are particularly valuable . these data sets reduce the upper limits by more than an order of magnitude relative to using only the hudf09 fields , therefore indicating that the @xmath0 lf at @xmath164 drops by a factor @xmath165 with respect to the observed lf at @xmath12 . however , from these shallow data sets , no constraints can be obtained on any accelerated evolution of the uv lf around @xmath166 . this only becomes apparent at @xmath167 ( corresponding to @xmath51 mag ) . such faint limits are only probed by the hudf09 data set . in particular , at @xmath168 , the upper limit on the lf is a factor @xmath169 below the expectation ( fig . [ fig : lfevol ] ) . thus , it is clear that data reaching to deeper than @xmath170 mag will be necessary to further constrain the drop in the lf in the future ( which is beyond the reach of the current mct programs ) . in order to quantify the change in the lf from @xmath0 to @xmath12 more robustly , we consider two possible scenarios . first , we assume the accelerated lf evolution occurs only in @xmath151 , at a constant rate since @xmath15 , and we fit the poissonian likelihood for the observed number of sources . this can be written as @xmath171 @xmath172 , where @xmath173 runs over all fields , and @xmath174 runs over the different magnitude bins , and @xmath175 is the poissonian probability . our extrapolation of the uv lf is a modification of the fitting formulae of @xcite . we thus use @xmath176 mpc@xmath177const . and @xmath178const . , and assume : @xmath179 . we then fit for @xmath180 , finding @xmath181 , which results in an estimate for @xmath182 mag . alternatively , we assume @xmath183const . ( as derived for @xmath12 from our empirical extrapolation ) , @xmath184 , and we fit only for an evolution in the normalization with redshift relative to the @xmath12 lf . this results in @xmath185 mpc@xmath186 , with best fit @xmath187 . thus , using this extrapolation , the normalization of the uv lf from @xmath0 to @xmath12 is expected to increase by a factor 12 . these results are summarized in table [ tab : summary ] . lf from our combined data set , evaluated in 0.5 mag bins . the upper limits correspond to @xmath163 poissonian limits including the additional uncertainty of 50% cosmic variance per pointing . it is clear that the luminosity function evolves strongly from @xmath12 to @xmath0 , as our upper limits are a factor @xmath12 - 5 below the measured @xmath12 lf of ( orange data ; * ? ? ? the expected @xmath0 lf as extrapolated from fits to lower redshift lbg lfs is shown as a dashed red line . using this lf , we would expect to detect six sources in the full data set ( @xmath188 in the hudf , and @xmath189 in each of the hudf09 parallels ; see figure [ fig : nexp ] ) . for comparison also the @xmath31 and @xmath15 lfs are plotted @xcite , showing the dramatic build - up of uv luminosity across @xmath190 gyr of cosmic time . the light gray vectors along the lower axis indicate the range of luminosities over which the different data sets dominate the @xmath0 lf constraints . ] above @xmath191 mag ( @xmath192 ) . the filled circle at @xmath193 is the luminosity density directly measured for our only @xmath0 galaxy candidate . the two connected dots at @xmath74 show the range of possible ld values , given the two simple , accelerated extrapolations of the uv luminosity function described in section [ sec : lfconstraints ] . the red line corresponds to the empirical lf evolution from @xcite . its extrapolation to @xmath130 is shown as dashed red line . the @xmath194 data at @xmath11 is taken from @xcite . as can be seen , @xmath194 increases by more than an order of magnitude in the 170 myr from @xmath0 to @xmath12 , indicating that the galaxy population at this luminosity range evolves by a factor @xmath195 more than expected from low redshift extrapolations . the predicted @xmath194 evolution of the semi - analytical model of @xcite is shown as dashed blue line , and the theoretical model prediction of @xcite is shown as blue solid line . these reproduce the expected luminosity density at @xmath0 remarkably well . ] ccccccc @xmath196 ( fixed ) & @xmath197 & @xmath198 ( fixed ) & @xmath199 & @xmath200 + @xmath201 & @xmath202 ( fixed ) & @xmath198 ( fixed ) & @xmath203 & @xmath204 + & @xmath205 & @xmath206 the quantity most easily comparable between observations and simulations is the observed luminosity density ( @xmath194 ) above a given limiting magnitude , which is shown in figure [ fig : ldevol ] . the ld from the @xmath0 candidate alone amounts to @xmath207 erg@xmath208s@xmath47hz@xmath47mpc@xmath186 . however , a more realistic estimate of the luminosity density can be obtained from the two possible extrapolations of the uv lf we derived in the previous section , which include the contribution from galaxies at @xmath209 mag that are currently undetected . assuming the lf evolution only occurs in the characteristic luminosity , we find for the @xmath0 luminosity density @xmath210 erg@xmath208s@xmath47hz@xmath47mpc@xmath186 , while assuming the evolution to be driven by a normalization of the schechter function only , we find @xmath211 erg@xmath208s@xmath47hz@xmath47mpc@xmath186 . these different estimates are also summarized in table [ tab : summary ] . given that the observed luminosity density at @xmath12 is @xmath212 erg@xmath208s@xmath47hz@xmath47mpc@xmath186 @xcite , the inferred increase in luminosity density in the 170 myr from @xmath0 to @xmath12 amounts to more than an order of magnitude . this is a factor @xmath195 higher than what would have been inferred from the empirical relation for the uv lf evolution , which predicts an increase by only a factor @xmath169 . note however , that such a rapid increase in luminosity density is actually predicted by many theoretical models . in figure [ fig : ldevol ] we also show the luminosity densities at @xmath213 mag as derived from the semi - analytical model ( sam ) of @xcite , and from the theoretical model of @xcite . although there is still some discrepancy on the exact shape of the @xmath214 uv lf between the @xcite sam and the observations , the integrated luminosity density and its evolution is remarkably well reproduced across the full redshift range @xmath11 ( see also discussion in * ? ? ? the semi - analytic model predicts a constant growth in @xmath194 with cosmic time at somewhat faster pace than observed , leading to some discrepancy between the observations and the model at @xmath105 and @xmath12 , where the observational data shows higher luminosity densities . however , the model reproduces our estimates of the @xmath0 ld remarkably well . similar conclusions are reached for the purely theoretical model of @xcite . in particular , this model predicts the uv ld to evolve at an accelerated rate at @xmath130 , being only @xmath215 erg@xmath208s@xmath47hz@xmath47mpc@xmath186 at @xmath0 , in excellent agreement with our observed estimates . since the model is only based on the evolution of the underlying dark matter mass function , this indicates that an accelerated evolution in the galaxy population can be explained even without the need for a change in the physical mechanisms of galaxy formation . for further theoretical model predictions , see also e.g. @xcite , or @xcite . it is also interesting to note that the star - formation rate densities ( see table [ tab : summary ] ) inferred from our data , are more than an order of magnitude too low to account for the stellar mass densities observed at @xmath105 in systems of similar brightness . with constant star - formation over @xmath216 myr from @xmath0 to @xmath105 the observed galaxy population would only produce a stellar mass density of @xmath217mpc@xmath186 , compared to @xmath218mpc@xmath186 as estimated by , e.g. , @xcite . if the inferred mass densities and sfrs are correct , this suggests that the majority of the stars found in @xmath105 galaxies down to @xmath219 mag have to be formed in systems below our detection limit at @xmath0 or are younger than 300 myr . in this paper , we have extended our search for @xmath0 galaxies to @xmath220 arcmin@xmath2 of public wfc3/ir data obtained around the goods south field ( see figure [ fig : fields ] ) . these data sets have been acquired through the three surveys hudf09 , ers , and candels , and reach to varying depths , from @xmath68 to @xmath221 . based on strict optical non - detection requirements and a color cut of @xmath6 , we search these fields for lyman break @xmath7-dropout galaxies , which are expected to lie at @xmath80 . a total of 17 sources satisfy these criteria . however , 16 out of these sources show strong irac detections which rule out their being at such very high redshifts . rather , these galaxies are found to have best - fit photometric redshifts in the range @xmath222 ( see section [ sec : contaminants ] and appendix ) . they remain undetected in the optical due to their evolved stellar populations with non - negligible dust obscuration . this shows how important spitzer irac data is for removing contaminating lower redshift galaxies . interestingly , these contaminants are essentially only detected with magnitudes in the range @xmath223 mag . this is @xmath190 mag brighter than the detection limits of our bright surveys , which suggests that such evolved and dusty galaxies follow a peaked luminosity function at these wavelengths . if confirmed by future wide - area wfc3/ir data sets , this would indicate that such extremely red systems are not as much of a problem for @xmath28 searches at fainter levels as has been expected to date . however , the existence of such galaxy populations will make it challenging to use large - area wfc3 surveys such as pure - parallel fields ( e.g. * ? ? ? * ) for constraining the bright end of the @xmath0 uv lf without additional follow - up observations to validate the candidates . even with our expanded search area , the only @xmath224 detected galaxy with a color limit of @xmath101<1.6 $ ] ( @xmath76 ) and thus the only possible @xmath0 galaxy candidate is the same source that we reported already in @xcite . in appendix [ sec : possiblez10 ] we additionally note one other lower s / n candidate that is suggestive of being at comparable redshift , but requires confirmation from further hst wfc3/ir data . interestingly , we would have expected to detect six @xmath0 galaxies in our data , if the uv lf evolved to @xmath0 as expected from lower redshift trends ( see [ sec : nexp ] ) . thus , the galaxy population appears to evolve at an accelerated rate beyond @xmath130 . we infer that the uv luminosity density increases by more than an order of magnitude in only 170 myr from @xmath0 to @xmath12 , and we are thus likely witnessing the first massive build - up of the galaxy population at these early epochs in the reionization era . the fact that theoretical models based on the evolution of dark matter halos do , in fact , predict such an accelerated increase in the ld , indicates that these rapid changes are mainly driven by an accelerated evolution of the underlying dark matter mass function , rather than due to a change in star - formation properties of these early galaxies ( see e.g. , * ? ? ? * ; * ? ? ? the accelerated evolution of the galaxy population also has interesting consequences for cosmic reionization by galaxies brighter than @xmath225 mag , the current detection limits . with such a sharp decrease in the luminosity density above @xmath12 , it is impossible for such bright galaxies alone to create a reionization history in agreement with the high optical depth measurement of wmap and the vast majority of the ionizing flux has to be created by fainter galaxies ( see also * ? ? ? * ) . unfortunately , our knowledge of galaxies at @xmath130 still remains very uncertain . however , wfc3/ir offers unique opportunities to make significant progress in expanding the number of galaxies at @xmath79 and addressing some of the key issues related to early galaxy formation and its impact on reionization , even before the advent of jwst . in figure [ fig : nrequired ] , we show the depth required to detect ten @xmath27 and ten @xmath0 galaxies for a given survey area . such future data sets will have to reach significantly fainter than @xmath226 mag to accomplish this goal , even for large surveyed areas of 50 wfc3/ir fields . this is deeper than currently planned wide - area wfc3/ir data ( including mct programs ) . in fact , to characterize the luminosity function at luminosities below l@xmath227 and to set better constraints for reionization , surveys to fainter than 29 ab mag are really needed . note that with only one wfc3/ir field , a magnitude of @xmath228 has to be reached to detect a significant @xmath0 population , which is fainter than is practical with hst . based on the current lf constraints , we find that imaging a few fields provides the best chance to improve on current @xmath0 constraints . furthermore , in order to further constrain the possible accelerated evolution of the uv lf , the @xmath27 regime offers the best opportunity . a sizable population of @xmath27 galaxies is expected to be seen already down to @xmath229 mag over multiple wfc3/ir fields , which can be achieved with the efficient f140w filter . thus , it is likely that already with wfc3/ir , we can soon push the frontier of statistical galaxy samples with wfc3/ir from @xmath12 another @xmath230 myr back out to @xmath79 . galaxies ( blue shaded region and dashed line ) and ten @xmath0 galaxies ( red shaded region and dashed line ) as a function of survey area . these regions show the expected number of sources calculated using our lf extrapolations to @xmath0 from section [ sec : lfconstraints ] , i.e. assuming only evolution in @xmath151 or only in @xmath155 to reproduce the detection of our one @xmath0 candidate . the dashed lines show the same , but assuming the standard lower - redshift extrapolation of the uv lf @xcite . in particular , at @xmath0 the estimated depths differ significantly . the range of @xmath231 values of these extrapolations is indicated with an errorbar in the lower left . as can be seen , surveys reaching significantly deeper than @xmath226 mag will be required to detect a significant population of @xmath130 galaxies . the vertical dashed lines indicate the area of one and 50 wfc3/ir fields , respectively . a survey with only one pointing would need to reach to @xmath228 mag in @xmath96 to significantly constrain the @xmath0 galaxy population , which is out of reach with hst wfc3/ir . therefore , multiple fields are favorable for searching for @xmath0 galaxies . additionally , comparably deep optical data ( @xmath232 mag ) and deep irac imaging would be required over such fields in order to robustly exclude low - redshift contaminants . the identification of @xmath27 galaxies would benefit from imaging in different filters ( e.g. , f140w ) than adopted in current deep wfc3/ir fields . ] facilities : . , s. v. w. , et al . 2006 , , 132 , 1729 , e. , & arnouts , s. 1996 , , 117 , 393 , r. , broadhurst , t. , & illingworth , g. 2003 , , 593 , 640 , r. j. , illingworth , g. d. , blakeslee , j. p. , broadhurst , t. j. , & franx , m. 2004 , , 611 , l1 , r. j. , illingworth , g. d. , franx , m. , & ford , h. 2007 , , 670 , 928 , r. j. , illingworth , g. d. , thompson , r. i. , & franx , m. 2005 , , 624 , l5 , r. j. , et al . 2009 , , 705 , 936 . 2010 , , 709 , l133 . 2010 , , 708 , l69 . 2011a , , 469 , 504 . 2011b , , 737 , 90 . 2011c , arxiv e - 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[ fig : stampscontamin ] ) . the best estimate for its color is @xmath101=1.6\pm0.5 $ ] , which is too red for a likely @xmath28 source . interestingly , six of these 16 sources are present in catalogs of potential passive high redshift galaxies of @xcite and @xcite . one of these is also detected in x - ray emission and classified as a so - called exo source hosting a potentially highly obscured agn , particularly if it is an evolved galaxy at intermediate redshift , as pointed out by @xcite . we have derived photometric redshifts and mass estimates for these galaxies by complementing our hst photometry with the irac fluxes from the goods music catalog @xcite or from the simple images . based on sed fits with @xcite models , these galaxies are confirmed to be evolved , reddened systems with stellar masses around @xmath233 @xmath118 at redshifts @xmath234 . note that with the exception of four sources these are all detected individually in mips 24@xmath100 data . . the surface densities are extremely low . only @xmath10.4 sources are expected per wfc3/ir pointing . the rapid disappearance of such sources beyond @xmath235 , almost one mag brighter than the detection limits of our shallow fields from ers and candels , suggests that such sources may have a peaked lf at the wavelengths of interest in these surveys and so could be less of a problem for contamination for fainter @xmath0 galaxy samples . ] in figure [ fig : contaminmz ] we additionally show the surface density of these evolved intermediate redshift sources . even at their peak surface density , they are only found at 0.06 sources per magnitude per arcmin@xmath2 . therefore , only @xmath236 sources are expected per wfc3/ir pointing , providing an explanation for why none of these bright sources are found in the three deep hudf09 fields . the dearth of such galaxies faintward of @xmath237 , at @xmath190 mag brighter than the completeness limit of our bright surveys , suggests that such extremely red sources become even less frequent ( i.e. , that their lf may have peaked at these magnitudes ) and so they could be less problematic for contamination of fainter @xmath19 dropout searches . this will have to be confirmed , however , with future wide - area wfc3/ir data , searching explicitly for this type of galaxies . note that our finding of a disappearance of passive intermediate redshift galaxies to fainter magnitudes is in agreement with @xcite , who found an absence of passive , red galaxies at @xmath238 at masses below @xmath239 @xmath118 . lcccccccccc + jd-2162843432 & 03:32:16.28 & -27:43:43.2 & @xmath240 & @xmath241 & 14.6 & 0.65 & 70081 & + jd-2188742241 & 03:32:18.87 & -27:42:24.1 & @xmath242 & @xmath243 & 12.2 & 0.29 & 70040 & + jd-2226644214 & 03:32:22.66 & -27:44:21.4 & @xmath242 & @xmath244 & 10.9 & 0.26 & 70104 & + + jd-2487849357 & 03:32:48.78 & -27:49:35.7 & @xmath245 & @xmath246 & 11.4 & 0.33 & 70314 & + jd-2532547516 & 03:32:53.25 & -27:47:51.6 & @xmath247 & @xmath248 & 11.2 & 0.26&70236 & 4 + jd-2304648166 & 03:32:30.46 & -27:48:16.6 & @xmath249 & @xmath250 & 29.4 & 0.23 & 70258 & 1 + jd-2387448399 & 03:32:38.74 & -27:48:39.9 & @xmath251 & @xmath252 & 14.9 & 0.33 & 70273 & 1,2,4 + jd-2412344008 & 03:32:41.23 & -27:44:00.8 & @xmath253 & @xmath254 & 11.0 & 0.22 & 70092 & + jd-2158349541 & 03:32:15.83 & -27:49:54.1 & @xmath255 & @xmath256 & 19.1 & 0.38 & 70316 & 1 + jd-2249748085 & 03:32:24.97 & -27:48:08.5 & @xmath257 & @xmath258 & 7.5 & 0.33 & 70252 & + jd-2080646581 & 03:32:08.06 & -27:46:58.1 & @xmath259 & @xmath260 & 6.1 & 0.16 & & + + jd-2489152264 & 03:32:48.91 & -27:52:26.4 & @xmath245 & @xmath261 & 12.3 & 0.26 & 70442 & + jd-2358952367 & 03:32:35.89 & -27:52:36.7 & @xmath262 & @xmath263 & 7.2 & 0.35 & 70455 & 1 + jd-2351353198 & 03:32:35.13 & -27:53:19.8 & @xmath264 & @xmath258 & 5.8 & 0.35 & 70484 & + jd-2331152057 & 03:32:33.11 & -27:52:05.7 & @xmath265 & @xmath266 & 5.8 & 0.20 & 70429 & 1,3 + jd-2211356269 & 03:32:21.13 & -27:56:26.9 & @xmath255 & @xmath267 & 7.9 & 0.44 & & mag , but with strong irac detections . this rules out the possibility that these sources are at @xmath28 . the images are 7.5 arcsec on a side , and show from left to right ( 1 ) a stack of @xmath40 , @xmath41 , @xmath42 , @xmath43 , ( 2 ) @xmath7 , ( 3 ) @xmath4 , ( 4 ) spitzer irac [ 3.6 ] , and ( 5 ) irac [ 8.0 ] . all these sources are clearly detected in all irac bands , including [ 8.0 ] . thus , their @xmath268 $ ] colors are greater than @xmath269 , which is very different from the relatively flat color expected for a real @xmath0 source . note that irac flux of the source jd-2080646581 is heavily contaminated by neighboring sources . in the [ 3.6 ] band , we therefore show the cleaned image , after subtraction of the contaminating flux , which reveals its clear detection . ] as pointed out in section [ sec : discussion2 ] , due to the apparent paucity of such sources , it will be difficult to extend the @xmath0 candidate lists to large numbers of reliable sources in the near future with hst . in cycle 18 , an additional 128 orbit imaging survey was granted ( pi : ellis ) to further study the @xmath25 galaxy population over the hudf data . however , given that only one single field will be targeted and that only a relatively small amount of @xmath4 data is currently planned to be taken ( 22 new orbits , relative to the @xmath158 already available from the hudf09 and candels programs ) , we can forecast what this program will bring in terms of @xmath0 galaxy science . essentially , every 5@xmath3 source that will be obtained as a result of this new program should already be present as a 4.2@xmath3 source in the current @xmath4 band data . we therefore systematically searched for lower significance sources ( down to 4@xmath3 detections ) that satisfy our @xmath0 dropout criteria . essentially , the only additional candidate that we found is already detected at 4.8@xmath3 in the current data , and is located at ra@xmath27003:32:43.01 , dec@xmath270 - 27:46:53.3 ( see also * ? ? ? note , however , that the @xmath42-band data shows very faint positive flux ( @xmath119 ) exactly at the location of this source , which increases the chance that it is at lower redshift . this demonstrates the value of even deeper optical data over these fields for more robust high redshift galaxy selections . given the scarcity of low significance candidates in the hudf , it is clear , however , that it will be very important to image at least one additional field to similar depth in order to further constrain the accelerated evolution in the cosmic star - formation rate density ( see fig . [ fig : nrequired ] ) .

we search for @xmath0 galaxies over @xmath1160 arcmin@xmath2 of wfc3/ir data in the chandra deep field south , using the public hudf09 , ers , and candels surveys , that reach to 5@xmath3 depths ranging from 26.9 to 29.4 in @xmath4 ab mag . @xmath5 galaxy candidates are identified via @xmath6 colors and non - detections in any band blueward of @xmath7 . spitzer irac photometry is key for separating the genuine high - z candidates from intermediate redshift ( @xmath8 ) galaxies with evolved or heavily dust obscured stellar populations . after removing 16 sources of intermediate brightness ( @xmath9 mag ) with strong irac detections , we only find one plausible @xmath0 galaxy candidate in the whole data set , previously reported in bouwens et al . ( 2011 ) . the newer data cover a @xmath10 larger area and provide much stronger constraints on the evolution of the uv luminosity function ( lf ) . if the evolution of the @xmath11 lfs is extrapolated to @xmath0 , six @xmath0 galaxies are expected in our data . the detection of only one source suggests that the uv lf evolves at an accelerated rate before @xmath12 . the luminosity density is found to increase by more than an order of magnitude in only @xmath13 myr from @xmath0 to @xmath12 . this increase is @xmath14 larger than expected from the lower redshift extrapolation of the uv lf . we are thus likely witnessing the first rapid build - up of galaxies in the heart of cosmic reionization . future deep hst wfc3/ir data , reaching to well beyond 29 mag , can enable a more robust quantification of the accelerated evolution around @xmath0 .

You are an expert at summarizing long articles. Proceed to summarize the following text: field theories defined on a nc geometry are highly fashionable , in particular because they arise from a low energy limit of string theory @xcite . a nc space may be defined by the non commutativity of some of its coordinates @xmath2 = i\theta_{\mu\nu } \,\overset{\text{2d}}{=}\ , i \theta \epsilon_{\mu \nu}\,.\ ] ] for a review of nc field theories , see ref . the extension of actions of commutative field theories to their nc counterparts can be realized by replacing all products between fields by the _ star product _ @xmath3 in the nc @xmath0 model this replacement leads to the action @xmath4\,,\nonumber\ ] ] where only the interaction term requires the star product . in perturbation theory the one loop contribution to the 1 pi two point function splits into two parts coming from the planar and the non planar graphs . the planar terms are proportional to their ( uv divergent ) commutative counterparts @xcite . in the case of the non planar graphs the momentum cut off @xmath5 is replaced by an effective cut off @xmath6 @xcite , @xmath7 where @xmath8 denotes the incoming momentum . the cut off @xmath5 may be safely send to infinity , leading to a uv finite contribution . however , the uv divergences reappear as ir divergences in the limit @xmath9 . this mixing of uv and ir effects still causes serious problems in a perturbative treatment of nc field theories beyond one loop . we studied the mixing of divergences non perturbatively in the 3d model , with a commutative time direction . , see refs . @xcite . ] to avoid the ( cpu ) time consuming lattice version of the star product , we mapped the system on a dimensionally reduced model @xcite . here the scalar fields @xmath10 defined on a @xmath11 lattice are mapped on @xmath12 hermitian matrices @xmath13 . their action reads @xmath14&\hspace{-.5cm}=\vspace{-.1cm}n\textrm{tr}\sum_{t=1}^{t } \biggl [ \frac{1}{2}\sum_\mu\left(\gamma_\mu\hat{\phi}(t)\ , \gamma_\mu^\dagger\!-\hat{\phi}(t)\right)^2\\ & \hspace{-.5cm}+\frac{1}{2}\left(\hat{\phi}(t\!+\!1)-\hat{\phi}(t)\right)^2\!\!+ \!\frac{m^2}{2}\hat{\phi}^2(t)+\frac{\lambda}{4}\hat{\phi}^4(t ) \biggl]\nonumber\,,\end{aligned}\ ] ] where the twist eaters @xmath15 are defined by @xmath16 this implies @xmath17 , where @xmath18 is the lattice spacing . for this action we studied the phase diagram and the dispersion relation . based on the momentum dependent order parameter @xmath19 , with is the spatial fourier transform of @xmath10 . ] @xmath20 we studied the phase diagram in the @xmath21@xmath22 plane . our results for various values of @xmath23 are shown in fig . [ fig : phase ] . model . the connected symbols show the separation line between disordered and ordered regime , and the vertical lines mark the transition region between uniformly ordered and striped phase . _ _ ] we identify a clear separation line ( connected symbols ) between the disordered phase and the ordered regime . the ordered regime splits into a uniformly ordered phase and a striped phase , where the transition region is marked by two vertical lines for each value of @xmath24 . to illustrate the striped phase we present in fig . [ fig : snapshot ] snapshots of single configurations , which represent the ground state in this phase in the @xmath25@xmath26 plane at fixed time @xmath27 at a certain time @xmath27 , for @xmath28.__,title="fig : " ] at a certain time @xmath27 , for @xmath28.__,title="fig : " ] for @xmath29 at @xmath30 and @xmath31 . the dotted areas indicate @xmath32 and in the blank areas @xmath33 is negative . here we show configurations with two diagonal stripes resp . four stripes parallel to the @xmath25 axis . at smaller values of the coupling @xmath21 or smaller system size @xmath24 we also find two stripes parallel to one of the axis @xcite . these results agree qualitatively with the conjecture by gubser and sondhi , who predicted the occurrence of a striped phase @xcite . to complete the agreement the striped phase has to survive the continuum limit , where the number of stripes should diverge , such that the width of the stripes remains finite . the star product breaks explicitly the lorentz symmetry , which leads to a deformation of the standard dispersion relation . the one loop result for this relation reads @xcite @xmath34 where @xmath6 is defined in eq . ( [ eq : eff - cutoff ] ) . the deformation causes a shift in the energy minimum from @xmath35 to non zero momenta . we investigated numerically the energy momentum relation in the disordered phase . the energy @xmath36 can be computed from the correlator @xmath37 where the physical momenta are given by @xmath38 . this correlator behaves like a cosh @xmath39 and the study of its decay allows to extract the energy . in fig . [ fig : disp1 ] ( on top ) the system is close to the uniformly ordered phase transition . here the square of the energy is linear in @xmath40 as in a lorentz invariant theory . close to the striped phase ( fig . [ fig : disp1 ] below ) the situation is changed . we see a clear deviation from lorentz symmetry . the minimum of the energy is now at the lowest non zero ( lattice ) momentum and thus there will be two stripes parallel to the axes in the non uniform phase . in fig . [ fig : disp2 ] the results at very large coupling @xmath21 , far outside the phase diagram in fig . [ fig : phase ] , are shown . now the energy minimum is shifted to larger momenta , leading to the more complicated patterns in the striped phase as in fig . [ fig : snapshot ] . in figs . [ fig : disp1 ] and [ fig : disp2 ] the solid lines are fits to the one loop result for the energy in eq . ( [ eq : deformation ] ) . . close to the uniform phase at @xmath41 ( on top ) and close to the striped phase at @xmath42 ( below).__,title="fig : " ] + . close to the uniform phase at @xmath41 ( on top ) and close to the striped phase at @xmath42 ( below).__,title="fig : " ] we studied numerically the effects of uv / ir mixing in the 3d nc @xmath0 model . for the phase diagram we found that the ordered regime is split into an ising type phase for small coupling @xmath21 and a striped phase for larger coupling . the patterns in the striped phase become more complex when @xmath21 or the system size @xmath24 is increased . these results are in qualitative agreement with the conjecture of gubser and sondhi , if this type of stripes survives the large @xmath24 limit . the energy momentum relation behaves as predicted from one loop perturbation theory . this is a remarkable result , since due to the uv / ir mixing there could be strong effects from higher order calculations . our results imply that such effects do not change the results qualitatively . however , for final conclusions one has to perform the continuum limit @xcite .

the uv / ir mixing in the @xmath0 model on a non - commutative ( nc ) space leads to new predictions in perturbation theory , including hartree fock type approximations . among them there is a changed phase diagram and an unusual behavior of the correlation functions . in particular this mixing leads to a deformation of the dispersion relation . we present numerical results for these effects in @xmath1 with two nc coordinates .

You are an expert at summarizing long articles. Proceed to summarize the following text: in recent years simulations of babcock - leighton type flux - transport ( hereafter blft ) solar dynamos in both 2d and 3d @xcite demonstrated the crucial role the sun s global meridional circulation plays in determining solar cycle properties . time variations in speed and profile of meridional circulation have profound influence on solar cycle length and amplitude . the recent unusually long minimum between cycles 23 and 24 has been explained by implementing two plausible changes in meridional circulation , ( i ) by implementing the change from a two - celled profile in latitude in cycles 22 to a one - celled profile in cycle 23 @xcite , and ( ii ) by performing a vast number of simulations by introducing a flow - speed change with time during the declining phase of each cycle @xcite . accurately knowing the speed and profile variations of the meridional circulation would greatly improve prediction of solar cycle features . the meridional circulation has been observed in the photosphere and inside the upper convection zone in the latitude range from the equator to @xmath1 in each hemisphere @xcite . however , the speed , pattern and time variations of the circulation at high latitudes and in the deeper convection zone are not known from observations yet . theoretical models of meridional circulation @xcite provide some knowledge , but the flow patterns derived from model outputs vary from model to model , primarily because of our lack of knowledge of viscosity and density profiles and thermodynamics in the solar interior , which are essential ingredients in such models . as differential rotation does not change much with time compared to meridional circulation , in this first study we focus on time variation of meridional flow - speed , using a set - up similar to that used previously @xcite . since the meridional circulation is a specified parameter in kinematic blft dynamos and the dynamo solutions depend sensitively on the spatio - temporal patterns of this circulation , we ask the question : can we infer the meridional circulation ( in both space and time ) from observations of the magnetic field ? the purpose of this paper is to describe an ensemble kalman filter ( enkf ) data assimilation in a 2d blft solar dynamo model for reconstructing meridional flow - speed as a function of time for several solar cycles . a subsequent paper will investigate the reconstruction of spatio - temporal patterns of meridional circulation in the solar convection zone . data assimilation approaches have been in use for several decades in atmospheric and oceanic models , but such approaches have been implemented in solar and geodynamo models only recently . @xcite introduced a variational data assimilation system into an @xmath2-@xmath3 type solar dynamo model to reconstruct the @xmath2-effect using synthetic data . very recently @xcite applied a variational data assimilation method to estimate errors of the reconstructed system states in a stratified convection model . a detailed discussion of data assimilation in the context of the geodynamo can be found in @xcite . in a sequential data assimilation framework , a set of dynamical variables at a particular time defines a `` model state '' , which is the time - varying flow speed in the context of the present paper . scalar functions of these state variables that can also be observed using certain instruments are called `` observation variables '' , which are magnetic fields here . more detailed terminology for identifying data assimilation components with solar physics variables is given in 2 . in brief , the goal of sequential data assimilation is to constrain the model state at each time - step to obtain model - generated observation variables that are as close to the real observations as possible . the basic framework is based on statistical multidimensional regression analysis , a well - developed method that has been applied in atmospheric and oceanic studies ( see @xcite for details ) . the enkf sequential data assimilation framework also allows adding model parameters to the set of model states and estimating values of these parameters that are most consistent with the observations . it is a common practice to perform an `` observation system simulation experiment '' ( osse ) in order to validate and calibrate the assimilation framework for a particular model . an osse generates synthetic observations from the same numerical model that is used in the assimilation . in this case the numerical model is a simple blft dynamo model containing only a weak nonlinearity in the @xmath2-quenching term ; thus adding gaussian noise to model - outputs for producing synthetic observations works well . in a more realistic situation for a large system with highly nonlinear processes , such as in numerical weather prediction models , it may be necessary to use a non - gaussian ensemble filter ( see , e.g. @xcite ) . a few examples of predicting model parameters using sequential data assimilation techniques have been presented by @xcite and @xcite in the context of estimating neutral thermospheric composition , and most recently by @xcite for estimating thermospheric mass density . an enkf data assimilation framework has recently been applied to a 3d , convection - driven geodynamo model for full state estimation using surface poloidal magnetic fields as observations @xcite . we implement enkf sequential data assimilation to reconstruct time - variations in meridional flow - speed for several solar cycles , using poloidal and toroidal magnetic fields as observations . we note certain differences in our case compared to the cases described above , namely , unlike neutral thermospheric composition and thermospheric mass density , the meridional flow - speed is not governed by a deterministic equation . in order to describe the enkf data assimilation methodology , we first identify the data assimilation components with solar physics variables . a physical blft dynamo model , which generates magnetic field data for a given the meridional flow - speed , is called the `` forward operator '' . the time - varying meridional flow - speed at a given time is the `` model state '' and will be estimated by the enkf system . the enkf requires a prediction model that generates a forecast of the meridional flow - speed at a later time given the value at the current time . here , that prediction model simply adds a random draw from a gaussian distribution to the current meridional flow - speed , because the blft dynamo model we are using here is a kinematic dynamo model . in the future , the results can be further refined by imposing additional physical conditions using a dynamical dynamo model . it is important not to confuse the prediction model with the blft dynamo model that acts as a forward operator for the data assimilation process . figure 1 schematically depicts the data assimilation framework considering an `` ensemble of three members '' of the prediction model state ( meridional flow - speeds ) . sequential assimilation can usually be described as a two - step procedure , a forecast followed by an analysis each time an observation is available . in figure 1 , @xmath4 near the label ( a ) denote three different realizations of initial flow - speeds , which are input to the prediction model . we generate `` prior '' estimates of the model state ( denoted by @xmath5 near the label ( b ) ) by using the prediction model to advance the estimates of the flow - speed from the initial time ( @xmath6 ) to the time ( @xmath7 ) at which the first observations of magnetic field are available . in this case , the prediction model just adds a different draw from the gaussian noise distribution to each initial ensemble estimate of the flow speed ( @xmath8 along the solid green arrows in figure 1 ) . the resulting ensemble of flow speeds is referred to as a prior ensemble estimate . the central equation for estimating the prior state is a stochastic equation given by , @xmath9 in which , @xmath10 is a function for generating normalized gaussian random numbers with unit amplitude and unit standard deviation , and the amplitude of the prediction model noise is governed by @xmath11 . thus the evolution of the system from label ( a ) to ( b ) through equation ( 1 ) , denoted by eq1 in figure 1 , constitutes the first step of the two - step procedure in sequential assimilation . the second step includes ( i ) producing observations from outputs of the forward operator and ( ii ) estimating posterior flow - speeds by employing regression among these observations , real observations ( synthetic in osse ) and prior flow - speeds . the forward operator ( blft dynamo in this case ) is denoted by @xmath12 along the solid black arrows , which uses the prior estimates of flow speed to produce a prior `` ensemble of observation estimates '' which are magnetic field outputs . three realizations of magnetic fields from the forward operator are denoted by @xmath13 , @xmath14 , @xmath15 in figure 1 . note that the statement , `` forward operator operating on three model states generates three prior observation estimates '' , is equivalent to the statement , `` blft dynamo running with three meridional flow speeds produces three sets of model outputs of magnetic fields '' . to elucidate the second step of assimilation , we describe the function of the `` filter '' . for real prediction using enkf data assimilation , we would have real data ( observations ) from instruments ; in our osse case it is synthetic , denoted by @xmath16 in figure 1 . synthetic observations are generated by applying the forward operator to a specified time series of meridional flow - speeds ( as shown in figure 2a ) . our goal is to apply an enkf to obtain an improved distribution of estimated flow - speeds ( i.e. `` posterior states '' , @xmath17 ) using the prior ensemble and the observation of magnetic field . the enkf ( black - dotted box in the diagram ) first compares the prior ensemble of observation - estimates to the actual observation and computes increments to the prior observation - estimates . these observation - increments are then regressed using the joint prior - ensemble distribution of flow - speed and magnetic field observations to compute increments for the prior - ensemble of flow - speeds . the enkf can also produce a posterior - ensemble of magnetic field observations ( @xmath18 ) , which can be used for diagnostic purposes . the posterior - ensemble distribution of flow - speeds is the best estimate of the flow - speed distribution at time @xmath7 given the available observations . mean flow - speed at time @xmath7 can be calculated by taking the average over all ensemble members . to proceed with the reconstruction of flow - speeds at the time of the next observation , @xmath19 , the reconstructed flow - speeds at time @xmath7 are used as the input to the prediction model ( equation ( 1 ) ) , and the same procedure described in the previous paragraphs is repeated . random gaussian noise through equation ( 1 ) prevents degeneration of ensemble . thus after many time steps , the entire time series of the ensemble distribution of flow - speeds can be constructed . time series of the mean flow - speeds can be calculated by taking the average over all ensemble members at each time . however , it may produce a better reconstruction in some cases if one ensemble member , which produced observation ( @xmath20 ) closest to real observation ( @xmath16 ) , is chosen . in order to perform an osse , it is now necessary to define the `` true state '' flow - speed as a function of time . synthetic observations are generated at selected times by applying the forward operator ( blft dynamo model ) to the true state flow - speed at the appropriate time and adding on a random draw from a specified observational error distribution to simulate instrumental and other errors . as noted above , the forward operator is a kinematic blft dynamo model , described in detail in @xcite . the dynamo ingredients are a solar - like differential rotation , a single - celled meridional circulation , a babcock - leighton type surface @xmath2-effect and a depth - dependent diffusivity ; the mathematical forms are prescribed in @xcite . dynamo equations , computation domain and boundary conditions are used as given in @xcite . at @xmath21 we start the integration of the dynamo - dart system over the first assimilation - step by initializing the forward operator ( ftd model ) with a converged solution for a flow - speed of @xmath22 . in the subsequent assimilation - steps , the solution at the end of previous assimilation - step for each ensemble - member is used as initial condition . we construct a time - varying flow - speed for a span of 35 years ( see figure 2(a ) ) , guided by observations @xcite , which has a natural variation of 20 - 40% with respect to the mean flow considered here , i.e. 14 @xmath23 , as shown by the thin black line in figure 2(a ) . this specified time - varying flow - speed is referred to as the true model state . thus , keeping the spatial pattern of meridional circulation fixed , we consider here reconstructing the time - series of the scalar flow - speed . to generate synthetic magnetic field data , we incorporate the time - varying true meridional flow - speed in our blft dynamo , and simulate the time series of idealized magnetic fields in the entire computation domain . then we construct the time series of synthetic magnetic observations by adding synthetic observational error to the simulated idealized magnetic data . figure 2(b ) shows a single observation , which is created from the simulated poloidal field by extracting from the location , @xmath24 and @xmath25 . however , note that the dynamo simulation in @xmath26 grids can give us as many as 20202 synthetic magnetic observations of poloidal and toroidal fields . considering only one observation , as shown in figure 2(b ) , we perform assimilation runs with 16 ensemble members with an observational error of @xmath27 , which means an error of @xmath27 about the ideal magnetic field generated using the true meridional flow speed . to estimate the prior states of flow - speed , we use equation ( 1 ) , in which we set @xmath28 . if the meridional flow - speed varies up to @xmath29 ( i.e. @xmath30 for a mean flow - speed of @xmath31 ) during six months , the variation in 15 days can be @xmath32 . thus we chose @xmath28 so that it is large enough to capture the variation in flow - speed within our selected updating time - step of 15 days and also large enough to avoid ensemble collapse , but not so large as to produce unusual departures from cyclic behavior in a flux - transport dynamo . we show in figure 3 the reconstructed meridional flow - speed as a function of time ( panel ( a ) ) and the estimates of the observation computed from the flow speed ensemble after data assimilation ( panel ( b ) ) . we see in figure 3(a ) that the reconstruction is reasonably good except for the time windows between @xmath33 to 18 years and 33 to 35 years during which we find @xmath34 error in the reconstructed flow speed . observations indicate @xmath35 error in the measurement of meridional flow speed @xcite . the inset of figure 3(a ) reveals , for an initial guess , far - off from the true - state , the reconstructed states asymptotically converges towards the true - state . even though they oscillate around the truth with large amplitude , the oscillations damp with time . figure 3(b ) shows histograms for the normalized distribution of flow - speeds before ( in cyan ) and after ( in magenta ) the analysis stage , along with true - state ( blue ) , for the time instances of 5 , 6.9 , 10.1 and 27.5 years ( marked by vertical lines in figure 3(a ) ) during assimilation . for this case with 16 ensemble members we chose 10 bins for an optimal display . four time instances are chosen in such a way as to present the following four different phases of reconstructions : ( i ) the distribution of prior and posterior states has small overlap ( top left frame of figure 3(b ) ) , ( ii ) distribution is sharply peaked in one bin each for prior and posterior ( top right frame ) , ( iii ) distribution is broad and has significant overlap ( bottom left frame ) and ( iv ) distribution has no overlap at all ( bottom right frame ) . however , we can clearly see the successful performance of the enkf which reveals that in all cases the analysis phase brings the posterior distribution ( magenta bars ) closer to true - state ( blue ) . figure 3(c ) reveals that the assimilated magnetic observation ( blue solid curve ) is well - reproduced when compared with real observation ( red - dashed curve ) . this is not surprising , because it has already been noted that short - term small fluctuations in flow speed do not significantly influence the overall evolution of global magnetic fields generated by a babcock - leighton dynamo ( see , e.g. @xcite ) . we also plot the innovation ( @xmath36 ) for this observation ( black curve ) and the cumulative innovation ( orange curve ) . the innovation at the @xmath37 analysis - step ( @xmath36 ) is defined as the signed difference between real and reconstructed observations , whereas the cumulative innovation ( @xmath38 ) at @xmath37 analysis - step is the normalized sum of norm of innovation vectors over all the previous analysis - steps . @xmath36 and @xmath38 being small , both have been ten - fold magnified to superimpose on observations . we see that , at three different time instances ( @xmath39 , 17 and 27 years ) , the innovation is relatively large ; this is because at @xmath21 the initial guess is far - off from the truth , and at @xmath40 and 27 years there are relatively sharp changes in flow - speed . the cumulative innovation asymptotes to zero as expected , implying no bias of the system . to investigate the possibility of further improvement in the quality of the enkf reconstruction , we examine the consequences of three important aspects of the enkf : variation in observational error , size of ensemble and number of observations . we perform convergence tests by estimating the error in the reconstructed state as functions of observational errors , size of ensemble and number of observations . we define the error as the root mean square of the difference between the reconstructed state and the true - state . the assimilation interval is chosen to be 15 days in the present case ; denoting every 15 days by the index @xmath41 , the true state and the reconstructed state at the @xmath37 assimilation step by @xmath42 and @xmath43 respectively , we define the error as , @xmath44 , in which @xmath45 is the total number of indices for the 15-days assimilation intervals during the entire time - span of 35 years . figure 4 shows the rms errors in reconstructed flow speed as functions of observational error ( figure 4(a ) ) , size of ensemble ( figure 4(b ) ) and number of observations ( figure 4(c ) ) . we see in figure 4 that the error decreases systematically and asymptotes for certain values of the observational errors ( 1% ) , size of ensemble ( 192 members ) and number of observations ( 180 ) . in the case of more than one observation , we include more poloidal field observations ( synthetic ) from various locations at and near the surface , and more toroidal field observations at and near the bottom of the convection zone . while we vary observational errors in panel ( a ) , we use @xmath46 observational error in panels ( b ) and ( c ) . panels ( a , b ) show convergence to typical hyperbolic patterns , as is often seen in numerical convergence tests ( see , e.g. figure 3 of dikpati ( 2012 ) ) . but panel ( c ) shows that the reconstruction can be improved systematically only when there are more than a certain number of observations ( four in our case ) . in all these experiments , we used the same enkf scheme . bias or systematic error in an osse reconstruction may arise primarily because of the following assumptions made in the assimilation system : ( i ) the evolution of the ensemble spread is linear , ( ii ) the ensemble is sufficiently large , ( iii ) the forward operators are linear . though these assumptions are roughly valid , they are not strictly true . but in general , the resulting systematic errors are small unless the assimilation is applied in a large , highly nonlinear system like a numerical weather prediction model . with the knowledge gained from the convergence experiments shown in figure 4 , we consider a case with 192 ensemble members and 180 magnetic observations , consisting of equal numbers of poloidal and toroidal magnetic fields at various latitude and depth locations . each of these observations has an observational error of @xmath46 . we present the reconstructed flow - speed from this assimilation in figure 5 . the reconstructed flow - speed ( red curve ) matches very well with the true state , and thus the true state plotted in blue is essentially hidden behind red and green curves . it is not realistic to expect an observational error of as small as 1% ; figure 5 presents here an illustrative example of one of the best possible reconstructions . in fact , several additional assimilation runs indicate that the reconstruction is still good if the observational error does not exceed 40% , and reasonably good when 90 out of 180 observations have up to @xmath47 errors . but the reconstruction fails when all observations have more than 40% error . what we have demonstrated so far is that the time - dependent amplitude of meridional circulation , having one flow - cell per hemisphere , can be reconstructed by implementing enkf data assimilation with synthetic data . in reality we do not know from observational data whether they were produced by a dynamo operating with a single - celled flow in each hemisphere , or with a more complex flow profile , or with a combination of complex time - variations in all possible dynamo ingredients . in order to investigate the outcome from this method when the assumption made about the flow profile is wrong , we carry out an experiment to reconstruct flow - speed assuming a one - cell flow , while using observations of magnetic fields produced from a flow pattern that has two cells in latitude in each hemisphere ( see @xcite for prescription of a two - celled flow ) . we obtain synthetic data for a case with two flow cells in latitude . using 192 ensemble members and 180 magnetic observations with 1% error in each of these observations , we estimate the time variation in flow - speed by assuming a single - celled flow , and plot in figure 6(a ) . we find that the reconstruction is relatively poor , as expected . however , with a closer look we can see that the reconstructed speed is trying to approach the true - state from a lower value for the first 12 years and from a higher value for the next 13 years . when the trend in the true - state reverses near the year 27 , the osse has greater difficulty in converging on it . figure 6(b ) shows the innovation for two typical observations of poloidal fields near the surface , at @xmath25 ( i.e. at low latitude near the equator ) and at @xmath48 ( at high latitude near the pole ) . much larger innovation at high latitude than at low latitude reflects the fact that the spatial pattern is more erroneous at polar latitudes . the cummulative innovation ( orange curve ) does no longer asymptote to zero ; this implies bias in the system , due to incorrect assumption of flow - pattern . in the future , we will extensively explore the reconstruction of spatial pattern of meridional circulation . we have demonstrated through osses that enkf data assimilation into a babcock - leighton flux - transport dynamo can successfully reconstruct time - varying meridional circulation speed for several cycles from observations of the magnetic field . to obtain the best reconstruction , we have fed the assimilation system of 192 ensemble members with 180 observations with 1% observational error . however , a reasonably good reconstruction can be obtained when all observations have up to @xmath49 error , or half of the observations have up to @xmath50 errors , but the rest of them have much smaller errors . @xcite noted that the response time of a babcock - leighton dynamo model to changes in meridional flow is @xmath51 months , so the relatively poor reconstruction with only one observation with 33% observational error and 16 ensemble members can be improved if this information about the dynamo model s response time to flow changes can be exploited . a forthcoming paper will investigate how to use this response time during assimilation . throughout this paper , we used an assimilation interval of 15-days . the reason for this choice is as follows . recently @xcite have done a very thorough assessment of the predictability of solar flux - transport dynamos . predictability of a model refers to the time it takes for two solutions that start from slightly different values of either initial conditions or input parameters to diverge from each other to the point that they forecast substantially different outcomes . @xcite found an @xmath52-folding time of about 30 years . the flux - transport dynamo model we use here is physically very similar to theirs . therefore we judge that the @xmath52-folding time for our model will be similar . it is clear that the time interval for updating the data in our osse s should be much less than the predictability limit of the model , but it should be long compared to the integration time step ( a few hours ) , and consistent with the time scale for changes in axisymmetric solar observations , which is one solar rotation . it should also be shorter than the `` response time '' of our dynamo model ( @xmath51 months , see @xcite ) to a sudden change in inputs . we have therefore chosen our updating interval to be 15 days ; we plan to test the sensitivity of assimilation to changes in the updating interval from 15 days to longer ( for instance , 30 days ) and shorter ( @xmath53 days ) . in this study we have demonstrated what it takes to reconstruct the amplitude variations with time of a one - celled meridional circulation of fixed profile with latitude and radius . in reality the sun s meridional circulation may not be a one - celled pattern it may undergo changes both in profile and speed with time . we have demonstrated that the innovation in observation - forecast can be very large when the assumption about the spatial structure of the flow - pattern is incorrect , and it can be even larger where the departure in assumed spatial pattern from actual pattern in flow is larger . thus an obvious next step with our data assimilation system would be to attempt to reconstruct the spatio - temporal pattern of meridional flow . our ultimate goal is to perform assimilation runs from actual observations instead of synthetic data , in cases of reconstruction as well as future predictions . from this study we can build confidence about the power of enkf data assimilation for reconstructing not only the flow speed but also the profile of meridional circulation in the entire convection zone of the sun in the future . we thank nancy collins and tim hoar for their invaluable help with assimilation tools and software . we extend our thanks to two reviewers for many helpful comments and constructive suggestions , which helped significantly improve the paper . the dart / dynamo assimilation runs have been performed on the yellowstone supercomputer of nwsc / ncar under project number p22104000 , and all assimilation tools used in this work are available to the public from @xmath54 . this work is partially supported by nasa grant nnx08aq34 g . dhrubaditya mitra was supported by the european research council under the astrodyn research project no . 227952 and the swedish research council under grant 2011 - 542 and the hao visitor program . national center for atmospheric research is sponsored by the national science foundation . , j. l. , wyman , b. , zhang , s. & hoar , t. , assimilation of surface pressure observations using an ensemble filter in an idealized global atmospheric prediction system , j. atmos . , 62 , 2925 - 2938 , 2005 , a. , hulot , g. , jault , d. , kuang , w. , tangborn , a. , gillet , n. , canet , e. , aubert , j. & lhuillier , f. , an introduction to data assimilation and predictability in geomagnetism , space sci rev , 155 , 247 - 291 , 2010 , j. , bogart , r. s. , kosovichev , a. g. , duvall , t. l. , jr . , hartlep , h. , detection of equatorward meridional flow and evidence of double - cell meridional circulation inside the sun , apj lett . , 774 , l29 , 1 - 6 , 2013

accurate knowledge of time - variation in meridional flow - speed and profile is crucial for estimating a solar cycle s features , which are ultimately responsible for causing space climate variations . however , no consensus has been reached yet about the sun s meridional circulation pattern observations and theories . by implementing an ensemble kalman filter ( enkf ) data assimilation in a babcock - leighton solar dynamo model using data assimilation research testbed ( dart ) framework , we find that the best reconstruction of time - variation in meridional flow - speed can be obtained when ten or more observations are used with an updating time of 15 days and a @xmath0 observational error . increasing ensemble - size from 16 to 160 improves reconstruction . comparison of reconstructed flow - speed with `` true - state '' reveals that enkf data assimilation is very powerful for reconstructing meridional flow - speeds and suggests that it can be implemented for reconstructing spatio - temporal patterns of meridional circulation .

You are an expert at summarizing long articles. Proceed to summarize the following text: gamma - ray emitting pulsar wind nebulae ( pwne ) are excellent test grounds for studying pulsars relativistic winds and particle acceleration that takes place presumably at termination shocks . especially , pwne which accompany the diffuse synchrotron x - ray emission show the important clues to understand the time evolution of the pwne and how the accelerated particles escape from the shocks . while young pwne with characteristic ages of less than 10 kyr are relatively well studied in the past observations with _ chandra _ and _ xmm_-_newton _ @xcite , the sample of middle aged pwne remains small . the kookaburra region includes a middle aged pwn ( psr j1420 - 6048 ) and a plausible pwn ( rabbit ) , making it a suitable target for the detailed study of the diffuse emission with the _ suzaku _ x - ray observatory . _ suzaku _ , characterized by the low detector background compared to _ chandra _ and _ xmm - newton _ , is crucial for the analysis of faint and diffuse x - ray emission . the complex of compact and extended radio / x - ray sources , called kookaburra ( designated by @xcite ) , spans over about one square degree along the galactic plane around @xmath8 . the h.e.s.s . galactic survey revealed two very high energy ( vhe ) sources in this region ; the brighter of the two sources , hess j1420 - 607 , is centered at the position of ( r.a . = ( 14@xmath920@xmath1009@xmath11 , -60@xmath1245@xmath1336@xmath14 ) with an intrinsic extension of @xmath15 , and the slightly less bright second source , hess j1418 - 609 , is centered at the position of ( r.a . , = ( 14@xmath918@xmath1004@xmath11 , -60@xmath1258@xmath1331@xmath14 ) with an intrinsic extension of @xmath16 ( a major - axis of @xmath17 and a minor - axis of @xmath18 fitted with an elongated gaussian shape ) . the kookaburra region is also bright in the gev band and registered as a _ @xmath0-ray source @xcite . there is an energetic pulsar psr j1420 - 6048 at the south of hess j1420 - 607 . it is located at ( r.a . , dec . ) = ( 14@xmath920@xmath1008@xmath19 , -60@xmath1248@xmath1317@xmath20 ) , which is @xmath21 offset from the center of the vhe @xmath0-ray emission . psr j1420 - 6048 is a radio / x - ray pulsar with a period of @xmath22 ms and period derivative of @xmath23 @xcite . the distance of the pulsar was estimated as @xmath24 kpc based on the pulsar s dispersion measure with the ne2001 galactic electron - density model @xcite . the characteristic age and the spin - down luminosity are @xmath25 kyr and @xmath26 erg s@xmath27 , respectively . the strength of surface magnetic field is @xmath28 g. _ chandra _ observations revealed a faint and narrow diffuse x - ray emission in k3 nebula ( designated by @xcite with the radio observation ) around psr j1420 - 6048 @xcite . spectral fitting with an absorbed power - law model to the k3 nebula ( @xmath29 radius from the pulsar ) yielded an absorption column density of @xmath30 with a photon index of @xmath31 . the rabbit nebula is located at the eastern edge of hess j1418 - 609 with a distance of @xmath32 to the best central position of the vhe emission . @xcite found two point - like sources with the _ xmm - newton _ data , labeled as r1 and r2 ; the brighter source ( r1 ) is located at the edge of the rabbit , while the fainter source ( r2 ) at ( r.a . = ( 14@xmath918@xmath1039@xmath33 , -60@xmath1257@xmath1356@xmath34 ) , which appears embedded in the narrow diffuse emission . timing analysis with the epic pn data revealed a @xmath35 ms pulsation with a period derivative of @xmath36 from r2 , although not highly significant . assuming these values are correct , the characteristic age can be estimated as @xmath37 kyr . fitting the spectrum from the whole rabbit nebula ( @xmath38 radius ) with an absorbed power - law model , they found an absorption column density of @xmath39 and a photon index of @xmath40 . the two vhe sources are considered to be pwne @xcite , given the x - ray and tev results reported so far . generally , detailed observations of the synchrotron x - ray nebulae are crucial to understand spatial and spectral distributions of accelerated electrons . however , previous x - ray observations did not have enough sensitivity to detect faint x - ray nebulae with sizes comparable to the tev nebulae . in this paper , we report new detections of the diffuse x - ray emission of the kookaburra complex using the _ suzaku _ satellite . we observed the kookaburra region with the _ suzaku _ satellite in 2009 february . the _ suzaku _ observations were performed with the x - ray imaging spectrometer ( xis : @xcite ) in 0.3 12 kev and the hard x - ray detector ( hxd : @xcite ) in 13 600 kev . the xis , located at the focal plane of the x - ray telescopes ( xrt : @xcite ) , consists of one back - illuminated ccd camera ( xis1 ) and three front - illuminated ccds ( xis0 , 2 , and 3 ) . one of the front - illuminated ccds , xis2 , was not available at the time of our observations , since it suffered from a fatal damage on 9 november 2006 , and unusable since then . the xis instruments were operated in a normal full - frame clocking mode ( a frame time of 8 s ) with spaced - row charge injection ( sci ) @xcite . the hxd consists of the silicon pin photo diodes ( hereafter pin ) capable of observations in the 13 70 kev band and the gso crystal scintillators ( hereafter gso ) which cover the 40 600 kev band . since we could not find significant signals in the hxd data after subtracting the non - x - ray background ( nxb ) , cosmic x - ray background ( cxb ) , and galactic ridge components , we focused on the xis data analysis in this paper . we used data sets processed by a set of software of the _ suzaku _ data processing pipeline ( version 2.2.11.24 ) . the telemetry saturating time was excluded in the pipeline processing . basic analysis was done using the heasoft software package ( version 6.4 ) . we made use of cleaned event files , in which standard screening was applied . the standard screening procedures include event grade selections , and removal of time periods such as spacecraft passage of the south atlantic anomaly ( saa ) , intervals of small geomagnetic cutoff rigidity ( cor ) , and those of a low elevation angle . specifically , for the xis , the elevation angle larger than 5@xmath12 above the earth and larger than 20@xmath12 from the sunlit earth limb are selected . the total exposure from the two observations amounts to 72 ks after standard data screening . table [ tab : kookaburra_obs ] gives the log of the _ suzaku _ observations . i d & coord . ( j2000 ) & exposure & date + & r.a . , dec . & xis & + 503110010 & 215.0292 , -60.8156 & 50.3 & 01/11/2009 + 503071010 & 214.6625 , -60.9675 & 21.3 & 02/14/2009 + [ tab : kookaburra_obs ] in order to combine two different pointing images , we extracted the photon events in the energy range of 210 kev from each sensor . the data between 5.73 and 6.67 kev were removed from the image to exclude the calibration sources . we corrected the vignetting effect by dividing the image with a flat sky image simulated in the energy range of 28 kev using the xrt + xis simulator @xcite . the image was binned to 8 @xmath41 8 pixels and smoothed with a gaussian function of @xmath42 . a combined _ suzaku _ xis ( 0 + 1 + 3 ) image in the kookaburra region is show in fig . [ fig : image_merge ] . * both hess j1420 - 607 and hess j1418 - 609 are reported as diffuse tev-@xmath0-ray emitting sources @xcite . the intrinsic extensions of the diffuse emissions are indicated as white circles in the figure . the overlaid contours are taken from the h.e.s.s . observation . we can clearly see that the separate x - ray sources locate with some offsets from the @xmath0-ray peak positions and are coincident with the vhe sources . * we extracted the photon events in the energy ranges of 12 kev and 210 kev . the xis images are shown in fig . [ fig : k3image ] . we can see several thermal sources in the lower energy image , which are invisible in the higher energies . the bright x - ray emission is well coincident with the location of psr j1420 - 6048 and extended diffuse emission surrounds the bright pulsar . the diffuse component has an elongated shape which extends from the pulsar to the @xmath0-ray peak . * in order to determine the extension of non - thermal x - ray emission , we created a surface brightness profile from the enclosed region shown as `` rect '' in fig . [ fig : k3image ] ( a ) along the north to south direction . the surface brightness in 210 kev vs. relative coordinate is shown in fig . [ fig:1d_k3pos ] . the profile was fitted with a gaussian function plus constant as a background to derive the size of the diffuse emission , using the sigma of the gaussian ( @xmath43 ) . since a gaussian profile often provides a reasonable approximation for the surface brightness profiles @xcite , we used relative coordinates between [email protected] and [email protected] as a fitting range in consideration for the non - symmetrical profile . we excluded a range between [email protected] and 4@xmath44 as a narrow pulsar component . this is to avoid the contribution from the bright pulsar , whose image is smeared by the point - spread function of xrt . we defined the source size as three times the @xmath43 . the fitted result shows @xmath45 . if we assume the distance as 5.6 kpc , the physical size of the extended diffuse emission corresponds to @xmath46 pc . the intrinsic emission size of the bright pulsar was also estimated by fitting the image with a two - dimensional gaussian function , however , the calculated size was smaller than the attitude fluctuation of the satellite ( see * ? ? ? * ) . * ( a ) . ] fig . [ fig : raimage ] shows the xis images of the rabbit nebula . similar to k3/psr j1420 - 6048 , thermal sources are visible in the lower energies . the x - ray peak is coincident with the point - like sources reported by @xcite , which are indicated as black cross points in fig . [ fig : raimage ] . according to @xcite , these bright sources are embedded in narrow diffuse emission . compared with the _ xmm_-_newton _ , the diffuse emission in fig . [ fig : raimage ] ( b ) looks more extended than the previous image , and another diffuse structure which is clearly separated from the point - like sources can be seen in the southeast direction . in order to determine the size of the diffuse emission , we made a projected profile with the same procedure as in k3/psr j1420 - 6048 . the extracted area is indicated as `` rect '' in fig . [ fig : raimage ] ( a ) . the 1d projected result is shown in fig . [ fig:1d_rapos ] . we excluded a range between @xmath47 and @xmath48 from the fitting as the central pulsar region to avoid its influence . the projected profile was well represented with a gaussian function plus constant background component . the size of the diffuse component is estimated as @xmath49 from the 1d profile . the physical size of the diffuse emission can be estimated as @xmath50 pc . we assumed the distance as 5 kpc . ( a ) . ] firstly , we checked the background level extracted from the `` bgd '' region in fig . [ fig : k3image ] ( b ) . we compared its background spectrum with public blank sky observations on lockman - hole ( obs . i d : 102018010 ) and also with aso 0304 ( obs . i d : 504054010 ) , which locates near the kookaburra region , i.e. ( r.a . , dec.)=(213.3355 , -62.0808 ) . the background levels are consistent within 10% and the galactic ridge emission is ignorable . we thus concluded that the `` bgd '' region is a source - free in the following spectral analysis . in order to determine the absorption column density around psr j1420 - 6048 , we extracted the photon events within an angular distance of 1@xmath51 from the pulsar , which corresponds to `` ring1+ring2 '' in fig . [ fig : k3image ] ( b ) . we have co - added the data from xis0 and xis3 to increase statistics . the response ( rmf ) files and the auxiliary response ( arf ) files were produced using _ xisrmfgen _ and _ xissimarfgen _ , respectively . [ fig : spec_k3_src ] shows the spectral result with the joint - fitted xis0 + 3 ( black ) and xis1 ( red ) data . the spectral shapes are well represented by an absorbed power - law model ( ` wabs@xmath41pow ' ) . from the spectral fitting , the absorption column density was estimated as @xmath52 , which is consistent with the previous observations . to determine the total flux from the pulsar and diffuse emission , we used the `` reg1-reg2 '' as the extracted region . the flux in the energy range of 210 kev is @xmath53 erg@xmath54 . the systematic uncertainty coming from the difference of the effective area between the arf responses for diffuse and point sources was less than 10% . in order to investigate the spatial dependence of the spectral shapes , we separated the extraction region into four annular regions , i.e. `` ring1 '' : 0@xmath55 , `` ring2 '' : @xmath55@xmath56 , `` ring3 '' : @xmath56@xmath57 , `` ring4 '' : @xmath57@xmath58 . to reduce the uncertainty of the column density with limited statistics , we fixed the absorption column density at @xmath59 determined from the `` ring1+ring2 '' region . [ fig : k3_4specs ] shows the spectra ( xis0 + 3 ) for each circular region . in order to exclude the thermal contamination , we used the 2 - 10 kev band for the spectral fitting . in the `` ring3 '' spectrum , we can see a weak trend of the iron emission line around 6.7 kev . this might come from unresolved thermal sources . the photon index vs. relative coordinate is shown in fig . [ fig : k3_index_change ] . quoted errors are at the @xmath60 confidence level . we can see that the photon index slightly changes according to the distance from the bright pulsar region . the contamination effect from a point source in `` ring1 '' on the outer region `` ring4 '' was estimated as @xmath615% . the fitted parameters are summarized in table [ tab : fit ] . as for the hxd - pin , we could not detect any significant pulse in timing analysis , thus there is no excess in the hard x - ray band from the pulsar . ( b ) . ] we performed the same analytical procedures to the photon events from the rabbit nebula as described in k3/psr j1420 - 6048 . [ fig : spec_ra_src ] shows the spectra extracted within an angular distance of @xmath62 from the x - ray peak , which corresponds to the `` ring1+ring2 '' region in fig . [ fig : raimage ] ( b ) . the background spectrum was consistent with that of the blank sky data . the spectral shapes are well represented by an absorbed power - law model . the absorption column density was determined as @xmath63 by the spectral fitting . for the spatial dependence of the spectral shapes , we chose the annular regions as `` ring1 '' : 0@xmath64 , `` ring2 '' : @xmath64@xmath62 , `` ring3 '' : @xmath62@xmath65 , `` ring4 '' : @xmath65@xmath66 . the spectral results are shown in fig . [ fig : rabbit_spec_4circulars ] . to exclude the thermal contamination , we used the energy range of 210 kev for spectral fittings . the photon index vs. relative coordinate is shown in fig . [ fig : rabbit_index_change ] . the spectral shape becomes softer according to the distance from the inner bright core to the outer region , which is a similar trend as in the k3//psr j1420 - 6048 region . the systematic error of the psf in `` ring1 '' to `` ring4 '' was less than 5% . the best - fit parameters are summarized in table [ tab : fit ] . the total flux in the energy range of 210 kev is estimated as @xmath67 erg@xmath54 determined from the extraction region of `` reg1-reg2 '' . ( b ) . ] rrrrr object & region & @xmath68 ( wabs ) & photon index & flux [ 2 - 10 kev ] & @xmath69 + k3/psr j1420 - 6048 , ring1+ring2 & 4.4 @xmath70 0.3 & 2.00 @xmath70 0.10 & 1.53 @xmath70 0.03 & 1.01 ( 72 ) + reg1-reg2 & 4.4 ( fix ) & @xmath71 & @xmath72 & 0.83 ( 33 ) + + ring1 & & @xmath73 & & 0.86 ( 32 ) + ring2 & & @xmath74 & & 1.01 ( 32 ) + ring3 & &@xmath75 & & 0.86 ( 35 ) + ring4 & &@xmath76 & & 0.94 ( 35 ) + rabbit , ring1+ring2 & 2.7 @xmath70 0.2 & 1.82 @xmath70 0.10 & 2.65 @xmath70 0.05 & 0.87 ( 101 ) + reg1-reg2 & 2.7 ( fix ) & 2.00 @xmath70 0.06 & 6.27@xmath70 0.13 & 0.80 ( 52 ) + + ring1 & & @xmath77 & & 1.06 ( 46 ) + ring2 & & @xmath78 & & 1.09 ( 35 ) + ring3 & & @xmath79 & & 1.10 ( 46 ) + ring4 & & @xmath80 & & 1.08 ( 33 ) + [ tab : fit ] we have detected the diffuse x - ray emission around k3/psr j1420 - 6048 and rabbit , which were unnoticed in the previous observations , in the kookaburra region with the _ suzaku _ satellite . the x - ray peaks of the two sources are both within the error circles of the tev sources and each of them has an offset from the @xmath0-ray peak with @xmath81 for k3/psr j1420 - 6048 and @xmath82 for rabbit . the xis spectra of the two sources were well reproduced by an absorbed power - law model with a photon index of @xmath83 2.3 . if tev / x emission comes from the same object , the origin of the vhe @xmath0-rays can be explained by the inverse compton ( ic ) scattering of the cmb photons by high - energy electrons , while the x - ray emission via synchrotron radiation in a mean magnetic field @xmath84 . in this case , typical energies of responsible electrons are @xmath85 tev for ic @xmath0-rays ( in the thomson regime ) at a photon energy @xmath86 , and @xmath87 tev for x - rays at @xmath88 , respectively . the synchrotron and ic spectra produced by the relativistic electrons obeying an energy distribution of @xmath89 , have the same spectral shape , @xmath90 as long as ic scattering occurs in the thomson regime . if the high - energy electrons are continuously injected into the radiation zone at a constant rate and loosing energy predominantly by synchrotron or ic ( thomson ) losses , the electron energy spectrum becomes one power of @xmath91 steeper above the cooling break @xmath92 that is determined by the cooling timescale and the age of the source . as a result , the synchrotron and ic spectra from the cooled electrons become softer by @xmath93 . our _ suzaku _ results clearly show that the photon index smoothly changes according to the distance from the bright center regions , and the x - ray photon indices near the @xmath0-ray peaks are roughly consistent with those in the tev energies in both objects ( @xmath94 for hess j1420 - 607 and @xmath95 for hess j1418 - 609 , @xcite ) . in addition to that , the difference of the photon index between the bright center and diffuse emission is @xmath96 . the smooth spectral steepening of the synchrotron x - ray emission likely reflects synchrotron burn - off of the accelerated electrons . such effects have been observed for some young pwne @xcite , but we now find an interesting example of the smoothly steepening of the synchrotron x - ray emission in a middle - aged pwn of k3/psr j1420@xmath976048 ( @xmath98 kyr ) . in order to estimate the mean magnetic field strength , we modeled the spectral energy distribution . since information about the spatial dependence of the tev emission is not available , we assumed a simple one - zone synchrotron + ic model as the first order approximation , in which a single population of relativistic electrons in tev energy range emit both x - rays through synchrotron radiation and tev @xmath0-rays through ic scattering off the cmb . the electrons energy distribution is formally assumed to be an exponentially cutoff power - law of the form @xmath99 , where @xmath100 . we do not specify the physical meaning of @xmath101 , which may account for both the maximum electron energy and the cooling break . this spectral form is invoked only for the purpose of an estimate of the magnetic field . * [ fig : sed ] shows the spectral energy distribution of hess j1420 - 607 and hess j1418 - 609 at the x - ray and @xmath0-ray bands ; the model curves are obtained with a combination of @xmath102 ( hess j1420 - 607 ) or @xmath103 ( hess j1418 - 609 ) and @xmath104 for both objects . the black thick lines overlaid on the _ suzaku _ data points indicate the absorbed power - law models with best - fit parameters as shown in table [ tab : fit ] . we note that the `` ring1 '' region mainly comes from bright pulsars while the `` reg1-reg2 '' region includes both emissions from pulsar and diffuse components . the deviation between the spectrum of the `` reg1-reg2 '' region and the model curve in higher x - ray energies comes from the pulsar s intrinsic hard spectrum since the intrinsic diffuse emission can be approximately estimated as `` reg1-reg2 '' - `` ring1 '' . * the magnetic fields can be estimated as @xmath105 g for hess j1420 - 607 and @xmath7 g for hess j1418 - 609 . these values are in agreement with the magnetic field strength that is expected for middle - aged pwne , which lends support to the pwn scenarios for both objects . the diffuse x - ray emission of k3/psr j1420 - 6048 has asymmetry ( a circular shape of the radius @xmath106 pc + a tail - like component ) and its tail - like component extends to the @xmath0-ray peak . on the other hand , the diffuse emission in the rabbit seems to be more concentrated near the bright central source with higher surface brightness and relatively symmetrical shape of @xmath107 pc . the angular sizes of the vhe @xmath0-ray sources are larger than those of the x - ray emission regions , e.g. @xmath108 pc ( hess j1420 - 607 ) and @xmath109 pc ( hess j1418 - 609 ) . for the estimated magnetic field of @xmath110 g , the energy loss rate due to synchrotron radiation is just comparable to the loss rate via ic scattering off the cmb in the thomson regime , which is given by @xmath111 the lifetime of the electrons emitting synchrotron x - rays at 1 kev is @xmath112 kyr for the estimated magnetic field strength . on the other hand , the energy loss timescale of the electrons emitting ic @xmath0-rays at 1 tev is @xmath113 kyr ( see e.g. , * ? ? ? the tev emitting electrons have a longer lifetime than the cooling time of the x - ray emitting electrons , which is shorter than the characteristic age of psr j1420@xmath976048 and probably also shorter than that of the pulsar in the rabbit . therefore the morphological differences between the two bands can be expected , though details may depend largely on the pwn evolution @xcite . according to @xcite , @xmath0-ray to x - ray energy flux ratio of pwne is proportional to the characteristic age . the energy flux ratios for both objects are @xmath114 for psr j1420 - 6048 and @xmath115 for the rabbit nebula ( @xmath116 ) values were taken from @xcite ) . if we assume that the characteristic ages for both objects are correct , i.e. @xmath25 kyr for psr j1420 - 6048 , and @xmath37 kyr for rabbit , the flux ratios roughly follow the mattana s relation . the high - energy electrons are thought to be transported via convection and/or diffusion due to the magnetic fields randomized at the termination shock . let us assume that diffusion is the dominant transport mechanism . indeed the pulsar wind slows down from relativistic to relatively low expansion velocities at some distance after the pulsar wind has been shocked . the diffusion coefficient can be estimated as @xmath117 where @xmath118 is the observed size of the synchrotron x - ray nebula . on the other hand , the diffusion coefficient can be written as @xmath119 where @xmath120 is the mean free path of scattering , parameterized by a gyrofactor @xmath121 and the electron gyroradius @xmath122 . by equating eq . ( 2 ) with eq . ( 3 ) , * we can estimate the gyrofactor as @xmath123 for both objects , which indicates that the quite efficient acceleration has occurred in these pwne . * interestingly , diffusion of electrons is slow , nearly at the the bohm limit of @xmath124 . the presumed configuration of the toroidal magnetic field in pwne means that the direction of the magnetic field in the post - shock flow should be essentially perpendicular to the radial flow direction . therefore , tev electrons can be confined quite efficiently in the absence of a radial field component . in this regard , the slow diffusion is not unexpected for pwne . if instead diffusion of electrons occurs along a radial magnetic field line , the magnetic field in the post - shock flow must be highly turbulent so that the bohm limit is realized . to further investigate diffusion processes , we need to incorporate spatial and temporal evolution of the physical parameters of the post - shock flow into morphological and spectral modeling . we expect that the next - generation cherenkov telescopes , cherenkov telescope array ( cta ) , will make it possible to perform spatially - resolved spectroscopy at the tev @xmath0-ray band , allowing us to constrain the spatial and temporal evolution of the pwne by combining x - ray and tev data ( see * ? ? ?

we report on the results from _ suzaku _ x - ray observations of the radio complex region called kookaburra , which includes two adjacent tev @xmath0-ray sources hess j1418 - 609 and hess j1420 - 607 . the _ suzaku _ observation revealed x - ray diffuse emission around a middle - aged pulsar psr j1420 - 6048 and a plausible pwn rabbit with elongated sizes of @xmath1 and @xmath2 , respectively . the peaks of the diffuse x - ray emission are located within the @xmath0-ray excess maps obtained by h.e.s.s . and the offsets from the @xmath0-ray peaks are @xmath3 for psr j1420 - 6048 and @xmath4 for rabbit . the x - ray spectra of the two sources were well reproduced by absorbed power - law models with @xmath5 . the spectral shapes tend to become softer according to the distance from the x - ray peaks . assuming the one zone electron emission model as the first order approximation , the ambient magnetic field strengths of hess j1420 - 607 and hess j1418 - 609 can be estimated as 3 @xmath6 g , and @xmath7 g , respectively . the x - ray spectral and spatial properties strongly support that both tev sources are pulsar wind nebulae , in which electrons and positrons accelerated at termination shocks of the pulsar winds are losing their energies via the synchrotron radiation and inverse compton scattering as they are transported outward .

You are an expert at summarizing long articles. Proceed to summarize the following text: we are concerned with the evolution of compact hypersurfaces @xmath0 moving according to the general non - local law of propagation @xmath1 where @xmath2 is the normal velocity of @xmath3 which depends , through the evolution law @xmath4 , on time , on the position of @xmath5 , on the set @xmath6 enclosed by @xmath3 , on the unit normal @xmath7 to @xmath3 at @xmath8 pointing outward to @xmath6 and on its gradient @xmath9 which carries the curvature dependence of the velocity . when such motion is local , i.e. , when @xmath4 does not depend on @xmath6 , and satisfies _ the inclusion principle _ or _ geometrical monotonicity _ , i.e. , when , at least formally , the inclusion @xmath10 at time @xmath11 implies @xmath12 for any @xmath13 , it is proved by souganidis and the first author @xcite that the motion can be defined and studied by the _ level set approach _ , which was introduced by osher and sethian @xcite for numerical calculations and then developed , from a theoretical point of view , by evans and spruck @xcite for the mean curvature motion and by chen , giga and goto @xcite for general velocities . this approach replaces the geometrical problem with a degenerate parabolic partial differential equation called the _ geometric _ or _ level set equation . _ this equation is designed to describe the desired evolution via the 0-level set of its solution . more precisely , the existence and uniqueness of the level set solution @xmath14 allows to define @xmath3 as being the set @xmath15 . in recent years , there has been much interest on the study of front propagations problems in cases when the normal velocity of the front depends on a non - local way of the enclosed region like . this interest was motivated by several types of applications like dislocations theory or fitzhugh - nagumo type systems or volume dependent velocities that we describe below . it is worth pointing out that the level set approach still applies for motions with nonlocal velocities provided that the inclusion principle holds , following the ideas of slepcev @xcite . but , in many of the above mentioned applications , one faces non - monotone surface evolution equations . for such class of problems , the level set approach can not be used directly since the classical comparison arguments of viscosity solutions theory fail and therefore , the existence and uniqueness of viscosity solutions to these equations become an issue . though the existence properties for such motions seem now to be well understood ( see @xcite ) , this is not the case for uniqueness . in particular , there are not many uniqueness results for curvature dependent velocities . as far as the authors know , there are only two works by forcadel @xcite and forcadel and monteillet @xcite which investigate the motion arising in a model for dislocation dynamics which is included by our general equations . the aim of this article is to consider cases where we have , at the same time , a non - local velocity which induces a non - monotone evolution together with a curvature dependence ( we explain later on the state of the art for such problems and why the curvature dependence creates a specific difficulty ) . more specifically , we describe a method to show short time uniqueness results for the general motion . we now describe some typical applications we have in mind . we first consider a model for dislocation dynamics @xmath16 where @xmath17 denotes the indicator function of a subset @xmath18 of @xmath19 and @xmath20 , @xmath21}\to{\mathbb{r}}$ ] are given functions and we write @xmath22 here , @xmath23 denotes the @xmath24-dimensional unit sphere . the term @xmath25 is called the _ anisotropic _ ( or _ weighted _ ) _ mean curvature _ of @xmath3 at @xmath8 ( in the direction of @xmath7 ) . see for instance giga @xcite . typically , the reasonable assumptions in this context are the following : @xmath26 is a positive and bounded function , @xmath27 are bounded , continuous functions which are lipschitz continuous in @xmath8 variable ( uniformly with respect to @xmath28 variable ) , @xmath29;l^{1}({\mathbb{r}}^n))$ ] and @xmath30 is a positively homogeneous function with degree @xmath31 . the surface evolution equation without the last term in the right hand side is well - known as typical models of the dislocation dynamics ( see @xcite for a derivation and the physical background ) . we next consider asymptotic equations of a fitzhugh - nagumo type system as an example of interface dynamics coupled with a diffusion equations , @xmath32 where @xmath33 are bounded and lipschitz continuous with @xmath34 . this system has been investigated by giga , goto and ishii @xcite and soravia and souganidis @xcite . finally , we consider equations depending on the measure of the fronts like @xmath35 where the function @xmath36 is lipschitz continuous . a typical example is @xmath37 for some @xmath38 which has been investigated by chen , hilhorst and logak in @xcite ( see also @xcite ) . as we already mentioned it above , these examples are not only nonlocal but also non - monotone surface evolution equations . indeed , in , the kernel @xmath39 may change sign and , in and in , the functions @xmath40 , @xmath41 may be non - monotone . we also refer to @xcite for some monotone non - local geometric equations . by using the framework which we present in this paper , we give short time uniqueness results for , and . there are many results of existence and uniqueness for the simplest case of motions of , i.e. , @xmath42 and @xmath43 without a curvature term . a short time existence and uniqueness result was first obtained in @xcite . but then most of the results were obtained for curvature - independent velocities ( @xmath44 ) : long time existence and uniqueness results were obtained when the velocity is positive , i.e. , @xmath45 by alvarez , cardaliaguet and monneau in @xcite and by the first two authors in @xcite by different methods . the first two authors with cardaliaguet and monneau in @xcite presented a new notion of weak solutions ( see definition [ def weak sol ] ) of the level set equation for without a curvature term , gave the global existence of these weak solutions and analysed the uniqueness of them when holds . a similar concept of solutions already appeared in @xcite . in the companion paper @xcite , the first two authors with cardaliaguet and monteillet proposed a new perimeter estimate for the evolving fronts with uniform interior cone property and by using this , they extended the uniqueness result for dislocation dynamics equations and provided the uniqueness result for asymptotic equations of a fitzhugh - nagumo type system , still under the positiveness assumption . in this paper , we do not use the perimeter estimate in an essential way but either elementary measure estimates or , in the most sophisticated cases , the interior cone property ( see lemma [ lem - for - unique ] ) . since the studies by @xcite , it is now well - known that estimates on _ lower gradient bound _ and perimeter of @xmath31-level sets of viscosity solutions of associated local equations are key properties to obtain existence and uniqueness results for nonlocal equations derived from , and . let us describe the main difficulty of our problem and , to do so , we consider the level set equations of the simplest case of or here . considering the non - local part as a given function , we are led to the study the ( local ) initial value problem @xmath46 where @xmath47 and @xmath48})$ ] are bounded and lipschitz continuous with respect to the @xmath8 variable . one of our main results is a short time lower gradient bound estimate for the viscosity solution of , i.e. , @xmath49 for first - order eikonal equations , lower gradient bound comes naturally from the barron - jensen s approach ( see @xcite ) . for second - order equations like , it is affected by the _ diffusion _ " term and the _ non - empty interior _ difficulty and therefore we can not expect that the property holds generally and for long - time . indeed , in @xcite , see also @xcite , they consider the simple example of with @xmath50 and smooth @xmath51 such that @xmath52 on the initial front @xmath53 they prove that , up to choose suitable @xmath51 , fattening may occur for arbitrary @xmath54 i.e. , the front may develop an interior . it is precisely this reason which implies that there are not many results on the nonlocal second - order equations like the level sets equations of , and and a short - time result is optimal . existence results were obtained by @xcite but they concern merely existence of weak solutions defined by definition [ def weak sol ] . as stated above , there are two works @xcite which give uniqueness results for the motion . the difference between our results and theirs is that , in @xcite , only the evolution of hypersurfaces which can be expressed by graphs of functions is considered while , in @xcite , the arguments are based on minimizing movement for and they are completely different from our arguments which are based on the theory of viscosity solutions . moreover , for the existence of minimizing movement for the simplest case of , the assumptions that @xmath55 is symmetry and @xmath27 are smooth enough are essentially used . therefore , uniqueness results for the examples and are not covered by @xcite . another difference with existing results in the literature is that @xmath4 is allowed to change sign in , contrary to @xcite where is one of the main assumption to get uniqueness . it may give rise of fattening , see ( * ? ? ? * proposition 4.4 ) , and it is another explanation of the short time result . finally , we explain the key idea to obtain for viscosity solutions of . in order to get it , we make the following assumption on @xmath56 . there exist constants @xmath57 , @xmath58 and @xmath59 such that @xmath60 for all @xmath61 $ ] . then we prove that such a property is preserved for the solution of , at least for short time , i.e. , @xmath62 for all @xmath63 $ ] , @xmath64 $ ] and some @xmath65 , @xmath66 $ ] , where @xmath67\to[0,\infty)$ ] is a non - increasing continuous function such that @xmath68 the assumption on @xmath51 is inspired by ( * ? ? * theorem 4.3 ) , where it is formulated only for the sign - distance function . a similar result to may be found in @xcite where it is used to prove uniqueness results for the mean curvature motion for entire graphs . if @xmath51 is a smooth function with @xmath69 on the compact hypersurface @xmath70 , then the assumption is satisfied with @xmath71 and if there exists a smooth solution of the level set equation , then holds for short time . but , on one hand , the general degenerate parabolic and nonlinear equations we consider do not have classical solutions in general , and on the other hand , the above assumption on @xmath51 is valid in cases when @xmath72 is not a smooth hypersurface , which is also an important point here . the proof of uses in a crucial way the geometric property of and a _ continuous dependence result _ for parabolic problems ( which is , by the way , of independent interest ) . we refer to @xcite and references therein for the detail of the continuous dependence result for elliptic and parabolic problems . we derive lower gradient estimate from formally here . we have @xmath73 for all @xmath64 $ ] with @xmath74 as @xmath75 . dividing @xmath76 in the above and taking a sufficiently small @xmath77 $ ] , we get the lower estimate . we also obtain the interior cone property of fronts by . the paper is organized as follows : in section 2 , we state a continuous dependence result for a class of equations which encompasses level set equations associated to . in section 3 , we obtain the key estimate and derive the lower - bound gradient and perimeter estimates of @xmath31-level sets of viscosity solutions of local equations . in section 4 , we consider the _ level set _ equation of and give the proof for the short time uniqueness result ( theorem [ uniqueness ] ) . section 5 is devoted to existence and uniqueness results for the level set equations of , and as applications of theorem [ uniqueness ] . * notations . * for some @xmath78 , we denote by @xmath79 the @xmath80-dimensional euclidean space equipped with the usual euclidean inner product @xmath81 , and by @xmath82 the space of @xmath83 symmetric matrices . we write @xmath84 for @xmath85 @xmath86 and @xmath87 for @xmath88 . the symbols @xmath89 and @xmath90 denote the @xmath80-dimensional lebesgue and hausdorff measures , respectively . we write @xmath91 for the transpose of the matrix @xmath92 and @xmath93 . finally , for @xmath38 , we write @xmath94 and @xmath95 . * acknowledgements . * we are grateful to a. chambolle , e. jakobsen and l. rifford for their comments and advice . this work was done while h. mitake was visiting the laboratoire de mathmatiques et physique thorique , universit de tours . his grateful thanks go to the faculty and staffs . in this section , we are concerned with the equation @xmath96 @xmath97 , @xmath98 is the unknown function , @xmath99 , @xmath100 and @xmath101 stand respectively for its time and space derivatives , and hessian matrix with respect to @xmath8 variable . we use the following assumptions . * @xmath102\times({\mathbb{r}}^{n}\setminus\{0\ } ) \times{\mathcal{s}}^{n})$ ] . * the equation is _ degenerate parabolic _ , i.e. , @xmath103 for any @xmath104\times({\mathbb{r}}^{n}\setminus\{0\})$ ] and @xmath105 with @xmath106 , where @xmath107 stands for the usual partial ordering for symmetric matrices . * for any @xmath108}$ ] , @xmath109 , where @xmath110 ( resp . , @xmath111 ) is the upper - semicontinuous envelope ( resp . , lower semicontinuous envelope ) of @xmath112 . * there exist @xmath113 , @xmath114 such that @xmath115 for any @xmath116 , @xmath117 , @xmath118 , @xmath119 , @xmath105 and some @xmath120 satisfying @xmath121 where @xmath122 with @xmath123 . we note that , in this section , we do not assume that @xmath112 is geometric . [ continuous dependence ] let @xmath124 be functions on @xmath125\times({\mathbb{r}}^{n}\setminus\{0\ } ) \times{\mathcal{s}}^{n}$ ] satisfying assumptions ( a1)(a4 ) . let @xmath126})$ ] be , respectively , a bounded viscosity subsolution and viscosity supersolution of with @xmath127 for @xmath128 . assume that there exists @xmath129 such that @xmath130\ ] ] for either @xmath131 or @xmath132 , and that there exists @xmath117 such that @xmath133\ ] ] for both @xmath131 and @xmath134 then there exists @xmath135 which depends only on @xmath136 , @xmath137 , @xmath138 and @xmath139 such that @xmath140 for all @xmath141 $ ] . an assumption like ( a4 ) is natural in viscosity theory to obtain continuous dependence results of the type and the regularity of the solution ( cf . ) is a key ingredient too , see @xcite . in example [ ex-1 ] below , we show that ( a4 ) holds in the cases we are interested in . note that are not restrictive assumptions when dealing with front propagation problems , see @xcite . let @xmath142 and @xmath143 . we shall later fix @xmath144 . consider @xmath145 } \{u_1(x , t)-u_2(y , t)- \frac{|x - y|^4}{{\varepsilon}^4}-kt\}.\ ] ] noting , it is clear that the supremum is attained at @xmath146 $ ] for small @xmath147 . we consider the case where @xmath148 $ ] . in view of ishii s lemma , for any @xmath149 , there exist @xmath150 and @xmath151 ( see @xcite for the notation ) such that @xmath152 where @xmath18 is the matrix defined by . the definition of viscosity solutions immediately implies the following inequalities : @xmath153 hence we have @xmath154 using that @xmath155 or @xmath156 is lipschitz continuous with respect to @xmath8 variable , we get , by standard estimates , @xmath157})}$ ] , @xmath158})}$ ] and @xmath137 . we now distinguish two cases : ( i ) for any @xmath142 , @xmath159 ; ( ii ) there exist @xmath160 such that @xmath161 as @xmath162 , @xmath163 for any @xmath164 . we first consider case ( i ) . in view of ( a4 ) , we have @xmath165 sending @xmath166 , we get @xmath167 in case ( ii ) , we have @xmath168 . due to , we have @xmath169 , @xmath170 and @xmath171 . by ( a2 ) , we have @xmath172 therefore , we get @xmath173 set @xmath174 and then the two above cases can not hold ; this means that necessarily we have @xmath175 . therefore , for any @xmath176 $ ] , @xmath177 an optimization with respect to @xmath147 yields @xmath178 for some @xmath179 . [ ex-1 ] we consider the functions @xmath180}\times({\mathbb{r}}^n\setminus\{0\ } ) \times{\mathcal{s}}^{n}$ ] defined by @xmath181 for @xmath128 , where @xmath182 are compact metric space and @xmath183 , @xmath184 are , respectively , real - valued functions and @xmath185 matrix valued functions for some @xmath186 on @xmath125\times({\mathbb{r}}^{n}\setminus\{0\})$ ] with a possible singularity at @xmath187 we assume that the functions @xmath188 , @xmath184 satisfy the following conditions by replacing @xmath4 by @xmath188 , @xmath184 for any @xmath189 , @xmath190 , @xmath128 , respectively : @xmath4 are continuous on @xmath191}$ ] and for some @xmath192 ( independent of @xmath193 ) , @xmath194 for all @xmath195 , @xmath141 $ ] , @xmath196 . let @xmath116 , @xmath118 , @xmath197 , @xmath105 satisfy for some @xmath117 and let @xmath18 be the matrix given by . we omit the dependence of @xmath193 for simplicity of notation . we calculate that @xmath198 where @xmath199 and @xmath200 for some @xmath201 , where @xmath202 is the canonical basis of @xmath19 , @xmath203 and @xmath204 . due to , we have @xmath205 from the above computations , it follows that the inequality ( a4 ) holds by replacing @xmath206 and @xmath207 by @xmath208 and @xmath209 , respectively . therefore , if the @xmath210 s are solutions of with @xmath127 given by , @xmath211 then the conclusion of theorem [ continuous dependence ] holds and reads @xmath212 for all @xmath141.$ ] finally , note that , applying theorem [ continuous dependence ] with @xmath213 gives comparison and uniqueness for . [ appr - arg ] for the applications we have in mind , a continuous dependence result for equations with a measurable dependence in time will be needed . we do not state a precise result here but we mention that it can be obtained by an easy approximation argument . we consider the initial value problem in this section @xmath214 we make the following assumption on @xmath56 throughout this section . * @xmath215 and @xmath216 for all @xmath217 and there exists @xmath218 such that @xmath219 for all @xmath220 . * there exist constants @xmath221 , @xmath58 and @xmath222 such that @xmath223 , \end{aligned}\ ] ] where @xmath224 . [ rmk - c1-lisse - geom ] without loss of generality , we may assume that @xmath225 is a smooth bounded lipschitz continuous function and henceforth we will assume it from now on . indeed , let @xmath226 for @xmath142 be an approximate function of @xmath225 , then we have @xmath227 if @xmath228 is enough small , then we have @xmath229.\ ] ] let @xmath230 such that @xmath231 then , for @xmath232 and @xmath233 enough small , @xmath52 in @xmath234 and , setting @xmath235 we have @xmath236 where @xmath237 is a modulus of continuity of @xmath238 in @xmath239 and @xmath240 therefore ( i2 ) holds for @xmath241 and @xmath242 $ ] for @xmath243 enough small . moreover , the implicit function theorem implies that @xmath72 is a @xmath244 hypersurface . conversely , assume that @xmath72 is a @xmath244 hypersurface with the _ unique nearest point property _ ( that is , there exists a neighborhood @xmath245 of @xmath72 such that , for all @xmath246 there exists a unique @xmath247 such that @xmath248 ) . then the signed distance function @xmath249 to @xmath72 is @xmath244 ( see @xcite ) . it follows that ( i1 ) , ( i2 ) hold with @xmath51 such that @xmath250 in a neighborhood of @xmath72 and @xmath51 is a suitable regularization of @xmath249 elsewhere . more generally , when considering front propagation problems , it may be convenient to have a characterization of ( i2 ) in geometrical terms . such a result does not seem obvious . however , we have partial results in the following lemma , the proof of which is given in the appendix with additional comments . a subset @xmath251 is _ star - shaped _ with respect to @xmath252 if , for every @xmath253 the segment @xmath254 belongs to @xmath255 it is _ star - shaped with respect to a ball _ @xmath256 if @xmath18 is star - shaped with respect to every @xmath257 [ i2geom ] let @xmath258 be an open bounded set with boundary @xmath259 * ( star - shaped with respect to a ball domains ) the set @xmath260 is star - shaped with respect to a ball , i.e. , there exists a compact subset @xmath261 and @xmath262 such that @xmath263 } { \overline}{b}(\alpha x , ( 1-\alpha)r_0),\ ] ] if and only if there exists @xmath264 such that @xmath265 and ( i1 ) , ( i2 ) hold in @xmath245 with @xmath266 in this case , @xmath72 is locally the graph of a lipschitz continuous function . * if there exists @xmath143 such that @xmath72 is locally the graph of a lipschitz continuous function with constant @xmath267 then there exists @xmath51 such that , ( i1 ) and ( i2 ) hold . hereinafter , we set @xmath268 from remark [ rmk - c1-lisse - geom ] , we may assume that @xmath225 is a smooth bounded lipschitz continuous function and , replacing @xmath243 by a smaller constant in order that @xmath269 we obtain that @xmath270 is a @xmath244-diffeomorphism in @xmath19 with @xmath271 we assume ( a1)(a3 ) and make the following additional assumptions on @xmath112 throughout this section . * the function @xmath112 is _ geometric _ , i.e. , @xmath272 for all @xmath273 , @xmath274 , @xmath275}\times({\mathbb{r}}^{n}\setminus\{0\})\times{\mathcal{s}}^{n}$ ] . * there exists @xmath276 such that @xmath277 for any @xmath278}\times({\mathbb{r}}^{n}\setminus\{0\})$ ] and @xmath105 . * for any @xmath117 , there exists @xmath279 such that @xmath280 for any @xmath281 $ ] and for any @xmath282 , @xmath118 , @xmath141 $ ] , @xmath119 , @xmath105 satisfying and @xmath283 given by . * there exists at least one viscosity solutions of which satisfies @xmath284\ ] ] for some @xmath285 . let us make some comments about these new assumptions : ( a5 ) is needed to use the level set approach to describe front propagation ( see @xcite for instance ) . assumption ( a6 ) is satisfied for a wide class of quasilinear equations under interest in this paper , see example [ ex-2 ] . a consequence of ( a5 ) and ( a6 ) is : for any @xmath117 , there exists @xmath286 such that @xmath287\times ( { b}(0,r)\setminus \{0\ } ) \times{\mathcal{s}}^{n},\ ] ] which is a crucial property to obtain hlder continuity in time for the solutions of , see proposition [ regularity ] . assumption ( a7 ) is a natural condition to obtain a preservation of the initial property ( i2 ) during the evolution . this condition is related to ( a4 ) ; it is worthwhile to notice , as it was done at the end of example [ ex-1 ] , that such a condition gives uniqueness for the solutions of . existence of solutions to is assumed in ( a8 ) because it is not the point in this paper , see @xcite for some conditions which guarantee existence . more precisely , we have the following result about solutions of and the proof is given in appendix : [ regularity ] there exists a unique viscosity solution @xmath288})$ ] of and we have @xmath289 , @xmath290 $ ] , where @xmath291 are positive constants which depend only on @xmath292 and @xmath293 respectively . now , we state the main result of this section . [ key estimate ] there exist @xmath294 @xmath295 ( @xmath243 is given by ( i2 ) and satisfies and a non - increasing continuous function @xmath67\to[0,\infty)$ ] which depend only on @xmath296 @xmath297 @xmath298 @xmath299 @xmath300 @xmath301 @xmath302 @xmath303 such that @xmath304 and @xmath305 satisfies @xmath306,\ ] ] where @xmath307 in the appendix , we can extend in @xmath308 @xmath309,\end{aligned}\ ] ] where @xmath310 and @xmath311 are introduced in lemma [ psi ] . we first prove @xmath312 for all @xmath108}$ ] , @xmath313 $ ] , some constant @xmath314 , which is depends only on @xmath315 @xmath316 and @xmath317 ( note that @xmath318 does not depend on @xmath303 contrary to @xmath310 which depends on @xmath303 through @xmath243 because of ) . fix @xmath313 $ ] . set @xmath319 and @xmath320 for all @xmath108}$ ] . since @xmath112 is geometric and @xmath311 is a nondecreasing function , the functions @xmath321 satisfy v_t+ h(_(x),t , d_(_(x))^tdv(x , t ) , + d_(_(x))^td^2v(x , t)d_(_(x ) ) + d^2_(_(x))dv(x , t ) ) = 0 & in @xmath322 , + v(x,0)=u_0(_(x ) ) & in @xmath19 , w_t+h(x , t , dw , d^2w)=0 & in @xmath322 , + w(x,0)=(u_0(x)+_0 ) & in @xmath19 in the viscosity sense ( see ( * ? ? ? * theorem 4.2.1 ) for instance ) . let @xmath323 be the constant in ( a8 ) and recall that @xmath324 in lemma [ psi ] . for any @xmath325 $ ] , @xmath326 , which implies that @xmath327 . therefore , we only need to show that for any @xmath328 $ ] inequality holds . note that @xmath329 by assumptions ( a6 ) , ( a7 ) and theorem [ continuous dependence ] with @xmath330 and @xmath331 , we get , for any @xmath332 $ ] , @xmath333 for all @xmath141 $ ] , which implies . setting @xmath334 and @xmath335 for all @xmath64 $ ] , we obtain the conclusion . the first important consequence is a lower - gradient bound estimate on the front . [ lower bound ] we have @xmath336 where @xmath337 and @xmath338 are given in theorem [ key estimate ] . theorem 3.1 in @xcite implies that we can not expect global in time lower gradient estimates for solutions of with general initial data like ( i1 ) , ( i2 ) , even if we assume some positiveness assumptions on the velocity like in @xcite . before giving the proof of this result , we continue by stating another consequences of theorem [ key estimate ] . we need to introduce some notations . for any @xmath141 $ ] , @xmath339 $ ] , we set @xmath340 and define the cone with vertex @xmath341 , axis @xmath342 and parameters @xmath343 by @xmath344 } { \overline}{b}(z+ae , a\,\frac{\rho}{\theta } ) \\ { } & = \ , \{z+ae+a\,\frac{\rho}{\theta}\xi \mid a\in[0,\theta],\xi\in{\overline}{b}(0,1)\}. \end{aligned}\ ] ] the following result means that the evoluting fronts have the interior cone property . [ cone ] for any @xmath345 $ ] and @xmath64 $ ] , @xmath346 where @xmath347 when @xmath18 is a subset of @xmath348 we will write , by abuse of notation , @xmath349 for the perimeter of @xmath18 . notice that it does not always correspond to the usual definition of perimeter . the two definitions coincide for instance when the boundary is locally the graph of a lipschitz function , which is often the case in our applications . for further details , see ( * ? ? ? * section 5 and remark p.183 ) or @xcite . [ estimate perimeter ] there exists a constant @xmath350 which depends only on the constants appearing in theorem [ key estimate ] such that @xmath351 for all @xmath345 $ ] and @xmath352.$ ] we turn to the proofs . take any function @xmath353 satisfying @xmath354 and @xmath355 for all @xmath356 for some @xmath357 , where @xmath337 is given by theorem [ key estimate ] . by theorem [ key estimate ] and mean - value theorem , we have @xmath358 dividing by @xmath359 in the above and letting @xmath77 $ ] go to @xmath31 , we obtain the conclusion . fix @xmath345 $ ] , @xmath64 $ ] and @xmath360 . by theorem [ key estimate ] , we have @xmath361.\ ] ] set @xmath362 . for any @xmath363 , we have @xmath364 which implies that @xmath365 for any @xmath313 $ ] . therefore , we have @xmath366 } { \overline}{b}(z+{\lambda}\nu(z ) , { \lambda}\frac{\rho(t)}{{\overline}{{\lambda } } } ) = \bigcup_{a\in[0,\theta(z ) ] } { \overline}{b}(z+a\frac{\nu(z)}{|\nu(z)| } , a\frac{\rho(t)}{\theta(z ) } ) \subset { \overline}{{\omega}}_{t}^{r}.\ ] ] before doing the proof of corollary [ estimate perimeter ] , we recall the following lemma . [ perimeter ] let @xmath367 be a compact subset of @xmath19 having the interior cone property of parameters @xmath368 and @xmath369 . then there exists a positive constant @xmath370 such that for all @xmath117 , @xmath371 set @xmath372 . let @xmath373 be the functions in corollary [ cone ] and set @xmath374 and @xmath375 } \theta(z)$ ] . by theorem [ key estimate ] and corollary [ cone ] , we see that @xmath376 and we have , @xmath377.\ ] ] due to lemma [ perimeter ] , there exists @xmath378 such that @xmath379 for all @xmath380 $ ] , @xmath345 $ ] . we end this section with an application . [ ex-2 ] we consider the function @xmath381 where the functions @xmath382 and @xmath383 satisfy for all @xmath189 , @xmath190 , respectively . we add the following assumptions on @xmath384 : @xmath385 for all @xmath386 , @xmath108}$ ] , @xmath387 and some @xmath388 . these assumptions are related to ( a5 ) . a typical example is @xmath389 and then the second - order term is the so - called mean curvature term . we claim that the function @xmath112 satisfies ( a1)(a8 ) . it is easy to check that the function @xmath112 satisfies ( a1)(a5 ) . we check that the function @xmath112 satisfies ( a7 ) . note that , by , , we have @xmath390 for @xmath391 small enough and any @xmath392 . by abuse of notations , we write @xmath393 instead of @xmath384 for any @xmath394 . we compute @xmath395 and @xmath396 for some @xmath397 and any @xmath398 and @xmath281 $ ] . by using the same computations as example [ ex-1 ] , we have @xmath399 for some @xmath400 and any @xmath282 , @xmath118 , @xmath141 $ ] , @xmath197 with @xmath401 , @xmath105 satisfying and @xmath283 given by , which implies that @xmath112 satisfies ( a7 ) . we finally check that @xmath112 satisfies ( a8 ) . at first , the constant function @xmath402 is obviously a subsolution of . we set @xmath403 with @xmath404 and define the function @xmath405}\to{\mathbb{r}}$ ] by @xmath406 where @xmath407 . we prove that @xmath408 is a viscosity supersolution of . it is easily seen that @xmath409 on @xmath19 . indeed , for all @xmath410 , we have @xmath411 and , for all @xmath412 , @xmath413 ( see ( i1 ) ) . we have @xmath414 , @xmath415 and @xmath416 for any @xmath417 and @xmath418 . note that @xmath419 for all @xmath141 $ ] . set @xmath420 . we calculate that @xmath421 set @xmath422 and take @xmath423 for @xmath424 so that @xmath425 is an orthonormal basis . then we have @xmath426 and @xmath427 for @xmath424 . therefore , @xmath428 since @xmath429 and @xmath430 by . moreover , @xmath431 and @xmath402 is obviously a supersolution on @xmath432 . setting @xmath433 , we see that ( a8 ) is satisfied in view of the comparison theorem for viscosity solutions of . in this section , we consider the initial value problem of the nonlocal and non - monotone geometric equations which is derived from , through the _ level set approach _ ( see @xcite ) , @xmath434(x , t , du , d^{2}u)=0 & & \textrm{in } \ { \mathbb{r}^n\times(0,t ) } , \\ & u(\cdot,0)=u_0 & & \textrm{in } \ { \mathbb{r}}^{n}. \end{aligned } \right.\ ] ] for any function @xmath435},[0,1])$ ] , @xmath436 $ ] denotes a real - valued function of @xmath437\times({\mathbb{r}}^n\setminus\{0\})\times{\mathcal{s}}^{n}$ ] . for almost any @xmath438 $ ] , @xmath439(x , t , p , x)$ ] are continuous functions on @xmath440 with a possible singularity at @xmath163 . for all @xmath441 , @xmath442(x , t , p , x)$ ] are measurable functions . for any @xmath435},[0,1])$ ] , @xmath436=h$ ] satisfies ( a2 ) , ( a3 ) , ( a5 ) . furthermore , we make the following assumptions ( h1)(h5-(i ) ) or ( h5-(ii ) ) and ( i1 ) , ( i2 ) on @xmath56 throughout this section . * for any @xmath435},[0,1])$ ] , equation has a bounded uniformly continuous @xmath443-viscosity solution @xmath444 $ ] . moreover , there exist constants @xmath445 independent of @xmath435},[0,1])$ ] such that @xmath446 for all @xmath108}$ ] and @xmath447 for all @xmath448 $ ] . * for any @xmath449 $ ] and @xmath450;l^{1}({\mathbb{r}}^n))$ ] such that @xmath451 is compact for any @xmath452 $ ] , @xmath436\in c({\mathbb{r}}^n\times[0,\tau ] \times({\mathbb{r}}^{n}\setminus\{0\})\times{\mathcal{s}}^{n})$ ] . * the functions @xmath436 $ ] satisfy ( a6 ) with @xmath453 $ ] uniformly for any @xmath435},[0,1])$ ] . * the functions @xmath436 $ ] satisfy ( a7 ) with @xmath453 $ ] uniformly for any @xmath435},[0,1])$ ] . * for any @xmath117 , there exists @xmath279 such that @xmath454(y , t , p , y)| \nonumber\\ \le \ , & c_{h}\bigl(\frac{|x - y|^{4}}{{\varepsilon}^{4}}+ { \kappa}_{\chi_1 , \chi_2}(x , t ) + \frac{{\kappa}_{\chi_1 , \chi_2}^{2}(x , t)|x - y|^{2}}{{\varepsilon}^{4 } } + \rho\|a^{2}\| \bigr ) \label{assump main ineq}\end{aligned}\ ] ] for any @xmath455},[0,1])$ ] and for any @xmath282 , @xmath118 , @xmath141 $ ] , @xmath197 and @xmath105 satisfying and @xmath283 given by , where @xmath456 * one has inequality by replacing @xmath457 by @xmath458 where @xmath459 is the green function defined by @xmath460 * for any @xmath435},[0,1])$ ] , if @xmath461 , then the nonlinearity @xmath462(x , t , p , x)\ ] ] satisfies ( a1)-(a4 ) and @xmath463 \to u[\chi]$ ] uniformly in @xmath464 $ ] as @xmath465 . assumptions ( h1)(h4 ) are modifications of ( a1)(a7 ) in order to be able to deal with the nonlocal equation . while ( h5-(i ) ) and ( h5-(ii ) ) are specially designed to encompass dislocation type equations or fitzhugh - nagumo type systems . finally ( h6 ) is the assumption which allows to use theorem [ continuous dependence ] through an approximation argument ( cf . remark [ appr - arg ] ) . further detailed examples are given in section [ sec : applic ] . we use the following definition of weak solutions introduced in @xcite which is inspired by @xcite . [ def weak sol ] let @xmath466}\to{\mathbb{r}}$ ] be a continuous function . we say that @xmath305 is a weak solution of if there exists @xmath435},[0,1])$ ] such that * @xmath305 is an @xmath443-viscosity solution of @xmath467(x , t , du , d^2u)=0 & \textrm{in } \ { \mathbb{r}^n\times(0,t ) } , \\ & u(\cdot,0)=u_0 & \textrm{in } \ { \mathbb{r}}^n , \end{aligned } \right.\ ] ] * for almost every @xmath417 , @xmath468 moreover , we say that @xmath305 is a classical solution of if in addition , for almost all @xmath141 $ ] , @xmath469 [ weak is classical ] if there exists a weak solution @xmath288})$ ] of , then @xmath305 is classical in @xmath470 for some @xmath65 which depends on @xmath296 @xmath297 @xmath298 @xmath299 @xmath300 @xmath301 @xmath302 @xmath303 . let @xmath471},[0,1])\times c({\mathbb{r}^n\times[0,t]})$ ] be an @xmath443-viscosity solution of . we prove that @xmath472 for some @xmath65 . we use ( h6 ) and set @xmath473 $ ] . we recall that @xmath474 is the viscosity solutions of with @xmath475 for all @xmath476 . by the comparison theorem for local equations , proposition [ regularity ] we have @xmath477 , @xmath290 $ ] and some @xmath478 which is independent of @xmath479 . in view of ascoli - arzel theorem , the stability ( see @xcite ) and the uniqueness ( see @xcite ) of @xmath443-viscosity solutions of , we have @xmath480 locally uniformly on @xmath191}$ ] for @xmath288})$ ] which is the @xmath443-viscosity solution of . moreover , in view of corollary [ lower bound ] , we have @xmath481 for some @xmath65 . by the usual stability result of viscosity solution , we get @xmath336 which implies that @xmath482 for a.e . @xmath483 in view of ( * ? ? ? * corollary 1 in p. 84 ) . therefore , we get . by proposition [ weak is classical ] , we have @xmath484;l^{1}({\mathbb{r}}^n))$ ] for any weak solution @xmath305 of . in view of ( h2 ) , we see that @xmath485(x , t , p , x)$ ] is continuous on @xmath486 $ ] for any @xmath487 . we state our main result . [ uniqueness ] if there exist weak solutions of the initial - value problem , they are classical and unique in @xmath488,$ ] where @xmath337 is given by theorem [ key estimate ] . we formulate the main ingredient of the proof of the above theorem as a lemma . [ lem - for - unique ] let @xmath489 and @xmath338 be a continuous function on @xmath490 $ ] such that @xmath491 for any @xmath492 $ ] and @xmath493\to{\mathbb{r}}$ ] be a bounded lipschitz continuous function with respect to @xmath8 variable which satisfies , and on @xmath490.$ ] then we have @xmath494 for any @xmath495 , \end{gathered}\ ] ] where @xmath496 is a constant depending on @xmath497 and @xmath498 is a constant depending on @xmath499 , @xmath233 , @xmath500 , @xmath137 , @xmath501 , @xmath502 . we first prove the estimate . we have @xmath503 for @xmath504 , since @xmath505 we claim that @xmath506 for @xmath507 $ ] and @xmath508 small enough . we recall that @xmath509 is a @xmath244-diffeomorphism when @xmath76 satisfies . to prove the claim , let @xmath510 $ ] such that @xmath511 and set @xmath512 we distinguish two cases . if @xmath513 then , by , @xmath514 for @xmath515 if @xmath516 then , by , @xmath517 for @xmath518 @xmath519 and @xmath492.$ ] finally , holds if @xmath508 is such that @xmath520 by a change of variable , we have @xmath521 for small @xmath522 and therefore small @xmath76 , since @xmath523 from and , it follows @xmath524 by . we next prove the estimate . note that implies the lower gradient estimate @xmath525 from the increase principle of ( * ? ? ? * lemma 2.3 ) , we get @xmath526 where @xmath527 . therefore , noting that implies that @xmath6 has a interior cone property as we can see in the proof of corollary [ cone ] , by ( * ? ? ? * lemma 4.4 ) for some @xmath528 depending on @xmath499 , @xmath233 , @xmath500 , @xmath529 , @xmath137 , @xmath501 , @xmath502 , we have @xmath530 where @xmath531 suppose that there exist viscosity solutions @xmath155 and @xmath156 of . let @xmath532 $ ] which will be fixed later and set @xmath533}|(u_{1}-u_{2})(x , t)|.\ ] ] in view of theorem [ continuous dependence ] and ( h5-(i ) ) or ( h5-(ii ) ) , we have @xmath534 where @xmath535 } \int_{{\mathbb{r}}^n } \mathbf{1}_{\{u_{2}(\cdot , t)\ge0\}}(y)|\,dy \label{dislocation}\\ \textrm{and } \nonumber\\ { \overline}{{\kappa}}_\tau:= \sup_{x\in{\mathbb{r}}^{n } , t\in[0,\tau ] } \int_{0}^{t}\int_{{\mathbb{r}}^n } g(x - y , t - s)|\mathbf{1}_{\{u_{1}(\cdot , s)\ge0\}}(y ) -\mathbf{1}_{\{u_{2}(\cdot , s)\ge0\}}(y)|\,dyds . \label{fn}\end{gathered}\ ] ] note that @xmath536 we fix @xmath537 where @xmath337 is given by theorem [ key estimate ] . take @xmath538 small enough in order that @xmath539 the lower - bound gradient estimate ( corollary [ lower bound ] ) holds on @xmath540 $ ] and , for all @xmath541,$ ] @xmath542 moreover , take @xmath543 such that @xmath544,\ ] ] where @xmath367 is the constant give by proposition [ regularity ] . by continuity of @xmath545 , @xmath546 which achieve the same initial condition @xmath51 , it is always possible to find @xmath547 small enough in order that the above condition holds . by lemma [ lem - for - unique ] , we have @xmath548 for some @xmath549 depending on @xmath550 and @xmath551 for some @xmath552 depending on @xmath553 . therefore , we get @xmath554 for some constant @xmath555 which is independent of @xmath556 . for @xmath556 small enough , we have @xmath557 . it follows @xmath558 on @xmath559 $ ] . we consider @xmath560\}$ ] . if @xmath561 , then we can repeat the above proof from time @xmath562 instead of @xmath31 . finally , we have @xmath558 on @xmath563 $ ] for all @xmath564 which gives the conclusion . in the companion paper @xcite , the framework to show existence of weak solutions of is given and as applications , existence results for weak solutions of level set equations appearing in dislocations theory and in the study of fitzhugh - nagumo systems are presented ( see ( * ? ? ? * sections 3.2 , 4.2 ) ) . in this section , we give uniqueness results for viscosity solutions of such equations . we consider the level set equation of the evolution of hypersurfaces : @xmath565 where we use the notations in introduction . here @xmath566 , @xmath567}\to{\mathbb{r}}$ ] , @xmath568 are given functions which satisfy the following assumption ( a ) : * @xmath569 , @xmath570 for all @xmath571 , where @xmath572 and @xmath573 is lipschitz continuous ; * @xmath574 ; l^{1}({\mathbb{r}}^n))$ ] , @xmath575})$ ] , @xmath576;l^{1}({\mathbb{r}}^n))$ ] ; * there exist constants @xmath577 such that , for any @xmath578 and @xmath141 $ ] @xmath579 * @xmath580 for some positively homogeneous function @xmath581 of degree @xmath582 , i.e. , @xmath583 for @xmath273 , @xmath196 , which satisfies @xmath584 @xmath585 the function @xmath30 is called the _ cahn - hoffman _ vector and the last term in the right hand side of is the anisotropic curvature of @xmath3 at @xmath8 given by @xmath586 . we refer the reader to the monograph by giga @xcite and the references therein for more details . for reader s convenience , we derive the level set equation of , see @xcite . we have @xmath587 since @xmath588 is positively homogeneous of degree @xmath582 , @xmath589 is positively homogeneous of degree @xmath31 for all @xmath590 , i.e. , @xmath591 for all @xmath273 , @xmath571 . differentiating in @xmath40 , setting @xmath592 and noting that @xmath593 yields @xmath594 where @xmath595 . equality yields @xmath596 where @xmath597 and @xmath598 where @xmath599 . introducing an auxiliary function @xmath466}\to{\mathbb{r}}$ ] such that @xmath600 on @xmath3 and @xmath601 in @xmath6 for all @xmath141 $ ] , we note @xmath602 and it follows @xmath603 in the above equalities , we used the homogeneity of degree @xmath31 of @xmath30 and of degree @xmath402 of @xmath604 . set @xmath605(x , t , p):= m(-\hat{p})\bigl(c_0(\cdot , t)\ast\chi(\cdot , t)(x ) + c_1(x , t)\bigr ) , \\ { \sigma}(p):= \sqrt{m(-\hat{p } ) } \bigl(d^{2}{\gamma}(-\hat{p})\bigr)^{1/2 } r_{p } , \\ h_{d}[\chi](x , t , p , x):= -c[\chi](x , t , p)|p| -{{\rm tr}\,}\bigl({\sigma}(p){\sigma}^{t}(p)x\bigr ) , \end{gathered}\ ] ] for any @xmath435},[0,1])$ ] , @xmath275}\times({\mathbb{r}}^n\setminus\{0\})\times{\mathcal{s}}^{n}$ ] . the level set equation of is the equation @xmath606 ( x , t , du , d^{2}u)=0 \quad\textrm{in } \ { \mathbb{r}^n\times(0,t)},\ ] ] which is a particular case of . [ dislocation thm ] under assumptions ( a ) , ( i1 ) and ( i2 ) , the initial value problem with @xmath607 has at least a weak solution in @xmath191}$ ] . moreover , weak solutions are classical and unique in @xmath563 $ ] for some @xmath608 $ ] which depends only on @xmath609 @xmath316 and @xmath317 . we refer to @xcite for the short time existence and uniqueness of the solution of a dislocation dynamics equation with a mean curvature term under different assumptions . the existence of weak solutions is proved in ( * ? ? ? * theorem 3.3 ) . by using theorem [ uniqueness ] , we prove a short time uniqueness . it is easy to check that ( h2 ) , ( h3 ) are satisfied . due to the arguments in example [ ex-2 ] and assumptions ( a ) ( i ) , ( iv ) , we see that ( h4 ) is satisfied . we prove that @xmath610 satisfies ( h1 ) and ( h5-(i ) ) . we first check ( h5-(i ) ) . set @xmath611(x , t , p)$ ] for @xmath278}\times({\mathbb{r}}^n\setminus\{0\})$ ] , @xmath612},[0,1])$ ] and @xmath128 . note that @xmath613 we finally check ( h1 ) . let @xmath614})\times{l^{\infty}}({\mathbb{r}^n\times[0,t]},[0,1])$ ] be a @xmath443-viscosity solution of . we extend the functions @xmath615(x,\cdot , p)$ ] to be equal @xmath31 when @xmath616 and @xmath617 . we set @xmath618(x,\cdot , p)\ast\zeta_{n}(t ) , \end{gathered}\ ] ] for all @xmath278}\times({\mathbb{r}}^n\setminus\{0\})$ ] , where @xmath619 for @xmath620 and @xmath621 is a standard mollification kernel . then we have @xmath622}\times({\mathbb{r}}^n\setminus\{0\}))$ ] , @xmath623 locally uniformly in @xmath624 as @xmath625 . moreover , the @xmath626 s are lipschitz continuous with respect to @xmath8 variable with the constant @xmath627 and bounded ( independently of @xmath479 ) . let @xmath628 be the viscosity solutions of @xmath629 for all @xmath476 . by example [ ex-2 ] and proposition [ regularity ] , we have @xmath630 } , \quad u_{n}(x , t)=-1 \ \textrm{on } \ ( { \mathbb{r}}^n\setminus b(0,r(t)))\times[0,t ] , \\ & |u_{n}(x , t)-u_{n}(y , t)|\le c|x - y| , \quad \end{aligned}\ ] ] for all @xmath578 , @xmath290 $ ] and some @xmath201 , where @xmath631 is the function introduced in example [ ex-2 ] . by ascoli - arzel theorem , the uniqueness of @xmath443-viscosity solutions of with @xmath436=h_{d}[\chi]$ ] and ( * ? ? ? * theorem 1.1 ) , we have @xmath480 locally uniformly on @xmath191}$ ] and @xmath305 still satisfies properties . the above mollification argument can be an alternative way of getting the approximation property we need ( cf . ( h6 ) ) by a regularization of @xmath436 $ ] by @xmath436(x,\cdot , p , x)\ast\zeta_{n}(t)$ ] instead of @xmath632(x , t , p , x)$ ] . we consider the following system : u_t=((v)+div())|du| & in @xmath322 , + v_t - v= g^+(v)_\{u0}+ g^-(v)(1-_\{u0 } ) & in @xmath322,[fn asymp ] + u(,0)=u_0 , v(,0)=v_0 & in @xmath19 , which is obtained as the asymptotic as @xmath633 of the following fitzhugh - nagumo system arising in neural wave propagation or chemical kinetics ( see ( * ? ? ? * theorem 4.1 ) ) : @xmath634 where @xmath635 the functions @xmath636 and @xmath637 appearing in are associated with @xmath408 and @xmath638 . we make the following assumptions ( b ) : * the function @xmath639 is bounded and of class @xmath640 with @xmath641 ; * the functions @xmath642 are lipschitz continuous on @xmath643 with a lipschitz constant @xmath644 and there exist @xmath645 such that @xmath646 * @xmath647 , @xmath648 for all @xmath649 and some @xmath650 . for @xmath435},[0,1])$ ] , we write @xmath651 for the solution of @xmath652 and set @xmath615(x , t):={\alpha}(v(x , t))$ ] . then problem reduces to @xmath653(x , t ) + { { \rm div}\,}\bigl(\frac{du}{|du|}\bigr)\bigl)|du(x , t)| = 0 & & \textrm{in } \ { \mathbb{r}^n\times(0,t ) } , \\ & u(\cdot,0)=u_0 & & \textrm{in } \ { \mathbb{r}}^{n } , \end{aligned } \right.\ ] ] which is a particular case of . [ fn thm ] under assumptions ( b ) , ( i1 ) and ( i2 ) , the initial value problem has at least a weak solution in @xmath191}$ ] . moreover , it is classical and unique in @xmath563 $ ] for some @xmath654 which depends only on @xmath655 @xmath316 and @xmath317 . the existence of weak solutions is proved in ( * ? ? ? * theorem 3.4 ) . see also @xcite . it is easy to check that ( h2 ) and ( h3 ) are satisfied . due to similar arguments as those in the proof of theorem [ dislocation thm ] , we see that ( h1 ) and ( h4 ) are satisfied . we prove that ( h5-(ii ) ) is satisfied . for @xmath656},[0,1])$ ] , the solutions of are given by @xmath657 for @xmath128 . by the proof of ( * ? ? ? * theorem 4.1 ) , we have @xmath658(x , t)-c[\chi_{2}](x , t)|\le l_{{\alpha}}({\overline}{g}-{\underline}{g})e^{3l_{g}t } \int_{0}^{t}\int_{{\mathbb{r}}^n } g(x - y , t - s ) @xmath108}$ ] . this completes the proof . we consider the following evolution of hypersurfaces : @xmath659 where the function @xmath36 is lipschitz continuous . a typical example is @xmath37 for some @xmath38 which has been studied by chen , hilhorst and logak in @xcite ( see @xcite also ) . the authors prove that the limiting behaviour of the following reaction - diffusion equation u_t = u+f(u,_u ) & in @xmath660 , + = 0 & on @xmath660 , + u(,0)=g^ & in @xmath661 is characterized by the motion of hypersurface with @xmath37 . the level set equation of is the nonlocal equation : @xmath662 for some positive constant @xmath663 it is worthwhile to mention that we can not expect the global existence of weak solutions without any restriction of the growth of @xmath41 , because the front can blow up at a finite time . a growth condition to ensure a global existence result is given below in ( c-(ii ) ) , see also ( * ? ? ? * section 4.2 ) . we can easily check that the equation satisfies assumptions ( h1)-(h4 ) and ( h5-(i ) ) . so , the following result holds . under assumptions ( i1 ) and ( i2 ) , for any @xmath664 , there exists a constant @xmath665 $ ] and at least a weak solution @xmath614})\times{l^{\infty}}({\mathbb{r}^n\times[0,t]},[0,1])$ ] of the initial value problem for such that @xmath666 for any @xmath667 $ ] . moreover , the weak solution is classical and unique in @xmath563 $ ] for some @xmath668 $ ] which depends only on the lipschitz constant of @xmath41 , @xmath669 , @xmath670 , @xmath671 @xmath316 and @xmath317 . now , adding the assumption ( c ) : * the assumption ( i2 ) holds with @xmath672 in @xmath245 , * there exist @xmath673 such that @xmath674 we can show a global existence and uniqueness result of solutions of for small @xmath588 . more precisely , we get [ measure global thm ] under assumptions ( i1 ) , ( i2 ) and ( c ) , there exists a positive constant @xmath675 such that , for any @xmath676 , there exists a unique viscosity solution of in @xmath191}$ ] . [ measure key estimate ] let @xmath305 be the viscosity solution of u_t = c(t)|du| + tr((i-)d^2u ) & in @xmath322 , + u(,0)=u_0 & in @xmath19 , where @xmath588 is a positive constant and @xmath677)$ ] is a nonnegative given function . then we have , for any @xmath678 $ ] , @xmath679,\ ] ] where @xmath136 is a positive constant which depends only on @xmath670 and @xmath112 and @xmath310 is the constant given by lemma [ psi ] ( we may assume that @xmath680 ) . the proof of lemma [ measure key estimate ] is very similar to that of theorem [ key estimate ] , but since we would like to explain how to use the positiveness of @xmath681 and note the dependence of @xmath136 in lemma [ measure key estimate ] , we give it here . let @xmath311 be the function given by lemma [ psi ] and fix @xmath313 $ ] . setting @xmath682 and @xmath683 for all @xmath108}$ ] , we consider @xmath145 } \{w(x , t)-v(y , t)- \frac{|x - y|^4}{{\varepsilon}^4}-\tilde{k}t\}.\ ] ] for @xmath684 . by similar arguments as those used in theorem [ continuous dependence ] , there exist @xmath685 $ ] for some @xmath285 and small enough @xmath147 such that the supremum attains at @xmath686 and @xmath687 and @xmath688 satisfy . note that the lipschitz constant of @xmath305 is @xmath316 in this case . we have @xmath689 . the definition of viscosity solutions immediately implies that we have @xmath690 it follows @xmath691 we note that @xmath692 is a positive definite bounded matrix and @xmath693 by . in view of the positiveness of @xmath694 we get , for some constant @xmath695 which may change line to line , @xmath696 sending @xmath166 and taking @xmath697 , we necessarily have @xmath175 . thus we have for any @xmath108}$ ] @xmath698 setting @xmath699 we get @xmath700 where @xmath701 depends only on @xmath702 this implies a conclusion . in ( * ? ? ? * section 4 ) , the global existence result for weak solutions of is given . due to lemma [ measure key estimate ] , we see that if @xmath703 then @xmath704.\ ] ] therefore , we get @xmath705 , \ { \lambda}\in[0,{\overline}{{\lambda}}].\ ] ] a careful review of the proof of theorems [ uniqueness ] gives the conclusion . [ star perimeter ] inequality implies that the @xmath706-level set of @xmath707 for @xmath706 close to 0 , are star - shaped domains with respect to a ball with center @xmath708 see lemma [ i2geom ] . in particular , they are locally lipschitz continuous graphs . then we can get perimeter estimates without using lemma [ perimeter ] . indeed , noting that , from the lower gradient estimate ( corollary [ lower bound ] ) and the increase principle ( * ? ? ? * lemma 2.3 ) , for small @xmath147 , @xmath709 we have , for any @xmath64 $ ] , @xmath710 in view of the co - area formula and proposition [ regularity ] . since the @xmath711 s are locally lipschitz continuous , the @xmath24-hausdorff measure and the perimeter in geometric measure theory coincide ( see @xcite ) . since this latter perimeter is lower - semicontinuous with respect to the hausdorff convergence , sending @xmath633 , we get @xmath712 for all @xmath141 $ ] . we give the proof of lemma [ i2geom ] , proposition [ regularity ] and lemma [ psi ] . let @xmath260 be defined by and consider the function @xmath713 it is not difficult to see that , ( i1 ) and ( i2 ) hold with @xmath266 conversely , suppose that ( i1 ) and ( i2 ) with @xmath672 hold for some @xmath714 we claim that @xmath715 is star - shaped with respect to the ball @xmath716 with @xmath717 let @xmath718 and @xmath719 and define @xmath720 it suffices to show that @xmath721 on @xmath722.$ ] from ( i1 ) , ( i2 ) , we have @xmath723 for @xmath724.$ ] let @xmath725 : g>0 \ { \rm on } \ [ 0,{\lambda}]\}.\ ] ] if @xmath726 then the proof is complete . otherwise , let @xmath147 small enough such that @xmath727 from ( i1 ) , ( i2 ) , we have @xmath728 which is a contradiction . it completes the proof of the claim . the proof of the fact that a star - shaped with respect to a ball domain has a locally lipschitz continuous boundary may be found in ( * ? ? ? 2.4.4 and theorem 2.4.7 ) or ( * ? ? ? * lemma p.20 ) . we turn to the proof of ( ii ) . we need to recall some notations and definitions and we refer the reader to @xcite for further details . the clarke generalized directional derivative at @xmath8 in the direction @xmath4 of @xmath729 is @xmath730 the clarke generalized derivative at @xmath8 is the closed convex set @xmath731 which is nonempty when @xmath408 is locally lipschitz continuous at @xmath732 the clarke tangent cone to @xmath733 at @xmath734 is the convex cone @xmath735 and the clarke normal cone is the polar of the latter , i.e. , @xmath736 the proof is divided in several steps . _ we claim that , for any @xmath734 and @xmath737 there exists @xmath738 such that _ @xmath739 we fix @xmath740 @xmath741 and consider neighborhoods @xmath742 of @xmath743 and @xmath744 of @xmath745 such that there exists a @xmath367-lipschitz continuous function @xmath746 with @xmath747 and @xmath748 let @xmath749 by definition , for all @xmath750 @xmath751 by ( * ? ? ? * theorem 2.5.7 ) , since @xmath408 is lipschitz continuous , @xmath752 it follows that , for any @xmath753 @xmath754 still belongs to @xmath755 from , we get @xmath756 a first consequence is that @xmath757 necessarily . moreover @xmath758 indeed , if @xmath759 then holds for all @xmath620 and therefore @xmath760 for all @xmath761 therefore , using that @xmath408 is @xmath367-lipschitz continuous , we get @xmath762 which leads to a contradiction for @xmath763 using again ( * ? ? ? * theorem 2.5.7 ) , we have @xmath764 since @xmath408 is @xmath367-lipschitz continuous , it follows @xmath765 using that @xmath766 we obtain @xmath767 taking @xmath768 we obtain easily . we denote by @xmath769 the distance to @xmath733 and by @xmath770 the signed distance to @xmath771 which is negative in @xmath772 for any set @xmath773 @xmath774 is the closure of the convex hull of @xmath255 _ 2 . we claim @xmath775 $ ] for all @xmath734 . _ ( notice it means that the generalized derivative of the signed distance does not contain 0 . ) let @xmath776 and @xmath777 a sequence of points which converges to @xmath771 such that @xmath770 is differentiable at @xmath778 assume that @xmath779 converges to @xmath780 suppose first that , up to extract a subsequence , @xmath781 then , since @xmath782 in @xmath783 and @xmath769 is differentiable at @xmath784 it means that @xmath777 has a unique closest point @xmath785 and @xmath786 but @xmath787 is the convex hull of the cone generated by such limits ( ( * ? ? ? * exercise 8.5 , p.96 ) ) . thus @xmath788 from ( * ? ? ? * theorem 2.8.1 ) , @xmath789 is the convex hull of such @xmath790 which completes the proof of the claim . there exists an open bounded neighborhood @xmath791 of @xmath771 and @xmath792 such that , for all @xmath793 @xmath794 is a nonempty compact convex subset of @xmath795 _ the subsets @xmath796 are clearly convex , closed and bounded for any @xmath797 and open subset @xmath798 it remains to prove that there are nonempty for some @xmath799 it is true for @xmath800 with @xmath801 by claims 1 and 2 . to extend this property in a neighborhood @xmath802 of @xmath803 we notice the following facts : since @xmath804 is upper - semicontinuous ( ( * ? ? ? * proposition 2.1.5 ) ) and @xmath805 there exists @xmath806 such that @xmath807 for all @xmath808 by clarke s implicit function theorem ( * ? ? ? * proposition 3.3.6 ) , the @xmath809-level sets of @xmath770 are lipschitz continuous in @xmath810 the lipschitz constant is controlled by the distance from 0 to @xmath811 up to take @xmath706 small , it depends only on @xmath812 then , we can repeat the previous arguments and obtain the result in @xmath802 ( up to take @xmath813 smaller than @xmath814 ) . we then find @xmath791 by compactness of @xmath815 _ 4 . the multi - valued map @xmath816 is lower - semicontinuous . _ let @xmath817 let @xmath818 and @xmath819 since @xmath804 is upper - semicontinuous , by definition , there exists @xmath820 such that , if @xmath821 then @xmath822 ( where @xmath823 ) . any @xmath824 can be written @xmath825 where @xmath826 and @xmath827 using that @xmath828 it follows @xmath829 this proves that @xmath830 which is the definition of the lower - semicontinuity of a multi - valued function . michael s continuous selection theorem and end of the proof . _ from steps 3 and 4 , we can apply michael s continuous selection theorem ( @xcite ) : there exists a continuous map @xmath831 such that @xmath832 i.e. , for all @xmath833 and @xmath834 we have @xmath835 according to remark [ rmk - c1-lisse - geom ] , up to decrease @xmath836 we may choose @xmath837 which is smooth , bounded and lipschitz continuous . using the lipschitz continuity of @xmath770 and @xmath838 we may earn some uniformity in up to reduce @xmath839 more precisely , for every @xmath840 there exists @xmath841 and @xmath842 such that @xmath843 for all @xmath844 and @xmath845.$ ] we conclude by compactness of @xmath771 ( note that we can modify the signed distance function far from @xmath771 in order to have a bounded function ) . star - shaped property is not sufficient to ensure ( i2 ) , see the counterexample of figure [ dessins - ens ] . there exist some sets which satisfy ( i2 ) but they do not have a locally lipschitz boundary , see figure [ dessins - ens ] . existence of a solution to is given by assumption ( a8 ) . we prove the uniqueness and the lipschitz continuity regularity in @xmath8 of the solutions . let @xmath288})$ ] be a solution of which satisfy . let @xmath846 and set @xmath847 } \{u(x , t)-u(y , t)-e^{kt}\frac{|x - y|^{4}}{{\varepsilon}^{4}}-\eta t\}.\ ] ] let @xmath26 be attained at @xmath848 $ ] . by ( a8 ) , we may assume that @xmath849 $ ] . we first consider the case where @xmath148 $ ] . in view of ishii s lemma , for any @xmath149 , there exist @xmath850 and @xmath851 such that @xmath852 where @xmath853 the definition of viscosity solutions immediately implies the following inequalities : @xmath854 we have @xmath855 therefore , sending @xmath166 and setting @xmath859 , necessarily we have @xmath175 . we get for all @xmath578 , @xmath141 $ ] @xmath860 we have used the young inequality in the third inequality . setting @xmath861 we get @xmath862 sending @xmath863 , we have @xmath864 on the one hand , by taking @xmath865 in the above inequality , we get @xmath866 and obtain the uniqueness of the solution . on the other hand , by choosing @xmath867 we obtain . we now prove . set @xmath868 , where @xmath367 is the constant given by proposition [ regularity ] . recalling , in view of lemma 9.1 in @xcite , there exists a constant @xmath869 such that @xmath870 for all @xmath871 , @xmath872 $ ] . noting that @xmath873 on @xmath874 $ ] , we get @xmath870 for all @xmath875 , @xmath290 $ ] . it is easy to see inequality if @xmath881 in view of ( i2 ) . in the case where @xmath882 , we have @xmath883 which implies inequality if @xmath313 $ ] with @xmath884 finally , we consider the case where @xmath885 . by replacing @xmath310 by a smaller constant if necessary , we may assume that @xmath886 for all @xmath313 $ ] . then we have @xmath887 , which yields a conclusion . m. bourgoing , _ viscosity solutions of fully nonlinear second order parabolic equations with @xmath888 dependence in time and neumann boundary conditions_. discrete contin . 21 ( 2008 ) , no . 3 , 763800 . m. bourgoing , _ viscosity solutions of fully nonlinear second order parabolic equations with @xmath443 dependence in time and neumann boundary conditions . existence and applications to the level - set approach_. discrete contin . 21 ( 2008 ) , no . 4 , 10471069 . e. r. jakobsen and k. h. karlsen , _ continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations _ , electron . j. differential equations ( 2002 ) , no . 39 , 10 pp . ( electronic ) .

we describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non - monotone second - order geometric equations arising in front propagation problems . our method is based on some lower gradient bounds for the solution . these estimates are crucial to obtain regularity properties of the front , which allow to deal with nonlocal terms in the equations . applications to short time uniqueness results for the initial value problems for dislocation type equations , asymptotic equations of a fitzhugh - nagumo type system and equations depending on the lebesgue measure of the fronts are presented .

You are an expert at summarizing long articles. Proceed to summarize the following text: recent cmb observations @xcite indicate that the early universe has passed through an inflationary period with hubble parameter @xmath9 . the standard theoretical setup for inflationary models generically involve some scalar inflaton fields slowly rolling down their potential . the slow - roll that is needed to ensure a resolution to the problems of standard big bang cosmology , and consistency with the cmb results , demands @xmath10 and @xmath11 to be smaller or of order @xmath12 @xcite . in the context of simple single scalar models with potential @xmath13 , @xmath14 and @xmath6 are a measure of flatness of the potential and , specifically , @xmath15 , where @xmath16 is the effective mass of the inflaton field . therefore , the theoretical framework invoked for inflationary model building should also provide mechanisms to protect the potential and its flatness against quantum and/or quantum gravity corrections . in physics models we generically associate smallness and protection of a quantity like @xmath6 with an approximate symmetry , such that when the symmetry is exact this parameter ( here the effective mass of inflaton ) is zero . supersymmetry , for example , can be such a symmetry . it turns out that in the presence of gravity , as in our case where we are dealing with inflation models , symmetries protecting @xmath6 are all broken , inducing an inflaton mass term of order the hubble scale @xmath17 of the background . we hence end up with an order one @xmath6 , i.e. , the @xmath6-problem . from the above discussion it is seen that the @xmath6-problem may appear in two ways : in a top - down approach where we invoke a theory of quantum gravity like string theory for inflationary model building . or in a bottom - up approach where we take the usual field theory setup of einstein gravity plus scalar inflaton fields , assuming that this framework is valid up to planck scale @xmath18 . in the top - down approach the @xmath6-problem appears as a classical ( not loop ) effect , usually due to the interaction of the inflationary sector with the moduli stabilization sector , see , e.g. , @xcite . in these models it turns out to be easy to have small @xmath14 with controlled back - reactions on it , but @xmath6 receives order one corrections . intuitively , in these top - down setups , the @xmath6-problem can be understood as follows . with a vacuum energy of order @xmath19 all scalars , including the inflaton , will be endowed with soft masses of order @xmath20 since the supersymmetry breaking scale is not below @xmath17 . in this work , however , we will focus on the @xmath6-problem in the bottom - up approach . in the bottom - up approach the @xmath6-problem arises from _ quantum _ loop corrections to the tree - level graviton - scalar - scalar vertex . despite being non - renormalizable , one can still apply the ( wilsonian ) effective field theory techniques to the einstein gravity theory and consider loop corrections , e.g. see @xcite . in the presence of a scalar field @xmath21 minimally coupled to einstein gravity , as in generic inflationary models , these corrections at one loop level generically involve a @xmath22 term , a term whose presence was noted long ago @xcite . as we will review below , such a term is quadratically divergent and in the one loop effective action appears as @xmath23 where @xmath24 is the uv cutoff of the theory and @xmath25 is an order one coefficient . assuming a planckian cutoff scale , @xmath26 , in an inflationary background where @xmath27 , this term yields a correction of order @xmath28 to the inflaton mass , causing the @xmath6-problem . the @xmath6-problem , or the hubble scale mass term for the effective inflaton field , seems quite generic and one may put forward the idea of _ kinematically _ reducing the coefficient @xmath29 . in this letter we explore this possibility . one obvious possibility is to choose the cutoff @xmath24 , the scale where quantum gravity effects become important , to be one or two orders of magnitude smaller than @xmath18 @xcite . in this case the coupling constant of gravitons will be reduced like the momentum uv cutoff and the @xmath6 problem persists . alternatively , one may explore the idea that @xmath25 is a kinematical factor that for some reason is not of order one , while the bare cutoff @xmath24 is @xmath18 . in fact , similar suppressions are already very well known in the context of large @xmath0 gauge theories @xcite : the nonplanar part of a given feynman diagram comes with powers of @xmath5 suppression compared to the planar part of the same diagram . as we will show similar analysis can be repeated for the theories involving large number of scalar fields minimally coupled to gravity . in particular , if we have @xmath0 number of light fields , lighter than @xmath30 , @xmath25 turns out to have a @xmath5 suppression factor . in a sense , as if , the diagram leading to is a nonplanar diagram . this observation is closely related to the species dressed gravity cutoff scale ideas discussed in @xcite , in light of which the @xmath31 result may be interpreted as dealing with a dressed cutoff @xmath32 while @xmath33 . inflationary models with many scalar fields have recently got attention in view of their success in providing a natural explanation for the smallness of the inflaton self - couplings ( the issue of steepness of the potential ) and for the super - planckian excursion of the effective inflaton in the field space @xcite . this idea is not exotic to string theory motivated inflationary settings where it is quite common to have an abundant number of fields / degrees of freedom with masses below the dressed cutoff @xmath30 , see , e.g , @xcite . even though in some of these setups , like n - flation , the individual field excursion is greater than the dressed uv cutoff , some , like gauged m - flation @xcite or multiple m5-branes inflation @xcite , remain immune to thisbeyond - the - cutoff problem . in this work we examine the above proposed @xmath5 resolution to the @xmath34-problem . we assume that there is a hierarchy of scales between @xmath17 , the dressed gravity cutoff @xmath35 and @xmath18 : @xmath36 which is easily achieved by e.g. @xmath37 . this provides a window where one can safely use the standard techniques of quantum field theory and effectively deal with a system that could be described by einstein - hilbert gravity , the inflaton sector , and other heavy remnants of the theory of quantum gravity whose masses @xmath38 falls below the new gravity cutoff , i.e. @xmath39 . the outline of this work is as follows . we consider a system of @xmath0 light scalars minimally coupled to einstein gravity and work out basic feynman rules of the theory and compute the quadratically - divergent part of the one loop contributions to the graviton propagator and graviton - scalar - scalar vertices . we show that one loop graviton two - point function has a linear @xmath0 parametric dependence while the graviton - scalar - scalar vertex has no @xmath0 dependence . therefore , if we ( re)normalize the graviton two - point function , the vertex will have a factor of @xmath40 . this latter leads to @xmath31 ( _ cf . _ ) . we discuss how this can resolve the @xmath6-problem in the context of many - field models like n - flation @xcite or m - flation . consider the action of @xmath0 scalars minimally coupled to gravity @xmath41 where @xmath42 is the number of scalar fields , and summation over repeated @xmath43 indices is assumed . one or some of these scalars play the role of inflaton(s ) , and @xmath44 could be any potential that realizes slow - roll inflation at the classical level , while the rest exhibit possible remnants of the underlying quantum gravity theory . we assume the mass of these remnants to be below our dressed cutoff @xmath45 . the action once quantized will receive all possible corrections compatible with the symmetries of the system , in particular an @xmath46 correction which appears at one loop level . as discussed if this term comes with an order one coupling can cause the @xmath34-problem . in what follows we show by carrying out explicit one loop calculations involving gravitons , that this term is suppressed by factors of @xmath5 , providing a setting to resolve the @xmath6-problem in the context of multi - field models of inflation . in our analysis in this section and section 3 we ignore the loops involving scalar self - interactions . as we will discuss in the discussion section these diagrams do not change our main result . to perform the one loop analysis , as in any quantum field theory , we need to work out basic tree level feynman diagrams of propagators and interaction vertices . to do so for the gravity sector , following @xcite , we introduce the tensor densities , @xmath47 to bring the gravitational part of the action to the goldberg s form @xcite @xmath48 we will decompose the density metric to the flat part and the deviation from the flat space part , @xmath49 where @xmath50 is defined as @xmath51 the inverse of @xmath52 is given by @xmath53 where on the r.h.s . the indices are raised and lowered by the flat minkowski space metric @xmath54 . perturbing the action up to third order in @xmath55 , one obtains @xmath56 where @xmath57 note that @xmath58 is written to lowest order in @xmath55 and so it is independent of @xmath55 . grp ( 0,0)(0,5 ) ( 40,30 ) + spr ( 0,5)(0,5 ) ( 40,30 ) + hpp0 ( 0,10)(0,20 ) ( 80,80 ) + hhpp ( 0,20)(0,20 ) ( 80,80 ) + hhh0 ( 0,20)(0,0 ) ( 80,80 ) from this interaction term , and dropping the last term in which is inessential for our purposes , one can read the vertex @xmath59 to be @xmath60 , where @xmath61 and @xmath62 are two external four - momenta on the @xmath63 particles . to work out the basic feynman graphs of the theory we need to gauge - fix the diffeomorphism invariance . this may be done through gauge fixing term @xmath64 in figure [ hphiphi1 ] we have plotted the basic feynman graphs in this gauge . having the tree level theory we now proceed to the one loop analysis and revisit the one loop propagator calculations as well as graviton - scalar vertex . since we are interested in the quadratically divergent term , it is appropriate to use cutoff regularization ; dimensional regularization , which is very well suited in capturing the logarithmic divergences and already used in @xcite , can not be employed here . hh1 ( 10,10)(30,0 ) ( 100,100 ) hh2 ( 30,0)(10,0 ) ( 100,100 ) hh3 ( 10,10)(30,10 ) ( 100,100 ) hh4 ( 30,10)(10,10 ) ( 100,100 ) t90 ( 10,0)(10,0 ) ( 80,80 ) as depicted in figure [ 1-loop - gr ] , there are five feynman diagrams contributing to the one loop graviton propagators . the first two are coming from the pure einstein gravity sector and the other three involve scalar fields running in the loop . since we are only interested in the @xmath0 dependence of the diagrams we focus on the ones with scalar fields in the loop . in electrodynamics , the gauge invariance enforces the photon self - energy to be transverse . this reduces the degree of divergence from two to zero . however , in gravity the gauge invariance does not do so . it only relates the three diagrams that involve the scalar field in the loops . thus the quartic divergence , @xmath65 , which corresponds to the cosmological constant term , @xmath66 , remains . this difference is due to the fact that @xmath66 is still gauge - invariant whereas @xmath67 is not . this is the famous cosmological constant problem which we are not intended to deal with here . next - to - leading divergent part diverges like @xmath68 and this is the part that renormalizes the graviton propagator . in particular , the diagram that involves two graviton - scalar three - vertices is @xmath69 as long as @xmath70 . the diagram involving the graviton - scalar four vertex is of the form @xmath71 the leading part of this integral is quartic in the uv cutoff and just renormalizes the cosmological constant . the next - to - leading order is quadratic in the uv cutoff and is proportional to @xmath72 , assuming that the masses @xmath38 are all much smaller than the undressed ( bare ) uv cutoff @xmath24 , which is taken to be @xmath18 . the last diagram in figure [ 1-loop - gr ] has only a quartic divergence and does not contribute to the renormalization of the graviton propagator at all . thus we see that the one loop graviton propagator is proportional to number of fields @xmath0 , as well as @xmath73 . this term may be viewed as the correction to the newton constant or @xmath18 . that is , the quantum gravity effects become important when this term becomes of the same order as the classical tree level value . this happens if the cutoff momentum @xmath24 is of order @xmath74 which is the _ species dressed uv cutoff_. besides this `` perturbative '' argument , the fact that one should use this reduced cutoff instead of @xmath18 in presence of large number of species has also been backed up by black hole physics and hawking radiation from black holes in theories with large number of light species @xcite . one may also compute one loop correction to the scalar propagator . again there are diagrams involving only scalars and two diagrams involving gravitons . it is immediate to see that the latter two diagrams have no parametric dependence on the number of scalars @xmath0 . as depicted in figure [ scalar - gravitons - exchange ] , there are three diagrams contributing to scalar - graviton vertex at one loop level . [ c]cccccccccccccccccccccccc hpp1 ( 40,0)(40,0 ) ( 100,100 ) hpp2 ( 0,0)(0,0 ) ( 100,100 ) hpp3 ( 30,30)(40,0 ) ( 100,100 ) & & & & & & & & & & & & & & & & & & & & & & & our interest in these diagrams are twofold : _ i ) _ we read the correction to the tree level graviton - scalar three - vertex depicted in figure 1 and , _ ii ) _ compute the coefficient of the @xmath46 term which appears at one loop level from these diagrams . the details of the loop calculations , which are straightforward , are given in the appendix , here we just quote the result . since we are mainly interested in the @xmath0 dependence of the loop expressions we only focus on that issue here . there are no @xmath0 dependence appearing in any of the diagrams in figure [ scalar - gravitons - exchange ] , and these diagrams , compared to the tree level results , are proportional to @xmath75 . this in particular implies that the coefficient in front of the effective @xmath46 term ( up to numeric factors of @xmath76 ) is proportional to @xmath77 and if @xmath24 is the species dressed cutoff @xmath78 this term is suppressed by the number of light species @xmath0 . to summarize , the one loop correction to graviton propagator is dressed with a power of @xmath0 , while the graviton - scalar - scalar vertex is @xmath0 independent . this result is very similar to the well - established t hooft @xmath5 expansion @xcite , that if we normalize the two point function to one , the interaction term has @xmath5 suppression . so far we have shown that in a theory with @xmath0 number of species with masses lighter than dressed cutoff @xmath78 , the coefficient of the @xmath46 term generated at one loop level is @xmath80 , where @xmath81 is an order one @xmath82-number . the above analysis was carried out in a flat space background and should be revisited for inflationary ( almost de sitter ) backgrounds . it is readily seen , however , that the basic argument behind the factors of @xmath0 does not depend on the background geometry . also , the presence of the new scale @xmath83 should not affect our argument in any qualitatively important way . we still expect that species lighter than the high energy cutoff @xmath78 in general will contribute to @xmath0 . the only change concerns the very lightest species , with masses roughly below the hubble scale @xmath17 , where momenta at super - hubble scales will not contribute , as we will now show . we recall that the equation of motion of a free massive scalar on an inflationary background is @xmath84 where dot denotes derivative with respect to the cosmic comoving time and @xmath85 is the scale factor . the relevant observation is that the modes contributing to @xmath0 are the _ quantum modes _ , e.g. those with oscillatory ( as opposed to exponentially damping or growing ) behavior . to be able to solve the above equation , let us drop the time - dependent piece @xmath86 for the moment and consider the equation @xmath87 whose solution is of the form @xmath88 with @xmath89 . to have a quantum mode @xmath90 should have an imaginary part . this latter implies that @xmath91 . note that this result is @xmath92 independent and that addition of the @xmath86 term will only slightly modify this result , as it is positive definite : all the modes with @xmath93 , regardless of their @xmath92 , are always quantum modes , while modes with @xmath94 are _ classical _ for large wavelengths ( i.e. super - hubble physical momenta @xmath95 ) , and quantum mechanical for sub - hubble momenta . note also that the damping coefficient @xmath96 is removed in the process of canonical quantization as canonical momentum conjugate to @xmath63 is @xmath97 . this is of course the standard established result in inflationary cosmic perturbation theory and quantum field theory on curved ( de sitter ) space time @xcite . since we are only interested in the uv behavior of the loop integrals , we can instead of integrating over @xmath92 all the way from zero to @xmath98 restrict the integral to go from @xmath17 to @xmath99 . in this way we avoid the unnecessary complication with super - hubble modes . in summary , all the modes lighter than @xmath99 , with both super - hubble or sub - hubble masses , contribute to the @xmath0 in the loop integral . in other words , as long as @xmath100 , @xmath0 is the same for inflationary and flat space and @xmath101 where @xmath0 is the number of species lighter than the cutoff , @xmath99 . in particular , the coefficient of the @xmath46 term generated at one loop will become @xmath80 , with @xmath81 of order one . we are now ready to address the @xmath6-problem . to this end , we recall that the one loop corrected action is @xmath102 hence , the slow - roll parameter @xmath103 , where @xmath104 , is@xmath105 to have a successful slow - roll inflationary period we usually demand @xmath106 , and if we assume @xmath107 , quantum corrections to @xmath6 will be suppressed enough for @xmath108 . so that this effective inflaton field is canonically normalized . the @xmath46 term , being quadratic in the @xmath63 s , will not receive any normalization factors due to the @xmath0 scaling relating canonically normalized inflaton and the original fields . ] the bottom - up @xmath6-problem seems quite generic to all models of inflation that involve a scalar field minimally coupled to gravity . even if the parameters of the inflaton potential are chosen meticulously at tree - level , the loop corrections that arise from interactions of the graviton with the scalar field create the quadratically divergent conformal mass term which leads to the @xmath6-problem , if the uv cutoff of the theory is of order @xmath18 . this kind of @xmath6-problem is of course different from the `` top - down '' @xmath6-problem arising within the string theory setups in which the volume modulus stabilization often resurrects the @xmath6-problem . the precision that should be enforced upon the tree - level parameters are often not needed to sustain inflation , but to match the observed density of perturbation @xcite . in this letter we examined the possibility of circumventing the @xmath6-problem , in the former sense discussed above , in many - field models of inflation that are minimally coupled to gravity . these many - fields whose masses are assumed to be smaller than @xmath3 , have a natural appearance in effective low energy field theory description of quantum gravity models . as we argued @xmath109 will resolve the @xmath6-problem . even though it is not necessary for our argument , the scalar fields should be non - interacting to realize inflation . one example of such many - field models is n - flation @xcite which has the @xmath110 symmetric potential @xmath111 . as stated above , a few hundred scalar fields will be enough to circumvent the @xmath6-problem . like its chaotic counterpart , the mass parameter @xmath16 has to be @xmath112 which is smaller than @xmath45 unless one resorts to an unnaturally large number of scalar fields , _ i.e. _ @xmath113 . this of course comes at the price of exposing the model to quantum instability of the type discussed in @xcite . namely , the quantum fluctuations of these light fields may dominate over the classical evolution of the inflaton . assisted model with quartic potential @xmath114 is another possibility . to have an observationally viable model , the effective coupling must be around @xmath115 and thus with @xmath116 , one needs around @xmath117 scalar fields . this scenario also suffers from the above quantum instability with such a large number of massless scalar fields . another disadvantage of both these two models is that the physical excursion of the fields is larger than @xmath45 during the required @xmath118 e - folds of inflation . another explicit example is m - flation @xcite or its gauged version @xcite where the inflaton potential is constructed by three @xmath119 non - commutative hermitian matrices whose action is invariant under @xmath120 . the classical dynamics is simplified considerably in the @xmath121 sector where these three scalar fields are proportional to the generators of the @xmath121 algebra . gauged m - flation , in addition to the above ingredients , has an extra @xmath122 field , associated with `` center of mass '' @xmath123 . m - flation in the @xmath121 sector besides the inflaton field contains some number of `` spectator fields '' which do not contribute to classical inflationary trajectory while can be excited quantum mechanically and appear in the loops . for the gauged m - flation there are @xmath124 such scalar modes and @xmath125 massive vector modes . these modes have a hierarchical spectrum , _ i.e. _ they can be lighter or heavier than the hubble scale @xmath17 , for the explicit masses see @xcite . not all these modes are light enough to be counted in the dressed cutoff . as discussed in @xcite , the number of `` contributing species '' @xmath126 varies between @xmath127 and @xmath128 , depending on the region of the potential inflation happens . hence , the species dressed uv cutoff is @xmath129 . consequently the @xmath22 term is suppressed by a factor of @xmath130 and could be safely ignored . gauged m - flation could be motivated from the branes dynamics in an appropriate flux where the above matrices correspond to three of the perpendicular directions of a stack of @xmath0 d3-branes which are scalars in the adjoint representation of the @xmath120 . as such , although the quantum " @xmath6-problem is resolved for m - flation , embedded within string theory , one should still deal with the `` stringy @xmath6-problem '' ( cf . introduction for further references and discussions ) . finally , the term leading to the @xmath6-problem , @xmath4 , is a one - loop but marginal operator . one may naturally worry about other loop corrections , that enhancement factors of @xmath0 will dominate over @xmath131 suppressions . it is straightforward to show that the largest such @xmath0 enhancement factor appears in the graviton two point function ( at higher loops ) for which this factor is @xmath132 , where @xmath133 is number of loops . all the other diagrams will be of the form @xmath134 with @xmath135 . hence our one loop result seems to be also valid to all orders . a.a . was supported by the gran gustafsson foundation . . is supported by the swedish research council ( vr ) and the gran gustafsson foundation . we would like to thank liam mcallister for helpful discussions . now let us focus on all the diagrams which may generate the @xmath22 term , or equivalently @xmath136 , at the one - loop order . the first one , is given by the left diagram in figure [ hphiphi1 ] , where a graviton exchange between two external scalars modifies the scalar form factor . the diagram is proportional to @xmath137 \frac{i(\eta^{\alpha\gamma}\eta^{\beta\kappa}+\eta^{\alpha\kappa}\eta^{\beta\gamma}-\eta^{\alpha\beta}\eta^{\gamma\kappa})}{(p - k)^{2}+i\epsilon } \right . \nonumber\\ & \left . \frac{i}{k^{2}+i\epsilon } \frac{1}{m_{\mathrm{pl } } } \left [ p^{\prime}_{\gamma}(k+q)_{\kappa}-\eta_{\gamma\kappa}p^{\prime}\cdot ( k+q)\right ] \frac{1}{(k+q)^{2}+i\epsilon}\right . \nonumber\\ & \left . \frac{1}{m_{\mathrm{pl } } } \left [ k_{\mu}(k+q)_{\nu}-\eta_{\mu\nu } k\cdot(k+q)\right ] \right ) .\end{aligned}\ ] ] what we are interested in is the divergent part of the above integral which multiplies the generated @xmath22 term . there are six momenta in the numerator , only four of which are internal . thus the integral will be divergent as @xmath138 the other diagram that contributes to the @xmath139 vertex is the middle one in figure [ scalar - gravitons - exchange ] . to estimate the leading divergent part of this diagram , one should note that the vertex that involves three @xmath140 is proportional to @xmath141 . more specifically , @xmath142 \ , .\nonumber\end{aligned}\ ] ] sum is over symmetrization on the index pairs @xmath143 , @xmath144 and @xmath145 and also the six permutation done over momentum index triplets @xmath146 , @xmath147 and @xmath148 . again , this diagram has six momenta in the numerator , two of which are external . thus the divergent part of diagram behaves as times the terms among which @xmath22 exists . finally , let us look at the right one - loop diagram in fig . ( [ scalar - gravitons - exchange ] ) . the vertex @xmath149 is proportional to @xmath150 . in more details : @xmath151 \,.\nonumber\end{aligned}\ ] ] the diagram contains two propagators in the denominator and four momenta in the numerator , two of which are external . thus the leading correction will be , again , of order times the terms among which @xmath22 exists . note that besides the conformal mass term , there are also other higher dimensional operators , whose explicit coefficients could be obtained by exactly calculating the amplitudes . for example , from usual tensorial analysis , terms like @xmath152 are expected to be generated from such one loop diagrams . however , all these terms are suppressed by extra powers of the cutoff and also by slow - roll parameters in an inflationary background . the other point which is worth mentioning is that the existence of higher order self - interactions for the scalar fields will not disturb our argument . for example inclusion of mass term in the potential for the inflaton , will modify the scalar field propagator and also add corrections proportional to @xmath153 in the @xmath154 vertex . such terms would at most introduce correction of order @xmath155 to the leading contribution . thus , as long as @xmath156 , the effect of such terms are very small . other forms of potential for the scalar field induce vertices that lead to more loops whose effect is more suppressed in comparison with the one - loop diagrams . b. s. dewitt , `` quantum theory of gravity . 1 . the canonical theory , '' phys . rev . * 160 * , 1113 - 1148 ( 1967 ) ; `` quantum theory of gravity . 2 . the manifestly covariant theory , '' phys . rev . * 162 * , 1195 - 1239 ( 1967 ) . a. ashoorioon , h. firouzjahi , m. m. sheikh - jabbari , `` m - flation : inflation from matrix valued scalar fields , '' jcap * 0906 * , 018 ( 2009 ) . [ arxiv:0903.1481 [ hep - th ] ] ; `` matrix inflation and the landscape of its potential , '' jcap * 1005 * , 002 ( 2010 ) , [ arxiv:0911.4284 [ hep - th ] ] .

we observe that the dominant one loop contribution to the graviton propagator in the theory of @xmath0 ( @xmath1 ) light scalar fields @xmath2 ( with masses smaller than @xmath3 ) minimally coupled to einstein gravity is proportional to @xmath0 while that of graviton - scalar - scalar interaction vertex is @xmath0 independent . we use this to argue that the coefficient of the @xmath4 term appearing at one loop level is @xmath5 suppressed . this observation provides a resolution to the @xmath6-problem , that the slow - roll parameter @xmath6 receives order one quantum loop corrections for inflationary models built within the framework of scalar fields minimally coupled to einstein gravity , for models involving large number of fields . as particular examples , we employ this to argue in favor of the absence of @xmath6-problem in m - flation and n - flation scenarios . + @xmath7 _ institutionen fr fysik och astronomi uppsala universitet , + box 803 , se-751 08 uppsala , sweden _ + @xmath8 _ school of physics , institute for research in fundamental sciences ( ipm ) , + p.o.box 19395 - 5531 , tehran , iran _ +

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Proceed to summarize the following text: the first extensive survey of the far - infrared sky was made by the _ infrared astronomy satellite _ ( iras ) launched in 1983 , more than two decades ago . iras provided point source catalogs , as well as infrared sky maps , for almost the entire sky , revealing the infrared view of the universe . the iras products became a standard dataset , not only for infrared astronomy but also for many other fields . with the widespread use of the iras dataset , observations in the infrared are now considered an irreplaceable tool . compared with the datasets at other wavelengths , the iras dataset appears rather shallow , although it provides an unbiased and wide area coverage survey . however , a next - generation infrared dataset is needed to push the frontier of astrophysics . the far - infrared surveyor ( fis ) on the akari satellite @xcite was developed to provide this new dataset in the far - infrared region , taking advantage of recent technology , including the cryogenics @xcite . the fis was designed to perform an all - sky survey in the far - infrared region with higher spatial resolution and higher sensitivity than the iras . a combination of the infrared camera ( irc ) @xcite on the akari satellite also enables a wider wavelength coverage than iras . the _ spitzer space telescope _ ( sst ) @xcite , which is used extensively in the observation of various objects in the infrared region , has brought new insights into the universe with its high spatial resolution and high sensitivity . fis is a complementary instrument because of its capability of covering wide areas , and its multiple photometric bands . furthermore , the fis has the advantage of allowing far - infrared spectroscopy with a fourier transform spectrometer ( fts ) . the _ infrared space observatory _ ( iso ) @xcite demonstrated the potential of spectroscopy in the infrared region for diagnostics of the interstellar medium and radiation field . adopting two - dimensional array detectors , the fts of fis works as an imaging fts . consequently , it affords high efficiency observations of the spatial structure in spectra . this paper describes the design and operation of fis in two sections , and then discusses its flight performance and advantage . the fis is a composite instrument , consisting of a scanner and a spectrometer . adopting the newly developed large format array detectors , fis achieves high spatial resolution and sensitivity . fig.[fig : fis ] shows a picture of fis during final integration ; the top cover is removed to show the interior . fig.[fig : fis_optics ] illustrates the fis optical design . fis is about 50 cm along the major axis and weighs about 5.5 kg . to reduce the size and resources required , the scanner and spectrometer share some optical components and detector units . rotating the filter wheel selects the appropriate function . light coming from the telescope is focused near the fis input aperture . after passing through the input aperture , the beam is bent into the fis optical plane and led to the collimator mirror . the collimated beam goes to the filter wheel , which selects the scanner or the spectrometer by choosing the filter combination . in the scanner mode , it selects a the combination of an open hole and a dichroic beam splitter . the beam passes through the hole , and is reflected by a flat mirror to the dichroic filter on the filter wheel . longer wavelength photons ( @xmath1 ) pass through the filter and shorter wavelength photons are reflected . camera optics focus the beams onto the detector units according to their wavelength . a fourier transform spectrometer ( fts ) serves as the spectroscopic component . a polarizing michelson interferometer , the so - called martin puplett interferometer @xcite , is employed in the fts optics . this type of interferometer requires input and output polarizers to provide linear polarization . these two polarizers are on the filter wheel , and are selected in the spectrometer mode . the collimated beam is reflected by the input polarizer to the interferometer , while the remaining component is absorbed by a blocking wall . a beam splitter divides the linearly polarized beam into two beams . the beam splitter is also a polarizer , whose polarizing angle is rotated by 45 degrees relative to the incident linear polarized beam . all polarizers are wire - grid filters printed on thin mylar films supplied by qmc instruments ltd . the divided beams are reflected by roof - top mirrors ; one is fixed and the other is movable to change the optical path difference between two beams . after that , the beam splitter recombines the two beams , and the interfered beam goes to the output polarizer . finally , the elliptically polarized beam is separated into two axis components by the output polarizer . each component is focused on the corresponding detector unit . fis has a cold shutter at the input aperture to allow measurement of the dark current of the detector . for the calibration of the detectors , there are three light sources in the fis housing . one is in the light path of the scanner , and the others irradiate the front of each detector . the intensity and emitting duration of each lamp are independently controlled by commands independently . these lamps can also simulate the light curve of the point source in the all - sky survey . since the simulated light curve is quite stable and reproducible , the detector responses to a point source can be monitored at any time . in addition to these calibration lamps , there is a blackbody source , whose temperature is controllable up to 40k , opposite the interferometer input that is used to check that operation of the fts . furthermore , to improve the transient response of the long wavelength detector , a background light source , with controllable power , is placed on the detector unit to continuously irradiate detector pixels . at the current setting , the incident power corresponds to a sky brightness of @xmath2100 mjy / sr . we use two types of photoconductive detector array to cover the far - infrared wavelengths ( 50 180 @xmath0 ) . one is the direct - hybrid monolithic ge : ga array @xcite for the shorter wavelength of 50 to 110 @xmath0 ( labeled sw ) , which was developed by the national institute of information and communications technology ( nict ) . the ge : ga monolithic array is bump - bonded by indium with silicon - based cryogenic readout electronics @xcite . the other is a compact stressed ge : ga array for longer wavelengths of 110 to 180 @xmath0 ( labeled lw ) , which is an evolved version of a previous model @xcite . the lw detector also employs cryogenic readout electronics . all ge : ga chips were supplied by nict . there are two arrays in each detector unit for the different photometric bands wide - s and n60 for the sw detector and wide - l and n160 for the lw detector . the array formats are @xmath3 , @xmath4 , @xmath5 , and @xmath6 for wide - s , n60 , wide - l , and n160 , respectively , as shown in fig.[fig : array_format ] . several pixels did not work properly after fabrication ( shown by hatching in the figure ) ; however , no additional bad pixels appeared after the launch . the fields - of - view ( fovs ) of wide - s and wide - l , n60 and n160 detectors overlap , although their coverage differs . the pixel scales are designed to be comparable to the telescope s diffraction limits . the arrays are rotated by 26.5 degrees relative to the scanning direction . because of this configuration , the width of the stripe swept out on the sky is reduced to about @xmath7 of the array width , although the spatial sampling grid becomes half of the pixel pitch in the cross - scan direction . the fovs of the sw and lw detectors were measured in space by observing point sources . the measured fovs are shifted from the designed position by about 1 arcmin in both in - scan and cross - scan directions referred to the telescope s boresight ( see top panel of fig.[fig : array_format ] ) . in the spectrometer mode ( bottom panel of fig.[fig : array_format ] ) , the misalignment of each array is larger than in the scanner mode , which limits the observational efficiency of spectral mapping . the distortion of the fovs due to the fis optics is evaluated by optical simulation . although the actual distortion and magnification factor of the fis optics do not match precisely , no significant discrepancy with the design is indicated . the four photometric bands of fis are defined by the combination of the optical filters and the spectral response of the detectors , as the incident photons reach the detectors through the optical filters . the collimated beam passes through two blocking filters that block the mid- and near - infrared photons contributed mainly by stars . the dichroic filter then divides the beam in the frequency domain : higher frequency photons ( @xmath8 @xmath9 in wavenumber ) are reflected and lower frequency photons are transmitted . finally , two filters on the front of each detector shape the photometric bands . four photometric bands cover the 50 to 180 @xmath0 wavelength region ; two are wide bands ( wide - s and wide - l ) and the other two are narrow ( n60 and n160 ) . in the spectrometer mode , the dichroic beam splitter is replaced by a combination of three polarizers . the other optical filters are the same as for the scanner mode . only the wide bands ( wide - s and wide - l ) are used for spectroscopy . the spectrometer could in principle take interferograms with narrow bands , although some of the outer pixels of the arrays will vignette the telescope beam . fig.[fig : fis_bands ] shows the fis system s spectral response in the photometric mode . the responses are normalized at the peak . the filters and optics were measured in an end - to - end system configuration at room temperature , although the narrow band filters for n60 and n160 were measured individually at cryogenic temperature , since their properties depend on temperature . the spectral responses of the detectors were evaluated by a spectrometer , each pixel of the detector array having a different spectral response . this difference is larger in the lw detector , due to non - uniformity of the effective stress on the ge : ga chips . the plot in fig.[fig : fis_bands ] provides typical profiles . the spectral response of each pixel must be used to calculate precise pixel - to - pixel color corrections . the spectral response can also be measured by the fis spectrometer itself , using the internal and external blackbody sources at different temperatures . in orbit , we confirmed the system spectral response of fis in this manner . the spectra of the internal blackbody source taken in orbit are within @xmath10 of those measured in the laboratory after scaling the responsivity . since the filters and optics are considered stable , the system spectral response of each photometric band is expected to be similar to that measured in the laboratory as shown in fig.[fig : fis_bands ] . the blocking filters are expected to avoid leakage of the mid- and near - infrared photons . the blocking efficiency required from the scientific observations is @xmath11 at 10 @xmath0 and @xmath12 at 0.5 @xmath0 , which are realized optimally , and will in future be checked by observations of well - known stars . fis is designed primarily to perform an all - sky survey with four photometric bands . the goal is to observe the entire sky for at least two independent orbits . the first half year following the performance verification phase is dedicated to observations for the all - sky survey . during the remaining life , supplemental survey observations will fill in the incomplete sky areas , sharing the observation time with dedicated science programs . during the survey , the detectors are read out continuously with a constant sampling rate for each array , corresponding to about three samples in a pixel crossing time . detectors are reset to discharge the photo current at appropriate intervals of about 2 sec nominally , 0.5 sec for bright sky , and for each sampling ( correlated double sampling : cds ) at the galactic plane , whose reset intervals correspond to about 26 and 45 ms for the sw and lw detectors , respectively . calibration flashes with illuminator lamps are inserted periodically every minute while keeping the shutter open to trace the detector responsivity . near the ecliptic poles , where the detectors sweep frequently , a one - minute calibration sequence with the shutter closed is executed for nearly every orbit to monitor the long term trend of the detector responsivity . before launch of the akari satellite , the project science team members selected the core programs and target sources were selected . in addition to the core programs , some portion of the observation time was open to the community . for the scientific programs and the open time proposals , the akari instruments operate in a pointing mode . the fis observations are categorized in three major astronomical observation templates ( aots ) , two for photometry and one for spectroscopy . details of each aot are described below . the fis01 template is used to observe point - like or small scale sources . in this aot , for one pointing observation , the detectors sweep the sky two times in round trips . between two round trips , the scan path is shifted by either a few pixels ( 70 arcsec ) or half of the fov ( 240 arcsec ) in the cross - scan direction , which is selectable . the other adjustable parameters are the reset interval and scan speed . the scan speed is selectable from 8``/sec or 15''/sec , which are nearly 14 to 30 times slower than that of the all - sky survey . the detection limits should be improved by a factor of the exposure time or more using the charge integration amplifier . furthermore , using a slower scanning speed reduces the transient response effects of the detector response . an observation sequence takes about 30 minutes as shown in the top panel of fig.[fig : fis_aot ] . during about 10 minutes of maneuvering to the spacecraft to point toward the target , the standard calibration sequence is executed with the shutter closed , i.e. , measuring dark current , illuminating the calibration lamps continuously for about two min and flashing the calibration lamps several times . after opening the shutter and waiting 210 sec , the scanning sequence begins and continues for about 12 minutes . then , the shutter is closed again and the maneuver to the all - sky survey begins with the post observation calibrations . during scanning observations , at the scan turning points , the shutter is closed for 30 sec and the calibration lamps are turned on for about eight sec to monitor the drift of the detector responsivity during the pointing observation . fis02 is the template for wide area mapping . this aot executes only one round trip . the detectors sweep a longer strip than for fis01 , but the detection redundancy is reduced . overlapping scans are critical for high quality wide area mapping . by selecting the 15" / sec scan speed , the strip length reaches over one degree . the calibration sequences in the pre- and post observation phase and at the turning point are the same as for fis01 . the middle panel of fig.[fig : fis_aot ] illustrates the observation sequence . fis03 is the template for spectroscopic observations . the bottom panel of fig.[fig : fis_aot ] illustrates the observation sequence . in this aot , the target is locked on the detectors . to use the fts , the optics are switched to fts mode by rotating the filter wheel during the maneuver to the target position . the sampling sequence changes for the fts and the movable mirror starts to operate . important parameters in fis03 are the spectral resolution and array selection . users choose from two spectral resolution modes : a full resolution mode and a low resolution mode ( named sed mode ) , with spectral resolution of 0.19 @xmath9 and 1.2 @xmath9 without apodization , respectively . taking one interferogram in the full resolution mode takes four times longer than that in the sed mode . consequently , in one pointing observation , 15 full resolution or 59 sed mode interferograms can be taken . the other important parameter is the array selection , which was added after launch , because the fovs of the two detectors are misaligned as shown in fig.[fig : array_format ] . depending on the position of the target on the detectors , there are three choices : nominal , sw , and lw positions . during the maneuver to the target position , the internal blackbody source is turned on at the proper temperature , with the shutter closed , and interferograms are taken as reference spectra . a short calibration sequence using the calibration lamps is also conducted at the pre- and post observation phase . after the observation , the observation mode changes to photometry mode for the all - sky survey , during the recovery maneuver to the all - sky survey . parameters for each aot are summarized in table [ tab : aot_list ] . fis can operate in parallel with the irc observations . since the fovs of fis are separated from the fovs of the irc by about half a degree , they can not observe the same target . nevertheless , taking data with fis is interesting in many cases as a serendipitous survey . the nominal operation of the irc is to make observations with long exposure times to provide deep images or spectra . since the fis photometry is designed for scanning mode , the pointed observations of the photometry are ineffective due to the narrow array formats and larger pixel scales . therefore , fis is operated in the spectrometer mode , if the sky is bright enough to detect signals by the fts . the observation sequence for parallel observation is the same as for fis03 . these parallel observations are productive , especially for the large magellanic cloud , where irc is making systematic surveys , as well as toward bright complex regions like the galactic plane . specific calibration sequences operates during the remaining irc oriented observations . once a day , the calibration sequence to evaluate detector transient response is executed . about once every three days , the calibration lamp stability is also measured . these calibration data are used to track long term trends in detector performance . in orbit , all fis functions are working as designed . the cold shutter and the filter wheel have operated constantly while in orbit , more than ten thousand times and a few hundred times , respectively . all the functions are controlled by the onboard electronics without any difficulty . in the following subsections , the fis flight performance is described . the point spread functions ( psfs ) of fis were measured in the laboratory using a pin hole source . widths of the measured psfs were almost consistent with those expected from the optical simulation . in orbit , the system psfs including the telescope system are constructed from observations of bright point sources . the psfs are similar to the laboratory measurements . as shown in fig.[fig : fis_psf ] , the psfs conform to the estimations from the optical model at more than the half maximum of the peak . the full widths at the half maximum of the psfs , derived from gaussian fitting , are summarized in table [ tab : performance ] . at the tails of the psfs , there are significant enhancements , whose power is about 30% of the total power . this extended halo is a cause for the degradation of source detection . in the spectrometer mode , the psfs are evaluated by slow scan observations of the point sources with the spectrometer optics , and are about @xmath13 wider than in the photometer mode . it is , however , possible to take spectra for each pixel and images with nearly one arc minute spatial resolution , simultaneously . an additional factor degrades the imaging quality of the sw detector , namely , considerable cross talk between pixels in both axes of the arrays . one possibility to explain this phenomenon is that the incident far - infrared photons diffuse into the monolithic ge : ga array during multiple reflection on the front and back surfaces of the detector substrate . furthermore , both the sw and the lw detectors show significant ghost signals . the reason for the ghost is , presumably , electrical cross talk in the multiplexer of the cryogenic readout electronics . in this case , the ghost appears in other arrays of the same detector . the images of the asteroid _ ceres _ observed by fis01 are shown in fig.[fig : fis_ghost ] , as an example . to enhance the effect of the cross talk , the color level has been adjusted . since the position and strength of the cross talk are stable , it should be possible to remove it from the original . another possible degradation of the image quality comes from the detector s transient response . for pointed observations , the detector scans a source , both forward and backward on the same pixel . the effect of transient response has been evaluated for each observation , and is not significant for slow scan observations . the readout method for the fis detectors is based on a capacitive trans - impedance amplifier ( ctia ) using newly developed cryogenic devices @xcite . although the ctia has a wide dynamic range , the linearity is rather poor , and the effective bias on the detectors drifts . in laboratory measurements , the relation between the output signal and the amount of the stored charge was well calibrated , and the relation is confirmed in orbit . therefore , the photo current can be accurately reproduced from the output signal . charged particle hits are another important influence on detector performance . about once a minute , charged particles hit a pixel , and in some cases , the responsivity drifts for between several seconds and several minutes . near the south atlantic anomaly ( saa ) , the hit rate of charged particles is too high to observe the sky signal . after passing the saa region , the detector responsivity increases significantly and relaxes gradually with a decay on the order of hours . to cure the effect of the saa , bias boosting is applied just after passing through the saa . by increasing the bias voltage to breakdown for a short time , the detector responsivity quickly relaxes to a stable level . finally , the pixel - to - pixel variations of the detector responsivity are shown in fig.[fig : fis_flat_field ] , which are the relative detector signals in the observation of flat sky . the unevenness of the detector responsivity could come from the non - uniformity of the effective detector bias , due to the offset of the readout electronics . this is particularly an issue for the lw detector , because of its small bias voltage . in addition , due to variations of the spectral response for each pixel in the lw detector , it has poor flatness as compared to the sw detector . for absolute calibration , several kinds of astronomical sources are used well - modeled objects such as asteroids , planets , stars and galaxies , as well as the spectra of bright ir cirrus or interplanetary dust emission . to calibrate the absolute flux from point like sources , the aperture photometry procedure must be defined , and then applied to the observations of asteroids , stars and galaxies with a wide range of fluxes . the relation between the signal and the source flux for several sources has a good linear correlation as shown in fig.[fig : abs_flux_cal ] . the uncertainties of the current signal to flux calibration are no more than 20% for the n60 and wide - s bands , and 30% and 40% for the wide - l and n160 , respectively . our goal is to reach an absolute calibration accuracy of about 10% in all bands . we will achieve this by mitigation of various image artifacts described in this paper , and through analyses of many repeated observations of our network of well - known calibration sources , that are performed continually . comparing the signals of the sky brightness measured by fis and dirbe on cobe , which provide a well calibrated infrared sky map , we can obtain the absolute calibration for sky brightness . the observations of bright ir cirrus regions with no significant small scale structure are compared with values measured by dirbe to make an absolute calibration . the resulting calibration , however , disagrees with the calibration derived from the point like sources by a factor of about two . the absolute calibration for diffuse sources still has a large uncertainty due to the difficulty of baseline estimation . furthermore , the contribution of detector transient response differs between the two calibration methods . according to the absolute calibration derived from the compact sources , the nominal detection limits evaluated from the signal - to - noise ratio of the detected sources are listed in table [ tab : performance ] . the performance of the pointed observation is demonstrated by @xcite . all the spectrometer functions work as they did in the laboratory . after tuning the control sequence of the movable mirror , interferograms of the internal blackbody source and sky were measured . the fourier transformed spectra of the internal blackbody source are consistent with that taken in the laboratory , within a 10% error after scaling the responsivity , which means that the laboratory optical performance is reproduced in space . the data reduction to reproduce source spectra is difficult because the interferogram is distorted by the detector transient response as described above . furthermore , a channel fringe in the interferogram also causes complications . through observations of well - known bright sources , line sensitivity and reproducibility of spectra were evaluated . the fts system performance is almost same as the estimates from the laboratory measurements . the line spectrum observed by the fts is shown in the fig.[fig : fts_line ] as an example . the spectral resolution of the fts in full resolution mode is about 0.19 @xmath9 , which agrees well with the expected value of 0.185 @xmath9 . interim detection limits of the fts derived for on - source pixels from observations of bright sources are roughly 20 jy , 50 jy and 100 jy for continuum spectra in wavenumber of @xmath14 @xmath9 , @xmath15 @xmath9 and @xmath16 @xmath9 , respectively , and , @xmath17 @xmath18 and @xmath19 @xmath18 for line emissions of [ cii](158 @xmath0 ) and [ oiii](88 @xmath0 ) , respectively , which are @xmath20 values for one - pointed observation . fis strives to provide an improved version of the all - sky survey performed by iras more than two decades ago . the spectral coverage of fis is extended to longer wavelength by the wide - l and n160 bands , which cover up to 180 @xmath0 . the longer wavelength coverage allows determination of the contribution of cold dust components , which perform important roles in the interaction of the interstellar medium and radiation field . the advantage of longer wavelength capability has been demonstrated by the isophot serendipity survey ( isoss ) of iso @xcite . although the sky coverage of isoss is about @xmath21 of the whole sky , the isoss 170 @xmath0 sky atlas is utilized in variety of fields , especially , related to galaxies and cold galactic sources . this precursor survey indicates the benefit of the longer wavelength bands in the fis all - sky survey . the high spatial resolution of the fis all - sky survey is a great advantage for source detection and detailed mapping . as shown in above , the spatial resolution of the fis , which is about 0.7 acrmin for the n60 and wide - s bands , and about 1 arcmin for the wide - l and n160 bands , is more than five times better than iras even in the longer wavelength bands . the higher spatial resolution comes from the progress of detector technology , although the 60 cm diameter telescope of iras is comparable with the akari telescope @xcite . these advantages of the fis all - sky survey are demonstrated by @xcite and @xcite . the point source flux levels at signal - to - noise ratio of five for one scan are listed in table [ tab : performance ] . we processed the observed data using the preliminary version of the data - processing pipeline for the all - sky survey data . we estimated the system sensitivity based on a series of observations of asteroids as calibration sources in the all - sky survey . we used sussextractor , which is a point source extraction and photometry software dedicated for the akari all - sky survey @xcite , to make the photometry of the asteroids . we derived noise levels by observing dark areas of the sky . the estimated flux levels are significantly degraded from the detection limits estimated prior to the launch ( see table 3 in @xcite ) . we found several causes for the degradation . firstly , we observed several types of excess noise in orbit . the current version of the pipeline successfully removed the effects of some types of noises . secondly , we also observe the response to point sources is smaller than we expected . this is partly due to the psfs described above , and partly due to the detector ac response . thirdly , we reduced the bias voltage for wide - l and n160 in orbit to stabilize the behaviour of detectors after irradiation of high - energy particles . furthermore , the actual source detection is degraded by various effects , e.g. frequent glitches due to high energy particles and low - frequency baseline fluctuation due to the change of the detector responsivity , in the data reduction process . the potential performance of 90 @xmath0 band ( wide - s ) , however , is higher than that of the iras 100@xmath0 band . unfortunately , the detection limits of the lw detector , which provides new wavelength bands , are also degraded . the potential performance of the longer wavelength bands is comparable to or better than the isoss . the galaxy list of isoss has a 170 @xmath0 completeness limit of about 2 jy @xcite , and is useful for a cross calibration of the all - sky survey . the performance of the all - sky survey evaluated from an initial mini - survey will be discussed by @xcite . the performance of the detailed observations using the pointing observation mode should be compared with the recent instrument mips on sst . mips has advantages in spatial resolution and sensitivity , as a result of its larger telescope , small pixel scales , and long exposure capability . the advantage of fis is that it has four photometric bands between 50 and 180 @xmath0 , whereas mips has only two bands ( 70 @xmath0 and 160 @xmath0 ) in the corresponding wavelength range . multi - band photometry with fis is effective for determining a spectral energy distribution @xcite . the performance of the slow scan observation is demonstrated by @xcite , who reported the detection limit at 90 @xmath0 ( wide - s ) achieves 26 mjy ( 3@xmath22 ) using the observations at the lockman hole . in the paper , they discuss about the source counts , and point out that the number of sources detected at 90 @xmath0 is significantly smaller at the faint end compared to the expected values from the model , which explains the mips source counts well . this implies that fis is a complementary instrument to mips in the sed coverage . in the main observation phase , the low cirrus region near the south ecliptic pole ( sep ) is observed by fis intensively with almost the same sensitivity at the lockman hole for about 10 square degrees @xcite . the deep survey near the sep will become a legacy survey of fis for extragalactic studies . the spectroscopic capability in the far - infrared region is a unique feature of fis in contrast to mips . previously , lws on iso was available for far - infrared spectroscopy . the wavelength coverage and the spectral resolution of lws in the grating mode are comparable with the fis spectrometer . the sensitivity of fis in the spectrometer mode is not so excellent as mention above . however , the spectrometer of fis has the advantage of high observational efficiency . since it is an imaging fts , fis can take spectra with arcminute spatial resolution . for example , fis could map the m82 galaxy with spectra in several pointed observations , which correspond to about one hour exposure time . furthermore , the spectroscopic serendipity survey by parallel observations is expected to provide a unique dataset . the far - infrared surveyor was designed to survey the far - infrared region with four photometric bands within 50 180 @xmath0 , with high spatial resolution and sensitivity . additionally , a spectroscopic capability was installed as an imaging fourier transform spectrometer . all functions of fis work very well in orbit . fis performance is demonstrated in the initial papers of this volume . the all - sky survey is performed continuously and should provide a new generation all - sky catalog in the far - infrared . in addition to the all - sky survey , a large area deep survey ( @xmath23 square degrees ) near the south ecliptic pole and many scientific programs are being executed . to bring out the potential of the fis instrument , the data processing methods are continuously being improved . since calibration data are accumulated constantly by the end of the mission life , the fis data quality should be substantially better than that listed here . the akari project , previously named astro - f , is managed and operated by the institute of space and astronautical science ( isas ) of japan aerospace exploration agency ( jaxa ) in collaboration with the groups in universities and research institutes in japan , the european space agency , and korean group . we thank all the members of the akari / astro - f project for their continuous help and support . fis was developed in collaboration with isas , nagoya university , university of tokyo , national institute of information and communications technology ( nict ) , national astronomical observatory of japan ( naoj ) , and other research institutes . the akari / fis all - sky survey data are processed by the international team which consists of members from the iosg ( imprerial college , uk , open university , uk , university of sussex , uk , and university of groningen , netherlands ) consortium , seoul national university , korea , and the japanese akari team . the pointing reconstruction for the all - sky survey mode is performed by the pointing reconstruction team at european space astronomy center ( esac ) . we thank all the members related to fis for their intensive efforts toward creating new frontier . m. cohen s contribution to this paper was partially supported by a grant from the american astronomical society . we would like to express thanks to dr . raphael moreno for providing flux models of giant planets . doi y. , et al . , 2002 , adv . in space research , 30 , 2099 doi y. , et al . , 2007 , , 00 , 000 fujiwara m. , hirao t. , kawada m. , shibai h. , matsuura s. , kaneda h. , patrashin m.a . , & nakagawa t. , 2003 , , 42 , 2166 jeong w - s . , et al . , 2007 , , 00 , 000 kaneda h. , kim w. , onaka t. , wada t. ira y. , sakon i. , & takagi t. , 2007a , , 00 , 000 kaneda h. , et al . , 2007b , , 00 , 000 kessler m.f . , et al . 1996 , , 315 , l27 martin d.h . , & puplett e. , 1969 , infrared phys . , 10 , 105 matsuhara , h. , et al , 2006 , , 58 , 673 matsuura , s. , et al . , 2007 , , 00 , 000 murakami h. , et al . 2007 , , 00 , 000 nagata h. , shibai h. , hirao t. , watabe t. , noda m. , hibi y. , kawada m. , & nakagawa t. , 2004 , ieee trans . devices , 51 , 270 nakagawa t. , et al . , 2007 , , 00 , 0000 onaka t. , et al . 2007 , , 00 , 000 savage r. , et al . 2007 in preparation shibai h. , 2007a , adv . in space research , accepted for publication shibai h. , 2007b , in preparation stickel , m. , klaas , u. , and lemke , d. , 2007a , , 466 , 831 stickel , m. , krause , o. , klaas , u. , and lemke , d. , 2007b , , 466 , 1205 suzuki t. , kaneda h. , nakagawa t. , makiuti s. , doi y. , & shibai h. , 2007 , , 00 , 000 werner m.w . , 2004 , , 154 , 1 lccc & * fis01 * & * fis02 * & * fis03 * + observation mode & & pointing + observation target & compact source & area mapping & spectroscopy + + - target position & & source + - reset interval & & [ 0.1 s , 0.25 s , 0.5 s , 1.0 s , ( 2.0 s ) ] + - parameter 1 & & spectral resolution + & & [ full res . , sed ] + - parameter 2 & shift size & - & array selection + & [ 70 `` , 240 '' ] & - & [ mod , sw , lw ] + amount of data & & 49.8 mb + + lccccl & * n60 * & * wide - s * & * wide - l * & * n160 * & + band center & @xmath24 & @xmath25 & @xmath26 & @xmath27 & [ @xmath0 ] + effective band width & @xmath29 & @xmath30 & @xmath31 & @xmath32 & [ @xmath0 ] + pixel scale & 26.8 & 26.8 & 44.2 & 44.2 & [ arcsec ] + pixel pitch & 29.5 & 29.5 & 49.1 & 49.1 & [ arcsec ] + + - measured fwhm & 37 @xmath33 1 & 39 @xmath33 1 & 58 @xmath33 3 & 61 @xmath33 4 & [ arcsec ] + + - variation & 19% & 14% & 43% & 53% & + + - survey mode & 2.4 & 0.55 & 1.4 & 6.3 & [ jy ] + - pointing mode & 110 & 34 & 350 & 1350 & [ mjy ] + +

the far - infrared surveyor ( fis ) is one of two focal plane instruments on the akari satellite . fis has four photometric bands at 65 , 90 , 140 , and 160 @xmath0 , and uses two kinds of array detectors . the fis arrays and optics are designed to sweep the sky with high spatial resolution and redundancy . the actual scan width is more than eight arcmin , and the pixel pitch is matches the diffraction limit of the telescope . derived point spread functions ( psfs ) from observations of asteroids are similar to the optical model . significant excesses , however , are clearly seen around tails of the psfs , whose contributions are about 30% of the total power . all fis functions are operating well in orbit , and its performance meets the laboratory characterizations , except for the two longer wavelength bands , which are not performing as well as characterized . furthermore , the fis has a spectroscopic capability using a fourier transform spectrometer ( fts ) . because the fts takes advantage of the optics and detectors of the photometer , it can simultaneously make a spectral map . this paper summarizes the in - flight technical and operational performance of the fis .

You are an expert at summarizing long articles. Proceed to summarize the following text: the recent radial velocity detection of a planet in the habitable zone of the nearby m dwarf proxima centauri ( hereafter proxima b and proxima ) @xcite has spurred over a dozen theoretical papers speculating on the planet s atmosphere ( e.g. , * ? ? ? * ; * ? ? ? * ) , habitability ( e.g. , * ? ? ? * ; * ? ? ? * ) , and orbital and formation histories ( e.g. , * ? ? ? * ; * ? ? ? * ) as well as prospects for a direct detection or atmospheric characterization ( e.g. , * ? ? ? * ; * ? ? ? as proxima is the nearest neighbor to the solar system , it has been suggested as a target for future space missions , including those hoping to characterize its atmosphere and search for life ( e.g. , * ? ? ? * ; * ? ? ? in many of these studies , authors have assumed a rocky planet with a thin atmosphere or no atmosphere at all , and some have assumed a mass near or equal to the projected mass of @xmath7 @xmath1 , but little has been done to assign a degree of certainty to these assumptions . most notably , previous studies have revealed two distinct populations of exoplanets with super - earth radii : ` rocky ' planets composed almost entirely of rock , iron , and silicates with at most a thin atmosphere , and ` sub - neptune ' planets which must contain a significant amount of ice or a h / he envelope ( e.g. , * ? ? ? * ; * ? ? ? if there is a significant probability that proxima b is of the latter composition , then this should be taken into account when assessing its potential habitability or observability . in this letter , we generate posterior distributions for the mass of proxima b using monte carlo simulations of exoplanets with an isotropic distribution of inclinations , where the radii , masses , and compositions of the simulated planets are constrained by results from combined transit and radial velocity measurements of previously detected exoplanets . by comparing the posterior mass distribution to the composition of planets as a function of mass , we determine the likelihood that proxima b is , in fact , a rocky world with a thin ( if any ) atmosphere . radial velocity and transit studies of exoplanets have yielded mass and radius measurements for a statistically significant number of targets , thereby enabling the study of how the occurrence and composition of exoplanets varies with planet radii , orbital periods , and host star type . in this section , we review previous results which we will use to place stronger constraints on the mass and composition of proxima b. it can be shown ( e.g. , * ? ? ? * ) that the probability distribution of @xmath8 corresponding to an isotropic inclination distribution is @xmath9 since this distribution peaks at @xmath10 , the mass distribution of an rv - detected planet - assuming no prior constraints on the mass - peaks at the minimum mass @xmath11 . in their models of the possible orbital histories of proxima b , @xcite find that galactic tides could have inflated the eccentricity of the host star s ( at the time unconfirmed ) orbit around the @xmath12 cen binary , leading to encounters within a few hundred au and the possible disruption of proxima s planetary system . if so , this could affect the likely inclination of the planet in a non - isotropic way . however , @xcite have presented radial velocity measurements showing that proxima is gravitationally bound to the @xmath12 cen system with an orbital period of 550,000 years , an eccentricity of @xmath13 , and a periapsis distance of 4,200 au . at this distance , the ratio of proxima s gravitational field to that of @xmath12 cen at the planet s orbit ( @xmath14 au ) is greater than @xmath15 ; unless proxima s orbit was significantly more eccentric in the past , it seems unlikely that @xmath12 cen would have disrupted the system . @xcite provide up - to - date occurrence rates of planets around m dwarf stars from the _ kepler _ mission . the sample is limited to @xmath16 days , over which they find the occurrence rates to be mostly independent of the period . the binned rates and a regression curve , as well as their uncertainties , are presented in figure [ fig : occurrence_rates ] . _ kepler _ statistics for m dwarfs remain incomplete below 1 @xmath6 , but complete statistics for earlier - type stars suggest a flat distribution for @xmath17 @xmath6 @xcite . since mass - radius relationships typically find a strong dependence of mass on radius ( @xmath18 ) ( e.g. * ? ? ? * ; * ? ? ? * ) , we assume _ a priori _ that proxima b ( @xmath19 @xmath1 ) is larger than @xmath20 @xmath6 . therefore , for this letter we adopt the regression curve fitted to the binned data , but set the occurrence rates to be flat for @xmath21 @xmath6 . * occurrence rates for m dwarf planets + ( with @xmath22 days ) * ( dotted ) to be flat , since the sample is incomplete in this range . bottom : mass - radius relationships for the rocky ( blue ) and sub - neptune ( red ) populations . the plotted relationships are from @xcite ( solid ) and @xcite ( dashed).[fig : occurrence_rates ] [ fig : mass - radius_relationship ] ] * mass - radius relationships for @xmath23 @xmath24 * ( dotted ) to be flat , since the sample is incomplete in this range . bottom : mass - radius relationships for the rocky ( blue ) and sub - neptune ( red ) populations . the plotted relationships are from @xcite ( solid ) and @xcite ( dashed).[fig : occurrence_rates ] [ fig : mass - radius_relationship ] ] multiple works ( e.g. * ? ? ? * ; * ? ? ? * ) have determined the existence of two distinct populations of exoplanets smaller than neptune ( @xmath25 @xmath6 ) : a small radius population with densities consistent with an entirely iron and silicate composition ( hereafter ` rocky ' ) , and a large radius population with lower density planets which must have significant amounts of ice or a thick h / he atmosphere ( hereafter ` sub - neptunes ' ) . @xcite studies the abundance of planets of each composition as a function of radius . they define @xmath26 as the likelihood that a planet of radius @xmath27 is dense enough to be consistent with a rocky composition , and determine @xmath28 for a sample of planets with known masses and radii . they suggest fitting the data with a two - parameter linear model : @xmath29 they find a step function to best describe the data , with @xmath30 fixed at zero and @xmath31 @xmath6 . for the purposes of this letter , we prefer this fit , but will also vary @xmath32 and @xmath30 to see how they affect our results . we stress that a planet for which @xmath33 is only _ sufficiently _ dense to be rocky ; we still can not necessarily exclude an ice or volatile component . here , we will assume that all planets for which @xmath33 follow the low - radius m - r relationships given in the following section , which were empirically fitted without prior knowledge of the planets compositions . for simplicity , we refer to these as ` rocky ' planets , and the other population as ` sub - neptunes ' , but we will revisit this distinction later on . since proxima b is in the habitable zone , it receives an amount of stellar flux comparable to that received by earth , so we should bear in mind the possibility that the volatile envelope of a sub - neptune could be lost due to photoevaporation . @xcite model rocky planets with thick h / he envelopes in the habitable zones of m dwarfs , finding that planets with @xmath34 @xmath1 maintain their envelopes over gyr timescales and are therefore uninhabitable . the 2@xmath35 lower limit on the minimum mass of proxima b is @xmath36 @xmath1 , so it is unlikely that any h / he envelope on the planet would evaporate under this rule . however , we note that this study focuses on planets with a primarily rocky composition , so it may not be directly applicable to habitable zone sub - neptunes . additionally , @xcite empirically define boundaries for atmospheric evaporation as a function of stellar heating , escape velocity , and atmospheric composition . in particular , a planet receiving an earth - like flux must have an escape velocity above @xmath37 km / s in order to maintain an h@xmath38 atmosphere for 5 gyr . we will revisit this requirement in section [ sec : escape ] . empirically derived relationships between exoplanet masses and radii rely on radial velocity ( rv ) or transit - timing variation ( ttv ) measurements of transiting exoplanet masses . @xcite fit a mass - radius ( hereafter m - r ) relationship to a sample of 65 transiting exoplanets , in which they find evidence for the two populations discussed in section [ sec : compositions ] . through least - squares regression , they find the densities of the rocky planets to increase linearly with planet radius : @xmath39 while the rv - measured masses of sub - neptunes increase nearly linearly with planet radius : @xmath40 @xcite use an expanded version of this data set to fit power law m - r relationships using a more statistically robust bayesian method . for the rocky planets , they find @xmath41 and for the sub - neptunes with rv - measured masses , @xmath42 due to the larger sample size and more robust fitting procedure , we adopt equations [ equation : wolfgang_rocky ] and [ equation : wolfgang_rv ] as our preferred m - r relationships , but for completeness we consider the @xcite relationships as well . we find that the choice of m - r relationships has a minimal impact on our final results . both sets of relationships are plotted in figure [ fig : mass - radius_relationship ] . it is important to note that the above relationships for sub - neptunes exclude masses measured by ttv , since ttv masses have been found to be systematically lower than rv masses . this could indicate a selection bias or systematic error in the method used , but since proxima b was detected through rv measurements , we believe it is proper to exclude the ttv masses either way . it is also clear that there is a significant spread in the masses of the observed planets . @xcite suggest a spread of @xmath43 @xmath1 for the sub - neptune planets , which we adopt for our simulations . for rocky planets , the spread is noticeably smaller . there are too few planets to constrain this spread , but it should most likely increase with mass , so we arbitrarily define the spread to be 30% of the calculated mass . the fitted occurrence rates and their uncertainties ( @xmath44 ) are given in even bins in log - space . we use them to generate a random sample of radii , where the number of radii in each bin ( @xmath45 ) is selected from a normal distribution with mean value @xmath46 and standard deviation @xmath47 . we find that the results converge for 1,000 samples of the occurrence rates , with each sample containing @xmath48 radii . to each radius , we assign a composition ( ` rocky ' or ` sub - neptune ' ) based on the model of @xcite ( equation [ equation : rogers_linear ] ) , with @xmath49 @xmath6 and @xmath50 . we then assign a mass to each radius and composition from a gaussian distribution with mean value @xmath51 - calculated using our chosen m - r relationships ( equations [ equation : wolfgang_rocky ] and [ equation : wolfgang_rv ] ) - and a standard deviation @xmath52 which represents the spread . we choose a spread proportional to the calculated mass for rocky planets ( @xmath53 ) , but a constant spread for sub - neptunes ( @xmath54 @xmath1 ) . we also reject negative masses , which could in principle bias the assigned masses towards higher - than - average values - however , we find that only a negligible number of masses are rejected . finally , we assign a line - of - sight inclination parameter @xmath8 to each planet , drawn from the isotropic probability distribution discussed in section [ sini_distribution ] . the prior mass and radius distributions , @xmath55 and @xmath56 , can be derived directly from the simulated sample . factoring in the projected minimum mass @xmath11 , the posterior distributions @xmath57 and @xmath58 can be calculated from bayes formula : @xmath59 where @xmath60 represents mass or radius . since @xmath11 is known , @xmath61 is just a normalizing constant . taking @xmath62 @xmath1 as the projected mass of proxima b and the upper limit @xmath63 @xmath1 as its standard deviation , we calculate for each simulated planet @xmath64 then @xmath65 and @xmath66 are the average values of @xmath67 for each bin in mass or radius . the prior and posterior distributions are calculated for each sample of @xmath68 planets , and the final results are taken to be the mean result of 1,000 samples . the prior probability that a planet in a given mass bin is rocky is equal to the number of simulated rocky planets in that bin divided by the total number of planets in the same bin . since we want to know the likelihood that proxima b belongs to the ` rocky ' population , we multiply this prior composition probability distribution by the posterior mass distribution from the previous section and integrate over all masses . * prior mass distribution * . [ fig : prior_posterior_pdf ] ] * posterior mass distribution * . [ fig : prior_posterior_pdf ] ] * cumulative mass probability distribution * ( dashdot ) . the dotted lines intersect 68% and 95% confidence upper limits on the mass . [ fig : cumulative_pdf ] ] the prior and posterior mass probability distributions for proxima b are plotted in figure [ fig : prior_posterior_pdf ] . the shaded regions demonstrate the relative contributions of the populations at each mass . the prior distribution is valid for rv - detected planets around m dwarfs with intermediate periods ( @xmath69 days ) and radii ( @xmath70 @xmath6 ) , while the posterior distribution can be taken as the mass probability distribution for proxima b. for reference , we include the posterior distribution given _ no _ prior constraints on the mass ; that is , the distribution resulting from an isotropic @xmath8 distribution and the measured @xmath11 with its uncertainty . we find that this nearly matches our result , since both @xmath71 and @xmath55 are bottom - heavy . figure [ fig : cumulative_pdf ] shows the cumulative probability that @xmath72 for both of the considered m - r relationships ( section [ sec : mass - radius_relationships ] ) as well as for the case of no prior mass distribution . we find that there is little difference between the results for each m - r relationship . in order to verify that sub - neptune planets can maintain h@xmath38 envelopes in the habitable zone , we compare the escape velocities of our simulated sub - neptunes to the @xmath37 km / s cutoff for h@xmath38 atmospheric escape ( assuming an earth - like stellar flux ) defined by @xcite . in both the prior and posterior distributions of escape velocities , we find that fewer than 1% of the sub - neptunes have escape velocities below this threshold , with most having @xmath73 km / s . therefore , we do not believe that proxima b will be subject to significant atmospheric loss if it has a sub - neptune composition . table [ tab : cases ] lists the sets of parameters for which we run the simulation , including the mass spread @xmath52 for each composition and the central value ( @xmath32 ) and width ( @xmath30 , if nonzero ) of the transition region defined by equation [ equation : rogers_linear ] . the following results for each case are given : the probability @xmath74 that proxima b belongs to the ` rocky ' category of planets , i.e. that its density is _ consistent _ with a fully iron and silicate composition , and the expectation values @xmath75 and @xmath76 of the mass and radius under the assumption that it belongs to this population . case a is most consistent with the previous work we have cited , so we take it as our primary result . in this case , there is a @xmath2 probability that proxima b belongs to the ` rocky ' population , with an @xmath77 likelihood that it belongs to the ` sub - neptune ' population . in the case that it is rocky , the expectation values ( and 95% confidence intervals ) for the mass and radius are @xmath4 @xmath1 and @xmath5 @xmath6 . we investigate the effect of increasing ( case b ) and decreasing ( case c ) the mass spread for each composition , which results in lower and higher values of @xmath74 , respectively . this results from low - radius ( @xmath78 @xmath6 ) , low - mass sub - neptunes ; when @xmath52 is large , they can lie significantly below the m - r relation with masses between 1 and 2 @xmath1 , so that they are indistinguishable from the rocky planets in the mass domain . in cases d and e , we determine the effect of raising or lowering the threshold radius @xmath32 at which the rocky and sub - neptune populations are split . a 0.2 @xmath6 offset in either direction , which encompasses most of the values suggested in the literature , results in a @xmath79 to @xmath80 shift in @xmath74 , where higher threshold radii allow for more rocky planets and therefore a higher probability of a rocky composition . furthermore , allowing for a non - zero width @xmath30 to the cutoff region allows sub - neptunes to exist with lower radii and masses , thereby decreasing @xmath74 . in all cases , we find @xmath74 to be between @xmath81 and @xmath82 using the @xcite m - r relationship , and we find similar values using the @xcite relationship ( e.g. @xmath83 for case a ) , so this result does not vary substantially over the range of reasonable values for the input parameters . by considering occurrence rates from the _ kepler _ mission and empirically derived m - r relationships , we derive a posterior probability distribution for the actual mass of proxima b. if the planet has a rocky composition , i.e. if it obeys the low - radius m - r relationship of @xcite , then the expectation values of the mass and radius ( with 95% confidence intervals ) are @xmath4 @xmath1 and @xmath5 @xmath6 . in all of our simulations , we find a probability of 80% to 95% that proxima b belongs to the ` rocky ' population of planets defined in section [ sec : compositions ] . in our ` best guess ' scenario ( case a ) , this probability is 90% . critically , we note that we have assumed all planets with @xmath33 ( according to the @xcite criterion ) are rocky planets , while in reality their density is only consistent with such a composition . with this in mind , it is safest to say that there is _ at least _ a 10% chance that proxima b has a sub - neptune composition . if it is a sub - neptune , then its surface gravity is high enough that it could maintain a thick hydrogen atmosphere . for future theoretical work involving the habitability and detectability of proxima b , we advise caution regarding assumptions made about its mass or composition ; if proxima b does possess a thick h / he envelope , then it is likely not habitable in the traditional sense . even if the mass could be further constrained , sub - neptunes have been measured with masses as low as @xmath84 @xmath1 , so the composition can not be conclusively inferred from the mass alone . nevertheless , the rocky composition originally asserted by @xcite remains the most likely possibility . the results reported herein benefited from collaborations and/or information exchange within nasa s nexus for exoplanet system science ( nexss ) research coordination network sponsored by nasa s science mission directorate . we thank benjamin rackham and gijs mulders for their constructive advice and insights , and the anonymous referee for their comments . cccccccccc * case a * & & @xmath85 & * 1.9 @xmath86 * & * 1.5 @xmath24 * & * - * & & @xmath87 & @xmath88 @xmath86 & @xmath89 @xmath24 + case b & & @xmath90 & 3.8 @xmath1 & 1.5 @xmath6 & - & & @xmath91 & @xmath92 @xmath1 & @xmath93 @xmath6 + case c & & @xmath94 & 0.7 @xmath1 & 1.5 @xmath6 & - & & @xmath95 & @xmath96 @xmath1 & @xmath97 @xmath6 + case d & & @xmath98 & 1.9 @xmath1 & 1.7 @xmath6 & - & & @xmath99 & @xmath100 @xmath1 & @xmath101 @xmath6 + case e & & @xmath98 & 1.9 @xmath1 & 1.3 @xmath6 & - & & @xmath102 & @xmath103 @xmath1 & @xmath104 @xmath6 + case f & & @xmath98 & 1.9 @xmath1 & 1.5 @xmath6 & 1.2 @xmath6 & & @xmath105 & @xmath106 @xmath1 & @xmath107 @xmath6 + case g & & @xmath90 & 3.8 @xmath1 & 1.5 @xmath6 & 1.2 @xmath6 & & @xmath108 & @xmath109 @xmath1 & @xmath110 @xmath6 +

recent studies regarding the habitability , observability , and possible orbital evolution of the indirectly detected exoplanet proxima b have mostly assumed a planet with @xmath0 @xmath1 , a rocky composition , and an earth - like atmosphere or none at all . in order to assess these assumptions , we use previous studies of the radii , masses , and compositions of super - earth exoplanets to probabilistically constrain the mass and radius of proxima b , assuming an isotropic inclination probability distribution . we find it is @xmath2 likely that the planet s density is consistent with a rocky composition ; conversely , it is at least @xmath3 likely that the planet has a significant amount of ice or an h / he envelope . if the planet does have a rocky composition , then we find expectation values and 95% confidence intervals of @xmath4 @xmath1 for its mass and @xmath5 @xmath6 for its radius .

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Proceed to summarize the following text: galaxies that radiate most of their extremely large , quasar - like luminosities ( @xmath2 10@xmath3 ) as infrared dust emission the ultraluminous infrared galaxies ( ulirgs ; sanders & mirabel 1996 ) dominate the bright end of the galaxy luminosity function in the nearby universe @xcite . they have been used , extensively , to derive information on the dust - obscured star - formation rate , dust content , and metallicity in the early universe @xcite . understanding the nature of nearby ulirgs , and more particularly , determining whether they are powered by starbursts or active galactic nuclei ( agns ) , or both , is of great importance in modern extragalactic astronomy . spectroscopic observation of the thermal infrared wavelength range ( 320 @xmath0 m ) is currently one of the most powerful methods of determining the energy sources of ulirgs . at this wavelength range , dust extinction is lower than at shorter wavelengths ( @xmath42 @xmath0 m ) , so it becomes possible to detect and measure the emission from more highly obscured energy sources , with smaller uncertainty in dust extinction corrections . in addition , spectral features in this waveband can be used to distinguish between starburst and agn activity . polycyclic aromatic hydrocarbon ( pah ) emission is detected in starbursts but not in agns , making it a good indicator of starburst activity @xcite , while the presence of agns hidden behind dust can be recognized through absorption features . pah emission and absorption features , detected in ground - based slit spectra ( e.g. , dudley 1999 ; imanishi & dudley 2000 ; soifer et al . 2002 ) and in _ iso _ spectra taken with large apertures ( e.g. , tran et al . 2001 ) , have been utilized to investigate the energy sources of ulirgs . in ulirgs , there are two potentially energetically significant components . one is a weakly obscured ( a@xmath5 @xmath6 20 mag)m is insignificant , which is roughly the criterion that a@xmath7 @xmath6 1 mag , or a@xmath5 @xmath6 20 mag if the galactic extinction curve of @xmath8 @xcite is adopted . ] starburst in the extended ( kpc scale ) host galaxy and also at the nucleus . the other is a nuclear , compact ( less than a few hundred pc in size ) , highly obscured energy source , powered by an agn and/or a compact starburst . recent ground - based high spatial resolution imaging observations @xcite have shown that the weakly obscured ( @xmath9 mag ) starbursts are energetically insignificant , and that the nuclear , compact , highly obscured energy sources dominate in ulirgs @xcite . however , the nature of these nuclear sources is not easy to determine , because dust extinction can be significant even in the thermal infrared wavelength range . the observed pah - to - infrared luminosity ratios in ulirgs have been found to be smaller than in known starburst galaxies by a large factor @xcite . assuming that the luminosities of the pah emission are a good indicator of the absolute magnitudes of starbursts , the smaller ratios in ulirgs could be due either to an increase in the dust extinction of the pah emission , or a dominant contribution from agns to the infrared luminosities of ulirgs . determining which explanation is correct requires reliable estimates of the dust extinction toward the starbursts in ulirgs . however , these estimates become complicated if the assumption of a `` foreground screen '' dust geometry is not valid for the starbursts @xcite . to understand the nature of the compact , highly obscured energy sources in ulirgs , the spectral shapes of the nuclear compact emission can provide more insight than just the absolute luminosities of pah emission , because the equivalent widths of pah emission are , by definition , robust to the amount of dust extinction . if the nuclear compact emission of ulirgs displayed significantly lower equivalent widths of pah emission than typically seen in starburst galaxies , and , in addition , strong absorption features were detected , it would suggest that energetically important dust - obscured agns were present . since the attenuation of the extended , weakly obscured emission is much smaller than that of the emission from the compact , highly obscured energy sources , the observed 320 @xmath0 m spectra taken with large apertures can be strongly contaminated by the extended emission , even though the total infrared ( 81000 @xmath0 m ) luminosities of ulirgs are dominated by the compact energy sources . in this case , the signatures of absorption features toward the compact energy sources can be diluted . hence spectroscopy with a narrow ( less than a few arcsec ) slit comparable to the spatial extent of the compact component is better suited to probe the nature of the compact energy sources in ulirgs . ground - based slit spectroscopy in the @xmath10-band ( 2.84.1 @xmath0 m ) has several important advantages for investigating the nature of dust - enshrouded agns : 1 . the dust extinction curve is fairly grey ( wavelength - independent ) at 310 @xmath0 m @xcite . 2 . the dust around an agn has a strong temperature gradient , in that the inner ( outer ) dust has a higher ( lower ) temperature . the temperature of the innermost dust is determined by the dust sublimation temperature ( t @xmath11 1000 k ) . in the case of blackbody radiation ( @xmath12 [ @xmath0 m ] @xmath13 t [ k ] @xmath11 3000 ) , the 34 @xmath0 m continuum emission is dominated by dust with @xmath141000 k located in the innermost regions , while the @xmath1110 @xmath0 m continuum emission is dominated by dust with @xmath15 k located further out . thus , the dust extinction toward the @xmath1110 @xmath0 m continuum emission regions , as estimated using the optical depth of the 9.7 @xmath0 m silicate dust absorption feature , is much smaller than the dust extinction toward the agn ; the dereddened agn luminosity will be underestimated if the extinction correction is made based on @xmath1110 @xmath0 m data alone . by contrast , the dust extinction toward the innermost , 34 @xmath0 m continuum emission regions , as estimated using 34 @xmath0 m data , is a better indicator of the dust extinction toward the agn itself . in the case that an agn is the only energy source , the 34 @xmath0 m continuum emission predominantly originates in the innermost , 8001000 k dust . the dust extinction toward the 34 @xmath0 m continuum emission can therefore be reliably fitted with a foreground screen dust model . in a real galaxy , weakly obscured starburst emission may contribute to the observed nuclear 34 @xmath0 m flux . however , as long as the agn is sufficiently luminous that agn - powered continuum emission contributes significantly to the observed 34 @xmath0 m flux , evidence for an agn can be found , provided that the starburst contribution is subtracted properly , even if the absorption optical depth toward the agn is larger than unity at 34 @xmath0 m ( @xmath16 mag or @xmath17 mag ; rieke & lebofsky 1985 ) . we have previously demonstrated this in two ulirgs : iras 08572 + 3915 ( @xmath18 mag ; imanishi & dudley 2000 ) , and ugc 5101 ( @xmath19 mag ; imanishi et al . 3 . at 2.84.1 @xmath0 m , in addition to the 3.3 @xmath0 m pah emission , 3.4 @xmath0 m carbonaceous dust absorption and 3.1 @xmath0 m h@xmath20o ice absorption are present . as we will show later in this paper , all these features can be used in a complementary manner to constrain the properties of the compact energy sources . they are simultaneously observable with recently available spectrographs that cover the whole @xmath10-band atmospheric window ( 2.84.1 @xmath0 m ) , if the target sources are at @xmath21 . in this redshift range , a statistically significant number of ulirgs have been found with _ iras_. 4 . in the near future , slit spectra of ulirgs at 5.340 @xmath0 m will be obtainable using the _ sirtf _ irs . by combining 34 @xmath0 m and 5.340 @xmath0 m slit spectra , the presence of a strong temperature gradient in the dust ( a signature of an agn ) can be investigated through the comparison of the optical depths of absorption features at different wavelengths at 320 @xmath0 m @xcite . in the study of ulirgs , it is particularly important to detect and quantify elusive agns that are deeply buried in dust along all sightlines ( hereafter buried agns ; imanishi et al . ulirgs that are classified optically as liners and have cool far - infrared colors ( sanders et al . 1988 ) have generally been taken to be starburst - powered @xcite . however , @xcite argued that liner ulirgs may contain buried agns ( see also antonucci 2001 ) . in this paper 2.84.1 @xmath0 m ground - based slit spectra of liner ulirgs with cool far - infrared colors are reported , to search for observational evidence for energetically important , buried agns . throughout this paper , h@xmath22 @xmath23 75 km s@xmath24 mpc@xmath24 , @xmath25 = 0.3 , and @xmath26 = 0.7 are adopted . the seven liner ulirgs in table [ tbl-1 ] were observed . table [ tbl-1 ] summarizes their infrared emission properties . all have cool far - infrared colors . the optical classifications as liners are based on @xcite and @xcite . table [ tbl-2 ] summarizes the observing log . the 3.14.0 @xmath0 m slit spectrum of ugc 5101 has already been presented in @xcite . descriptions of the observations and data analysis were offered in that letter . to summarize , the observations were made at the irtf on mauna kea , hawaii , using the grism mode of nsfcam @xcite . although the wavelength coverage of the nsfcam grism mode is 2.84.1 @xmath0 m , results at @xmath27 3.1 @xmath0 m were not presented in @xcite because our main aim in that study was to investigate the 3.3 @xmath0 m pah emission and 3.4 @xmath0 m carbonaceous dust absorption features , and because the earth s atmospheric transmission at 2.83.1 @xmath0 m is poorer than at @xmath28 3.3 @xmath0 m . however , after the publication of @xcite , @xcite reported the detection of 6.0 @xmath0 m h@xmath1o ice absorption in ugc 5101 , and suggested that the 3.1 @xmath0 m h@xmath1o ice absorption feature might be present in our 3.14.0 @xmath0 m spectrum . therefore we carefully analyzed the spectrum at 2.83.1 @xmath0 m to investigate the 3.1 @xmath0 m h@xmath1o ice absorption feature , whose profile is very broad , extending over at least @xmath29 2.83.3 @xmath0 m in the rest - frame ( smith , sellgren , & tokunaga 1989 ) . the 2.84.1 @xmath0 m spectra of the other ulirgs were obtained with the infrared spectrograph and camera ( ircs ; kobayashi et al . 2000 ) at the subaru telescope on mauna kea , hawaii . the sky was clear during the observations of these objects and the seeing at k was measured to be 0@xmath3050@xmath308 in full - width at half maximum . a 0@xmath309-wide slit and the @xmath10-grism were used with a 58-mas pixel scale . the achievable spectral resolution is @xmath11140 at 3.5 @xmath0 m . a standard telescope nodding technique , with a throw of 7 arcsec along the slit , was employed to subtract background emission . ulirgs and corresponding standard stars were observed with an airmass difference of @xmath40.1 to correct for the transmission of the earth s atmosphere , and to provide flux calibration . standard data analysis procedures were employed , using iraf . initially , bad pixels were replaced with interpolated signals from the surrounding pixels . bias was subsequently subtracted from the obtained frames and the frames were divided by a spectroscopic flat frame . finally the spectra of ulirgs and standard stars were extracted . wavelength calibration was performed taking into account the wavelength - dependent transmission of the earth s atmosphere . the spectra of ulirgs were divided by the observed spectra of standard stars , multiplied by the spectra of blackbodies with temperatures appropriate to individual standard stars ( table [ tbl-2 ] ) , and then flux - calibrated . appropriate binning of spectral elements was performed , particularly at @xmath31 @xmath0 m in the observed frame , so as to give an adequate signal - to - noise ratio in each element . the resulting spectral resolution @xmath32 at @xmath31 @xmath0 m is @xmath6100 . although the earth s atmospheric transmission curve at 2.83.3 @xmath0 m is highly wavelength - dependent if observed at high spectral resolution of @xmath33 ( figure [ fig1]a ) , it is fairly smooth at lower spectral resolutions , with r @xmath6 100 ( figure [ fig1]b ) . thus , even if the net positions of the target object and standard star on the slit differ slightly ( on the sub - pixel scale ) along the wavelength direction , the standard data analysis described above is expected to produce no significant spurious features in spectra with r @xmath6 100 . figure [ fig2 ] shows flux - calibrated 2.84.1 @xmath0 m spectra of the seven ulirgs . since our slit spectra cover physical scales larger than a few hundred pc in all the ulirgs , emission from both agns and compact ( less than a few hundred pc ) starbursts should be fully detected . all the ulirgs clearly show the 3.3 @xmath0 m pah emission . to estimate its flux , luminosity , and rest - frame equivalent width , we make the reasonable assumption that the profiles of the 3.3 @xmath0 m pah emission in these ulirgs are similar to those of galactic star - forming regions and nearby starburst galaxies ; the main emission profile then extends between @xmath34 @xmath23 3.243.35 @xmath0 m @xcite . the flux excess above the solid lines in figure [ fig2 ] should thus be ascribed to the 3.3 @xmath0 m pah emission . table [ tbl-3 ] summarizes the results for the 3.3 @xmath0 m pah emission in these seven ulirgs . the spectrum of ugc 5101 shows clear 3.4 @xmath0 m carbonaceous dust absorption with an observed optical depth of @xmath35 = 0.65 @xcite . in other ulirgs , there is no clear indication of this feature , partly because this absorption feature is intrinsically not so strong ( @xmath35/a@xmath5 @xmath11 0.0040.007 ; pendleton et al . 1994 ) . in addition to the 3.3 @xmath0 m pah emission feature , there is another important common feature of the spectra of five ulirgs ( ugc 5101 , iras 00188@xmath360856 , iras 03250 + 1606 , iras 16487 + 5447 , and iras 21329@xmath362346 ) : the continuum emission is concave in these spectra . at the shorter wavelength side of the 3.3 @xmath0 m pah emission , the continuum flux level initially decreases with decreasing wavelength , but then begins to increase again at @xmath37 @xmath0 m . in iras 00188@xmath360856 and iras 03250 + 1606 , the spectra become flat at the shortest wavelength parts ( @xmath34 @xmath4 2.7 @xmath0 m ) . this behavior is quite similar to that of the infrared - luminous galaxy ngc 4945 , in which 3.1 @xmath0 m h@xmath1o ice absorption was clearly detected in an _ iso _ spectrum @xcite . earlier ground - based @xmath10-band spectroscopy has revealed that many bright galactic sources behind substantial columns of dust and molecular gas show broad 3.1 @xmath0 m h@xmath1o ice absorption , with a main profile peaking at @xmath34 @xmath11 3.05 @xmath0 m and extending at least to @xmath34 @xmath23 2.83.3 @xmath0 m ( smith et al . 1989 ; smith , sellgren , & tokunaga 1993 ; smith 1993 ) . a recent _ iso _ spectrum of ngc 4945 suggests that the absorption may extend to wavelengths as short as @xmath38 @xmath0 m @xcite . in addition to this main profile , a weak longer - wavelength wing at @xmath393.7 @xmath0 m can also be distinguished @xcite . the concave - shaped continuum emission found in the spectra of these five ulirgs is consistent with the presence of this broad 3.1 @xmath0 m h@xmath1o ice absorption . some seyfert 2 galaxies with lower infrared luminosities were observed on the same nights as these ulirgs , but no signatures of strong 3.1 @xmath0 m h@xmath1o ice absorption were found . the spectra of standard stars observed before these ulirgs on the same nights were divided by other standard stars observed after , but the resulting spectra also never showed such signatures . the strong 3.1 @xmath0 m h@xmath1o ice absorption signatures found only in the spectra of the ulirgs are thus believed to be genuine and intrinsic to the ulirgs , and not an artifact caused by insufficient cancellation of the earth s atmospheric transmission . in fact , for ugc 5101 and iras 00188@xmath360856 , the clear detection of h@xmath1o ice absorption at 6.0 @xmath0 m was also reported by @xcite . to estimate conservative lower limits for the optical depths of the broad 3.1 @xmath0 m h@xmath1o ice absorption feature , the concave quadratic dashed lines in figure [ fig2 ] are adopted as the lowest plausible continuum levels . in iras 23234 + 0946 and iras 23327 + 2913 , the presence of strong 3.1 @xmath0 m h@xmath1o ice absorption is not clear in the case of the adopted concave continuum . for the remaining five ulirgs , the observed optical depths of the 3.1 @xmath0 m h@xmath1o absorption ( @xmath40 ) as well as the 3.4 @xmath0 m carbonaceous dust absorption ( @xmath35 ) are summarized in table [ tbl-4 ] . these @xmath40 values should be regarded as strict lower limits because : ( 1 ) linear continuum levels that connect data points at the shortest and longest wavelength parts of the spectra would give rise to larger values of @xmath40 , and ( 2 ) the data points at the shortest wavelength parts , used to determine continuum levels , may still be affected by the 3.1 @xmath0 m h@xmath1o ice absorption feature in some cases . compared to ugc 5101 , iras 00188@xmath360856 and iras 21329@xmath362346 , the presence of the broad h@xmath1o ice absorption feature is apparently less clear in iras 03250 + 1606 and iras 16487 + 5447 . however , as long as the spectral shapes of the 3.3 @xmath0 m pah emission in these ulirgs do not differ significantly from those observed so far in starburst galaxies , the continuum flux levels on the shorter wavelength side of the 3.3 @xmath0 m pah emission are clearly lower than on the longer wavelength side , which is consistent with the presence of the broad 3.1 @xmath0 m h@xmath1o ice absorption . the very broad emission - like features that extend between @xmath41 3.63.9 @xmath0 m and 3.43.8 @xmath0 m in iras 03250 + 1606 and iras 16487 + 5447 , respectively , can not be ascribed solely to the 3.3 @xmath0 m pah emission ; instead , they are caused by a combination of 3.1 @xmath0 m h@xmath1o ice absorption , 3.3 @xmath0 m pah emission , and ( possibly ) 3.4 @xmath0 m carbonaceous dust absorption . the observed 3.3 @xmath0 m pah to infrared luminosity ratios for these seven ulirgs are l@xmath42/l@xmath43 @xmath11 13 @xmath4410@xmath45 ( table [ tbl-3 ] ) , which are 3 to 10 times smaller than the ratios for starburst galaxies ( l@xmath42/l@xmath43 @xmath11 10@xmath46 ; mouri et al . 1900 ; imanishi 2002 ) . it can therefore be concluded that _ weakly obscured ( a@xmath47 mag ) nuclear _ starbursts are not contributing significantly to the infrared luminosities of these ulirgs . since the l@xmath42 values are based on slit spectroscopy measurements , the luminosities of _ extended _ ( kpc scale ) weakly obscured starbursts are not taken into account . however , as mentioned in @xmath481 , such extended starbursts have been shown to be energetically insignificant in ulirgs @xcite . for ugc 5101 , @xcite estimated , based on the _ iso _ s large aperture measurement , a 6.2 @xmath0 m pah emission flux of @xmath49 w @xmath50 , so that its luminosity is @xmath51 ergs s@xmath24 in our adopted cosmology . the resulting @xmath52 , which is only @xmath53 of the ratio typically seen in starburst galaxies ( @xmath54 : fischer 2000 ) . thus , weakly obscured extended starbursts are unlikely to be the dominant energy source in ugc 5101 , either . the dominant energy sources should be compact and heavily obscured ; for the five ulirgs , the presence of such buried energy sources is supported by the detection of strong absorption features in our 2.84.1 @xmath0 m spectra . the detection of strong ( @xmath40 @xmath55 0.6 ) 3.1 @xmath0 m h@xmath1o ice absorption can , in principle , provide stringent constraints on the nature of the buried energy sources at the nuclei of the five ulirgs . the 3.4 @xmath0 m carbonaceous and 9.7 @xmath0 m silicate dust absorption features are detectable if an adequate amount of dust in the _ diffuse inter - stellar medium _ is present in front of the continuum - emitting energy sources . however , in order for h@xmath1o ice absorption to be detected , dust grains _ covered with an ice mantle _ must be present in the foreground . such a situation can arise only when the dust grains are located in molecular clouds , and are sufficiently shielded from the uv radiation from any energy sources either inside or outside the molecular clouds @xcite . in other words , the detection of strong h@xmath1o ice absorption requires that a large amount of dust in _ molecular clouds _ be present in front of the continuum - emitting energy sources . foreground molecular clouds in the host galaxies could be responsible for the strong 3.1 @xmath0 m h@xmath1o ice absorption detected toward the nuclei if the host galaxies were seen from nearly edge - on . however , there is no strong evidence that the host galaxies of these ulirgs are edge - on @xcite . @xcite reported that 6.0 @xmath0 m h@xmath1o ice absorption is systematically stronger in ulirgs than in less infrared - luminous seyfert 2 galaxies . these seyfert 2 galaxies also generally show no detectable or weak ( @xmath56 ) 3.1 @xmath0 m h@xmath1o ice absorption ( imanishi et al . 2002 , in preparation ) . the systematically larger optical depths of h@xmath1o ice absorption toward the nuclei of ulirgs can reasonably be explained by a larger amount of highly concentrated molecular gas ( and dust ) in their nuclei @xcite . we thus regard it as likely that the strong ( @xmath40 @xmath55 0.6 ) 3.1 @xmath0 m h@xmath1o ice absorption detected toward the nuclei of these five ulirgs are caused by nuclear concentrations of molecular gas and dust . note that , although the host galaxy of ngc 4945 is viewed edge - on @xcite , a significant fraction of the strong 3.1 @xmath0 m h@xmath1o ice absorption ( @xmath57 ; spoon et al . 2000 ) is likely to originate in a highly concentrated nuclear component of ice - covered dust grains , on a scale of less than a few hundred pc @xcite . in the case of _ normal _ starbursts , it seems reasonable to assume that the continuum - emitting sources and molecular gas and dust are spatially well mixed @xcite . a detailed infrared study of the nearby starburst galaxy m82 suggests that the mixed - dust geometry is a better fit to the observed data than a foreground screen dust distribution @xcite . since the oscillator strengths of species are fixed , there are upper limits on the absorption optical depths that can be produced by the mixed - dust geometry ( unless unusual abundances are considered ) . this is because the foreground , less - obscured , and therefore less - attenuated , emission ( which shows only weak absorption features ) will dominate the observed fluxes . in practice , nearby starburst galaxies such as m82 and ngc 253 show weak ( @xmath58 @xmath11 0.2 ) or undetectable absorption by h@xmath1o ice @xcite or dust grains @xcite . this issue will be discussed quantitatively below . in galactic molecular clouds , the optical depth of the h@xmath1o ice absorption and dust extinction ( @xmath59 ) are correlated , with a relation given by @xmath60 . the value of @xmath61 has been found to be almost constant , @xmath110.06 , in galactic molecular clouds , which exhibit various different levels of star - forming activity @xcite . this is because it reflects the abundance of h@xmath1o ice relative to dust grains in the regions where an ice mantle can survive ; it is therefore unlikely to differ substantially from the galactic value , even in starburst galaxies . the a@xmath62 value in the above relation is the threshold required for dust grains to be sufficiently shielded from the ambient uv radiation . this value can vary dramatically , from 26 mag in quiescent molecular clouds ( taurus and serpens ; whittet et al . 1988 ; eiroa & hodapp 1989 ; murakawa et al . 2000 ) to 15 mag in active star - forming molecular clouds ( @xmath63-oph ; tanaka et al . 1990 ) . in starburst galaxies with a more intense uv radiation field than the @xmath63-oph molecular clouds , @xmath64 is almost certainly larger than 15 mag , but its exact value is unknown . thus the relation @xmath65 is adopted in starburst galaxies , where @xmath66 mag , or @xmath67 where @xmath68 is the fraction of dust that is covered with an ice mantle . figures [ fig3]a and [ fig3]b show schematic diagrams of a mixed - dust geometry . in this geometry , the observed flux i(@xmath69 ) is given by @xmath70 where @xmath71 is an unattenuated intrinsic flux and @xmath69 is the optical depth at each wavelength , which takes different values inside and outside the absorption features . for 34 @xmath0 m continuum emission outside the 3.1 @xmath0 m h@xmath1o ice and 3.4 @xmath0 m carbonaceous dust absorption features , @xmath72 @xcite , where , by definition , @xmath73 . at the peak wavelength of the 3.1 @xmath0 m h@xmath1o ice absorption feature , @xmath74 where the first and second terms give the flux attenuation for 34 @xmath0 m continuum and the h@xmath1o ice absorption feature , respectively . thus , the optical depth of the h@xmath1o ice absorption feature in an observed 34 @xmath0 m flux is @xmath75 \\ & = & \ln [ ( 1+f ) \frac{1 - e^{-0.06 \times a_{\rm v}}}{1 - e^{-0.06 \times a_{\rm v } \times ( 1+f ) } } ] \end{aligned}\ ] ] this last formula implies that , for a fixed @xmath68-value , @xmath40 increases with increasing a@xmath5 , but saturates at a certain value . the @xmath68-value can be constrained based on observations of the nearby starburst galaxy m82 . based on a mixed - dust model , the dust extinction toward the farthest parts of the starburst region is estimated to be a@xmath5 @xmath11 55 mag @xcite . the observed optical depth of the 3.1 @xmath0 m h@xmath1o ice absorption is @xmath76 @xcite . based on these values , @xmath77 is obtained . in other words , in the starburst galaxy m82 , @xmath78 of dust grains are covered with an ice mantle . if the cores of ulirgs are simply a scaled - up version of normal starbursts with a mixed - dust geometry and the @xmath68-value is @xmath110.3 , then @xmath40 should be @xmath110.2 . however , the @xmath68-value is strongly dependent on the fraction of active star - forming regions relative to quiescent molecular gas in molecular clouds . if the cores of ulirgs are powered by starbursts with a mixed - dust geometry , the @xmath68-value in ulirgs should be even smaller than in less infrared luminous starburst galaxies @xcite . here , however , @xmath68 @xmath11 0.3 is adopted for ulirgs in order to determine a conservative upper limit for @xmath40 within the framework of a mixed - dust geometry . even if this conservative value is adopted , equation ( 7 ) shows that @xmath40 can not be larger than 0.3 . therefore , it can be concluded that _ the observed @xmath40 values of @xmath550.6 in these five ulirgs are clearly incompatible with mixed - dust geometry and f @xmath79 0.3_. when the contribution from any weakly obscured starburst emission to the observed 34 @xmath0 m fluxes is subtracted , the @xmath40 values toward the buried energy sources will be even larger , making this conclusion more robust . the same argument is applicable to the 3.4 @xmath0 m carbonaceous dust absorption in ugc 5101 @xcite . for the 3.4 @xmath0 m carbonaceous dust absorption feature , @xmath80 is obtained @xcite , where the term @xmath81 , the fraction of dust grains not covered by an ice mantle , is included because the 3.4 @xmath0 m carbonaceous dust absorption feature may be suppressed if the dust grains are covered with an ice mantle @xcite . based on the same arguments as above , it can be quantitatively demonstrated that @xmath35 can not be larger than 0.2 in the mixed - dust geometry . the observed large optical depth of @xmath82 in ugc 5101 @xcite is again incompatible with a mixed - dust source geometry . in order to explain the strong absorption features in these five ulirgs , dust in a foreground screen must be obscuring the 34 @xmath0 m continuum - emitting energy sources this configuration requires that the energy sources be centrally concentrated and deeply buried in nuclear dust and molecular gas , as shown in figure [ fig3]c . the centrally - concentrated nature of the buried energy source in ugc 5101 is naturally explained by a buried agn @xcite . furthermore , as mentioned in @xmath481 , the observed rest - frame equivalent widths of the 3.3 @xmath0 m pah emission in starburst galaxies should be little changed by dust extinction . the significantly lower equivalent width in ugc 5101 ( table [ tbl-3 ] ) than in starbursts ( @xmath11120 nm ; moorwood 1986 ; imanishi & dudley 2000 ) also supports the presence of agns that produce strong 34 @xmath0 m continuum emission , but essentially no pah emission . when our data are combined with ground - based high spatial resolution mid - infrared ( 825 @xmath0 m ) imaging data @xcite , we can give more insight into the nature of the buried energy source . @xcite estimated the size of the spatially - unresolved core of ugc 5101 at 12.5 @xmath0 m to be @xmath4200 pc . if the attenuation of the mid - infrared flux from the spatially - unresolved compact core in ugc 5101 is negligible , the surface brightness of the core emission becomes @xmath551.0 @xmath13 10@xmath83 kpc@xmath84 @xcite . however , the detection of strong 3.4 @xmath0 m carbonaceous dust and 3.1 @xmath0 m h@xmath1o ice absorption toward the nucleus of ugc 5101 requires that 1 . the actual size of the buried centrally - concentrated energy source is substantially smaller than that of the dust and molecular gas , and 2 . the dust extinction toward the buried energy source is @xmath85 mag @xcite . on point ( 1 ) , since dust around a centrally - concentrated energy source shows a temperature gradient ( see @xmath481 ) , the size of the cooler dust region , which is also responsible for the foreground dust extinction , is larger than that of the @xmath11250 k dust traced by the 12.5 @xmath0 m emission ( @xmath4200 pc ) . however , the size of the cool dust emission ( traced by the 1.3 mm emission ) at the core of the ulirg arp 220 has been shown to be compact ( @xmath4200 pc ; sakamoto et al . furthermore , the sizes of the high density molecular gas regions at the cores of ulirgs have been estimated to be no more than @xmath11500 pc in diameter @xcite . thus , it seems reasonable to assume that the actual size of the buried centrally - concentrated energy source in ugc 5101 is only a fraction of this , @xmath11100 pc or less . on point ( 2 ) , the dust extinction of @xmath19 mag means that the flux attenuation at 12.5 @xmath0 m is more than factor of 25 @xcite . when these corrections are taken into account , the actual surface brightness of the centrally - concentrated buried energy source in ugc 5101 is @xmath86 kpc@xmath84 . this exceeds the peak surface brightness of star clusters in starburst galaxies ( @xmath115 @xmath13 10@xmath87 kpc@xmath84 ; meurer et al . 1997 ; soifer et al . 2000 ) by a factor of more than two . the high surface brightness is best explained by the presence of an energetically important buried agn in ugc 5101 . if a buried agn is present at the nucleus of ugc 5101 , the dust is expected to show a strong temperature gradient ( see @xmath481 ) . are the available data consistent with its presence ? @xcite could not explicitly find the signature of a temperature gradient based on their imaging data of ugc 5101 . here , it is searched for using spectroscopic data . as mentioned in @xmath481 and @xcite , if a strong temperature gradient is present in the dust , the optical depth of the 3.4 @xmath0 m carbonaceous dust absorption feature probes deeper into the dust distribution than that of the 9.7 @xmath0 m silicate dust absorption . thus , the @xmath35/@xmath88 ratio toward the nucleus of ugc 5101 is expected to be significantly larger than the value seen when no significant temperature gradient is present ( @xmath35/@xmath890.07 ; roche & aitken 1984 , 1985 ; pendleton et al . 1994 ) . after subtracting the contribution from the weakly obscured starburst emission to the observed 34 @xmath0 m flux , the @xmath35 value toward the buried agn at the core of ugc 5101 was estimated to be @xmath110.8 @xcite , while only a lower limit for the observed @xmath88 value ( @xmath551.5 ) was given by @xcite based on a measurement with a large aperture using _ iso_. although the @xmath88 value is a lower limit , its actual value is unlikely to exceed 2.0 , judging from the actual spectrum of ugc 5101 ( spoon et al . 2002 ; their figure 8) . thus the @xmath35/@xmath88 ratio is @xmath90 . the actual @xmath88 value toward the buried agn in ugc 5101 could be larger than the observed value , because weakly obscured starburst emission is likely to contribute significantly to the observed @xmath1110 @xmath0 m _ iso _ flux . however , since the equivalent width of the 6.2 @xmath0 m pah emission in the _ iso _ spectrum of ugc 5101 is less than half that of starburst galaxies @xcite , at least half of the observed continuum flux at @xmath555 @xmath0 m should originate in the buried agn , implying that the @xmath35/@xmath88 ratio toward the buried agn is @xmath91 . thus , these spectroscopic data are consistent with the presence of a temperature gradient of dust at the core of ugc 5101 . if the dust around an agn has a torus - like geometry , low - density clumpy clouds which are directly illuminated by the central agn can exist at relatively large distances from the agn along the axis of the torus - geometry dust ( the so - called the narrow line regions or nlrs ) . nlrs produce the strong high - excitation forbidden emission lines seen in agn . however , in a buried agn , x - ray dissociation regions ( maloney et al . 1996 ) , rather than nlrs , are expected to be produced around the agn , so that high - excitation forbidden emission lines will be weak , consistent with the observed data of ugc 5101 @xcite . @xcite and @xcite found that the infrared to hcn ( j=1 - 0 ) luminosity ratios in ulirgs , including ugc 5101 @xcite , roughly follow the relation established in less infrared - luminous starburst galaxies . by regarding the hcn ( j=1 - 0 ) luminosities as a tracer of the mass of high density ( @xmath5510@xmath92 cm@xmath46 ) molecular gas where star - formation actually occurs , @xcite argued that the infrared luminosities of ulirgs are powered by starbursts . however , in the luminous inner regions ( central few hundred pc ) of ulirgs , the gas pressures - and therefore molecular gas densities - are inevitably extremely high , whether the energy sources are starbursts or agns @xcite . the relative constancy of the infrared to hcn luminosity ratio from less infrared luminous starburst galaxies to ulirgs simply indicates that much of the infrared emission arises from dust in dense molecular clouds , which is inevitable whatever the sources of the luminosities of ulirgs . for the two ulirgs which do not show strong 3.1 @xmath0 m h@xmath1o ice absorption ( iras 23234 + 0946 and iras 23327 + 2913 ) , there is no strong observational evidence for the presence of buried agns . however , for the other four ulirgs which show strong 3.1 @xmath0 m h@xmath1o absorption ( iras 00188@xmath360856 , iras 03250 + 1606 , iras 16487 + 5447 , and iras 21329@xmath362346 ) , the centrally - concentrated nature of their buried energy sources and systematically ( roughly a factor of two ) lower - equivalent - width pah emission than starbursts ( table 3 ) are also naturally explained by the presence of energetically important , buried agns . assuming that approximately half of the observed 34 @xmath0 m fluxes originate in buried agns , the @xmath40 and @xmath35 toward the buried agns are twice as large as the observed values . these @xmath40 and @xmath35 values reflect the column density of foreground dust , toward the buried agns , covered with and without an ice mantle , respectively ( @xmath484.2 ) . based on the relation of @xmath40/a@xmath5 @xmath11 0.06 , @xmath35/a@xmath5 @xmath11 0.0040.007 ( @xmath48 4.2 ) , and @xmath94 @xmath11 0.06 ( @xmath48 1 ) , and on a foreground screen dust model ( @xmath48 1 ) , it is found that the dereddened 34 @xmath0 m dust emission luminosities powered by the agns could be comparable to the infrared luminosities of these four ulirgs . it is sometimes argued that ulirgs with cool far - infrared colors can not be powered by agns because agns show warm far - infrared colors ( e.g. , downes & solomon 1998 ) . indeed , many known seyfert 2 galaxies show warm far - infrared colors @xcite . however , as mentioned in @xmath484.2 , the systematically higher detection rate of h@xmath1o ice absorption in ulirgs , as compared to less infrared luminous seyfert 2 galaxies , is naturally explained by a higher column density of nuclear dust in ulirgs . this explanation is supported by imaging data at @xmath1110 @xmath0 m : while emission from the nuclei of less infrared luminous seyfert 2 galaxies is argued to be optically thin at @xmath1110 @xmath0 m @xcite , that from the nuclei of ulirgs is estimated to be optically thick @xcite . the higher column densities of dust around buried agns in ulirgs can naturally produce infrared spectral energy distribution with cooler color than in less infrared luminous seyfert 2 galaxies . it is quite possible that many other ulirgs with cool far - infrared colors contain energetically important agns deeply buried in dust . @xcite have found that a 60 @xmath0 m luminosity function in the local universe , derived based on @xmath1115000 galaxies , shows a bump at @xmath95 , and suggested that agn activity begins to be energetically important at @xmath96 . this result also favors the buried agn scenario in the majority of nearby ulirgs . strong ( @xmath97 ) 3.1 @xmath0 m h@xmath1o ice absorption has been detected toward the nuclei of five ulirgs : ugc 5101 , iras 00188@xmath360856 , iras 03250 + 1606 , iras 16487 + 5447 , and iras 21329@xmath362346 . all these ulirgs have cool far - infrared colors and are classified optically as liners . this detection requires a foreground screen dust geometry , which is best explained if centrally - concentrated , compact energy sources are buried in nuclear dust and molecular gas . for all the ulirgs , the observed equivalent widths of the 3.3 @xmath0 m pah emission are significantly lower than those of starburst galaxies . these observational data are naturally explained by the presence of agns deeply buried in dust , and support our previous arguments that liner ulirgs with cool far - 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( 1 ) : object name . ( 2 ) : observed flux of the 3.3 @xmath0 m pah emission . ( 3 ) : observed luminosity of the 3.3 @xmath0 m pah emission . col . ( 4 ) : observed 3.3 @xmath0 m pah - to - infrared luminosity ratio . ( 5 ) : rest - frame equivalent width of the 3.3 @xmath0 m pah emission . those for starbursts are @xmath11120 nm ( see text ) . lcc ugc 5101 & @xmath550.8 & 0.65 + iras 00188@xmath360856 & @xmath551.6 & @xmath40.2 + iras 03250 + 1606 & @xmath550.6 & @xmath40.1 + iras 16487 + 5447 & @xmath550.8 & @xmath40.35 + iras 21329@xmath362346 & @xmath550.9 & @xmath40.2 +

ground - based 2.84.1 @xmath0 m slit spectra of the nuclei of seven ultraluminous infrared galaxies ( ulirgs ) that are classified optically as liners and have cool far - infrared colors are presented . all the nuclei show 3.3 @xmath0 m polycyclic aromatic hydrocarbon ( pah ) emission , with equivalent widths that are systematically lower than those in starburst galaxies . strong 3.1 @xmath0 m h@xmath1o ice absorption , with optical depth greater than 0.6 , is also detected in five nuclei , and 3.4 @xmath0 m carbonaceous dust absorption is detected clearly in one of the five nuclei . it is quantitatively demonstrated that the large optical depths of the h@xmath1o ice absorption in the five sources , and the 3.4 @xmath0 m absorption in one source , are incompatible with a geometry in which the energy sources are spatially mixed with dust and molecular gas , as is expected for a typical starburst , but instead require that a large amount of nuclear dust ( including ice - covered grains ) and molecular gas be distributed in a screen in front of the 34 @xmath0 m continuum - emitting sources . this geometrical requirement can naturally be met if the energy sources are more centrally concentrated than the nuclear dust and molecular gas . the low equivalent widths of the pah emission compared to starbursts and the central concentration of the nuclear energy sources in these five ulirgs are best explained by the presence of energetically important active galactic nuclei deeply buried in dust and molecular gas .

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Proceed to summarize the following text: it is now generally agreed that the x - ray variability properties of @xmath3few solar mass black holes in x - ray binary systems ( xrbs ) and supermassive black holes in active galactic nuclei ( agn ) are similar . with a view to providing an independent method for determining physical parameters such as black hole mass and accretion rate , there is considerable interest in determining how these parameters might scale with characteristic observable x - ray variability parameters . x - ray variability can be quantified via the power spectral density ( psd ) and , to first order , the x - ray psds of agn are similar to those of xrbs @xcite , particularly to xrbs in the high - soft state ( e.g. * ? ? ? * ; * ? ? ? * ) . in this state the power , @xmath4 , at frequency , @xmath5 , is given by @xmath6 where @xmath7 at low frequencies bending , above a frequency @xmath8 , to a slope @xmath9 . many authors have shown that the timescale @xmath10 , corresponding to the frequency @xmath8 , scales approximately linearly with mass ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) although the relationship shows considerable scatter . however much of the scatter can be explained by an inverse scaling of with accretion rate ( where is the accretion rate in units of the eddington accretion rate , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the other variability parameter which has attracted attention is the normalisation of the high frequency ( hf ) psd . this normalisation can be defined in a number of ways . for example , by fitting a power law to the hf psds , after subtraction of the poisson noise contribution , @xcite measured the power at a fixed frequency ( @xmath11hz ) . the normalised variability amplitude , nva , was then defined as the square root of the power , which was measured in units of ( counts s@xmath12)@xmath13 hz@xmath12 , divided by the average count rate . the nvas for a sample of agn observed by exosat were therefore derived . although black hole mass was assumed to be a driving parameter , in 1988 very few agn black hole masses were available so the nvas were plotted against luminosity as a proxy , showing a strong anticorrelation . @xcite , using ginga observations , measured the timescale at which the psd crossed a particular power level ( @xmath14 , in rms@xmath13 hz@xmath12 units ) . scaling this timescale with mass from cyg x-1 , they estimated masses for 8 agn , although they noted that these masses were one or two orders of magnitude lower than masses derived by other methods . @xcite defined an hf psd normalisation , @xmath15 , which is very similar to the nva of @xcite although , as most of their measurements were of xrbs , @xcite defined @xmath16 at the higher frequency of 1hz . they also assumed @xmath17 whereas @xcite and @xcite measured @xmath18 . within the xrb sample of @xcite there is no correlation of @xmath15 with mass ( their fig.7a ) , although the range of masses is small and mass uncertainties large . within their sample of agn ( their fig.7c ) there is an approximate inverse correlation of @xmath15 with mass , strengthened if ngc4395 is included ( fig.7b ) . however the scatter is too large to define the relationship precisely within that sample . however if it is assumed that agn and xrbs follow the same scaling relationship then @xcite find that @xmath19 . @xcite fit a mixed ornstein - uhlenbeck model to cyg x-1 and to agn psds . in this model the hf psd normalisation is defined by @xmath20 , the fractional amplitude of the driving noise field . the psd slope is again fixed at @xmath17 with @xmath21 . for a sample of 10 agn they find a strong correlation with @xmath22 ( 90 per cent confidence ) . however such a relationship would not extrapolate to the xrb observations and so it is important to investigate the relationship between agn hf psd normalisation and mass using other datasets . although less direct , the psd normalisation can be estimated from the normalised excess variance of the lightcurves , @xmath23 ( e.g. * ? ? ? a number of authors ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) have found an approximate inverse scaling of @xmath23 with mass . if all of the frequencies sampled by the lightcurves lie above , and the hf psd slope is known , then gives a reasonable estimate of the hf psd normalisation , although @xcite show that is a noisy quantity when the psd is steep . if is not known and lies within the frequencies sampled , or if the low frequency psd slope is unknown , then is a less good measure of hf psd normalisation . the best way , therefore , to determine whether hf psd normalisation varies with mass is to derive hf psd normalisations directly from the psds . the hf psd is best determined from continuous long observations which suffer less from distortion by the window function of the sampling pattern than do observations which are split into many segments . thus , which allows continuous observations of up to 130ksec , provides a better measurement of the hf psd ( e.g. * ? ? ? * ; * ? ? ? * and many others ) than do satellites in low earth orbits with maximum continuous observation lengths of @xmath24s , i.e. ginga ( e.g. * ? ? ? * ) , asca ( e.g. * ? ? ? * ) or rxte . exosat was also useful as it allowed continous observations of up to 280ksec . the exosat low energy ( le ) imaging telescope is not comparable in sensitivity at low energies ( 0.1 - 2 kev ) to . however the exosat medium energy proportional counter ( me ) is comparable at the higher energies ( approximately 1 - 9 kev ) where , also , the effects of absorption which may affect measurements of x - ray flux variations , are less . in addition , the me has made long observations of some bright agn ( eg 3c273 , cena , ngc5506 ) for which long observations by do not yet exist . in this paper the exosat me observations of agn are re - examined to determine whether they reveal any scaling of hf psd normalisation with mass . @xcite provided initial nva measurements for a sample of agn observed by exosat , but a more precise analysis of all exosat me agn observations longer than 20ksec ( 32 agn in total ) was carried out by @xcite . @xcite removed the small window artefacts from exosat psds using their own 1d version of the ` clean ' algorithm @xcite and hence derived nvas . errors were determined from simulations . @xcite confirmed the inverse scaling of nva with luminosity , suggesting that the lower nvas arose in larger mass objects from a larger emitting region , and also found a positive correlation of nva with photon energy index . where variability was not detected , ie the hypothesis that the lightcurve was constant could not be rejected at the 95 per cent confidence level at least 90 per cent of the time , @xcite provide an nva upper limit , deduced from the photon counting noise level of the lowest noise observation . the reader is referred to @xcite for full details . .nvas from @xcite for agn with detected variability in exosat me observations . black hole masses are from 1 @xcite , 2 @xcite , 3 @xcite , 4 @xcite , 5 @xcite , 6 @xcite , 7 @xcite . [ cols="<,>,^,^ " , ] in table [ tab : detect ] the nvas where variability was detected are listed together with the most accurate black hole masses available . masses derived from stellar dynamical observations ( ngc3227 and ngc4151 ) , which are not subject to the uncertainty regarding the ` f - factor ' used to convert dynamical products from reverberation observations to black hole masses , are taken preferentially . as dynamical measurements are rare , reverberation masses are next taken ( 3c120 , ngc3783 , ngc4051 , 3c273 and ngc4593 ; see the caption to table 1 ) . for mcg-6 - 30 - 15 the mass estimated by @xcite is used . this mass is the mean of the mass derived from the width of the stellar absorption lines and from the emission line width . for ngc5506 and ngc7314 the mass is derived from the stellar absorption line widths of 98 and 60 km s@xmath12 respectively @xcite using , as for mcg-6 - 30 - 15 , the @xmath25 relationships of @xcite and @xcite . the latter , particularly , has been shown by @xcite to provide a good mass estimate for low mass systems . the mean of the masses derived from these two relationships is taken and a 50 per cent uncertainty is assumed . no reliable mass estimator could be found for mcg+8 - 11 - 11 and so this agn is not considered further . in table [ tab : undetect ] the nva upper limits for sources where variability was not detected are listed . black hole masses are also given . a simple maximum likelihood ( ml ) analysis , using the function within the qdp plotting package which takes only the errors on nva , was applied to the agn with detected variability ( except for mcg+8 - 11 - 11 ) . this analysis shows that the relationship @xmath26 , with @xmath27 ( 90 per cent confidence ) is a good description of these data . the qdp ml w - var statistic , which is similar to @xmath28 , is 6.1 for 8 d.o.f . , indicating that the errors are overestimated . to determine whether @xmath2 varies with the method of fitting , a number of other methods were examined , particularly fitexy @xcite and bayesian regression @xcite . in the fitexy method , errors in both directions are included . the best value and uncertainty in @xmath2 were determined by measuring the minimum value of @xmath28 as a function of @xmath2 , with free normalisation ( fig . [ fig : slopes ] ) . the lowest overall @xmath28 of 4.24 corresponds to @xmath29 . for 8 d.o.f . , this @xmath28 value confirms that the errors are over - estimates . the 1@xmath30 ( ie @xmath31=1 ) uncertainty is 0.08 . the 90 per cent confidence uncertainty ( ie @xmath31=2.7 ) is 0.12 . the bayesian regression analysis of @xcite , as implemented within idl gives an almost identical value of @xmath32 . the standard deviation of 0.10 is slightly larger , probably because bayesian analysis takes account of the fact that the observed agn are just a random sample drawn from the parent distribution , a consideration addressed here in section [ sec : sample ] . thus the results do not depend on the fitting method . the best fit from the fitexy method is shown in fig . [ fig : nva ] . unless stated otherwise , fitexy results will be quoted hereafter . the nva upper limits are not included in this fit but are consistent with the fit . in principle the nva upper limits can be used to refine the value of @xmath2 . however the result depends on how the ` upper limits ' are interpreted and what probability density functions ( pdfs ) are assigned to them , eg whether they are hard limits above which there is no probability of finding the datum , or limits with an associated measurement error . eg the methodology of @xcite , as implemented within idl , treating the limits as hard , gives @xmath33 . allowing some error in the limit allows slightly flatter values . however as it is not clear how the limits of @xcite should be interpreted , it is concluded only that the upper limits are consistent with values of @xmath2 derived from detections , but not favouring values of @xmath2 much flatter than -0.5 . for almost all of the present sample it is now known that is well below @xmath34hz ( e.g. see * ? ? ? * ) and so the nvas reported by @xcite are a good measurement of the normalisation of the hf psd . in addition , where psd slopes could be reasonably measured , @xmath18 is mostly close to 2 , as expected for the hf part of the psd . however for ngc4051 @xmath35hz @xcite and the exosat observation length ( 207ks ) is long enough that a single power law fit would be noticeably flattened by the psd values below , where @xmath36 . re - examination of the exosat psd shows that the nva could be underestimated by up to 50 per cent . with a corrected nva , @xmath37 . parameterising log(nva)=@xmath2log(m ) + @xmath38 , then here @xmath39 . the only other source where a significant underestimation of the nva might be expected , based on its low mass , is ngc7314 . however the exosat observation is only 22ks and so the hf psd is unlikely to be noticeably distorted . even allowing for an extreme 50 per cent underestimate similar to that for ngc4051 , and also allowing for the same underestimate in ngc4051 , @xmath2 increases only to @xmath40 . measurement errors in the covariate , in this case mass , bias the slope towards zero ( e.g. * ? ? ? although not discussed in detail here , that bias can be addressed via simulations where each data point is represented by a 2d pdf . random selection of simulated data points from those pdfs gives values of @xmath2 which are typically flatter by 0.03 to 0.04 than those obtained by fitexy . as the simulations include the measurement error twice , i.e. once in the value of the data point itself and once in the distribution of the pdf , but fitexy only includes the error once , the true values of @xmath2 may be @xmath41 steeper than those given above . it is often questioned whether the x - ray emission mechanism in 3c273 is similar to that in seyfert galaxies , ie thermal compton scattering from a non - beamed corona , or whether it is synchrotron or synchrotron self - compton emission from a relativistic jet ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? 3c273 shows a bend in its long term psd which is consistent with arising from the same variability process as in seyfert galaxies @xcite but it is still unclear whether it should be included together with seyfert galaxies and non - blazar radio galaxies ( 3c120 ) , ie the remainder of table [ tab : detect ] , in any x - ray timing survey . the fit was therefore repeated without 3c273 , providing a similar slope ( @xmath42 from fitexy or @xmath43 from bayesian regression ) whose extrapolation passes well within the error range for 3c273 . these results suggest that the same process that drives x - ray variability in seyferts also drives variability in 3c273 although the x - ray emission location or emission mechanism need not be the same in both cases . with small samples such as that listed in table [ tab : detect ] , the results can depend on the choice of the values of the measured variables and/or of the sample content . for example for mcg-6 - 30 - 15 one might take the mass derived only from the stellar absorption lines ( @xmath44 , * ? ? ? * ) , which would not alter @xmath2 . or one might decide to take masses derived from reverberation for ngc3227 ( @xmath45 , * ? ? ? * ) and ngc4151 . the value of @xmath2 is again almost unchanged ( @xmath46 or @xmath47 if 3c273 is excluded ) although the fit , whilst remaining good , is very slightly worse . one might even decide to exclude mcg-6 - 30 - 15 , ngc5506 and ngc7314 altogether , giving @xmath48 . it is finally noted that @xcite propose a revision of black hole masses . using their ` classical ' ( ie all morphological type ) @xmath25 relationship , the masses of mcg-6 - 30 - 15 and ngc5506 would reduce by almost exactly a factor 2 , and that of ngc7314 by a factor 3 . all masses based on reverberation mapping would reduce also by a factor 2 due to their downward revision of the ` f - factor ' . as these changes are almost entirely systematic , the various fits for @xmath2 are almost unchanged with typical values flattening by only @xmath49 ( eg from -0.54 to -0.52 ) with , if anything , very slightly reduced dispersion . the overall conclusion is that @xmath2 is robust to minor changes in sample parameters with a value close to -0.54 . it is shown here that nva , defined by @xcite and derived more precisely from exosat me observations by @xcite , correlates very well , within a sample of agn alone , with black hole mass . for a relationship @xmath50 then , even allowing for different choices of mass or sample content , @xmath2 remains close to -0.54 with a dispersion of @xmath51 . the main contribution of the present work is thus to show that a close to inverse linear scaling of hf psd normalisation with mass , proposed by @xcite on the assumption that agn and xrbs follow the same scaling , occurs independently within a sample of agn alone . the large scatter within the agn @xmath15 values of @xcite may have arisen because their values of @xmath15 were derived from earlier measurements of x - ray excess variance by satellites with large orbital gaps in their lightcurves , ie asca and rxte . variations in accretion rate may perhaps add further scatter . within their sample of xrbs , @xcite show that @xmath15 is not constant for any given xrb , although they note that the range of variability with flux is not large . they do , however , note that the value of @xmath15 in the soft state of cyg x-1 is larger than in the hard state which may indicate some variation of hf psd normalisation with accretion rate or state . however the range of accretion rates within the sample listed in table [ tab : detect ] is not large . thus , as there is no evidence for dispersion in the present value of @xmath2 over and above that expected from the errors in the mass and nva measurements , possible variation of nva with accretion rate are not considered further here although they can not be ruled out . finally , the differences between the results of @xcite and those presented here are considered . the equivalent value of the slope @xmath2 derived by @xcite from bayesian regression is @xmath52 ( 90 per cent confidence ) whereas the value presented here is @xmath53 ( 1@xmath30 , or @xmath54 at 90 per cent confidence ) , or -0.57 if ngc4051 is corrected . possible reasons for the differences in slope include differences in analysis methods and in sample content . regarding analysis methods , for the present sample all regression methods produce the same value of @xmath2 . another analysis difference is that the nva values are derived from simple fits to the hf psds whereas the values of @xmath20 are derived from fitting a particular model , the mixed ornstein - uhlenbeck model , to the datasets . although parameters in multiparameter fits usually interact , there is no obvious reason why this model should result in larger values of @xmath2 . similarly fixing the hf psd slope at @xmath17 ( @xcite ) compared with measuring @xmath17 ( @xcite ) and then correcting the nva if @xmath18 is lower than 2 should have little effect . large differences in sample content may , however , have a large effect on the value of @xmath2 . only 6 of the 10 agn in each sample are in common . @xcite include the high accretion rate nls1s mkn766 and ark564 ( whose mass is uncertain ) . however these two nls1s are not included here where the spread in accretion rate is more limited . mkn766 and ark564 are both low mass agn and so have large leverage on the value of @xmath20 . if there is a slight increase of hf psd normalisation with accretion rate or ` state ' , as the observations of cyg x-1 by @xcite suggest , inclusion of these two agn would steepen @xmath20 . also some masses used here differ from those used by @xcite , taken from @xcite . with small samples , such differences in content can have a significant effect . further studies to determine @xmath2 for other samples of agn are clearly merited . the present results are also consistent with the work of @xcite who measure the excess variance , @xmath23 , for a sample of agn observed by . they find that @xmath55 where @xmath56 , which is very close to the relationship derived here , for the agn with 80ksec minimum duration observations or @xmath57 for agn with a minimum of 40 ksec duration . using nva measurements from @xcite it has been shown that , within a sample of agn observed by exosat , hf psd normalisation scales almost exactly inversely with black hole mass . these observations support the proposal of @xcite that hf psd normalisation scales exactly inversely from xrbs to agn . it is also noted that the quasar 3c273 fits well onto the scaling relationship derived for seyfert galaxies , suggesting that the same process which drives x - ray variability in seyfert galaxies also drives x - ray variability in 3c273 , even though the emission process or emission location may be different . i thank dimitrios emmanoulopoulos and christian knigge for extensive discussions about statistics and regression and i thank dimitrios for tuition in the use of mathematica . i thank liz bartlett for advice regarding idl and brandon kelly for discussions on regression and for advice as to how to run his bayesian regression code inside idl . i thank the anonymous referee for a useful and informative report .

the old exosat medium energy measurements of high frequency ( hf ) agn power spectral normalisation are re - examined in the light of accurate black hole mass determinations which were not available when these data were first published @xcite . it is found that the normalised variability amplitude ( nva ) , measured directly from the power spectrum , is proportional to @xmath0 where @xmath1 . as nva is the square root of the power , these observations show that the normalisation of the hf power spectrum for this sample of agn varies very close to inversely with black hole mass . almost the same value of @xmath2 is obtained whether the quasar 3c273 is included in the sample or not , suggesting that the same process that drives x - ray variability in seyfert galaxies applies also to 3c273 . these observations support the work of @xcite who show that an almost exactly linear anticorrelation is required if the normalisations of the hf power spectra of agn and x - ray binary systems are to scale similarly . these observations are also consistent with a number of studies showing that the short timescale variance of agn x - ray lightcurves varies approximately inversely with mass .

You are an expert at summarizing long articles. Proceed to summarize the following text: the energy gap formation is an ubiquitous phenomena in condensed matter systems . when the band structure appears in the one - particle hamiltonian with a periodic potential , the band gap is the region in the spectrum where there is no density of states . on the other hand , the repulsion interaction generates the energy gap in the fractional quantum hall systems . generally speaking , systems with an energy gap are more stable against perturbations . the systems with the energy gap are , however , not good nurturing cradles for the superconductivity , which arises in the systems with fermi surface ( gapless ) . because of the instability of the interaction with the phonons , the electrons pair up and condense to the superconducting state . however , there are some classes of superconductors which were obtained by doping the antiferromagnetic insulators with mobile carriers , for example high transition temperature superconductors in the cooper - based transition metal oxides ( cuprates ) @xcite . by the chemical doping , the systems enter the phase where the energy gap structure is anisotropic in the momentum space , before becoming the superconductors @xcite . the enigmatic gap phase has agonised condensed matter community for three decades . recently , one of us ( chern ) developed a weak - coupling theory based on the hubbard model for the gap formation in cuprates @xcite . introducing the spin berry s phase as the gauge interaction @xcite , the hubbard model in two dimensions can be formulated in the renormalizable theory in the continuous limit . considering the antiferromagnetic fluctuation additionally , the gauge field acquires the mass via the stckelberg mechanism . the 2 + 1 dimensional lagrangian density is given by @xmath2[(\frac{\vec { \nabla}}{i}-g\vec a){\psi}_{\sigma}(x ) ] \nonumber \\ -\frac{1}{4}f_{\mu\nu}f^{\mu\nu}+m_{0}(d_{0}{\phi}(x))^{\dagger}(d_{0}{\phi}(x))-m_{1}(\vec d{\phi}(x))^{\dagger}{\cdot}(\vec d{\phi}(x ) ) , \label{u1}\end{aligned}\ ] ] where @xmath3 are the electrons , @xmath4 are the gauge fields , @xmath5 is the gauge coupling , @xmath6 is the antiferomagnetic fluctuation , @xmath7 are the covariant derivatives , and @xmath8 and @xmath9 are the mass parameters . the antiferromagnetic fluctuation is parameterised by a complex phase field @xmath10 , where @xmath11 is the coupling between the gauge field and the antiferromagnetic fluctuation . in two dimensions , the @xmath6 field takes place an infinite order phase transition at the finite temperature , so called the berezinski - kosterlitz - thouless transition @xcite . combining with the gauge fields , the @xmath6 field becomes the longitudinal mode of the gauge fields . as the transition of the mass acquisition takes place , the electronic energy structure opens a gap without breaking the translational and the time reversal symmetry . the gap formation is not the patent for cuprates but has found in many other strongly - correlated electron systems , for example the iron pnictides and the heavy fermion systems @xcite . unlike the cuprates , the iron pnictides and the heavy fermion materials are the multi - band systems . it inspires us to generalise the current u(1 ) scheme to the su(@xmath12 ) cases , where the multiple @xmath12-flavours of electrons can be considered . furthermore , while the stckelberg mechanism works in the u(1 ) case , we generalise the mass acquisition scheme to the higgs mechanism . restricting ourself to the simplest fundamental representation for both electrons and the higgs , we found that there is always one flavour of the electrons which is not degenerate to the other for @xmath1 . this robust behaviour can be understood by the group theory . in this paper , the sections are organised as the following . in the second section , the su(2 ) case will be discussed . in the third section , the results of the su(@xmath12 ) cases are provided . the last section is the discuss and the conclusion . for a system with multi - flavours of electrons that are degenerate to each other , we can possibly consider the su(2 ) gauge symmetry . for simplicity , we consider the electrons to be in the su(2 ) fundamental representation . the u(1 ) lagrangian in eq . ( [ u1 ] ) can be generalised to the su(2 ) form , @xmath13^{\dagger}[(\frac{\vec { \nabla}}{i}-g\vec a){\psi}(x ) ] \nonumber \\ -\frac{1}{4}f_{\mu\nu}f^{\mu\nu}+m_{0}^{2}(d_{0}{\phi}(x))^{\dagger}(d_{0}{\phi}(x))-m_{1}^{2}(\vec d{\phi}(x))^{\dagger}{\cdot}(\vec d{\phi}(x ) ) , \label{lagrangian}\end{aligned}\ ] ] where @xmath14 , @xmath15 , @xmath16 , @xmath5 and @xmath17 are the gauge couplings for the electrons and the higgs boson respectively , and @xmath18 , @xmath4 , and @xmath19 are matrix - valued , @xmath20 where @xmath21 are the pauli spin matrices . the higgs field can be stabilised by the following terms @xmath22 where @xmath23 is the higgs mass and @xmath24 is self - interaction parameter . the total lagrangian density is given by @xmath25 . the mass generation of the su(2 ) gauge bosons via the higgs mechanism is a textbook story . for example , the mass acquisition of the gauge boson is related to the group representation of the higgs field . in the fundamental representation , three gauge bosons acquire the equal mass , and in the adjoint representation , only two gauge bosons obtain the mass . on the other hand , different from the high - energy physics , the condensed matter community cares more about the length scale . the gauge bosons of zero mass produce a long - ranged interaction , and the ones of finite mass produce a short - ranged interaction . in the condensed matter systems , the long - ranged interaction is often screened and becomes short - ranged . in the systems with the gauge symmetry , it corresponds to the gauge bosons of finite mass @xcite . as the gauge bosons acquire the mass , the short - ranged interaction modifies the electronic specturm , opening a gap - like structure in the non - relativistic band structure @xcite . in the condensed matter language , the notion of the energy gap is different from the mass , which is determined by the curvature of the dispersion relation . the nature of the phase transition to the gap phase is , however , different from the u(1 ) case . in the higgs mechanism given by eq . ( [ higgs ] ) , it favors a second - order phase transition . in the real materials , it may take place at the finite temperature , if the two dimensionality of the space is only an approximation . similar to the u(1 ) case , we compute the energy gap using the single - particle green s function . the leading diagrams contributing to the self - energy term @xmath26 are given in the fig . ( [ diagram ] ) . in the fundamental representation of the higgs mechanism , the electronic gap , the energy at the bottom of the band , is @xmath27 for both flavors of the electrons . although the diagram in fig . ( [ diagram]b ) modifies the dispersion relation , it does not contribute to the gap generation . on the other hand , in the adjoint representation of the higgs mechanism , it becomes @xmath28 for all flavors of the electrons . the su(2 ) theory may be realized in the condensed matter system with the non - abelian holonomy @xcite and the magnetism . the non - abelian holonomy plays the role of the su(2 ) gauge fields . on the other hand , the ferromagnetic or the antiferromagnetic fluctuations may serve as the higgs field . if the non - abelian holonomy is in the particle - hole channel of the degrees of freedom , for example the spin berry s phase , it may be able to couple to the ( anti)-ferromagnetic fluctuation and manifests the effect of the electronic gap generation . the mechanism of the non - relativistic gap generation can be generalized to the su(@xmath12 ) case . the formalism of the su(@xmath12 ) lagrangian is the same as the ones in eq . ( [ lagrangian ] ) and eq . ( [ higgs ] ) . in addition , the electrons are considered in the su(@xmath12 ) fundamental representation , namely @xmath29 . if the higgs field is also considered in the fundamental representation , the mass spectrum of the @xmath30 gauge bosons can be given as the following . @xmath31 where @xmath32 gauge bosons remain massless , and the rest of them become massive . among the massive gauge bosons , there is always one boson acquiring different mass . the self energy of the electrons is also computed using the diagram in fig . ( [ diagram ] ) . we obtain @xmath33 for @xmath34 , we reproduce the results of the su(2 ) case . different from the su(2 ) case , however , there is always one flavor of the electron that is not degenerate to the rest of the @xmath0 electrons . this robust structure may be considered as the signature of the su(@xmath12 ) gauge symmetry for @xmath35 . the current results can be understood by the group theory . before the symmetry breaking of the higgs field , the theory is su(@xmath12 ) symmetric . in the fundamental representation , there are @xmath36 degrees of freedom in the @xmath12 multiplet of the higgs field . after the spontaneous symmetry breaking , there are @xmath37 goldstone modes which combine with the gauge bosons and become the longitudinal modes of the massive bosons . consequently , in the @xmath38 gauge bosons , there are @xmath32 boson remaining massless as shown in eq . ( [ higgsn ] ) . interestingly , the remaining @xmath32 bosons preserve the su(@xmath0 ) symmetry . after the symmetry breaking , the remnant symmetry becomes su(@xmath0 ) . therefore , spectrum of the @xmath12 electrons splits into @xmath39 , reflecting the su(@xmath0 ) symmetry . the nonrelativistic gap formation is generalized from the u(1 ) gauge symmetry with the stckelberg mechanism to the su(@xmath12 ) gauge symmetry with the higgs mechanism . in the u(1 ) case , the phase transition is the berezinskii - kosterlitz - thouless - like transition at the finite temperature in the 2 + 1 dimensional spacetime . namely , there is no significant signature of the phase transition . on the other hand , in the su(@xmath12 ) case , the gap spectrum of the @xmath12-plet of the electrons splits into @xmath39 , as the consequence of the remnant su(@xmath0 ) symmetry . the su(@xmath12 ) theory may be applicable to the system with non - abelian holonomy . we are grateful for the stimulated discussions with chong - der hu and pei - ming ho . this work is supported by ministry of science and technology of taiwan under the grant : most 103 - 2112-m-002 - 014-my3 and by na- tional taiwan university under the grant : 103r7831 and 104r7831 .

we demonstrate that the non - relativistic fermions open the energy gap when the su(n ) gauge bosons , mediating the interaction between fermions , acquire the mass . surprisingly , even though there is the su(n ) gauge symmetry , there is always one fermionic energy gap which is not degenerate to the rest of the @xmath0 fermions for @xmath1 in the fundamental representation . energy gap , non - abelian gauge systems , strongly - correlated electrons

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Proceed to summarize the following text: `` materials by design '' , the ability to design and create a material with specified correlated electron properties , is a long - standing goal of condensed matter physics . superlattices , in which one or more component is a transition metal oxide with a partially filled @xmath0-shell , are of great current interest in this regard because they offer the possibility of enhancing and controlling the correlated electron phenomena known @xcite to occur in bulk materials as well as the possibility of creating electronic phases not observed in bulk.@xcite following the pioneering work of ohtomo and hwang,@xcite heterostructures and heterointerfaces of transition metal oxides have been studied extensively . experimental findings include metal - insulator transitions,@xcite superconductivity , @xcite magnetism @xcite and coexistence of ferromagnetic and superconducting phases.@xcite solid solution in plane of carrier concentration ( changed by sr concentration ) and tilt angle in @xmath1 structure but with all three glazer s angles nearly equal . dashed line indicates relation between carrier concentration and rotation amplitude in physically occurring bulk solid solution . from ref . . ] in this paper we consider the possibility that appropriately designed superlattices might exhibit ferromagnetism . our work is partly motivated by a recent report@xcite of room - temperature ferromagnetism in superlattices composed of some number @xmath2 of layers of lavo@xmath3 ( lvo ) separated by one layer of srvo@xmath3 ( svo ) , even though ferromagnetism is not found at any @xmath4 in the bulk solid solution la@xmath5sr@xmath6vo@xmath3 . our study is based on a previous analysis@xcite of the possibility of obtaining ferromagnetism in variants of the crystal structure of bulk solid solutions of the form la@xmath5sr@xmath6vo@xmath3 . a key result of the previous work was that ferromagnetism is favored by a combination of large octahedral rotations and large doping away from the mott insulating lavo@xmath3 composition . a schematic phase diagram is shown in fig . [ fig : bulkpd ] . however , as indicated by the dashed line in the figure , in the physical bulk solid solution , doping away from the mott insulating concentration reduces the amplitude of the octahedral rotations so that the physical materials remain far from the magnetic phase boundary . the motivating idea of this paper is that in the superlattice geometry , octahedral rotation amplitude may be decoupled from carrier concentration . the rotations can be controlled by choice of substrate while the carrier concentration can be controlled by choice of chemical composition and may vary from layer to layer of a superlattice . in effect , an appropriately designed superlattice could enable the exploration of different paths in fig . [ fig : bulkpd ] . in this study , we combine single - site dynamical mean field approximation@xcite with realistic band structure calculations including the effects of the octahedral rotations to determine the ferromagnetic - paramagnetic phase diagram in superlattices with the crystal structures believed relevant@xcite to the experiments of ref . . unfortunately we find that the experimentally determined crystal structure is in fact less favorable to ferromagnetism than the one found in the bulk solid solution , but we indicate structures that may be more favorable . the paper has following structure . the model and methods are described in sec . [ sec : model ] . [ sec : cubicsuperlattice ] establishes the methods via a detailed analysis of the phase diagram of superlattices with no rotations or tilts . in sec . [ sec : tiltedsuperlattice ] we present the magnetic properties of superlattices with octahedral rotations similar to those observed experimentally . section [ sec : conclusions ] is a summary and conclusion . this paper builds on a previous study of the magnetic phase diagram of bulk vanadates.@xcite the new features relevant for the superlattices studied here are ( i ) the change in geometrical structure , including the differences from the bulk solid solution in the pattern of octahedral tilts and rotations and ( ii ) the variation of electronic density arising from superlattice structure . in the rest of this section we briefly summarize the basic theoretical methodology ( referring the reader to ref . for details ) , define the crystal structures more precisely , explain the consequences for the electronic structure and explain how the variation of density appears in the formalism . we study superlattices composed of layers of srvo@xmath3 ( svo ) alternating with layers of lavo@xmath3 ( lvo ) . if we idealize the structures as cubic perovskites , then the layers alternate along the @xmath7 $ ] direction . in bulk , svo crystallizes in the ideal cubic perovskite structure,@xcite while lvo crystallizes in a lower symmetry @xmath1 structure derived from the cubic perovskite via a four unit - cell pattern of octahedral tilts . @xcite the crystal structure of bulk solid solutions la@xmath5sr@xmath6vo@xmath3 interpolates between that of the two end - members with the rotation amplitude decreasing as @xmath4 increases . in the superlattice , the presence of a substrate and the breaking of translation symmetry can lead to different rotational distortions of the basic perovskite structure and also to a difference between lattice constants parallel and perpendicular to the growth direction . octahedral rotations in perovskites can be described using glazer s notation.@xcite in the coordinate system defined by the three v - o bond directions of the original cubic perovskite , there are 3 tilt angles @xmath8 and @xmath9 with corresponding rotation axes @xmath10,[010]$ ] and @xmath7 $ ] . the tilt is in - phase if successive octahedra rotate in the same direction , and anti - phase if they rotate in opposite directions . rotational distortions of the cubic perovskite @xmath11o@xmath3 structure may be denoted by @xmath12 where @xmath13 can be @xmath14 or @xmath15 denoting in - phase , anti - phase or no tilting , respectively and @xmath16.@xcite bulk lvo ( @xmath1 ) is of the type @xmath17 with @xmath18 and @xmath19.@xcite for superlattices , substrate - induced strain may change the situation in a way which depends on the growth direction . experiments@xcite confirm that the growth direction for the experimentally relevant superlattices is @xmath7 $ ] ( in the ideal cubic perovskite notation ) and we focus on this case here . recent experimental studies of superlattices@xcite and of lavo@xmath3 thin films , which apparently have the same growth direction,@xcite suggest that the rotations are of the type @xmath20@xcite and indicate that the dominant rotation is around the axis defined by the growth direction : @xmath21 and @xmath22 . this distortion pattern is different from that occurring in bulk . to explore its effects we set @xmath23 and consider the consequences of varying @xmath9 . in bulk la@xmath5sr@xmath6vo@xmath3 , while the 4-sublattice @xmath1 structure implies a difference in lattice constants , all v - o bond lengths are the same.@xcite the difference in lattice constants arises from a difference in tilting pattern . superlattices are typically grown on a substrate , and in epitaxial growth conditions the lattice constants perpendicular to the growth direction ( which we denote here by @xmath24 ) are fixed by the substrate , while the lattice parameter along the growth direction ( @xmath25 ) is free to relax . the result is a @xmath26 ratio typically @xmath27 contributed by both tilting and anisotropy in v - o bond lengths and possibly varying from layer to layer of the superlattice . for the experimentally studied superlattices , @xmath28 . @xcite the v - o bond lengths have not been determined but , as discussed in more detail in the appendix , our studies indicate that all v - o bonds have essentially the same length . further we show that a few percent differences have no significant effect on our study of ferromagnetism . in the rest of the paper we therefore ignore these distortions , setting all v - o bond lengths to be equal . we study superlattices designed to be similar to the system studied in ref . . in these superlattices , units of @xmath2 layers of lavo@xmath3 are separated by one layer of srvo@xmath3 . to define the superlattice , we begin from lvo in the appropriate bulk structure , then break translation invariance along the @xmath7 $ ] ( @xmath29-direction ) by replacing every ( m+1)@xmath30 lao plane with an sro plane . fig . [ fig : cartoon]a shows such a superlattice with @xmath31 . we assume that the superlattice is grown epitaxially so that in - plane bond lengths and other aspects of the local structure including rotations are the same for all layers . we therefore take the electron transfer integrals which define the band structure to the be same for all layers . in this case the electronic structure of a superlattice is defined by adding the electrostatic potentials of the sr and la ions to the basic translationally invariant hopping hamiltonian describing the bulk materials . in our calculations we follow the common practice in studies of early transition metal oxides by assuming that the energy splitting between transition metal @xmath0-bands and oxygen @xmath32-bands is large enough to justify the use of a `` frontier orbital '' model focusing on the @xmath32-@xmath0 antibonding bands which are mainly composed of vanadium @xmath33-symmetry @xmath0-states . the hamiltonian for the superlattice is thus @xmath34 where @xmath35 describes the electron - ion interaction and electron - electron interaction between different sites and @xmath36 describes the @xmath0-@xmath0 interactions , which we take to be on - site . @xmath37 is a tight binding model , derived by using maximally - localized wannier function ( mlwf ) techniques@xcite to fit the @xmath33-derived antibonding bands . the detailed procedure is described in our previous work.@xcite ( color online ) ( a ) schematic of superlattice lattice structure ( lavo@xmath3)@xmath38(srvo@xmath3)@xmath39 with @xmath31 . vanadium sites indicated as circles with charge density indicated by shading : heavy shading ( black online ) indicating higher charge density and light shading ( yellow online ) indicating lower charge density . lao and sro planes are shown as solid and dashed lines respectively . nearest neighbor ( @xmath40 ) and next - nearest neighbor ( @xmath41 ) hoppings between vanadium sites indicated by arrows . the numbers on the right are vo@xmath42 layer indices . ( b ) inset : @xmath43 hopping between @xmath33 orbital and @xmath32-orbital . main panel : two - dimensional nearest neighbor hopping @xmath40 made of two @xmath43 hoppings from @xmath44 orbital of one vanadium site to oxygen @xmath45 or @xmath46 orbital , then to @xmath44 orbital of another vanadium site . ] the kinetic hamiltonian has the quadratic form @xmath47 where @xmath48 and @xmath49 are electron creation and annihilation operators in reciprocal space with wavevector @xmath50 . @xmath51 and @xmath52 are orbital and layer indices , and @xmath53 is the spin index . we assume that the interaction takes the standard slater - kanamori form @xcite which following ref . we write as @xmath54 where the values of the on - site interaction @xmath55 and the hund s coupling @xmath56 are @xmath57 and @xmath58ev so that lvo is an insulator in bulk while svo is a metal . in the approximation employed here , the superlattice is defined by the coulomb interaction between the la / sr ions and electrons . this , and the off - site part of the electron - electron interaction is contained @xcite in @xmath59 to construct @xmath60 , we assume that the whole ion charge of svo or lvo unit cell comes into the sr or la site . consider srvo@xmath3 , the valence of v is @xmath61 ( @xmath62 . if this one @xmath0-electron is removed , the svo unit cell will have charge @xmath63 , hence , in our model , sr site has charge @xmath63 . similarly , lavo@xmath3 has v@xmath64 ( @xmath65 ) , thus la site has charge @xmath66 . as a result , @xmath60 has the form @xmath67 where @xmath68 is electron - occupation operator at v - site @xmath69 , @xmath70 is the relative dielectric constant . the part @xmath71 is the inter - site coulomb interaction of vanadium @xmath0-electrons @xmath72 @xmath71 is treated in the hartree approximation . note that in eq . , @xmath73 is the operator giving the total @xmath0-electron occupation of site @xmath69 , while @xmath74 is the expectation value of @xmath0-electron occupancy at site @xmath75 , which is determined self consistently . from @xmath35 , the coulomb potential @xmath76 for site @xmath69 is calculated using ewald summation.@xcite the dielectric constant @xmath70 is an important parameter in eqs . ( [ eqn : hel - ion ] , [ eqn : hel - el ] ) . it accounts for screening on the scale of a lattice constant so bulk measurements are not directly relevant and an appropriate value has not been determined . values ranging from @xmath77 to @xmath78 have been reported in the literature for similar systems.@xcite because the appropriate value of @xmath70 has not been determined , we have studied several cases and present results mainly for @xmath79 . we treat the on - site interaction terms using single - site dynamical mean field theory ( dmft)@xcite with the hybridization expansion continuous time quantum monte carlo ( ctqmc ) solver.@xcite the superlattice effect is taken into account by the coulomb potential @xmath80 . we use the superlattice dynamical mean field theory introduced by potthoff and nolting @xcite in the form given in ref . . here each v site @xmath69 has a self energy ( site local but dependent on site ) determined from the solution of a quantum impurity model which has parameters fixed by the dmft self - consistency equation linking the site local term of the lattice green function @xmath81 to the quantum impurity model green function@xcite @xmath82^{-1}\right\}_{ii},\ ] ] where @xmath83 is a site dependent quantity , diagonal in spin and orbital indices but linking different sites , derived from eqs . ( [ eqn : hel - ion ] , [ eqn : hel - el ] ) . the layers are coupled by a self - consistency condition which as discussed in refs . fixes both the hybridization function of the quantum impurity model and the layer - to - layer variation in the charge density . as described in ref . , it is advantageous to perform a site - local rotation to align the orbital basis to the local v - o bond directions of each octahedron before solving the impurity model . this reduces the sign problem in the ctqmc impurity solver and restores in - plane translation invariance in the sense of making the self - consistency equations the same for all sites in a given plane . in a superlattice composed of @xmath84 layers , it is in principle necessary to solve @xmath84 dynamical mean field problems , coupled by the self - consistency condition . however , we find ( see section [ sec : cubicsuperlattice ] ) that the susceptibility for a given layer of the superlattice may be determined from a bulk computation at the same local density and crystal structure . because the layer dependent density has no significant dependence on the temperature or the many - body physics , it may be determined once from a band structure calculation and then bulk results with the appropriate density for a wide range of temperature may be used to infer the curie temperature , substantially reducing the computational burden . the curie temperature for ferromagnetism is determined by extrapolating the inverse susceptibility @xmath85 to @xmath15 based on curie - weiss law @xmath86 . the test for the reliability of this method for @xmath87 has been done in ref . . a similar approach can be found in literature.@xcite in this section , we demonstrate that the magnetic phase diagrams of superlattice systems may be inferred , to reasonable accuracy , from the study of appropriately chosen bulk systems . this enables a considerable reduction in the computation resources required . ( color online ) panel ( a ) : non - interacting density of states for bulk system at carrier density @xmath88 . panels ( b ) : non - interacting density of states for different layers of ( lvo)@xmath3(svo)@xmath39 superlattice for two different values of dielectric constant @xmath89 ( solid ) and @xmath90 ( dashed ) with hopping parameters @xmath91ev and @xmath92ev . sro plane is between layers @xmath15 and @xmath93 ( the index is defined in fig . [ fig : cartoon ] ) . the fermi energy is at @xmath15 . ] we begin with a study of `` untilted '' or `` cubic '' superlattices : those in which all v - o - v bond angles are @xmath94 . we focus specifically on @xmath7 $ ] superlattices in which the unit cell contains @xmath2 layers lvo and one layer svo , where @xmath95 . for orientation , we present the density of states ( dos ) of the non - interacting system in fig . [ fig : dos ] . in obtaining these densities of states we used the simple tight binding parametrization . the dos for the bulk system is shown in panel ( a ) . one sees the typical three - fold degenerate dos for @xmath33 band , the van hove singularity is visible as a peak near the upper band edge . it is at high energy because the next - nearest neighbor hopping @xmath96 . the remaining panels show the layer - resolved densities of states for the @xmath31 superlattice . the upper two panels show layers sandwiched by la on both sides ; the lower two panels show the layers adjacent to the sro plane . the superlattice - induced changes in the density of states are seen to be relatively minor : the main effects are a weak splitting of the van hove peaks reflecting the breaking of translational invariance in the @xmath29-direction , and a relative shift in the positions of the van hove peaks arising from band bending associated with the different charges of the sr and la ions . [ fig : slchi ] shows the layer - resolved charge density and inverse susceptibility @xmath97 plotted against temperature for three different superlattice structures corresponding to @xmath98 . as expected from electrostatic considerations , the charge is lower for the vo@xmath42 planes nearer the sro layer and the charge variation between layers is controlled by the dielectric constant . the magnetization @xmath2 at the v sites on each layer was computed at field @xmath99 and the inverse susceptibility was obtained as @xmath100 . linearity was verified by repeating the computation using @xmath101 ( not shown ) . for the @xmath102 case ( fig . [ fig : slchi]a ) , we extended the computation to the lower temperature @xmath103ev ; for the other two cases @xmath104ev was the lowest temperature studied . the inverse susceptibilities are approximately linear in temperature at higher temperatures and in all cases , extrapolation to @xmath105 reveals @xmath106 , implying absence of ferromagnetism . ( color online ) temperature - dependent layer - resolved inverse magnetic susceptibilities for symmetry - inequivalent layers of untilted ( lvo)@xmath38(svo)@xmath39 superlattice structures with different numbers of lvo layers @xmath107 and @xmath108 . layer @xmath15 is adjacent to sro and layers @xmath109 and @xmath110 are between two lao layers . the relative dielectric constant is @xmath90 , magnetic field @xmath99 . the @xmath85 obtained from solution of bulk cubic systems with charge density set to the density on the given layer are also shown . `` bulk l0 '' ( `` bulkl2 '' ) denotes a calculation performed for a bulk system with density the same as for @xmath111 ( @xmath112 ) layer density . inset : the electron layer density distribution corresponding to the susceptibility plot , @xmath4-axis is the layer index , @xmath113-axis is the layer density . on - site interactions @xmath114ev , @xmath58ev . ] especially for the layer nearest the sro plane the @xmath115 curves exhibit weak upward curvature at the lowest temperatures studied . as shown in ref . , the curvature is a signature that the system is entering a fermi - liquid coherence regime . the fermi liquid coherence temperature is highest for the layers nearest the sro because the charge in these planes is farther from the @xmath116 mott insulating state . to verify this we followed ref . and computed the wilson ratio @xmath117 for each layer of the superlattice for the case @xmath118 , finding ( not shown ) that for each layer the @xmath117 extrapolates to @xmath109 at low temperature . the approach to the low temperature value is faster for layers with low density ( near sro planes ) than for layers with high density ( far from sro planes ) . @xmath119 is the value for a kondo lattice , while ferromagnetism is characterized by an @xmath120.@xcite we therefore believe that for `` untilted '' superlattices , the differences in @xmath115 among layers arise from differences in quasiparticle coherence scale , there is no evidence for ferromagnetism in this system , consistent with the solution of the corresponding bulk problem . to gain insight into the physics underlying the layer dependence of @xmath115 we have computed @xmath85 for the cubic bulk system ( @xmath121 , @xmath37 is constructed from the two - dimensional dispersion @xmath122 ) for carrier densities equal to those on the different vo@xmath42 layers . in fig . [ fig : slchi ] , we present bulk calculations for @xmath123 and @xmath124 corresponding to the densities calculated for layer @xmath15 and @xmath109 of the superlattice for all cases @xmath98 . for @xmath124 , bulk @xmath115 at @xmath125 and @xmath126ev are very close to those of @xmath112 layer of @xmath31 superlattice , which has the same density . for @xmath127 superlattices , bulk @xmath124 , @xmath85 ( not shown ) almost coincides with those of @xmath112 layer . for bulk @xmath123 , the difference between bulk and superlattice @xmath111 layer is small . these calculations demonstrate a general rule : within the single - site dmft approximation , the layer - resolved properties of a superlattice correspond closely to those of the corresponding bulk system at a density equal to that of the superlattice . ( color online ) comparison between bulk lvo partial dos ( positive curves ) and ( lavo@xmath3)@xmath3(srvo@xmath3)@xmath39 superlattice layer dos of layers near sro ( negative curves ) derived from band structure calculations ( dft+mlwf ) . both systems have the same lattice structure for each case : untilted structure for the top panel and @xmath128 structure ( glazer s notation @xmath20 ) with @xmath129 and @xmath130 and @xmath131 for other panels . the dos of bulk system is shifted towards higher energy so that bulk carrier density is the same as layer density of superlattices for the layers near sro ( @xmath132 ) . the vertical dashed line marks the fermi level . ] the superlattices of experimental relevance have crystal structures which are distortions of the `` untilted '' one , involving in particular a @xmath128 structure characterized by a rotational distortion of the @xmath20 type@xcite involving a large rotation about an axis approximately parallel to the growth direction and much smaller rotations about the two perpendicular axes . [ fig : dos_bulk_sl ] compares the non - interacting dos of bulk and ( lvo)@xmath3(svo)@xmath39 superlattice systems ( both with the same @xmath128 structure ) calculated using dft and a mlwf parametrization of the frontier bands . the dos of bulk system is shifted so that it has the same carrier density as layers of the superlattices near sro plane . for three different structures ( untilted structure and @xmath128 structure with @xmath133 and @xmath131 ) , the basic features of the partial dos are similar between bulk and superlattice . the translation symmetry breaking in @xmath29-direction leads to small extra peaks in the superlattice dos . these differences are smoothed out by the large imaginary part of the dmft self energy . because the dmft equations depend only on the density of states it is reasonable to expect that , as in the untilted case , they will therefore give the same results in the superlattice as in the bulk material with corresponding density of states . ( color online ) comparison in temperature dependent inverse susceptibility between bulk lvo ( solid lines ) and ( lavo@xmath3)@xmath3(srvo@xmath3)@xmath39 superlattice ( dashed lines ) . both have the same lattice structure @xmath128 with tilt angle @xmath134 and @xmath135 . bulk system has the same densities as those of layers of superlattice near and far from sro planes ( @xmath136 ) . left column : the plots in wide temperature range . right column : the expanded views near zero temperature . ] to verify that this is the case we have also compared bulk and superlattice susceptibilities for tilted structures . the four vo@xmath137 octahedra in a unit cell are related by rotation , so an appropriate choice of local basis means that only one calculation needs to be carried out for a given layer . [ fig : bulkvslayertilted ] compares the inverse susceptibilities for an @xmath31 superlattice to calculations performed on a bulk system with the same @xmath128 structure . in these calculations , we choose @xmath138 and dielectric constant @xmath139 . we see that in this case , as in the `` untilted '' case , the superlattice inverse susceptibilities @xmath85 are almost the same as those for bulk system calculated at the same density , with differences only resolvable in the expanded view for the largest tilt angles . in this section we present and explain our results for the magnetic phase diagram of ( lvo)@xmath38(svo)@xmath39 superlattices with the @xmath128 structure ( glazer s notation @xmath20 ) reported for the experimental systems.@xcite in these structures in - plane rotation along the growth direction @xmath141 $ ] is large @xmath130 ( presumably because of the strain imposed by the substrate ) , while the out - of - plane rotation is small ( @xmath129 ) perhaps because the system is free to relax along the growth direction . we concentrate on the effect of the large rotation by fixing the in - plane angles to @xmath142 while varying the out - of - plane angles over a wide range from @xmath143 . ( color online ) partial dos derived from dft+mlwf for `` bulk '' @xmath128 structure ( glazer s notation @xmath20 ) with @xmath129 and @xmath144 changing from @xmath145 to @xmath131 . only @xmath33 bands are plotted because @xmath146 bands are negligible in this range of energy . ] based on the results of section [ sec : cubicsuperlattice ] we generate a phase diagram for the superlattice from calculations for a bulk system which is a @xmath128 distortion of the ideal cubic perovskite structure of chemical composition lavo@xmath3 . the bulk system results are presented as a phase diagram in the plane of carrier concentration and @xmath9-rotation . specific layers of the superlattice will correspond to particular points on the phase diagram , with the layer dependent density fixed by number of lvo layers @xmath2 and the dielectric constant @xmath70 and the rotation fixed by the substrate lattice parameter . we use dft+mlwf methods to obtain the frontier orbital band structure for the @xmath33-derived antibonding bands [ fig : pdos_p21 m ] presents representative results for the orbitally resolved local density of states . in this figure the orbitals are defined with respect to the local basis defined by the 3 v - o bonds of a given vo@xmath137 octahedron . we define @xmath141 $ ] as the axis ( approximately parallel to the growth direction ) about which the large rotation occurs . [ fig : pdos_p21 m ] shows that @xmath147 and @xmath148 orbitals are almost degenerate , while @xmath44 orbital is strikingly different . the dos of @xmath44 orbital maintains the shape of a two - dimensional energy dispersion with a van hove peak well above the chemical potential , similar to the bulk cubic structure ( see e.g. fig . [ fig : dos]a ) . there are noticeable differences only at very high rotation angles . on the other hand , @xmath147 and @xmath148 orbitals are spread out with two small peaks , because hoppings along @xmath4 or @xmath113 directions ( more distorted ) are different from those along @xmath29-direction ( less distorted ) . when the distortion gets larger , the @xmath0-bandwidth becomes smaller , the @xmath44 peak gets larger and slightly closer to the fermi level , and @xmath147 and @xmath148 peaks near the fermi level also develop . based on @xmath149 generated by dft+mlwf , we carry out dmft calculations for in - plane rotation angle @xmath9 to get @xmath115 curves whose extrapolations define the curie temperatures @xmath87 . [ fig : tc_evolve ] shows how @xmath87 evolves when the rotation angle @xmath9 increases from @xmath150 to @xmath135 . in this figure , we consider two different carrier densities @xmath151 and @xmath152 , corresponding to the band structure prediction for the layer densities of layers near and far from sro planes in the superlattice . @xmath87 for @xmath153 is a slow function of rotation and is always negative for the range of @xmath9 under consideration , while @xmath87 for @xmath151 increases faster , so that the system becomes ferromagnetic when @xmath9 is between @xmath154 and @xmath131 . ferromagnetism is therefore expected only in superlattices with very large rotations , and then only in the layers with large hole doping ( i.e. the layers closest to the sro planes ) . ( color online ) inverse susceptibility @xmath115 vs. temperature @xmath155 for bulk @xmath128 structure of lavo@xmath3 at densities @xmath151 ( black circle solid lines ) and @xmath153 ( red diamond dashed lines ) for rotation angle @xmath9 increasing from @xmath156 . on - site interaction @xmath114ev and @xmath58ev . left column : the circles and diamonds are data points , the solid and dashed lines are fitted from these data points . right column : expanded view at small @xmath115 region . the vertical dashed line marks zero temperature . ] from a range of calculations such as those shown in fig . [ fig : tc_evolve ] we have constructed the superlattice magnetic phase diagram shown in fig . [ fig : phase_diagram ] . similar to ref . , there are uncertainties in our extrapolation for curie temperature , we consider @xmath157ev as the error bar for positions on the phase diagram . thus , @xmath158ev is considered as @xmath159 within the error bar . we see that ferromagnetism is favored only for very large rotations , much larger than the @xmath160 determined experimentally , and only for carrier concentrations far removed from @xmath116 . we may locate the experimentally studied superlattices on this phase diagram . for an @xmath31 superlattice , band structure calculations indicate layer densities @xmath161 for layers near sro plane and @xmath152 for the other layers . the experimentally determined rotation angle is @xmath162 . these two points are indicated by squares in fig . [ fig : phase_diagram ] . ( color online ) the magnetic phase diagram with @xmath4-axis carrier density @xmath163 and @xmath113-axis tilt and rotation angle along @xmath141 $ ] direction @xmath9 for bulk system lvo with the same type of distortion as for ( lvo)@xmath38(svo)@xmath39 superlattices ( @xmath128 structure ) , in - plane tilt angles @xmath164 . on - site interactions @xmath114ev , @xmath58ev . the white regime indicates absence of ferromagnetism ( @xmath165 ) , the colored regime indicates ferromagnetism with @xmath87 indicated by the color bar . also indicated are results for bulk la@xmath5sr@xmath6vo@xmath3 in the @xmath1 structure , from ref . . note that in the calculations for the @xmath1 structure all three tilt angles are almost the same . ] it is interesting to compare our results to those previously obtained @xcite for the bulk solid solution la@xmath5sr@xmath6vo@xmath3 ( @xmath1 structure ) . the dashed line in fig . [ fig : phase_diagram ] shows the theoretically estimated phase diagram for the bulk solid solution . we see that the bulk structure is more favorable for ferromagnetism than the superlattice structure . an important difference between the @xmath1 structure and the @xmath128 of the superlattice is that in the former case all three tilt angles are of comparable magnitude whereas in the @xmath128 structure only one rotation is large . we believe that this difference is responsible for the difference in phase boundary . in this paper , we have studied the possibility of ferromagnetism in superlattice structures of vanadium oxides derived from lavo@xmath3 and srvo@xmath3 . our investigation was based on the idea that ferromagnetism depends on an interplay between carrier density and octahedral rotation , and while these are coupled in bulk ( see the solid solution curve in fig . [ fig : phase_diagram ] ) they may be decoupled in the superlattice . in particular , the charge density varies across the superlattice , being lowest near the sro planes , while the rotation angle is controlled by the substrate . thus in an appropriately designed superlattice at least some portions of the system might be moved closer to ( or perhaps into ) the ferromagnetic region . in several important aspects this idea is consistent with calculations . we find that the local carrier density determines the local magnetic susceptibility ( see section [ sec : cubicsuperlattice ] ) and the density / tilt angle relationship may be significantly altered ( see solid line and square points in fig . [ fig : phase_diagram ] ) . however , we find that the @xmath128 octahedral rotation pattern characteristic of experimentally discovered superlattices is in fact less favorable to ferromagnetism than the @xmath1 pattern characteristic of bulk materials ( compare the phase boundaries in fig . [ fig : phase_diagram ] ) . thus while the general idea that an appropriately designed superlattice might provide conditions favorable for ferromagnetism thereby providing a potential explanation for the remarkable experimental report of room - temperature ferromagnetism in ( lavo@xmath3)@xmath38(srvo@xmath3)@xmath39 superlattices with @xmath166 by lders et . al.,@xcite ( even though there is no ferromagnetism in the bulk solid solution ) , our detailed findings are not consistent with the experimental result . our results indicate that designing ferromagnetism into a vanadate superlattice will require both large amplitude rotations about the growth axis and also substantial rotations about the other two axes . rotations about the growth axis arise from substrate - induced strain , so choosing substrates with smaller lattice parameter would be desirable . introduction of rotations about the orthogonal axes may be done by replacing the la with a smaller counterion such as y. our study has certain limitations . the calculations employ a frontier orbital model which includes only the @xmath33-derived antibonding bands . dft+dmft calculations based on correlated atomic - like @xmath0-states embedded in the manifold of non - correlated oxygen states provide a more fundamental description . our previous work@xcite indicates that the two models give very similar results if both calculations are tuned so that bulk lavo@xmath3 is a mott insulator , but the implications of the full ( but computationally very heavy ) dft+dmft procedure for the superlattice problem remain an open problem for future research . further , our calculations are based on the single - site dmft approximation , which includes all local effects but misses inter - site correlations . while it is generally accepted that these calculations give the correct trends and qualitative behavior , the quantitative accuracy of the methods is not known . unfortunately , as yet cluster extensions of dmft are prohibitively expensive for the multiband models considered here . the experimental results of lders et . al.@xcite therefore provide an interesting challenge to materials theory . they indicate that superlattices display ferromagnetism when the corresponding bulk solid solutions do not , whereas the present state of the art of real materials dynamical mean field calculations suggests that superlattices should be less likely to display magnetism than the corresponding bulk solid solutions . this discrepancy requires further investigation . we thank u. lders and j. okamoto for helpful conversations . we acknowledge support from doe - er046169 . htd acknowledges partial support from vietnam education foundation ( vef ) . we acknowledge travel support from the columbia - sorbonne - science - po ecole polytechnique alliance program and thank ecole polytechnique ( htd and ajm ) and jlich forschungszentrum ( htd ) for hospitality while portions of this work were conducted . a portion of this research was conducted at the center for nanophase materials sciences , which is sponsored at oak ridge national laboratory by the scientific user facilities division , office of basic energy sciences , u.s . department of energy . we use the code for ct - hyb solver@xcite written by p. werner and e. gull , based on the alps library.@xcite cccccccc atom & @xmath4 & @xmath113 & @xmath29 & atom & @xmath4 & @xmath113 & @xmath29 + la@xmath167 & 0 & 0.25 & 0 & la@xmath168 & 0.5 & 0.25 & 0.5 + v@xmath167 & 0.5 & 0 & 0 & v@xmath168 & 0 & 0 & 0.5 + o@xmath169 & 0.4662 & 0.25 & 0.0660 & o@xmath170 & 0.0392 & 0.25 & 0.4392 + o@xmath171 & 0.7638 & -0.0138 & 0.2362 & o@xmath172 & 0.2652 & -0.0493 & 0.2652 + + & @xmath173()&@xmath174( ) & @xmath175( ) & @xmath176 & @xmath24 ( ) & @xmath25 ( ) & @xmath26 ratio + exp . lvo thin film@xcite & 5.55 & 7.82 & 5.55 & @xmath177 & 3.91 & 3.945 & 1.008 + exp . superlattice@xcite & na & na & na & na & 3.88 & 3.95 & 1.018 + calculated with @xmath178 & 5.5988 & 7.8290 & 5.5821 & @xmath179 & 3.915 & 3.988 & 1.019 + calculated with @xmath180 & 5.5512 & 7.7623 & 5.5346 & @xmath179 & 3.881 & 3.954 & 1.019 + in this appendix , we present a more complete discussion of the strain - induced lattice distortions . the in - plane lattice constant of a superlattice epitaxially grown on a substrate matches that of the substrate and may therefore be different from the lattice constant preferred in a free - standing film or bulk material . the out - of - plane lattice constant is typically free to relax , and in the presence of an in - plane strain may also be different from that found in bulk materials . a difference in v - v distance may arise from a change in v - o bond length or from a difference in buckling of v - o bonds . we consider both possibilities here , but first remark that the main differences in structure between bulk and experimentally studied superlattices arise from differences in octahedral rotation . in the experimentally - studied superlattices , the in - plane v - v distance is in fact slightly less than the v - v distance in lvo . the v - o bond lengths have not been measured for the superlattice , but to a high degree of accuracy we are able to reconstruct the measured superlattice using the measured tilt angles given from experiments@xcite structure , assuming that all v - o bond lengths are equal . assuming the @xmath128 structure , we varied the in - plane and out - of - plane v - o bond lengths to fit the experimental data and found that @xmath181 only when the mean bond length @xmath0 is found in the range from @xmath182 to @xmath109 depending on which experimental result is fit but in all cases the v - o bond lengths are found to be equal to within an accuracy of @xmath183 . therefore , we believe that all the v - o bond lengths should , to a good approximation , be the same . the structure used in our calculations is presented in table . [ table : p21m_struct ] . although there are slight mismatches in in - plane angle and lattice constants , the @xmath181 ratio and bond angles are compatible with the experiment . ( color online ) partial dos for bulk lvo with @xmath184 with the @xmath26 ratio due to a change in v - o bonds ( panel ( a ) ) and to @xmath128 lattice structure with @xmath185 ( similar to superlattice structure)(panel ( b ) ) . the dashed blue curve is the @xmath44 orbital , the solid red curve is the degenerate @xmath147 ( or @xmath148 ) orbital . the dashed vertical line marks the fermi level . ] changing the amount of rotation has a different effect on the electronic structure than does changing the ratio of v - o bond lengths . [ fig : bondlength_pdos ] compares the partial dos for the two cases , using as example a hypothetical lavo@xmath3 crystal with @xmath184 . the upper panel presents the dos for the untilted structure with straight v - o - v bonds and the @xmath26 ratio induced by a difference in in - plane and out - of - plane v - o bond lengths . the lower panel presents the case of all equal v - o bonds , with the @xmath26 ratio produced by octahedral rotations about the @xmath29 axis . the densities of states are quite different , but can be understood from the simple energy dispersion @xmath186 where @xmath187 and @xmath188 are the in - plane and out - of - plane nearest neighbor hopping integrals and @xmath189 are the second neighbor hoppings . the lower band edge is assumed to be the same for all orbitals but we assume that the lattice distortions lead to different values for the in - plane and out of plane hoppings . ( color online ) inverse susceptibility vs. temperature for cubic structure of bulk hole - doped lvo . the in - plane and out - of - plane bondlengths are changed so that the octahedral volume is unchanged : tensile strain ( @xmath190 - black lines ) , no strain ( @xmath191 - red lines ) and compressive strain ( @xmath184 - blue lines ) . two levels of hole doping are considered : @xmath151 ( solid lines ) and @xmath153 ( dashed lines ) . these lines are linear fits for the data points . ] the lower band edge is defined to be zero and is independent of the distortion . the energy of the upper edge of the @xmath44 band is @xmath192 and of the @xmath193 bands is @xmath194 . the positions of the van hove singularities are at @xmath195 or @xmath196 . for @xmath44 band there is only one van hove peak , at @xmath197 ; while for @xmath198 band , there are two van hove peaks at @xmath199 and @xmath200 . when @xmath188 is different from @xmath187 , the difference in bandwidth of @xmath44 and @xmath148 orbitals is @xmath201 , which is also the distance between the two van hove peaks of @xmath148 band @xmath202 . with these definitions , we are in a position to understand the changes in the band structure . when the v - o bond lengths change ( fig . [ fig : bondlength_pdos]a ) so that the @xmath29-bond is longer and the in - plane bond is shorter but the octahedral volume is unchanged , the band structure calculation indicates that @xmath188 decreases but @xmath187 increases slightly . the difference between the bandwidth of the @xmath44 and @xmath198 bandwidths is @xmath201 which is the same as the splitting between the van hove peaks in the @xmath193 bands . on the other hand , if the @xmath26 ratio is produced by rotation , ( fig . [ fig : bondlength_pdos]b ) , the change is opposite . the in - plane hopping @xmath187 decreases because of the buckled in - plane v - o - v bonds , while the out - of - plane hopping @xmath188 is unchanged . the @xmath44 band therefore narrows substantially relative to the @xmath193 bands . in addition the splitting of the van hove peaks is greater . from the bandwidth of @xmath44 and @xmath148 bands ( fig . [ fig : bondlength_pdos]b ) , @xmath203ev , @xmath204ev , the van hove peak distance is @xmath205ev , which is compatible with the peak positions shown in fig . [ fig : bondlength_pdos]b . we tested with dmft calculations for the curie temperatures with the v - o bondlength changed . [ fig : chi_bondlength ] is the temperature - dependent inverse susceptibility derived from dmft for the bulk cubic structure with the @xmath26 ratio changing from 0.98 ( tensile strain ) to 1.02 ( compressive strain ) . for all the levels of hole doping under consideration , the results are nearly the same for every case of @xmath26 ratio . we conclude that even when the v - o bondlength changes within the physical range , the ferromagnetism is not affected . however , we also found that when the v - o bondlength is such that @xmath206 or @xmath207 , there is large orbital polarization and the ferromagnetism can be largely affected . but that range is unphysical and can be neglected in the context of this work .

motivated by recent reports ( phys . rev . b**80 * * , 241102 ) of room - temperature ferromagnetism in vanadium - oxide based superlattices , a single - site dynamical mean field study of the dependence of the paramagnetic - ferromagnetic phase boundary on superlattice geometry was performed . an examination of variants of the experimentally determined crystal structure indicate that ferromagnetism is found only in a small and probably inaccessible region of the phase diagram . design criteria for increasing the range over which ferromagnetism might exist are proposed .

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Proceed to summarize the following text: small oscillations about the stable equilibrium of a many - body ground state are quantized as bosonic quasi - particles or bosons . in various physical contexts the linearized equations of motion for such excitations are known as the random phase approximation , or rpa equations for short @xcite . concrete examples are furnished by the vibrational excitations of a solid , the spin waves of a magnet , the electromagnetic modes of an optical medium , or the density oscillations of a bose - einstein condensate . constrained by the requirement of dynamical stability , the hamiltonian @xmath2 of any vibrational or quasi - boson system of the mentioned kind must lie in a positive cone , @xmath3 , of so - called elliptic symplectic generators . it should be stressed that although @xmath2 is hermitian as an operator in fock space , the quantum - to - classical mapping sends @xmath2 to an rpa generator @xmath4 which is in general neither hermitian nor anti - hermitian as a linear operator on the classical phase space . in view of this , a distinctive feature of the set of elliptic generators @xmath5 is that they can be brought to diagonal form ( with real frequencies , corresponding to stable oscillatory motion ) by real bogoliubov transformations , i.e. , by conjugating with elements of the real symplectic group @xmath6 . in this general setting , our goal is to investigate what happens with observables such as the spectral statistics and the transport properties when the bosonic system is strongly disordered . in particular , we wish to understand whether there exists some low - energy universality , possibly of an unusual type , due to the interplay between disorder and the geometry of the cone @xmath3 . ( for example , a high degree of low - temperature universality is known to be observed @xcite in strongly disordered solids as well as amorphous or glassy systems . ) motivated by this question , in the present paper we initiate the study of a class of semi - realistic random matrix models for disordered bosons . by construction , the probability measures of the models we propose are supported on @xmath3 . thus , unlike @xcite , the unphysical behavior of runaway motion associated with complex frequencies is excluded . our rpa generators @xmath7 have the particular feature of being sums of a deterministic and a random part . for simplicity we focus here on models without time - reversal symmetry , although tr - invariant models of a similar kind can be treated with little extra effort . by using a variant of the efetov - wegner supersymmetry method , we will derive an approximation for the density of states of mean - field or self - consistent type , reminiscent of the so - called ` coherent potential approximation ' ( cpa ) @xcite . in the longer term , the goal is to develop a description of our disordered boson models by field theories of the non - linear sigma model type . ( as is well known @xcite , such a description has proven very useful for the case of disordered fermions ) . in that formulation , universality ( if any ) is expected to emerge whenever the renormalization group flow gets attracted to a few - parameter manifold of renormalizable field theories . the plan of the paper is this . in section [ sect:2 ] we outline the basic setting and in particular , we review the notion of positive cone of elliptic symplectic generators . we also introduce the random models to be considered and give a summary of the analytical results obtained . section [ sect:3 ] is concerned with the derivation of the coherent potential approximation for our models . for pedagogical reasons , we first discuss the zero - dimensional case in some detail . the extension to @xmath1 dimensions is given in section [ sect:4 ] . there we also show some results for the numerical solution of the cpa equation . in this section , we start with some background on mathematical formulation and describe a class of random models which are tractable by the superbosonization variant of the efetov - wegner supersymmetry method . we then give a summary of our analytical results , relegating the presentation of numerical results to the end of the paper . let @xmath8 be a set of creation and annihilation operators for bosonic quasi - particles with quantum numbers @xmath9 . ( for example , @xmath10 might be a discrete set of momenta selected by periodic boundary conditions in a finite box . ) such operators span a hermitian symplectic vector space @xmath11 with symplectic form @xmath12 : \ ; w \times w \to \mathbb{c}$ ] defined by the canonical commutation relations @xmath13 = \delta_{k k^\prime } \ ; , \qquad [ a_k , a_{k^\prime } ] = 0 \ , , \qquad [ a_k^\dagger , a_{k^\prime}^\dagger ] = 0 \,.\ ] ] we now assume that we are given a linear hamiltonian dynamics on @xmath14 . this may be interpreted either as a linear hamiltonian flow on @xmath14 viewed as a classical phase space , or as a quantum time evolution on @xmath14 viewed as a subspace of the associative algebra of polynomials in @xmath15 , @xmath16 ( the so - called weyl algebra ) . in either case , the dynamical equations are @xmath17 ( the bar means complex conjugation . ) in order for the canonical commutation relations to be invariant under the dynamics , we require that @xmath18 and @xmath19 . thus @xmath20 are the matrix elements of an anti - hermitian matrix @xmath21 , while @xmath22 are those of a complex symmetric matrix @xmath23 . altogether , these conditions mean that @xmath24 is the generator of a symplectic transformation . more precisely , defining the lie algebra , @xmath25 , of the complex symplectic group by the linear condition @xmath26 @xmath4 lies in a non - compact real form @xmath27 determined by @xmath28 it should be mentioned that this description is appropriate in the absence of time - reversal invariance . if time reversal is a symmetry of the physical system , then the time - evolution generator @xmath4 is subject to additional complex anti - linear conditions . equations ( [ eq : rpa ] ) arise as the equations of motion for a system of non - interacting bosons with hamiltonian @xmath29 and dynamics @xmath30 $ ] . alternatively , one may imagine that they arise as an approximation to the collective motion of an interacting many - particle system ; as a particular example we mention density oscillations of a fluid . in the latter case , equations ( [ eq : rpa ] ) go under the name of random phase approximation ( rpa ) . the characteristic frequencies of the dynamical system ( [ eq : rpa ] ) or equivalently , the single - boson energies of the hamiltonian @xmath2 , can be computed as the eigenvalues of @xmath4 . owing to the symplectic condition @xmath31 the characteristic polynomial satisfies @xmath32 . the eigenvalues of @xmath4 therefore come as pairs @xmath33 . if @xmath4 lies at some random position in the real symplectic lie algebra @xmath34 , then its eigenvalues will typically be complex , since @xmath35 is neither hermitian nor anti - hermitian . in the present context , however , complex eigenvalues are forbidden , as they correspond to the unphysical situation of unstable motion . in fact , the physical requirement of stability of the rpa dynamics dictates that the spectrum of @xmath4 must lie on the imaginary axis , so that the normal modes of the bosonic system are vectors in @xmath14 with periodic time dependence ( @xmath36 ) . moreover , the second - quantized hamiltonian @xmath2 in ( [ eq:2ndq ] ) must have a ground state in fock space . by standard considerations of linear algebra , all these stability conditions are met if and only if @xmath4 lies in the set @xmath37 we refer to @xmath3 as the _ positive cone of elliptic generators _ in @xmath34 . it is a fact that every @xmath5 can be brought to diagonal form by a real bogoliubov transformation , i.e. an element @xmath38 of the real symplectic group @xmath6 , which is defined by the condition @xmath39 in the present paper we consider rpa generators @xmath4 , or equivalently hamiltonians @xmath2 , which are a sum of two parts : @xmath40 the term @xmath41 is the deterministic ( i.e. , non - random ) part of @xmath4 . while the formalism developed below can in principle handle any choice of @xmath41 , the explicit calculations presented in section [ sect : apply ] will be carried out for a simple concrete model of @xmath41 with unit mass matrix and elastic constants given by a discrete laplacian . a precise description of the concrete model for @xmath41 is as follows . let @xmath42 be a cubic lattice in @xmath1 space dimensions and associate with each site @xmath43 the operators @xmath44 and @xmath45 for boson creation and annihilation , respectively . we then take the second - quantized hamiltonian to be @xmath46 where the sum for the second term on the right - hand side is over nearest neighbor pairs of sites @xmath47 of @xmath48 . such a hamiltonian is easily diagonalized by fourier transforming to momentum space . the spectrum of single - boson energies @xmath49 as a function of the wave vector @xmath50 comes out to be @xmath51 note that @xmath52 for small @xmath53 , which tells us that the speed of sound in units of the lattice spacing is @xmath54 . by computing the rpa generator from the dynamical equation @xmath55 $ ] for @xmath56 and @xmath57 we obtain the expression @xmath58 where @xmath59 is the scaled lattice laplacian ( with diagonal part removed ) which has eigenvalue spectrum @xmath60 . next , we tensor up the model by introducing @xmath0 identical bands . mathematically speaking , we pass from the symplectic vector space ( for each @xmath43 ) @xmath61 to the tensor product @xmath62 and take the generator @xmath41 to be @xmath63 this means that creation operators @xmath64 and annihilation operators @xmath65 get an extra band index @xmath66 . note that in the physical setting of lattice vibrations a reasonable choice of @xmath0 in @xmath1 dimensions is @xmath67 due to the vector nature of lattice displacements . we turn to @xmath68 , the second term in ( [ eq : mz3 ] ) , which is random . a particular feature of our disordered model is that we take @xmath68 to be diagonal in the sites @xmath43 of the lattice . for simplicity we begin the discussion with the very special case of @xmath48 consisting of just a single site . the full model to be discussed later is obtained by repeating the single - site discussion at every site of @xmath42 . with the single site of the lattice we associate a hermitian vector space @xmath69 with symplectic structure @xmath70 . in order to implement the positivity condition [ see eq . ( [ eq : cone ] ) ] for @xmath4 to be in the cone @xmath3 , we let @xmath71 where @xmath72 is a rectangular linear operator @xmath73 mapping @xmath14 into an auxiliary vector space @xmath74 . the dimension @xmath75 is a parameter of our model . it may be bigger or smaller than @xmath76 . a special role is played by the choice @xmath77 , as this is the minimal dimension for the operator @xmath68 to have full rank . it is easy to see that for @xmath78 the symplectic condition @xmath79 holds if and only if @xmath72 satisfies the reality condition @xmath80 this condition fixes a real form , say @xmath81 , of the complex vector space @xmath82 . note that if @xmath83 has full rank then @xmath84 and @xmath85 . disorder is introduced by declaring the matrix elements of @xmath72 to be gaussian random variables . more precisely , we define the probability measure for @xmath86 as @xmath87 where @xmath88 is lebesgue measure on the normed vector space @xmath81 and @xmath89 is a normalization constant . the parameter @xmath90 is a measure of the disorder strength . we mention in passing that the model for @xmath91 with probability measure ( [ eq : meas ] ) ( and @xmath92 ) is equivalent to the random matrix model studied in @xcite by different methods . finally , we describe the generalization to an arbitrary lattice or graph @xmath48 . with each lattice site @xmath43 we associate one copy @xmath93 of the hermitian symplectic vector space @xmath94 . the total physical space then is the orthogonal sum @xmath95 . note that @xmath96 where @xmath97 denotes the number of sites of @xmath48 . the full generator of the dynamics is @xmath7 where the deterministic part @xmath41 may in principle be any element of the positive cone @xmath98 . for concreteness we let @xmath42 and take @xmath41 to be the generator described in section [ sect:2.2.1 ] . the random part @xmath68 is a sum @xmath99 of @xmath100 made from independent and identically distributed random operators @xmath101 . in other words , the distribution for @xmath68 is given by the product distribution @xmath102 while our interest will ultimately be in correlation functions and transport properties , we here take a first step by studying the average resolvent of the time - evolution generator @xmath103 @xmath104 where the symbol @xmath105 means the expectation value with respect to the probability measure ( [ eq : meas - lambda ] ) . notice that by the symplectic property @xmath106 the resolvent satisfies @xmath107 , so @xmath108 is an odd function of the frequency parameter @xmath109 . because the support of our probability measure is contained in the positive cone of elliptic elements , @xmath3 , the eigenvalue spectrum of the random operator @xmath4 is always imaginary and @xmath110 is analytic in the right and left halves of the complex @xmath111-plane . in the following we assume @xmath112 . it is a standard fact that the local density function @xmath113 of the characteristic boson frequencies @xmath114 can be computed from @xmath115 we now come to our main result . fixing the ratio @xmath116 we take the large-@xmath0 limit of the model with dynamical generator @xmath7 on @xmath42 as described above . we then claim that in this limit @xmath110 is expressed by @xmath117^d } \frac{d^d k } { z^2 + p^2 + p\nu(2-\delta_k ) + \nu^2 ( 1-\delta_k ) } \,,\ ] ] where the complex and energy - dependent quantity @xmath118 is a solution of the self - consistency equation @xmath119^d } \frac{d^d k}{(2\pi)^d}\ ; \frac{p + \nu ( 1 - \frac{1}{2}\delta_k ) } { z^2 + p^2 + p\nu(2-\delta_k ) + \nu^2 ( 1-\delta_k ) } \,.\ ] ] @xmath118 plays the role of a ` self energy ' or ` coherent potential ' . we briefly discuss some features of the solution in two extreme cases . there is only one relevant parameter , @xmath120 . in the limit of weak disorder ( @xmath121 ) one infers that @xmath122 and @xmath123^d } \frac{d^d k } { z^2 + \nu^2 ( 1-\delta_k)}\ ] ] is simply the cauchy transform of the local density of frequencies of the deterministic generator @xmath41 . on the other hand , for strong disorder ( @xmath124 ) the coherent potential @xmath125 becomes large and the system ( [ eq : g(z ) ] , [ eq : cpa ] ) simplifies to @xmath126 a special situation arises for @xmath127 . in this case it follows by a short computation from ( [ eq : rmt - limit ] ) that the scaled function @xmath128 satisfies an equation , @xmath129 which was derived and solved by lueck , sommers , and one of the authors @xcite . the analysis of @xcite shows that the density of states behaves as @xmath130 near @xmath131 in this case . numerical results for @xmath132 and for the more general situation of intermediate disorder strengths will be presented in sections [ sect : rmt - limit ] and [ sect : apply ] . in the sequel we explain how to arrive at our main equations ( [ eq : g(z ) ] ) and ( [ eq : cpa ] ) . for pedagogical reasons we describe the method first for the simple situation of a single site ( the zero - dimensional case ) . throughout this section we let @xmath69 and @xmath133 . our plan is to compute the average trace of resolvent ( [ eq : resolvent ] ) by a variant of the efetov - wegner supersymmetry method . the first step of this method is to express @xmath134 as a gaussian berezin ( super-)integral . to get started , we use the elementary identity @xmath135 and then write each of the two determinants as a gaussian integral using ordinary integration variables for the determinant in the denominator and anti - commuting variables for that in the numerator . in the case of the ordinary gaussian integral , there exists a convergence issue because the elements @xmath35 have indefinite real and imaginary parts in general . it is therefore crucial that all our generators @xmath4 , constrained to lie in the positive cone @xmath3 , satisfy the inequality @xmath136 . to take advantage of this positivity property , we express the determinant as follows : @xmath137 note that owing to @xmath112 and @xmath138 , the @xmath139 matrix of operators in ( [ eq:3.3 ] ) has positive real part . next , we introduce symmetric complex bilinear inner products @xmath140 for each of the two vector spaces @xmath74 and @xmath14 . these inner products are consistent with the hermitian structures of @xmath74 and @xmath14 in the sense that , e.g. for @xmath74 , the sesqui - bilinear form @xmath141 agrees with the hermitian scalar product of @xmath74 . we then express the reciprocal determinant @xmath142 as an integral over two complex vectors @xmath143 and @xmath144 @xmath145 where it is understood that we are integrating with the product of lebesgue measures for @xmath74 and @xmath14 . the normalization is chosen in such a way that @xmath146 . we emphasize that the integral ( [ eq : boson - det ] ) converges absolutely due to @xmath147 and @xmath148 . in the case of the determinant itself we integrate in the sense of berezin ( i.e. , we actually differentiate ) with respect to two independent vectors @xmath149 and @xmath150 whose components are grassmann variables : @xmath151 again , it is understood that we are integrating with the flat berezin form , i.e. , the product of all partial derivatives w.r.t . the grassmann variables . the bar in the present instance means nothing but independence , e.g. , of @xmath152 from @xmath149 . we now multiply the two gaussian integral formulas ( [ eq : boson - det ] ) and ( [ eq : fermion - det ] ) and take the disorder average inside the absolutely convergent integral to obtain @xmath153 this integral representation is a suitable starting point for further analysis . next , we compute the disorder expectation value in ( [ eq : mz-3.6 ] ) . for that we introduce the quadratic quantities @xmath154 where @xmath155 is meant as a linear transformation from @xmath14 to @xmath74 with coefficients in the even part of a grassmann algebra , and similar for @xmath156 with the roles of @xmath14 and @xmath74 reversed . we then have @xmath157 by completing the square and shifting variables . at this point we make the observation that @xmath158 depends on @xmath159 , @xmath160 , @xmath149 , @xmath152 only through scalar products such as @xmath161 , @xmath162 , @xmath163 . these share the feature of invariance under the group @xmath164 of real orthogonal transformations of @xmath133 . it will be useful to organize all these @xmath164-scalars into a supermatrix : @xmath165 two of the matrix entries vanish since @xmath166 and , similarly , @xmath167 . we also have @xmath168 . we further note the expression @xmath169 for the supertrace of @xmath170 . there exist certain linear dependencies amongst the matrix elements of @xmath170 . to describe them we need the operation @xmath171 of taking the supertranspose : @xmath172 with its help we can formulate the symmetries of @xmath170 as follows : @xmath173 we now arrange the remaining integration variables @xmath174 , @xmath175 , @xmath150 , and @xmath176 in the form of rectangular supermatrices : @xmath177 more precisely , @xmath178 is to be viewed as a linear mapping from @xmath14 into the superspace @xmath179 with grassmann - even resp . grassmann - odd matrix coefficients on the even resp . odd positions of @xmath180 . it is easy to check the identity @xmath181 which lets us re - express our disorder average as @xmath182 the next step is to carry out the integral over @xmath174 , @xmath175 , @xmath150 , and @xmath176 , thereby eliminating @xmath178 and @xmath184 from the calculation . this will be straightforward to do because the dependence on these variables is gaussian . as a preparatory step , we verify from @xmath185 the relation @xmath186 where @xmath187 by using the results ( [ eq : mz-3.10],[eq : mz-3.11 ] ) in equation ( [ eq : mz-3.6 ] ) we then arrive at our next formula : @xmath188 the integral on the right - hand side is still over the original variables @xmath159 , @xmath160 , @xmath149 , @xmath152 in @xmath170 and @xmath174 , @xmath175 , @xmath150 , @xmath176 in @xmath178 , @xmath184 . finally , by using a standard formula for gaussian berezin superintegrals we perform the integral over @xmath178 and @xmath184 . this results in @xmath189 the superdeterminant here is over the tensor product space @xmath190 . we recall that the superdeterminant of a supermatrix is defined by @xmath191 the integral above is still over the variables @xmath159 , @xmath160 , @xmath149 , and @xmath152 entering via their scalar products into the supermatrix @xmath170 . by scaling these integration variables so that @xmath192 , we obtain the following expression for the generating function of our problem : @xmath193 the symbol @xmath194 stands for the identity in superspace @xmath179 . superbosonization is a change of variables @xcite which lets us switch from integrating over a large number of vector - type variables , to integrating over a smaller number of matrix - type variables . in the present context these are the components of the vector variables @xmath159 , @xmath160 , @xmath149 , @xmath152 and the matrix elements of the supermatrix @xmath170 , respectively . such a reduction of the number of integration variables does not come for free but requires the integrand to be invariant under one of the lie groups @xmath195 , @xmath196 , or @xmath197 . there exists a version of superbosonization for each of these classical lie symmetries . as we have seen , our integrand is expressed in terms of quadratic invariants of the orthogonal group @xmath198 . therefore we now recall from @xcite the superbosonization identity for the case of @xmath164-symmetry . the @xmath164-superbosonization identity reads @xmath199 where on the left - hand side we integrate with the flat berezin form @xmath200 and on the right - hand side the berezin integration form is @xmath201 where @xmath202 is still the flat berezin form ( i.e. , the product of differentials for the even variables and partial derivatives for the odd variables ) . the domain of integration for the so - called boson - boson block @xmath203 [ see eq . ( [ eq : def - p ] ) ] is the space of positive hermitian @xmath204 matrices @xmath205 subject to @xmath206 . in the fermion - fermion sector , the integration domain is the space of unitary @xmath204 matrices @xmath207 subject to the symmetry relation @xmath208 . these matrix spaces are diffeomorphic to the symmetric spaces @xmath209 and @xmath210 respectively . by applying the superbosonization identity ( [ bosonize1 ] ) to the integral representation ( [ eq : om(z ) ] ) , we obtain our final result for the generating function : @xmath211 notice that a substitution @xmath212 was also made . by the relation @xmath213 this removes the multiplicative constant @xmath214 from ( [ eq : om(z ) ] ) . the result ( [ eq : final ] ) is exact and mathematically rigorous for @xmath215 . ( in the present case of @xmath164-symmetry the superbosonization identity fails for @xmath216 ; see @xcite . ) from it we get the average trace of resolvent by differentiating at coinciding points @xmath217 : @xmath218 to conclude this section we consider the special limit of vanishing deterministic generator @xmath219 . in that case our expression simplifies to @xmath220 where all superdeterminants and supertraces are over @xmath179 . recalling the parameter @xmath221 we see that our integral is of the form @xmath222 with @xmath223 we now investigate the random - matrix limit @xmath224 with @xmath225 held fixed . in this limit the integral for @xmath226 can be computed by the saddle - point or laplace method . by the principles of supersymmetry , the leading contributions to the integral at @xmath217 can be shown @xcite to come from saddle points which are multiples @xmath227 of the identity . we here omit the details of the calculation and present only the outcome . by execution of the saddle - point method we find that @xmath228 where @xmath118 is a solution of the saddle - point equation @xmath229 in section [ sect:2.3 ] [ see eq . ( [ eq : rmt - limit ] ) ] we already remarked that for @xmath127 this is equivalent to an equation analyzed and solved in @xcite . hence in what follows we focus on @xmath230 . we begin with the case @xmath231 . a plot of the density of states for @xmath232 is shown in figure [ fig : rmt ] . we see that there is a gap at low frequencies . this feature can be understood in the same way as the marcenko - pastur law @xcite for rectangular wishart matrices . indeed , recall that our random generator is @xmath78 where @xmath233 is rectangular of size @xmath234 . the non - zero eigenvalues of @xmath68 coincide with those of the operator @xmath235 but the latter has @xmath236 additional eigenvalues at zero by rank - nullity . in the large-@xmath0 limit the level repulsion due to this macroscopic number of zero modes produces a spectral gap of size proportional to @xmath237 . the gap closes as @xmath238 approaches unity , leading at @xmath127 to the situation investigated in @xcite . for @xmath239 it is the operator @xmath83 which by rank - nullity has @xmath240 zero modes , and the same goes for @xmath241 . therefore the density of states contains a dirac mass @xmath242 at zero in this case . a plot of the density of states for @xmath239 is shown in figure [ fig : rmt ] , where see that the dos approaches a finite value at zero frequency . ( the argument of macroscopic level repulsion does not apply here , as the operator @xmath68 is neither hermitian nor anti - hermitian . ) the discussion above is concerned with the so - called bulk scaling limit . another limit of interest is the edge - scaling limit at @xmath243 where one sends @xmath224 while keeping @xmath244 fixed . for @xmath127 this limit was thoroughly investigated in @xcite , while for @xmath231 the situation is trivial because of the absence of states at @xmath243 . for @xmath239 the edge - scaling limit was studied in @xcite . the hessian of the function @xmath245 at the saddle point @xmath246 has eigenvalues of order @xmath109 . therefore , in edge scaling @xmath247 this saddle point is not isolated and one has to work with a whole supermanifold of saddle points . ( technically speaking , the saddle - point supermanifold is a riemannian symmetric superspace @xmath248 of type @xmath249 . ) the law for the density of states in the limit @xmath250 turns out @xcite to be the universal law for systems of class @xmath251 in the symmetry classification of @xcite . we now turn to the @xmath1-dimensional model described in section [ sect:2.2 ] . the procedure of deriving the coherent potential approximation ( [ eq : g(z ) ] , [ eq : cpa ] ) for this model remains essentially the same as before . again , our first step is to express the determinants in ( [ eq : mz-3.1 ] ) as gaussian integrals over vector variables @xmath159 , @xmath160 , @xmath149 , @xmath152 for @xmath252 and @xmath174 , @xmath175 , @xmath150 , @xmath176 for @xmath253 . the gaussian integral representation has the effect of factorizing the independent random variables associated with different sites of the lattice . the disorder average can therefore be carried out for each site separately . by the local @xmath164 gauge symmetry of the model , the integrand after disorder averaging depends only on @xmath164 gauge invariant combinations of the fundamental variables @xmath159 , @xmath160 , @xmath149 , @xmath152 . these organize into supermatrices @xmath170 as before . thus we introduce such a supermatrix @xmath254 for each site @xmath43 and switch to integrating over @xmath254 by the superbosonization formula ( [ bosonize1 ] ) . because the dependence on the variables @xmath174 , @xmath175 , @xmath150 , @xmath176 is still gaussian , they can again be integrated out to produce a superdeterminant . in this way we obtain @xmath255 where @xmath256 denotes the orthogonal projector from @xmath257 onto @xmath258 . the only difference of any essence from our earlier result ( [ eq : final ] ) is that the integral now is over a field of supermatrices @xmath259 instead of a single supermatrix @xmath170 . the operator @xmath260 is diagonal on @xmath14 but ( for generic @xmath254 ) non - diagonal on @xmath179 . on the other hand , the operator @xmath261 is trivial on superspace but couples the sites of the graph @xmath48 . the inverse square root of @xmath262 is raised to the @xmath263 power because each of the @xmath0 bands of the deterministic limit contribute the same factor . we now face the task of analyzing the model ( [ eq : final ] ) by the field - theoretic methods of gradient expansion and renormalization . ( note that a closely related problem has already been tackled in @xcite . ) hoping to make progress with this in a future publication , we here take a first step by computing the local density of states . let us finally work out the mean - field solution of the model ( [ eq : final ] ) with deterministic generator @xmath264 as defined in ( [ eq : mz6 ] ) . writing the integrand as @xmath265 ( for @xmath266 ) we take the general variation of @xmath267 : @xmath268 for large @xmath0 we expect the field integral to be essentially given by a spatially homogeneous saddle point @xmath269 ( independent of @xmath43 ) and small fluctuations around it . therefore , after setting @xmath270 we look for solutions of @xmath271 of this very form . the variational equation @xmath271 then reduces to an equation of self - consistent mean - field type : @xmath272 where we have used the property that the laplacian @xmath59 is diagonal with eigenvalues @xmath273 in momentum space . by evaluating the trace of the matrix inverse we immediately arrive at equation ( [ eq : cpa ] ) . within this mean - field ( or coherent potential ) approximation scheme , we obtain the expression ( [ eq : g(z ) ] ) for the average resolvent trace @xmath110 . let us finish by showing some numerical results for the model in dimension @xmath274 . in this case the density of states from ( [ eq : mz-2.21 ] ) for the pure system ( @xmath275 ) is @xmath276 as is seen in figure [ fig : cpa ] , the van hove singularity at @xmath277 is still visible for @xmath278 ( and @xmath279 . as the disorder strength @xmath90 is increased , the bulk of the spectrum is pushed to higher frequencies and a peak begins to develop at small frequencies ( see the plot for @xmath280 ) . at values of @xmath90 much larger than the sound velocity @xmath281 we recover the random - matrix limit shown in figure [ fig : rmt ] . 99 p. ring and p. schuck , _ the nuclear many - body problem _ ( springer , new york , 1980 ) r.o . pohl , x. liu , and e. thompson , rev . * 74 * ( 2002 ) 991 x. barillier - peruisel , o. bohigas , and h.a . weidenmller , ann . phys . * 324 * ( 2009 ) 1855 j. korringa , j. phys . solids * 7 * ( 1958 ) 252 j.l . beeby , phys . * 135 * ( 1963 ) a130 p. soven , phys * 156 * ( 1967 ) 809 f. evers and a.d . mirlin , rev . * 80 * ( 2008 ) 1355 t. lueck , h .- j . sommers , m.r . zirnbauer , j. math . * 47 * ( 2006 ) 103304 p. littelmann , h .- j . sommers , m.r . zirnbauer , commun . * 283 * ( 2008 ) 343 s. schmittner , diploma thesis , universitt zu kln ( october 2010 ) v.a . marcenko and l.a . pastur , math . ussr - sb * 1 * ( 1967 ) 457 a. altland and m.r . zirnbauer , phys . b * 55 * ( 1997 ) 1142

a family of random models for bosonic quasi - particle excitations , e.g. the vibrations of a disordered solid , is introduced . the generator of the linearized phase space dynamics of these models is the sum of a deterministic and a random part . the former may describe any model of @xmath0 identical phonon bands , while the latter is a @xmath1-dimensional generalization of the random matrix model of lueck , sommers , and zirnbauer ( lsz ) . the models are constructed so as to exclude the unphysical occurrence of runaway solutions . by using the efetov - wegner supersymmetry method in combination with the new technique of superbosonization , the disordered boson model is cast in the form of a supermatrix field theory . a self - consistent approximation of mean - field type arises from treating the field theory as a variational problem . the resulting scheme , referred to as a coherent potential approximation , becomes exact for large values of @xmath0 . in the random - matrix limit , agreement with the results of lsz is found . the self - consistency equation for the full @xmath1-dimensional problem is solved numerically .

You are an expert at summarizing long articles. Proceed to summarize the following text: for its remarkable high tensile strength and ductility @xcite , mg - based long - period stacking ordered ( lpso ) structures are considered as light - weight structural alloy for next generation . in order to clarify relationship between thier formation process and resultant properties in terms of application for structural materials , considerable number of experimental as well as theoretical studies have been carried out . previous theoretical studies mainly address formation process and thermodynamic stability of mg - based lpso alloy , including ( i ) the tendency of phase separation confirmed by cluster variation method @xcite , ( ii ) in - plane ordering of clusters consisted of y , zn substitutional atoms @xcite , and ( iii ) systematic understanding of energetic stability with respect to variety of substitutional atoms into mg - based alloys @xcite . although these previous theoretical works partly clarify themodynamic stability of lpso phases , they did not sufficiently discuss about ( i ) relative stability in terms of disordered phases or ( ii ) effect of lattice vibration on stability of lpso . the former one is considered essentially important , since well - established ordering energy , determining the thermodynamic stability of ordered phase with respect to temperature , is typically reffered to the difference in mixing energy between ordered ( here , lpso ) and disordered phases . for the latter one , vibrational effects , their significant role on phase stability has been amply demonstrated for several binary alloys , such as ref . [ fig:18r ] is a schematic illustration of stacking sequence of 18r lpso structure , which includes mg hcp stacking and y , zn substitutional atom concentrated phase including l1@xmath1 cluster . to systematically evaluate thermodynamic stability of such structures , we need to consider energetics of multiple structures including different stacking sequence however , most calculations confine the structural model , so calculation that does nt confine experimentally reported structures is highly required . furthermore , it is fundamentally important to assess thermodynamic stability of lpso phase by competing disordered phase . in this study , based on dft calculation , we systematically study thermodynamic stability of lpso structure in terms of disordered phases . to comprehensively address energetics of multicomponent system with various composition , we need to calculate tremendous number of structures of the system and computational cost therefore become large . in this study , we employ special quasirandom structure ( sqs ) to assess thermodynamic stability of lpso phase competing with disordered phase . sqs is a special microscopic state whose multiple correlation functions are numerically identical to those in perfect random structure , which therefore provides physical properties for perfect random alloy @xcite . as following procedure , we calculated correlation function of sqs of ternary system . let us consider the system with @xmath2 lattice points for number of components , @xmath3 . @xmath4 is a variable which specifies the occupation of lattice point @xmath5 and @xmath6 can specify any atomic arrangement and we can represent structures in following equation , which is called correlation function ; @xmath7 here , @xmath8 is complete orthonormal basis function at lattice point @xmath5 and it is obtained by applying gram - schmidt technique to the linearly independent polynominal set @xmath9 . @xmath10 denotes a cluster included in the structure , @xmath11 is the index of basis function @xmath8 and @xmath12 is the set of index . in the case of ternary alloy system , the occupation of lattice point by each element , mg , y , and zn , is defined by @xmath13 respectively , leading to the basis function of eq.([eq : basis_tri ] ) @xcite : + @xmath14 by averaging @xmath15 over equivalent clusters in lattice , correlation function of sqs in mg - y - zn ternary system with compositions of each element , @xmath16 , @xmath17 and @xmath18 , @xmath19 , is represented by eq.([eq : sqs4 ] ) , @xmath20 where @xmath21 and @xmath22 denote dimension of clusters and the number of @xmath23 respectively . the simulation was performed so that correlation functions come closer to ideal sqs correlation functions up to 4-th nearest neighbor pair ( @xmath0 6 pair clusters on hcp ) finded by eq.([eq : sqs4 ] ) . we optimized correration functions for each clusters by performing numerical simulation @xcite and constructed sqss . to evaluate the accuracy of obtained correlation functions of sqs , we compared the simulated values of correlation functions with standard deviation of them in configurational space . then , most of the errors of correlation function are small enough compared with standard deviation . based on these obtained structures , we calculate formation free energy , @xmath24 , and bulk modulus , @xmath25 to evaluate thermodynamic stability of mg - y - zn system that has multiple stacking sequence with various composition for random mixing . we constructed structures with various composition on mutiple stacking sequence , whose composition is shown in figure . [ fig : phasediagram ] . the detail of calculation condition of structure is shown in appendix . @xmath24 is denoted by eq.([eq : form1 ] ) and we define @xmath26 by eq.([eq : form2 ] ) , @xmath27 @xmath28 where @xmath29 is total energy of @xmath30 , and we evaluate configurational entropy , @xmath31 , based on bragg - williams approximation . in order to evaluate bulk modulus , we calculate total energies of structures , whose volume is expanded at a rate of @xmath323% , @xmath326% , @xmath329% . by using @xmath33 based on debye - grneisen approximation , debye temperature , @xmath34 , can be described as @xcite @xmath35 where @xmath36 is atomic volume and @xmath12 denotes atomic mass . based on empirical debye model , @xmath37 can be estimated by eq.([eq : fvib ] ) ; @xmath38 where @xmath39 is debye function . we estimated the effect of phonon on mg - based alloy by calculating @xmath40 and the vibrational free energy of formation with respect to the pure constituents can be estimated by eq.([eq : vibform ] ) @xcite ; @xmath41 where @xmath5 denotes element and @xmath42 is a composition of @xmath5 . in order to evaluate the effect of stacking faults on stability , we constructed a sqs which has stacking faults on hcp ( stacking sequence is `` ababcababcababc '' ) . hereinafter , we call this structure `` mixed '' . the effect of stacking difference is quantified by interfacial energy , which is defined by @xmath43 , where @xmath44 is total energy of structure including stacking faults and @xmath45 is for structures on fcc and hcp stacking . @xmath46 denotes area of interface , which is 1.4@xmath47 @xmath48 in this calculation . additionally , to address stability of ordered phase competeing with disordered phase , we also calculated formation energy of structures including y - zn l1@xmath1 cluster and estimated ordering temperature . we employ first - principles calculations using a dft code , the vienna ab - initio simulation package ( vasp ) @xcite @xcite , to obtain the total energies for structures of mg - y - zn alloys . the calculation of total energy is carried out for the structures in table . [ table : data_type ] . all - electron kohn - sham equations are solved by employing the projector augmented - wave ( paw ) method @xcite @xcite . we select generalized - gradient approximation of perdew - burke - ernzerhof ( gga - pbe ) @xcite form to the exchange - correlation functional . the plane - wave cutoff energy is set at 350 ev throughout the present calculations . brillouin zone sampling is performed on the basis of the monkhorst pack scheme @xcite . k - point mesh is set 4@xmath494@xmath491 for structure , c3 , and 4@xmath494@xmath494 for others and smearing parameter is 0.15 ev @xcite . first , we evaluated formation free energy of structures with multiple compositions on hcp and fcc stacking , which is shown in fig . [ fig : formation free energy ] . at @xmath50 , @xmath24 of structures on hcp is negative at all through composition and possesses two extreme values . through calculation of formation free energy of sqss , mg - y - zn alloy exhibits phase separation into mg- and y - zn- rich phase and this result is consistent with previous research by iikubo @xmath51 @xmath52 @xcite , which suggests the validity of this simulation based on sqss . using these optimized sqss , we proceed our discussion to calculate bulk modulus , ordering temperature , and the effect of stacking difference on phase stability . cluster ( f4 , g2 ) and sqss on hcp ( f1 ) and fcc ( f2 , g1 ) stacking . the ratio of concentration of substitutional atom is denoted in each figure . structure indices are denoted in table . [ table : data_type],title="fig:",width=359 ] + next , in order to evaluate the effect of lattice vibration on phase stability of mg - y - zn system , we calculated bulk modulus , which is shown in figure . [ fig : bulkmodulus ] . dashed line represents linear averaged bulk modulus , @xmath53 . bulk modulus for a1 , e1 h1 , f1 ( on hcp stacking ) , e2 ( on fcc stacking ) and f4 ( ordered structure ) are about 35 gpa . the values of bulk modulus are smaller than @xmath53 through all compositions . calculated @xmath40 at @xmath54k at ( @xmath16,@xmath17,@xmath18)=(87.5,6.25,6.25 ) was found to be negligibly small within the calculated accuracy of @xmath33 . compared with avove results of @xmath24 , @xmath37 does not have significant influence on @xmath24 of mg - y - zn alloys with multiple stackings and compositions . additionally , bulk modulus does not show significant dependence of mg concentration , which therefore means that the effects of phonon do not play essential role on lpso phase stability . in this study , since effect of optical mode for multicomponent system is not considered , further study including optical mode is needed to quantitively clarify the effect of phonon for stability of mg - y - zn alloys . hereinafter , in evaluateing effects of ordering and stacking sequence difference on stability , we only consider the configurational effect . , where @xmath55 is bulk modulus of @xmath46 and @xmath56 , @xmath57 , @xmath58 denotes composition of each element . colors correspond to the ratio of concentration of substitutional atom.,title="fig:",width=264 ] + then , we examine the effect of ordering on stability . we constructed structure including l1@xmath1 cluster , f4 and g2 , whose atomic arrangements are shown in fig . [ fig : l12clusterarrange ] . formation energies of f4 and g2 are lower than sqs on hcp ( f1 ) and fcc ( g1,f2 ) stacking ( right side of fig . [ fig : formation free energy ] ) , which shows this system is stabilized by the effect of ordering . as we estimated ordering temperature , @xmath59 , from f2 to f4 and from g1 to g2 was about 1050 k and 930 k respectively . this results shows that transition temperature depends on the arrangement of l1@xmath1 cluster and @xmath59 of g2 is close to experimental results by okuda @xmath60 . @xcite taking that transition temperature can be overestimated based on bragg - williams approximation into consideration . moreover , this result indicates that mg - y - zn system form ordered phase up to melting point : mg - y - zn alloy can be regarded as intermetallic compounds . clusters . red and blue spheres represent y and zn atom respectively . f4 : meta - stable . g2 : stable.,title="fig:",width=264 ] + finaly , we address the effect of stacking sequence on stability using interfacial energy . [ fig : stackingdifference ] is interfacial energies of structures on different stacking ( f1 , f2 , f3 ) with respect to the energy of structure on fcc stacking , f2 . as shown in fig . [ fig : stackingdifference ] , interfacial energy of disordered phase that introduces stacking fault is lower than the energy for linear average of hcp and fcc stacking . this result indicate that when stacking fault is introduced to mg - rich hcp alloys to form `` l1@xmath1-like '' ordering , corresponding interface between original hcp and formed fcc region gains `` negative '' energy , which is contrary to the conventional tendency that interface gains positive energy . this specific characteristics for interface energy in mg - based allos would thus be one of the fundumental prerequisite to accelaration of forming lpso phases , which should be further investigated in the future work . in this study , we calculate preference of energetics of mg - y - zn ternary alloy system in terms of disordered phase stability . through calculation of formation free energy of sqss , mg - y - zn alloy exhibits phase separation into mg- and y - zn- rich phase . bulk modulus for sqss and ordered structure ranges about 35 gpa and they do not show significant dependence of mg concentration , which therefore means that the effect of phonon does not play essential role on lpso phase stability within acoustic mode . order - disorder transition temperature is estimated about 930 k and this results is colse to the melting point that is experimentaly reported and this suggests that mg - y - zn lpso alloy can be regarded as intermetallic compounds . the effect of stacking faults stabilized the lpso phase and this results indicate that there remains profound relationship between introducing stacking faults and the formation of long - priod stacking ordering . this work is supported by a grant - in - aid for scientific research on innovative areas ( 26109710 ) from the ministry of education , science , sports and culture of japan . 999 y.kawamura , k.hayashi , a.inoue and t.matsumoto , mater.trans . * 42 * , 1172 ( 2001 ) . s.iikubo , s.hamamoto and h.ohtani , mater . trans . * 54 * , 636 ( 2013 ) . h.kimizuka , n.fronzi and s.ogata , scripta mater . * 69 * , 594 ( 2013 ) . j.saal and c.wolverton , acta mater . * 68 * , 325 ( 2014 ) . i.a.abrikosov , yu.h.vekilov and a.v.ruban , phys . a * 154 * , 407 ( 1991 ) . z.w.lu , b.m.klein , a.zunger , j.phase . * 16 * , 36 ( 1995 ) . k.yuge , j.phys . : condens . matter . * 21 * , 415401 ( 2009 ) . k.yuge , j.phys . jpn * 84 * , 084801 ( 2015 ) . a. van de walle and g.ceder , rev . phys . * 74 * , 11 ( 2002 ) . a.zunger , s .- h.wei , l.g.ferreira and j.e.bernard , phys . rev . lett . * 65 * , 353 ( 1990 ) . k.yuge , a.seko , y.koyama , f.oba , and i.tanaka , phys . b * 77 * , 094121 ( 2008 ) . g.kresse and j.hafner , phys . rev . b * 47 * , r558 ( 1993 ) . g.kresse and j.furthmuller , phys b * 54 * , 11169 ( 1996 ) . g.kresse and d.joubert , phys . rev . b * 59 * , 1758 ( 1999 ) . p.e.blochl , phys . b * 50 * , 17953 ( 1994 ) . j.p.perdew , k.burke , and m. ernzerhof , phys . lett * 77 * , 3865 ( 1996 ) . h.j.monkhorst and j.d.pack , phys . b * 13 * , 5188 ( 1976 ) . m.methfessel and a.t.paxton , phys . b * 40 * , 3616 ( 1989 ) . v.l.moruzzi , j.f.janak and k.schwarz , phys . b * 37 * , 790 ( 1988 ) . v.ozolins , c.wolverton and a.zunger , phys . rev . b * 58 * , r5897 ( 1998 ) . h.okuda , t.horiuchi , s.hihumi , m.yamasaki , y.kawamura and s.kimura , metall . mater . trans . a * 45a * , 4780 ( 2014 ) . in this study , we calculated free energy and bulk modulus of structures shown in following data .

in order to clarify thermodynamic stability of mg - based long - period stacking ordered ( lpso ) structure , we systematically study energetic preference for alloys on multiple stacking with different composition for random mixing of constituent elements , mg , y , and zn based on special quasirandom structure ( sqs ) . through calculation of formation free energy of sqs , mg - y - zn alloy exhibits phase separation into mg- and y - zn rich phase , which is consistent with previous theoretical studies . bulk modulus of sqss for multiple compositions , stacking sequences , and atomic configurations ranges around 35 gpa , @xmath0 , they do not show significant dependence of mg concentration , which therefore means that the effects of phonon do not play significant role on lpso phase stability . introducing stacking fault to hcp stacking gains `` negative '' energy , which indicates profound relationship between introducing stacking faults and the formation of long - period stacking ordering .

You are an expert at summarizing long articles. Proceed to summarize the following text: the bardeen - cooper - schrieffer ( bcs ) theory @xcite or its advanced version the hartree - fock - bogoliubov ( hfb ) theory @xcite has long been used to treat pairing correlations @xcite in atomic nuclei . in the mean field we introduce the bogoliubov quasi - particles and write the ground state as a slater determinant of the latter . usually the variation principle is used to determine the structure of the quasi - particles . although enjoying great success , the method has disadvantages of breaking the exact particle number and a need for an unphysical minimum pairing - force strength @xcite . the theory could be improved by the `` variation after particle - number projection '' ( vapnp ) procedure , in which the bcs or hfb wavefunction is projected onto good particle number before the variation is done @xcite . effectively , the pair condensate wavefunction [ eq . ( [ gs ] ) ] is taken as the variational ground state . the method has been discussed @xcite and applied with success to ultrasmall metallic grains @xcite in the vapnp+bcs version , and to realistic nuclei @xcite in the vapnp+hfb version . however , the computing time cost is relatively large and there is only a limited number of nuclei on the nuclear chart that has been calculated by the vapnp + hfb method in large configuration spaces . recently we proposed @xcite a new criteria to determine the pair condensate wavefunction based on the heisenberg equations of motion for density matrix operators . the relevant equations have been solved for the bcs - type hamiltonian : each single - particle level has a distinct set of quantum numbers corresponding to the symmetries of the pairing operator , thus a level could only be paired with its time - reversed partner . the validity of the theory has been proved on a large ensemble with random interactions @xcite . in this work we consider the general situation of the hfb - type hamiltonian , where several single - particle levels could have the same set of quantum numbers and pairing among them is allowed . the method should cost much less time in computing compared with the traditional variational calculation . in the algorithm of the current theory only the one - body density matrices on the pair condensate are calculated , whereas the variation principle needs the two - body density matrices in computing the expectation value of a two - body hamiltonian . in sec . [ sec_formalism ] we give the general formalism to solve for the particle - number - projected hfb wavefunction with the pairing theory based on the generalized density matrix ( gdm ) formalism . then the method is tested in the simple two - level model with factorizable pairing interactions and in the semi - realistic model with the zero - range delta interaction , in sec . [ sec_two_level ] and sec . [ sec_delta_int ] , respectively . finally sec . [ sec_summary ] summarizes the work . the antisymmetrized ( two - body ) fermionic hamiltonian governing the dynamics of the system is written as @xmath0 in the presence of pairing correlations , the ground state of the @xmath1-particle system is assumed to be an @xmath2-pair condensate , @xmath3 where @xmath4 is the normalization factor , @xmath5 is the pair creation operator @xmath6 in eq . ( [ p_dag_alpha ] ) @xmath7 represents a subspace consisting of `` pair - indices '' whose dimension is half of that of the single - particle space . we could take , for example , @xmath7 to consist of those single - particle levels with a positive magnetic quantum number . @xmath8 is the time - reversed level of the single - particle level @xmath9 ( @xmath10 ) . the pair structure matrix @xmath11 is hermitian and block - diagonal . the time - reversal invariance of @xmath12 implies that @xmath13 ( @xmath14 and @xmath9 have the same set @xmath15 of quantum numbers related to the symmetries of @xmath5 . for example , in the spherical shell model @xmath16 is a collection of parity @xmath17 , angular momentum @xmath18 and its projection @xmath19 ; in the deformed nilsson mean field @xmath20 is a collection of parity @xmath17 and angular - momentum projection @xmath21 onto the symmetry axis . we introduce the unitary transformation @xmath22 between the original single - particle basis @xmath23 and the new single - particle basis @xmath24 as @xmath25 where elements of the transformation matrix @xmath22 are defined through dirac notation . properties of time - reversal operation imply that @xmath26 . the matrix @xmath22 is block - diagonal : @xmath27 vanishes unless @xmath28 and @xmath29 have the same quantum number @xmath15 . under the transformation ( [ eta ] ) the operator @xmath5 ( [ p_dag_alpha ] ) becomes @xmath30 with @xmath31 we choose the transformation @xmath22 that diagonalizes the hermitian matrix @xmath11 in eq . ( [ v_12 ] ) , consequently @xmath5 becomes @xmath32 we call this new single - particle basis the `` canonical basis '' . in the following the arabic numerals @xmath28 , @xmath33 , ... refer to single - particle levels in this basis unless otherwise specified . in the canonical basis the density matrices for the pair condensate ( [ gs ] ) are `` diagonal '' : @xmath34 the normalization factor @xmath35 ( [ gs ] ) , occupation numbers @xmath36 ( [ rho_def ] ) , and pair - transfer amplitudes @xmath37 ( [ kappa_def ] ) are functions of the pair structures @xmath38 ( [ p_dag_1 ] ) ; their functional forms , as the `` kinematics '' of the system , have already been given in eqs . ( 23 ) and ( 24 ) of ref . @xcite and are not repeated here . the hartree - fock mean field @xmath39 and pairing mean field @xmath40 are defined as @xmath41 @xmath39 and @xmath40 are block - diagonal matrix : @xmath42 and @xmath43 vanish unless @xmath28 and @xmath33 have the same quantum number @xmath15 . the hamiltonian parameters ( @xmath44 and @xmath45 ) in eqs . ( [ f_def ] ) and ( [ delta_def ] ) should be calculated from those in the original single - particle basis ( @xmath46 and @xmath47 ) through the transformation ( [ eta ] ) . the equation of motion for the density matrix @xmath48 has been derived in eq . ( 14 ) of ref . @xcite , @xmath49 where @xmath50 and @xmath51 are ground state energies for @xmath12 and @xmath52 , respectively . on the right - hand side , @xmath39 and @xmath40 are the mean fields ( [ f_def ] ) and ( [ delta_def ] ) , @xmath53 is the transpose of @xmath39 , and terms like ` @xmath54 ' are understood as matrix multiplication . in deriving eq . ( [ eom_k0 ] ) we have used the main approximation of the method , @xmath55 which says that on the pair condensate ( [ gs ] ) the two - body density matrix factorizes into products of one - body density matrices in both the particle - hole and particle - particle channels . equation ( [ eom_k0 ] ) is a block - diagonal matrix equation ; its @xmath56 matrix element vanishes unless @xmath28 and @xmath33 have the same quantum number @xmath15 . within each block we take the @xmath56 matrix element on both sides of eq . ( [ eom_k0 ] ) , when @xmath57 we get after simplification @xmath58 and the @xmath59 matrix element gives @xmath60 equations ( [ main_12 ] ) and ( [ main_11 ] ) are the main equations of the theory . below we show that the number of constraints from these two equations equals to the number of parameters in the pair structure @xmath11 ( [ p_dag_alpha ] ) ; thus the latter is fixed completely . we assume a single - particle space of dimension @xmath61 split by the quantum number @xmath15 as @xmath62 . the number of restrictions in eq . ( [ main_12 ] ) is @xmath63 ( there is a factor @xmath64 because the equation has real and imaginary parts ) . equation ( [ main_11 ] ) implies that the right - hand side is independent of the single - particle label @xmath65 , which gives @xmath66 constraints . hence the total number of constraints is @xmath67 . on the other hand , the number of independent parameters in the hermitian block - diagonal pair - structure matrix @xmath11 ( [ p_dag_alpha ] ) is @xmath68 ( there is a `` @xmath69 '' because an overall normalization factor does not matter ) . the number of restrictions indeed equals to the number of parameters . in practical calculations usually the hamiltonian parameters @xmath46 and @xmath47 ( [ h_f ] ) are real . in this case the pair structures @xmath11 ( [ p_dag_alpha ] ) could be taken as real numbers and the transformation @xmath22 ( [ eta ] ) is an orthogonal matrix . the number of restrictions from eqs . ( [ main_12 ] ) and ( [ main_11 ] ) is @xmath70 , which equals to the number of independent parameters in the real symmetric block - diagonal pair - structure matrix @xmath11 ( [ p_dag_alpha ] ) . in the following we assume that this is the case . an alternative parametrization may be more convenient in practice . the independent parameters could be taken as those in the orthogonal transformation @xmath22 ( [ eta ] ) and the pair structure in the canonical basis @xmath71 ( [ p_dag_1 ] ) . the orthogonal transformation for @xmath72 could be parameterized as @xmath73 for @xmath74 , @xmath75 and in general , @xmath76 we test the formalism in the simple model with two single - particle levels and a factorizable pairing interaction . the theory is solved analytically and the results are compared with the exact shell - model results . we assume the rotational invariance and a single - particle space of two @xmath18-levels each with degeneracy @xmath77 ( degenerate in the magnetic quantum number @xmath19 ) . the fermionic hamiltonian ( [ h_f ] ) takes the form @xmath78 where @xmath79 is the pairing operator @xmath80 in which @xmath81 and @xmath82 are `` diagonal '' and `` off - diagonal '' pairing strengths , respectively . the hamiltonian parameters in the canonical single - particle basis are calculated through the transformation ( [ tran2 ] ) , @xmath83 where @xmath84 and @xmath85 . \label{pi_12}\end{aligned}\ ] ] based on eqs . ( [ f_def ] ) and ( [ delta_def ] ) we compute the matrix elements of the mean fields in the canonical single - particle basis , @xmath86 , \nonumber\end{aligned}\ ] ] and @xmath87 , \nonumber \\ \delta_{2\tilde{2 } } = - \omega ( g - p \sin2\theta ) [ ( g + p \sin2\theta ) s_1 + ( g - p \sin2\theta ) s_2 ] , \nonumber \\ \delta_{1\tilde{2 } } = - \omega p \cos2\theta [ ( g + p \sin2\theta ) s_1 + ( g - p \sin2\theta ) s_2 ] . \nonumber\end{aligned}\ ] ] consequently eqs . ( [ main_12 ] ) and ( [ main_11 ] ) imply @xmath88 \nonumber \\ + p^2 \sin4\theta [ \omega ( n_1 + n_2 - 1 ) ( s_1 - s_2 ) - ( s_1 + s_2 ) ( n_1 - n_2 ) ] , \label{main_2l_12}\end{aligned}\ ] ] and @xmath89 \nonumber \\ - 2 g p \omega \sin2\theta s_1 s_2 + ( \omega - 1 ) ( g + p \sin2\theta)^2 n_1 s_1 s_2 - ( \omega - 1 ) ( g - p \sin2\theta ) ^2 n_2 \label{main_2l_11}\end{aligned}\ ] ] these two equations fix the two independent model parameters @xmath90 and @xmath91 ( @xmath36 and @xmath37 are functions of @xmath90 ) . then the density matrices in the original single - particle basis are calculated through the transformation @xmath22 ( [ tran2 ] ) , @xmath92 and @xmath93 we consider the range of model parameters in the realistic spherical nuclear shell model . usually the two single - particle levels @xmath29 and @xmath9 belong to different major shells and @xmath94 is bigger than @xmath95 mev in magnitude that is about the energy of two major - shell gaps ( adjacent major shells have opposite parity ) . based on the empirical pairing strength formula @xmath96 mev ( @xmath97 is the mass number ) @xcite , @xmath98 should be less than @xmath64 mev in medium and heavy nuclei having considered possible deviations from the constant pairing formula . the off - diagonal pairing strength @xmath82 is usually smaller than the diagonal pairing strength @xmath81 owing to the smaller overlap of the two single - particle wavefunctions involved . we test the model numerically in an ensemble consisting of examples with different model parameters . the single - particle angular momentum @xmath18 of the examples takes value from @xmath99 . the number of pairs is in the range @xmath100 ( from two particles to two holes in the model space ) . the energy gap between the two single - particle levels is selected to be the energy unit so that @xmath101 and @xmath102 . based on the estimations in the previous paragraph , we choose the range of the pairing strength to be @xmath103 and @xmath104 with step size @xmath105 . we scan the whole parameter space and the total number of examples in the ensemble is @xmath106 ( the factor of @xmath107 is the number of possible values of @xmath81 and @xmath82 ) . the results for the density matrices @xmath108 and @xmath48 in the original single - particle basis are shown in figs . [ fig_rho_line ] and [ fig_kappa_line ] , respectively . each point in these figures corresponds to one example in the ensemble with the horizontal coordinate being the exact result of the respective quantity and the vertical one being that from the gdm calculation , thus a perfect calculation would have all the points lying on the @xmath109 straight line . from figs . [ fig_rho_line ] and [ fig_kappa_line ] we see that in general the gdm theory reproduces the exact density matrices well . the root - mean - square deviations are @xmath110 @xmath111 and @xmath112 are equal because @xmath113 . the results for the ground state energies are shown in fig . [ fig_e_line ] . the horizontal coordinate of each point is the pairing correlation energy for the example @xmath114 , where @xmath115 is the exact ground state energy by the direct diagonalization , and @xmath116 or @xmath117 is the occupation number of the naive fermi distribution . the vertical coordinate is the ground state energy by the gdm calculation measured from the exact one , @xmath118 , where @xmath119 is calculated in the canonical single - particle basis using the hamiltonian ( [ h_12 ] ) together with the recursive formulas derived in refs . we see that in general the gdm calculation reproduces well the ground state energies : the errors are small compared with the pairing correlation energies . the average values of the errors and the pairing correlation energies for the ensemble are @xmath120 and @xmath121 , respectively . at last in fig . [ fig_derivations_p ] we show the accuracy of the method depending on the off - diagonal pairing strength @xmath82 . the @xmath122 examples in the ensemble are divided into five subgroups according to the value of @xmath82 , and deviations within each subgroup of various quantities are calculated . we see that in general the errors of the gdm calculation increase with @xmath82 . in realistic spherical nuclear shell model @xmath82 should be smaller than about @xmath123 based on the estimations made in the paragraph below eq . ( [ kappa_ori ] ) . in this region the gdm theory is rather accurate . in this section we further test the theory in a semi - realistic model with the pairing hamiltonian matrix elements calculated from the zero - range delta interaction . the model has only one species of nucleons ( for example , neutrons ) and the single - particle space consists of five levels : three of them ( @xmath124 , @xmath125 , @xmath126 ) are from the @xmath127 major shell and two ( @xmath128 , @xmath129 ) are from the @xmath130 major shell . the single - particle energies are determined by random number generator to be @xmath131 , @xmath132 , @xmath133 , @xmath134 , and @xmath135 mev ( the gap between the @xmath136 and @xmath130 major shells is taken to be around @xmath95 mev ) . the two - body pairing matrix elements [ @xmath137 with @xmath138 are calculated from the zero - range delta interaction @xmath139 , taking the single - particle wavefunctions ( @xmath29 , @xmath9 ... ) to be the harmonic oscillator ones . we perform six sets of calculations at different model parameters of the particle number @xmath1 and the pairing strength @xmath140 as listed in table [ tab_setpara ] . the unit for @xmath140 is @xmath141 , where @xmath142 is the length parameter for the harmonic oscillator wavefunctions ( @xmath143 is the mass of the nucleon ) . values of @xmath140 are chosen so that magnitudes of the resulting two - body pairing matrix elements are realistic . the last row of table [ tab_setpara ] shows the average value of the pairing matrix elements of a specific form , @xmath144 = -\sum_\alpha v_{\alpha\tilde{\alpha}\tilde{\alpha}\alpha } / ( 2\omega)$ ] , where @xmath145 runs over the entire single - particle space whose dimension is @xmath61 . the results are shown in table [ tab_res ] . we see that in general the gdm calculation reproduces the exact density matrices accurately , except for the very small value of @xmath146 in set5 . particularly , the abrupt change of @xmath48 values from set3 to set4 is captured very well . the off - diagonal parts of the density matrices ( @xmath147 , @xmath148 , @xmath149 , and @xmath150 ) , originating from the `` off - diagonal '' parts of the hfb - type pairing hamiltonian , are always well reproduced , ranging from a less - than - one - percent effect to a few . the errors of the gdm ground state energy relative to the exact ones are shown in the last column . we see that the errors are small and consistent with the statistical estimate made within a large random ensemble for the case of a bcs - type pairing hamiltonian in ref . in summary , we considered the solution of the gdm pairing theory with the hfb - type hamiltonian . in the algorithm only the one - body density matrix is computed , thus the method should be much faster than the vapnp + hfb method in which the two - body density matrix is needed in calculating the expectation value of a two - body hamiltonian . with the assumption that the parent and daughter nuclei were represented by pair condensates with the same pair structure , the pair - transfer amplitudes @xmath48 could be calculated in one run together with the occupation numbers @xmath108 . in contrast , the traditional variation method calculates the parent and daughter nuclei separately , each with a different pair structure . the formalism is tested in the simple two - level model with factorizable pairing interactions , and the semi - realistic model with the zero - range delta interaction . in both cases the gdm calculation reproduces quite well the exact density matrices and ground state energy within the physical range of parameters for the realistic spherical nuclear shell model . the errors are small and consistent with the statistical estimates made within a large random ensemble for the bcs - type hamiltonian in ref . @xcite . in the current form the theory is ready for the application to realistic nuclear systems . support is acknowledged from the hujiang foundation of china ( b14004 ) , and the startup funding for new faculty member in university of shanghai for science and technology . part of the calculations is done at the high performance computing center of michigan state university . .particle number @xmath1 and pairing strength @xmath140 as input parameters for the six sets of calculations . the quantity @xmath151 in the unit of @xmath140 is the length parameter for the harmonic oscillator wavefunctions . also shown is @xmath152 $ ] , the average value of the pairing matrix elements of a specific form . see text for details . [ cols="^,^,^,^,^,^,^",options="header " , ]

recently we proposed a particle - number - conserving theory for nuclear pairing [ jia , phys . rev . c * 88 * , 044303 ( 2013 ) ] through the generalized density matrix formalism . the relevant equations were solved for the case when each single - particle level has a distinct set of quantum numbers and could only pair with its time - reversed partner ( bcs - type hamiltonian ) . in this work we consider the more general situation when several single - particle levels could have the same set of quantum numbers and pairing among these levels is allowed ( hfb - type hamiltonian ) . the pair condensate wavefunction ( the hfb wavefunction projected onto good particle number ) is determined by the equations of motion for density matrix operators instead of the variation principle . the theory is tested in the simple two - level model with factorizable pairing interactions and the semi - realistic model with the zero - range delta interaction .

You are an expert at summarizing long articles. Proceed to summarize the following text: two distinct glasses have been predicted and identified in hard spheres with short range attractions : a repulsion driven glass at high density , and an attraction driven one , formed at low temperatures ( or high attraction strength ) @xcite . whereas the former one is caused by the steric hindrance of the particle cores and the so - called _ cage effect _ , the latter forms due to the bonding between particles . this system is realized experimentally by a colloid - polymer mixture , where the effect of the polymers is to induce an effective attraction between the colloids @xcite . both glasses have been indeed identified , although the attractive one , which at low concentrations is termed ` gel ' , often competes with ( and inhibits ) liquid - gas phase separation @xcite . dynamical heterogeneities ( dh ) have been found in the proximity of repulsion driven glass transitions by computer simulations , i.e. in lennard - jones mixtures @xcite , or hard spheres @xcite . in these cases , while the system is structurally liquid - like ( homogeneous ) , a population of particles of increased mobility is observed . as the glass transition is approached from the fluid side , the heterogeneities become more pronounced , but decrease again deeper in the glass @xcite . the role of these dynamical heterogeneities in the glass transition is as yet unclear ; whereas mode coupling theory focusses on averaged quantities and neglects them @xcite , the so - called facilitated dynamics theories give dh the central role for their description of the glass transition @xcite . in recent works , it has been shown that dh can be found also in attractive glasses , by studying the distribution of particle displacements in the system @xcite . in fluid states close to the transition two populations of particles were found , separated by a minimum in the displacement distribution . a similar feature has been found also in repulsive glasses , which could imply a common origin @xcite . however , the low density of the attractive glass , as low as @xmath0 , causes structural heterogeneities as well ; the system forms a percolating cluster of high density material , leaving voids with no particles . a correlation between structural and dynamical heterogeneities is thus possible , showing that ` fast ' particles are in the surface of the cluster , whereas the ` slow ' ones are mostly trapped in the inner parts of it @xcite . in this work , we study the dh inside the non - ergodic region , for two different states , and compare them with those of the equilibrium systems . only one population of particles can be identified from the distribution of particle displacements , and the distribution is narrower for the state with stronger attractions . moreover , as the systems age , they become more and more homogeneous , from the point of view of the dynamics . both results indicate that the strongest dh are obtained in the fluid side of the phase diagram , close to the glass transition . as a side remark , it must be noted that the structural heterogeneities mentioned above persist in the out - of - equilibrium systems , and thus are not the sole origin of the dh in attractive glasses . we have performed computer simulations of a system composed of @xmath1 soft core ( @xmath2 ) particles with attractive interactions given by the asakura - oosawa ( ao ) potential @xcite . it models a mixture of colloids with non - adsorbing polymers , and the range of attraction is set by the polymer size . in order to guarantee full access to the whole parameter space , phase separations have been inhibited . crystallization is avoided by polydispersity ( flat distribution , @xmath3 width ) , and liquid - gas demixing by a repulsive barrier extending to two mean diameters . further details of the interaction potential can be found in previous works @xcite . length is measured in units of the average radius , @xmath4 , and time in units of @xmath5 , where the thermal velocity , @xmath6 , was set to @xmath7 . equations of motion were integrated using the velocity - verlet algorithm , in the canonical ensemble ( constant ntv ) , to mimic the colloidal dynamics , with a time step equal to @xmath8 . every @xmath9 time steps , the velocity of the particles was re - scaled to assure constant temperature . the range of the attraction is set to @xmath10 . density is reported as volume fraction , @xmath11 , with @xmath12 the number density , and the attraction strength is measured in units of the polymer volume fraction @xmath13 ( at contact the ao attraction strength is @xmath14 ) . the attractive glass transition for this system has been studied previously @xcite . an mct analysis of the results ( diffusion coefficient , time scale and viscosity ) yields a transition point at @xmath15 for the colloid density @xmath16 . for the study of aging here , the systems were equilibrated without attraction ( @xmath17 ) at @xmath16 , and then instantaneously _ quenched _ to the desired @xmath13 at zero time , @xmath18 . two attraction strengths have been studied , @xmath19 and @xmath20 , lying beyond the nonergodicity transition . in both cases , @xmath21 independent simulations have been performed , and the evolution of the system has been followed as a function of the time elapsed since the quench , called waiting time , @xmath22 . correlation functions thus depend on two times : @xmath23 and @xmath24 . , @xmath25 , @xmath26 , @xmath27 , ... , and for different polymer fractions , @xmath13 , as labeled . note that as @xmath13 increases , two populations of particles with different mobilities appear in the system . the glass transition is located at @xmath15 , estimated from mct analysis ( power law fittings ) @xcite.,width=491 ] in fluid states close to the attractive glass , increasing dh have been found , the stronger the attraction @xcite . two populations of particles are observed to appear as the attraction is increased , one of mobile particles and another one of quasi - immobile particles ( see fig . [ fig1 ] ) @xcite . ( the minimum in the distribution of squared displacements allows for an unambiguous identification of almost every particle . ) these two populations are structurally well differentiated : the particles in the ` skin ' of the gel , with a small number of bonds , are mobile , and the particles in the inner parts of the gel , the ` skeleton ' , are quasi - immobile . the populations were observed to be stable for long times , although particles can change from one to the other . it is thus an equilibrium feature of the system , partly induced by the structural heterogeneity . and @xmath20 , as labeled , for different waiting times ; from left to right , @xmath28 , @xmath29 , @xmath30 , @xmath31 , @xmath32 , @xmath33 . the dashed line shows the msd for hard spheres at this density.,width=491 ] at a long time , @xmath34 . ( qualitatively similar snapshots are obtained for @xmath20.),width=415 ] all of the states presented in fig . [ fig1 ] are , nevertheless , fluid states . the structural properties do not depend on time , nor the dynamical ones on the initial time , and the correlation functions averaged throughout the whole system decay to zero . by further increasing the attraction strength , however , the system falls out of equilibrium and shows aging . in fig . [ msd ] , the mean squared displacement ( msd ) is presented for the two quenches studied here , @xmath19 and @xmath20 ; different lines are the msd for different waiting times , @xmath22 . as @xmath22 increases , bonds are formed between particles , as shown in fig . [ nneigh ] for both states , which hinders the motion of the particles , causing the dynamical arrest . in the msd , a plateau develops at short distances , signaling the localization length , clearer in the case of the higher @xmath35 , where the bonds are stronger and the _ cage _ ( network of bonds ) of the particles is tighter , as shown by the shorter localization length ( implying higher non - ergodicity parameters in the density correlation function ) . the mean number of bonds per particle is , however , lower . also , the aging is more dramatic and at long waiting times the msd hardly reaches the range of the attraction , contrary to previous works where the plateau in msd , or the density correlation function , was not observed @xcite . here we will concentrate on the dh , and a full analysis of the aging will be presented elsewhere @xcite . and @xmath20 and three waiting times , as labeled . the different lines represent the distribution at @xmath36 , @xmath25 , @xmath26 , @xmath27 , @xmath37 , @xmath38 , @xmath39 , @xmath40 and @xmath41 , from left to right , respectively.,width=415 ] , as a function of its mean number of neighbours during this time , for @xmath19 and @xmath42 . ( only ten simulations are considered , i.e. @xmath43 particles ) . note the vertical bands at integer numbers of neighbours ( especially for @xmath44 ) , due to particles that have not changed their neighbours.,width=491 ] a snapshot of the system with @xmath19 is presented in fig . the system forms an intricate structure , with voids and tunnels , similar to the fluid states presented above . therefore , the skeleton and skin " picture presented above , with two different populations of particles , could still be applicable . in fig . [ distributions ] we present the distribution of squared displacements at different times for both quenches . strikingly , the distribution is monomodal in all cases , and the peak evolves to larger displacements as time proceeds . the strong dh observed in the fluid states , have , therefore , disappeared in the glass side . moreover , the deeper quench , @xmath20 , shows narrower distributions at all waiting times than the quench at @xmath19 , indicating that the dynamics is more homogeneous the deeper the state is in the glass , in agreement with findings for lennard - jones ( repulsive ) glasses @xcite . at @xmath19 , however , a tail in the distribution to long distances can be observed , caused by some particles that can travel long distances ; the number of which decreases with @xmath22 . this feature is reminiscent of the population of fast particles observed in the fluid ( fig . [ fig1 ] ) . the origin of this population is studied in fig . [ nneigh - msd ] , where the squared displacement of every particle for a given time is correlated with the mean number of neighbours of the particle during this time . the plot shows that indeed the fastest particles in the system have less neighbours on average , whereas the particles with many neighbours , move very little . therefore , the simple picture of skeleton and skin " , can still be applied for the attractive glass close to the transition . the number of these fast particles is , nevertheless , decreasing with waiting time , as observed by comparing similar times @xmath45 for different waiting times @xmath22 in fig . [ distributions ] . accordingly , the mean number of bonds per particle increases ( see fig . [ nneigh ] ) , implying a compaction in the system , and thus having fewer particles in the skin , and even those , more tightly trapped . however , we can not state whether the population of fast particles will reach a steady state or if it will vanish eventually at very long waiting times . at @xmath20 , on the other hand , the tail of fast particles is absent in the distribution of squared displacements , although similar structural heterogeneities are observed . we have shown that the strong dh found in fluid states close to the attractive glass transition in colloids with short range attractions decrease again deep into the non - ergodic region . the distribution of squared displacements is monomodal and no particles with increased mobility are observed ( @xmath20 ) . however , close to the glass transition , in the glass side , aging is slower and some dh can still be detected : a tail in the distribution of squared displacements indicates fast particles , that can be identified with particles in the outer parts of the particle network . this feature is reminiscent of the population of fast particles found in the fluid states close to the transition . however , the stability of this population of fast particles in the glass can not be established , but should not be present in a truly arrested glassy state . the results presented here agree with those of repulsive glasses , contrary to other comparions between attractive and repulsive glasses @xcite . a.m.p . acknowledges financial support by the dgcyt ( project mat2003 - 03051-co3 - 01 ) . this work was funded in part ( m.e.c . ) by epsrc gr / s10377 . a.m.p . and m.f . were partially funded by ai - daad project no . ha2004 - 0022 . pham , a.m. puertas , j. bergenholtz , s.u . egelhaaf , a. moussaid , p.n . pusey , a.b . schofield , m.e . cates , m. fuchs , w.c.k . poon , science , * 296 * , 104 ( 2002 ) . f. sciortino , nature materials , * 1 * 145 ( 2002 ) . c. n. likos , 2001 _ phys . rep . _ * 348 * 267 ; s. asakura and f. oosawa , j. chem . phys . * 22 * 1255 ( 1954 ) . pham , s.u . egelhaaf , p.n . poon , phys . e * 69 * 011503 ( 2004 ) . w. kob , c. donati , s.j . plimpton , p.h . poole , s.c . glotzer ; phys . e * 79 * 2827 ( 1997 ) ; c. donati , s.c . glotzer , p.h . poole ; phys . lett . * 82 * 5064 ( 1999 ) . e. flenner , g. szamel , phys . e * 72 * 011205 ( 2005 ) . e.r . weeks , j.c . crocker , a.c . levitt , a. schofield , d.a . weitz ; science * 287 * , 627 ( 200 ) . b. doliwa , a. heuer , phys . lett * 80 * 4915 ( 1998 ) ; phys . e * 61 * , 6898 ( 2000 ) . k. vollmayr - lee , w. kob , k. binder , a. zippelius , j. chem . phys * 116 * 5158 ( 2002 ) k. vollmayr - lee , a. zippelius , cond - matt/0507107 . w. gtze in _ liquids , freezing and glass transition _ ed . hansen , d. levesque and j. zinn - justin , amsterdam , north - holland ( 1991 ) . l. berthier , j.p . garrahan , phys . e * 68 * 041201 ( 2003 ) ; j.p . garrahan , d. chandler , phys . lett . * 89 * 035704 ( 2002 ) . a.m. puertas , m. fuchs , m.e . cates , j. chem . phys . * 121 * 2813 ( 2004 ) a.m. puertas , m. fuchs , m.e . cates , j. phys . b * 109 * 6666 ( 2005 ) ; phys . e * 67 * , 031406 ( 2003 ) . reichman , e. rabani , p.l . geissler , j. phys . b * 109 * 14654 ( 2005 ) . a.m. puertas , e. zaccarelli , f. sciortino , j. phys . * 17 * l271 ( 2005 ) . e. zaccarelli , g. foffi , f. sciortino , and p. tartaglia , phys . rev . lett . * 91 * 108301 ( 2003 ) a.m. puertas , m. fuchs , m.e . cates . in preparation

the dynamical heterogeneities ( dh ) in non - ergodic states of an attractive colloidal glass are studied , as a function of the waiting time . whereas the fluid states close to vitrification showed strong dh , the distribution of squared displacements of the glassy states studied here only present a tail of particles with increased mobility for the lower attraction strength at short waiting times . these particles are in the surface of the percolating cluster that comprises all of the particles , reminiscent of the fastest particles in the fluid . the quench deeper into the attractive glass is dynamically more homogeneous , in agreement with repulsive glasses ( i.e. lennard - jones glass ) . , , 64.70.pf , 82.70.dd , 61.20.lc

You are an expert at summarizing long articles. Proceed to summarize the following text: the established evidence that neutrinos oscillate and possess small masses @xcite necessitates physics beyond the standard model ( sm ) , which could manifest itself at the cern large hadron collider ( lhc ) and/or in low energy experiments which search for the lepton flavour violation @xcite . consequently , models of neutrino mass generation which can be probed at present and forthcoming experiments are of great phenomenological interest . neutrinos may obtain mass via the vacuum expectation value ( vev ) of a neutral higgs boson in an isospin triplet representation @xcite . a particularly simple implementation of this mechanism of neutrino mass generation is the `` higgs triplet model '' ( htm ) in which the sm lagrangian is augmented solely by @xmath10 which is a @xmath1 triplet of scalar particles with hypercharge @xmath11 @xcite . in the htm , the majorana neutrino mass matrix @xmath12 ( @xmath13 ) is given by the product of a triplet yukawa coupling matrix @xmath14 and a triplet vev ( @xmath15 ) . consequently , the direct connection between @xmath14 and @xmath12 gives rise to phenomenological predictions for processes which depend on @xmath14 because @xmath16 has been restricted well by neutrino oscillation measurements @xcite . a distinctive signal of the htm would be the observation of doubly charged higgs bosons ( @xmath0 ) whose mass ( @xmath17 ) may be of the order of the electroweak scale . such particles can be produced with sizeable rates at hadron colliders in the processes @xmath4 @xcite and @xmath18 @xcite . the first searches for @xmath0 at a hadron collider were carried out at the fermilab tevatron , assuming the production channel @xmath4 and decay @xmath19 . the mass limits @xmath20 @xcite were derived , with the strongest limits being for @xmath21 @xcite . the branching ratios ( brs ) for @xmath19 depend on @xmath14 and are predicted in the htm in terms of the parameters of the neutrino mass matrix @xcite . detailed quantitative studies of br(@xmath19 ) in the htm have been performed in @xcite with particular emphasis given to their sensitivity to the majorana phases and the absolute neutrino mass i.e. parameters which can not be probed in neutrino oscillation experiments . a study on the relation between br(@xmath19 ) and the neutrinoless double beta decay can be seen in @xcite . simulations of the detection prospects of @xmath0 at the lhc with @xmath22 previously focussed on @xmath4 only @xcite , but recent studies now include the mechanism @xmath18 @xcite . the first search for @xmath0 at the lhc with @xmath23 @xcite has recently been performed for both production mechanisms @xmath4 and @xmath18 , for the decay channels @xmath19 and @xmath24 . in phenomenological studies of the htm , for simplicity it is sometimes assumed that @xmath0 and @xmath25 are degenerate , with a mass @xmath26 which arises from a bilinear term @xmath27 in the scalar potential . in this scenario the only possible decay channels for @xmath0 are @xmath19 and @xmath28 , and the branching ratios are determined by the magnitude of @xmath15 . however , quartic terms in the scalar potential break the degeneracy of @xmath0 and @xmath25 , and induce a mass splitting @xmath29 , which can be of either sign . if @xmath30 then a new decay channel becomes available for @xmath0 , namely @xmath31 . some attention has been given to the decay @xmath31 , and it has been shown that it can be the dominant channel over a wide range of values of @xmath32 and @xmath15 @xcite , even for @xmath33 . another scenario is the case of @xmath34 , which would give rise to a new decay channel for the singly charged scalar , namely @xmath2 . this possibility has been mentioned in the context of the htm in @xcite only . we will perform the first study of the magnitude of its branching ratio , as well as quantify its contribution to the production of @xmath0 at the lhc . has also been briefly mentioned in @xcite in the context of a model with an isospin 3/2 multiplet with hypercharge @xmath35 , which also includes triply charged higgs bosons . ] the decay rate for @xmath2 is easily obtained from the corresponding expression for the decay rate for @xmath31 , and thus one expects that @xmath2 will be sizeable over a wide range of values of @xmath32 and @xmath15 . we point out for the first time that the decay @xmath2 would give rise to an alternative way to produce @xmath0 in pairs ( @xmath36 ) , namely by the production mechanism @xmath18 followed by @xmath37 . production of @xmath36 can give rise to a distinctive signature of four leptons ( @xmath38 ) , and simulations and searches of this channel currently only assume production via the process @xmath4 . our work is organised as follows . in section ii we describe the theoretical structure of the htm . in section iii the decay @xmath2 is introduced . section iv contains our numerical analysis of the magnitude of the cross section for @xmath36 which originates from production via @xmath18 followed by the decay @xmath2 . conclusions are given in section v. in the htm @xcite a @xmath11 complex @xmath1 isospin triplet of scalar fields is added to the sm lagrangian . such a model can provide majorana masses for the observed neutrinos without the introduction of @xmath1 singlet neutrinos via the gauge invariant yukawa interaction : @xmath39 here @xmath40 is a complex and symmetric coupling , @xmath41 is the dirac charge conjugation operator , @xmath42 is the pauli matrix , @xmath43 is a left - handed lepton doublet , and @xmath10 is a @xmath44 representation of the @xmath11 complex triplet fields : @xmath45 a non - zero triplet vacuum expectation value @xmath46 gives rise to the following mass matrix for neutrinos : @xmath47 the necessary non - zero @xmath48 arises from the minimisation of the most general @xmath49 invariant higgs potential @xcite , which is written ) by using @xmath50 ^ 2 - \text{tr}[(\delta^\dagger \delta)^2]$ ] and @xmath51 . ] as follows @xcite ( with @xmath52 ) : @xmath53 ^ 2 + \lambda_3{\rm det } ( \delta^\dagger\delta ) \nonumber \\ & & + \lambda_4(\phi^\dagger\phi){\rm tr}(\delta^\dagger\delta ) + \lambda_5(\phi^\dagger\tau_i\phi){\rm tr}(\delta^\dagger\tau_i \delta)+\left ( { 1\over \sqrt 2}\mu(\phi^ti\tau_2\delta^\dagger\phi ) + \text{h.c . } \right ) . \label{higgs_potential}\end{aligned}\ ] ] here @xmath54 in order to ensure @xmath55 which spontaneously breaks @xmath56 to @xmath57 , and @xmath58 is the mass term for the triplet scalars . in the model of gelmini - roncadelli @xcite the term @xmath59 is absent , which leads to spontaneous violation of lepton number for @xmath60 . the resulting higgs spectrum contains a massless triplet scalar ( majoron , @xmath61 ) and another light scalar ( @xmath7 ) . pair production via @xmath62 would give a large contribution to the invisible width of the @xmath63 and this model was excluded at the cern large electron positron collider ( lep ) . the inclusion of the term @xmath64 ) @xcite explicitly breaks lepton number @xmath65 when @xmath10 is assigned @xmath66 , and eliminates the majoron . thus the scalar potential in eq . ( [ higgs_potential ] ) together with the triplet yukawa interaction of eq . ( [ trip_yuk ] ) lead to a phenomenologically viable model of neutrino mass generation . for small @xmath67 , the expression for @xmath15 resulting from the minimisation of @xmath68 is : @xmath69 for large @xmath26 compared to @xmath70 one has @xmath71 , which is sometimes referred to as the `` type ii seesaw mechanism '' and would naturally lead to a small @xmath15 . recently there has been much interest in the scenario of light triplet scalars ( @xmath72 ) within the discovery reach of the lhc , for which eq . ( [ tripletvev ] ) leads to @xmath73 . in extensions of the htm the term @xmath64 ) may arise in various ways : i ) it can be generated at tree level via the vev of a higgs singlet field @xcite ; ii ) it can arise at higher orders in perturbation theory @xcite ; iii ) it can originate in the context of extra dimensions @xcite . an upper limit on @xmath15 can be obtained from considering its effect on the parameter @xmath74 . in the sm @xmath75 at tree - level , while in the htm one has ( where @xmath76 ) : @xmath77 the measurement @xmath78 leads to the bound @xmath79 , or @xmath80 . production mechanisms which depend on @xmath15 ( i.e. @xmath81 and fusion via @xmath82 @xcite ) are not competitive with the processes @xmath4 and @xmath18 at the energies of the fermilab tevatron , but such mechanisms can be the dominant source of @xmath0 at the lhc if @xmath83 and @xmath84 . at the 1-loop level , @xmath15 must be renormalised and explicit analyses lead to bounds on its magnitude similar to the above bound from the tree - level analysis , e.g. see @xcite . the scalar eigenstates in the htm are as follows : i ) the charged scalars @xmath0 and @xmath9 ; ii ) the cp - even neutral scalars @xmath85 and @xmath7 ; iii ) a cp - odd neutral scalar @xmath8 . the doubly charged @xmath0 is entirely composed of the triplet scalar field @xmath86 , while the remaining eigenstates are in general mixtures of the doublet and triplet fields . however , such mixing is proportional to the triplet vev , and hence small _ even if _ @xmath15 assumes its largest value of a few gev . @xcite . ] therefore @xmath87 are predominantly composed of the triplet fields , while @xmath85 is predominantly composed of the doublet field and plays the role of the sm higgs boson . the scale of squared masses of @xmath88 are determined by @xmath89 with mass splittings of order @xmath90 @xcite : @xmath91 the degeneracy @xmath92 can be understood by the fact that the higgs potential is invariant under a global @xmath93 for @xmath10 ( @xmath65 conservation ) when one neglects the trilinear term proportional to @xmath94 . the mass hierarchy @xmath95 is obtained for @xmath96 , and the opposite hierarchy @xmath97 is obtained for @xmath98 . in general , one would not expect degenerate masses for @xmath88 , but instead one of the above two mass hierarchies . the sign of @xmath99 is not fixed by theoretical requirements of vacuum stability of the scalar potential @xcite , although @xmath100 is necessary to ensure that @xmath101 and @xmath102 in eq . ( [ eq : mh ] ) are positive . therefore the decays channels @xmath103 and @xmath104 are possible in the htm . the potential importance of the decay channel @xmath105 ( for @xmath106 ) has not been quantified in the htm . for this decay to be kinematically open @xcite one needs the mass hierarchy where @xmath0 is the lightest of the triplet scalars ( @xmath107 ) , which is obtained for @xmath96 . for the opposite mass hierarchy with @xmath98 ( @xmath97 ) the related decay @xmath108 was shown to be important in the htm in @xcite . the expression for the decay width of @xmath2 is easily obtained from the expression for @xmath108 by merely interchanging @xmath17 and @xmath109 . after summing over all fermion states for @xmath110 , excluding the @xmath111 quark , the decay rate is given by @xmath112 where @xmath113 and the analytical expression for @xmath114 can be found in @xcite ( see also @xcite ) . note that this decay mode does not depend on @xmath15 . in eq . ( [ hhwdecay ] ) we take @xmath115 and @xmath116 to be massless , which is a good approximation as long as the mass splitting between @xmath17 and @xmath109 is above the mass of the charmed hadrons ( @xmath117 ) . in our numerical analysis we will be mostly concerned with sizeable mass splittings , @xmath118 . the other possible decays for @xmath9 are @xmath24 , @xmath119 , @xmath120 ( where @xmath85 is the sm - like scalar field ) and @xmath121 . explicit expressions for the decay widths of these channels can be found in the literature ( e.g. @xcite ) and they are presented below . the decay width for @xmath24 is given by @xmath122 note that @xmath123 has no dependence on the neutrino mixing angles because @xmath124 , where @xmath125 ( @xmath126 ) are neutrino masses . the decay widths for the channels which are proportional to @xmath127 are expressed as follows : @xmath128 ^ 3 , \label{eq : gam_hp_wz}\end{aligned}\ ] ] @xmath129 ^ 3 , \label{eq : gam_hp_wh}\end{aligned}\ ] ] @xmath130 @xmath131 the decay @xmath120 is caused by two small mixings of scalar fields . one is the mixing angle @xmath132 between @xmath133 and @xmath134 , and the other is the mixing angle @xmath135 between @xmath136 and @xmath137 . if @xmath138 , then one has @xmath139 in eq . ( [ eq : gam_hp_wh ] ) . since we are interested in the case where the exotic scalars have masses of the electroweak scale , we do not take a very large @xmath26 . however , we assume @xmath139 for simplicity , which can be achieved by @xmath140 . the decay @xmath141 is mediated by the small @xmath133 component of @xmath9 through @xmath142 . for @xmath143 , @xmath144 is comparable to @xmath145 and @xmath146 . these three decay widths in eq . ( [ eq : gam_hp_wz])-([eq : gam_hp_tb ] ) are greater than @xmath147 for @xmath148 while @xmath147 dominates for @xmath149 . it has already been shown that the decay @xmath31 can be the dominant decay channel for the doubly charged scalar over a wide range of values of @xmath150 and @xmath15 @xcite , even for @xmath33 . hence we expect a similar result for the decay @xmath2 for the singly charged scalar . the branching ratio @xmath151 will be maximised with respect to @xmath15 if @xmath152 which is achieved for @xmath153 . a numerical study of the magnitude of @xmath151 is presented in the next section . we now emphasise an important phenomenological difference between the distinct scenarios of a sizeable branching ratio for the decay channels @xmath31 ( for @xmath154 ) and @xmath2 ( for @xmath96 ) . the decay @xmath31 is expected to weaken the discovery potential of @xmath0 at the lhc , because it would reduce the branching ratio of a channel like @xmath155 ( which is otherwise the dominant channel for @xmath156 , and enjoys low sm backgrounds ) . we note that there has been no simulation of the detection prospects of @xmath31 , and its signature would be different to that of the standard decay channels @xmath19 and @xmath28 . in contrast , we point out that the decay @xmath157 could actually _ improve _ the discovery potential of @xmath0 at the lhc . from the production mechanism @xmath18 the decay mode @xmath157 would give rise to pair production ( @xmath36 ) of doubly charged higgs bosons . we believe that this additional way to produce @xmath0 has not been discussed before . in this scenario @xmath0 is the lightest of the triplet scalars , and its only possible decay channels are @xmath19 and @xmath28 , with branching ratios determined by the magnitude of @xmath15 . these two branching ratios can be of the same order of magnitude for @xmath153 , as can be seen in fig . [ fig : br_hdoub ] where we fix @xmath158 and @xmath159 ( similar figures can be found in @xcite ) . in the range of @xmath160 , one has @xmath161 for @xmath162 , while for @xmath163 one has br(@xmath164 . in simulations of pair production of @xmath0 it is assumed that the production channel @xmath4 is the only mechanism . if @xmath156 then the decay channel @xmath155 is dominant , and four - lepton signatures ( @xmath165 ) would be possible . studies have shown that the standard model background for the @xmath165 signature @xcite is considerably smaller than that for the signature of @xmath166 @xcite , and at present it is assumed that the @xmath165 signature can only arise from @xmath4 . the importance of the production mechanism @xmath18 has been appreciated for the @xmath166 signature , in which the decay @xmath167 is assumed @xcite . for the case of a sizeable branching ratio for @xmath157 we point out that the production mechanism @xmath18 can also contribute to the @xmath165 signature , which is the signature with lowest background . searches for four leptons originating from @xmath36 have already been performed by the tevatron @xcite and lhc @xcite . if br(@xmath157 ) were sizeable we would expect a strengthening of the derived limit on @xmath17 . in this section we quantify the magnitude of the number of pair - produced @xmath36 arising from the process @xmath18 with decay @xmath157 , and make a comparison with the conventional mechanism @xmath4 . the important parameters for our analyses are @xmath15 , @xmath109 , and @xmath17 . we take @xmath168 or @xmath169 and show results as functions of @xmath15 and @xmath109 . the decay branching ratios of @xmath9 also depend on two undetermined parameters , @xmath170 and @xmath171 ( one of the neutrino masses ) . these are fixed as @xmath172 and @xmath173 in our numerical analysis . note that @xmath170 only enters through the decay width for @xmath174 . neutrino oscillation experiments @xcite provide a measurement of two neutrino mass differences , @xmath175 , and we use the following values : @xmath176 , @xmath177 . although @xmath178 ( referred to as `` normal mass ordering '' ) is also assumed in our analysis , our results do not change significantly for @xmath179 because the neutrino masses are almost degenerate for @xmath180 . the experimental bound @xmath181 gives a stringent constraint on @xmath182 and @xmath17 . @xcite or @xmath183 @xcite . see also @xcite . ] assuming naively @xmath184 for @xmath173 , the bound on @xmath185 can be translated into the constraint @xmath186 . therefore , we use @xmath187 in order to satisfy this constraint for @xmath168 . + in fig . [ fig : br_hsing ] we show the brs of @xmath9 decays into @xmath188 ( red solid ) , @xmath189 ( blue dashed ) , @xmath190 ( green dotted ) , @xmath191 ( magenta dot - dashed ) , and @xmath192 ( cyan dot - dot - dashed ) as a function of @xmath193 for various values of @xmath15 , fixing @xmath168 and @xmath194 . the range of @xmath109 in the figures corresponds to @xmath195 , which easily satisfies the perturbative constraint @xmath196 . very large mass splittings ( e.g. @xmath197 ) are constrained by measurements of electroweak precision observables , but the mass splittings in fig . [ fig : br_hsing ] are compatible with the analyses in @xcite ( which are for models with a @xmath198 triplet ) . in fig . [ fig : br_hsing](a ) we fix @xmath199 , for which @xmath200 . one can see that @xmath105 competes with @xmath201 , with all other decay channels being negligible . for @xmath202 , @xmath105 becomes the dominant decay channel . in fig . [ fig : br_hsing](b ) we fix @xmath158 , and @xmath105 becomes the dominant decay channel for much smaller mass splittings , @xmath203 . in fig . [ fig : br_hsing](c ) we fix @xmath204 , for which the competing decays are @xmath205 , @xmath206 and @xmath207 . in this scenario the decay @xmath105 becomes the dominant channel for @xmath208 . in fig . [ fig : br_contour ] we show contours of br(@xmath2 ) in the plane @xmath209 $ ] . the red solid , green dashed , and blue dotted lines correspond to contours of @xmath210 , 0.9 , and 0.99 , respectively . the br is maximised at around @xmath211 , as expected . it is clear from fig . [ fig : br_hsing ] and fig . [ fig : br_contour ] that the decay of @xmath9 into @xmath0 can be dominant in a wide region of the parameter space of the htm even if the two - body decay into @xmath212 ( for @xmath213 ) is forbidden kinematically . moreover , for @xmath214 ( i.e. when the four - lepton signal arising from the decay of @xmath36 is dominant ) the magnitude of @xmath215 becomes very large if @xmath216 . we now study the magnitude of the number of pair - produced @xmath36 which originate from @xmath217 followed by the decay @xmath2 . we define the variable @xmath218 as follows : @xmath219 in fig . [ fig : brsig ] we show the behaviour of @xmath220 with respect to @xmath109 for several values of @xmath15 . in fig . [ fig : brsig]a we take @xmath168 and @xmath221 , and in fig . [ fig : brsig]b we take @xmath222 and @xmath223 . we use cteq6l1 parton distribution functions @xcite . the range of @xmath109 in fig . [ fig : brsig]b corresponds to @xmath224 . the horizontal dot - dashed line corresponds to the case of @xmath225 , i.e. the magnitude of @xmath226 alone . the red solid , green dashed , and blue dotted lines are the results with @xmath214 , @xmath227 , and @xmath228 , respectively . the red solid line ( for which @xmath229 ) shows that the extra contribution from @xmath2 can enhance the number of four - lepton events by a factor of 2 ( at @xmath230 in fig . [ fig : brsig]a ) and 2.4 ( at @xmath231 in fig . [ fig : brsig]b ) . for @xmath211 , around which @xmath232 can still be sizeable ( see fig . [ fig : br_hdoub ] ) , the enhancement factor for pair - produced @xmath36 can be as large as 2.6 in fig . [ fig : brsig]a and 2.8 in fig . [ fig : brsig]b . for @xmath204 the enhancement of pair - produced @xmath36 is interpreted as an increase in the number of @xmath233 events , because @xmath234 . the shape of the curves is caused by the different dependence of the cross section and br on the mass splitting @xmath32 . as @xmath109 increases , the cross section of @xmath235 is unaffected but the cross section of @xmath217 decreases . however , a larger mass splitting is favourable from the point of view of the br . finally , we note that a pair of @xmath0 can also be produced from other production mechanisms , namely @xmath236 , @xmath237 , @xmath238 , and @xmath239 . although the contribution from @xmath240 in eq . ( [ eq : hpphm ] ) is the most important one because of the mass hierarchy @xmath241 and its linear dependence on br@xmath242 , the above mechanisms can give a significant contribution to the number of pair - produced @xmath0 , as will be described qualitatively below . naively , one would expect the next most important mechanism to be @xmath236 because its contribution to the production of @xmath36 scales as @xmath243 as follows : @xmath244 ^ 2 .\end{aligned}\ ] ] however , the couplings for @xmath245 and @xmath246 are about a half of those for @xmath247 and @xmath248 , respectively . the interference between @xmath249 and @xmath250 is destructive for @xmath236 production while it is constructive for @xmath36 production . tables [ tab : lhc7 ] and [ tab : lhc14 ] show that @xmath251 is smaller than @xmath226 by a factor of @xmath252 , even for @xmath253 ( see e.g. @xcite ) . moreover , @xmath254 is suppressed relative to @xmath218 by an extra factor of br when @xmath255 . therefore the contribution from @xmath251 to the production of @xmath36 is considerably less than the qcd @xmath256 factor for @xmath235 ( which is known to be around 1.25 at the lhc @xcite ) . it turns out that the production of @xmath237 and @xmath238 are numerically more important than @xmath257 , despite their contributions scaling as @xmath258 . the narrow width approximation for contributions from @xmath7 and @xmath8 with @xmath92 is rather complicated because of their interference . we define the variables @xmath259 and @xmath260 as follows : @xmath261 ^ 2 , \label{eq : hph0 } \\ x_3^\prime & \equiv & \left\ { \sigma({{pp\to w^ * \to h^+h^0 } } ) + \sigma({{pp\to w^ * \to h^-h^0 } } ) \right\ } \nonumber\\ & & \hspace*{50 mm } \times { \text{br}}_-\ , [ { \text{br}}(h^\pm \to h^{\pm\pm } w^*)]^2 , \label{eq : hph0 - 2 } \\ { \text{br}}_\pm & \equiv & { \text{br}}(h^0\to h^\pm w^ * ) + { \text{br}}(a^0\to h^\pm w^ * ) \nonumber\\ & & \hspace*{20 mm } { } \pm \frac { 4 { \text{br}}(h^0\to h^\pm w^ * ) { \text{br}}(a^0\to h^\pm w^ * ) } { { \text{br}}(h^0\to h^\pm w^ * ) + { \text{br}}(a^0\to h^\pm w^ * ) } , \end{aligned}\ ] ] where we used @xmath262 because @xmath263 . the interesting point is that @xmath260 is for the process which gives _ same - sign _ @xmath264 ( with @xmath265 ) and @xmath266 ( with @xmath267 ) while @xmath259 is for @xmath36 production . since @xmath260 arises as the breaking effect of the lepton number ( @xmath10 has @xmath268 ) , it vanishes for @xmath269 , for which the total decay widths satisfy @xmath270 , namely @xmath271 . this means that the _ same - sign _ @xmath272 would not give the _ same - sign _ @xmath165 signal because @xmath273 is small for a large @xmath15 where @xmath260 could be sizeable . a pair of @xmath0 ( _ same - sign _ or different sign ) is provided by @xmath274 , which is proportional to @xmath275 $ ] ; the factor of 2 compensates the fact that the sum of the cross sections in eq . ( [ eq : hph0 ] ) is a half of the sum in eq . ( [ eq : hpphm ] ) for @xmath276 as shown in tables [ tab : lhc7 ] and [ tab : lhc14 ] . although @xmath277 ( likewise for @xmath7 ) and the maximum value of each is @xmath278 , this is compensated by @xmath279 in @xmath274 . since the partial decay widths of @xmath7 and @xmath8 depend on the scalar masses and @xmath15 in a way which is very similar to the partial decay widths of @xmath9 ( see e.g. @xcite ) , the analogies of fig . [ fig : br_contour ] for @xmath280 and @xmath281 would show a similar quantitative behaviour as fig . [ fig : br_contour ] . and @xmath5 were also mentioned as a source of @xmath9 in @xcite . ] thus the main difference between @xmath218 and @xmath274 would be the phase space factor because we take @xmath282 . the contribution of @xmath274 to the production of a pair of @xmath0 would be sizeable for @xmath153 , where the relevant brs in eq . ( [ eq : hph0 ] ) could be very large for a small mass splitting . moreover , the contribution of @xmath274 would not be so small even for large mass splittings e.g. @xmath283 and @xmath284 ( which give @xmath285 ) , for which the brs in eq . ( [ eq : hph0 ] ) could be maximal . the last mechanisms ( which scale as @xmath286 ) are @xmath287 ^ 2 , \\ x_4^\prime & \equiv & \sigma({{pp\to z^ * \to h^0 a^0}})\ , { \text{br}}_+\ , { \text{br}}_-\ , [ { \text{br}}(h^\pm\to h^{\pm\pm } w^*)]^2 , \\ x_4^{\prime\prime } & \equiv & \sigma({{pp\to z^ * \to h^0 a^0}})\ , { \text{br}}_-^2\ , [ { \text{br}}(h^\pm\to h^{\pm\pm } w^*)]^2 .\end{aligned}\ ] ] note that @xmath288 gives a pair of _ same - sign _ @xmath0 ( being proportional to @xmath289 , like @xmath260 ) and its magnitude is negligible for small @xmath15 . although both of @xmath290 and @xmath291 give @xmath36 , @xmath291 also vanishes for @xmath269 because it is sensitive to @xmath292 i.e. it is quadratic in lepton number violation . the phase space suppression ( @xmath241 ) ensures that @xmath293 is much smaller than @xmath226 for the case of a large mass splitting with @xmath221 . therefore , for @xmath290 to be important a large mass splitting with @xmath223 or a small mass splitting for @xmath153 are preferred . we note that the detection efficiencies for the above mechanisms ( @xmath218 , @xmath254 , @xmath259 and @xmath290 ) would in general be different from that of the well - studied mechanism @xmath235 because of the extra @xmath294 . we defer a detailed study to a future work . . production cross sections of a pair of exotic higgs bosons ( @xmath295 ) from off - shell gauge bosons ( @xmath296 ) in the htm at the lhc with @xmath221 . we take @xmath284 and we use a relation @xmath297 ; @xmath298 , @xmath299 , @xmath300 for @xmath301 , @xmath302 , @xmath303 , respectively . [ cols="^,^,^,^,^,^,^,^ " , ] doubly charged higgs bosons ( @xmath0 ) , which arise in the higgs triplet model ( htm ) of neutrino mass generation , are being searched for at the tevatron and at the lhc . we showed that @xmath0 can be produced from the decay of a singly charged higgs boson ( @xmath25 ) via @xmath304 , which can have a large branching ratio in a wide region of the parameter space of the htm . from the production mechanism @xmath305 , the above decay would give rise to pair production @xmath36 , with a number of events which can be comparable to that from the conventional mechanism @xmath4 . current simulations and searches for @xmath36 at the tevatron / lhc assume production solely from @xmath4 . the contribution from @xmath18 with decay @xmath304 would be an additional source of pair - produced @xmath0 , which should enhance the detection prospects in this channel ( e.g. four - lepton signatures if the decay mode @xmath19 is dominant ) . we also pointed out that production mechanisms involving the neutral triplet scalars ( @xmath7,@xmath8 ) of the htm can contribute to pair production @xmath36 through the decay chain @xmath306 followed by @xmath304 . we advocate dedicated simulations of @xmath18 with the decay @xmath304 ( and the analogous mechanisms with neutral scalars ) , and a comparison with @xmath4 . we thank mayumi aoki and koji tsumura for useful discussions . a.g.a was supported by a marie curie incoming international fellowship , fp7-people-2009-iif , contract no . 252263 . the work of h.s . was supported in part by the sasakawa scientific research grant from the japan science society and grant - in - aid for young scientists ( b ) no . 23740210 . y. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . lett . * 81 * , 1562 ( 1998 ) . y. kuno and y. okada , rev . phys . * 73 * , 151 ( 2001 ) ; 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the existence of doubly charged higgs bosons ( @xmath0 ) is a distinctive feature of the higgs triplet model ( htm ) , in which neutrinos obtain tree - level masses from the vacuum expectation value of a neutral scalar in a triplet representation of @xmath1 . we point out that a large branching ratio for the decay of a singly charged higgs boson to a doubly charged higgs boson via @xmath2 is possible in a sizeable parameter space of the htm . from the production mechanism @xmath3 the above decay mode would give rise to pair production of @xmath0 , with a cross section which can be comparable to that of the standard pair - production mechanism @xmath4 . we suggest that the presence of a sizeable branching ratio for @xmath2 could significantly enhance the detection prospects of @xmath0 in the four - lepton channel . moreover , the decays @xmath5 and @xmath6 from production of the neutral triplet scalars @xmath7 and @xmath8 would also provide an additional source of @xmath9 , which can subsequently decay to @xmath0 .

You are an expert at summarizing long articles. Proceed to summarize the following text: the process of analysing a time series of astronomical images to measure the variations in brightness of an object over time and to calibrate those variations against reference sources is a very time - consuming task . an accurate understanding of the variation in brightness of an object over time , i.e. its photometric light curve , allows modelling of key characteristics of the object ( e.g. the shape of an asteroid , the type of a supernova or gamma - ray burst optical afterglow ) . the measurement process often varies only little depending on the type of object , whether it is an asteroid tumbling through space , a supernova or the rapidly fading optical afterglow from a gamma - ray burst . the process usually involves examining the images in the time series for at least one suitable reference source in order to calibrate the brightness . some processes require the reference sources to have particular characteristics , e.g. a specific spectral class or magnitude . other processes may need sources that have been identified as a photometric standard . if more than one calibration source is available , the precision can be improved enormously ; however , this is also much more time - expensive . the process must be repeated for each image and corrections made from one image to another . these corrections can incorporate predictable adjustments such as extinction due to changing air mass . in addition , often the corrections that need to be made in order to obtain a consistent light curve are variable , non - uniform and unpredictable , and are based on experience and intuition . the ability to quickly analyse a set of images and to produce a high quality light curve is particularly valuable on automated telescopes , e.g. the fully robotic zadko telescope in gingin , western australia @xcite . participation in programmes like the prompt follow - up of gamma - ray burst alerts for the detection of transient optical emissions as part of the tlescope action rapide pour les objets transitoires ( tarot ) network @xcite and the gaia follow - up network for solar system objects ( gaia - fun - sso , http://www.imcce.fr/gaia-fun-sso ) @xcite results in the creation of data sets containing transient sources that need rapid analysis . we have developed a high efficiency image detection & identification ( heidi ) pipeline software for processing images and analysing the variations in brightness of optical transients . we take the approach that all sources in an image are potential reference sources . the variations in the brightness of the sources in each image of a time series are assessed automatically . corrections are made to compensate for these variations while also taking into account that some sources may intrinsically vary on short time scales ( e.g. short period variable stars ) . to facilitate production of photometric light curves solar system objects are identified automatically during the image processing . as a result the automated analysis of an image set and the production of a calibrated light curve is achieved in a matter of minutes . heidi thus fulfils an emerging need for the rapid analysis of time series image data . heidi is implemented in the programming language python ( http://www.python.org ) for linux platforms . it processes astronomical images in fits format and requires only a minimal amount of user - supplied detail . it can also be installed on linux virtual machines . this makes the software extremely versatile . preprocessing tasks such as astrometric calibration and image alignment are accomplished using existing applications as described in [ section : fitsalign ] . the main functions of identification and selection of calibration sources , identification of targets and production of light curves are accomplished using newly developed python code as described in [ section : correlstar ] and [ section : photometry ] . heidi has been benchmarked on a pentium class system and a virtual machine , i.e. low - performance , low - cost systems . the first step in the image analysis process is to perform the astrometric calibration . the amount of information contained in fits image headers depends heavily on the system in place at the observing site . while an integrated system would typically produce fits images which include calibration information in the image headers , some sites may not even include the pointing information . we have therefore adopted a standard approach of uploading one image from the set to the astrometry.net server in order to obtain a new astrometric calibration @xcite . by using the blind astrometric calibration system provided by astrometry.net , heidi thus removes any reliance on the images having already had astrometric calibrations or having to verify the pointing information , orientation or scale . after obtaining the astrometric calibration , heidi assumes that all the images in the time series have the same pointing and updates the fits image headers . the world coordinate system ( wcs ) information from astrometry.net is included in the headers of the updated fits images . the images are then processed through sextractor in order to build a catalogue of all objects found in the image set . the sextractor catalogue is subsequently processed through scamp @xcite which produces image headers that are ready to be used by swarp @xcite . swarp aligns the images and includes the appropriate header information from any translation or rotation applied by swarp during the alignment process . most of the processing time is taken up with the sextractor - scamp - swarp workflow . on a single - cpu pentium - class system , the astrometric calibration and image alignment stage for a set of 50 images takes about 10 min ( vs. 1.25 min for the automated production of the photometric light curve by heidi , [ section : correlstar ] , [ section : photometry ] ) . in the correlation stage the images from the astrometric calibration and alignment stage are reprocessed through sextractor to produce catalogues of all the sources that have been detected across the entire image set . these catalogues are correlated by heidi to identify the sources which are common to every image in the time series . these common sources are the candidate reference sources for the photometric calibration of the image set . heidi calculates the natural logarithm of the flux value for each candidate source as a representation of relative ( instrumental ) magnitude . we refer to the natural logarithm of the flux hereafter simply as log flux . examination of the log flux value for each candidate source may show some variation across the image set . heidi tests the log flux variation of each candidate source and rejects sources as candidates if the variation is greater than a predetermined threshold . an examination of the log flux values for the candidate sources in a time series of 50 images for asteroid 939 isberga shows a variation common to all candidate sources ( figure [ fig : figure1 ] ) . as this variation appears to affect every source in a consistent manner , we conclude that this variation can be analysed and corrected . the selected stars from observations of asteroid 939 isberga show a variation in the log flux values across the time series . as this variation appears to affect every source in a consistent manner it can be analysed and a correction applied . ] for a given time series of @xmath0 images , heidi finds all sources that are common to all @xmath0 images in that time series . these @xmath1 sources are selected as candidates for photometric calibration and are tested for variability over the time series . step 1 : for the @xmath0 images and @xmath1 sources , heidi reads the flux values @xmath2 , @xmath3 , @xmath4 from the sextractor catalogues and calculates the natural logarithm of the flux . we will denote the natural logarithm of the flux as @xmath5 , @xmath3 , @xmath4 and refer to it hereafter as log flux . the initial log flux set is hence @xmath6 , @xmath7 and contains @xmath8 values . step 2 : for each image @xmath3 in the time series , heidi calculates @xmath1 log flux means : for each source @xmath4 , the source @xmath9 is excluded ( denoted by @xmath10 ) and the log flux mean of the remaining sources @xmath11 is determined : @xmath12 step 3 : heidi determines the global maximum of the log flux : @xmath13 for all @xmath3 , @xmath4 the difference between the global maximum log flux @xmath14 and the log flux means @xmath15 ( determined in step 2 ) are calculated : @xmath16 this will be our correction factor . step 4 : to smooth out the variations across the image set , the log flux values @xmath5 , @xmath3 , @xmath4 are adjusted , yielding an adjusted log flux set : @xmath17 step 5 : for each source @xmath4 heidi determines the adjusted log flux mean , the maximum adjusted log flux and the minimum adjusted log flux of this source across all @xmath0 images : @xmath18 @xmath19 @xmath20 step 6 : for each source @xmath4 heidi tests whether the deviation of the adjusted log flux values @xmath21 from the adjusted log flux mean @xmath22 is within a predetermined tolerance @xmath23 . this test rejects sources that are varying in brightness on short time scales ( e.g. short period variable stars ) or are otherwise not varying in accordance with the other candidates in the set ( e.g. edge effects or vignetting ) . only sources that pass this test will remain in the candidate list : @xmath24 sources that do not pass this test will be excluded from the candidate list . if @xmath25 is the number of excluded sources then the reduced candidate list consists of @xmath26 candidates . this concludes the first pass and reduces the original candidate list @xmath6 to @xmath27 : @xmath28 step 7 : repeat step 2 to step 6 for the reduced candidate list @xmath29 . this process of filtering variable sources is applied iteratively . the tolerance @xmath23 for the deviation test of the log flux ( step 6 ) is 0.5 , 0.2 and 0.1 for the first , second and third pass , respectively . we found that additional iterations with finer tolerances do not significantly improve the result as one reaches the limit of measurement precision . the candidate stars that survive this filtering process are used in the photometric calibration of the image set for producing the light curve for the target as described in [ section : photometry ] . in addition , the candidates that fail the test are recorded for further study . figure [ fig : figure2 ] shows a sample image from a time series of @xmath30 images for asteroid 939 isberga . this image contains exactly @xmath31 sources that are common to all images in the time series , marked with a cross @xmath32 . these sources are selected as candidate sources and are tested for their suitability for use as calibration sources . circles @xmath33 indicate the reduced set of 37 sources that were found to be suitable for use as calibration sources by heidi . the position of the target , asteroid 939 isberga is indicated by a square ( @xmath34 ) . sample image from the time series for asteroid 939 isberga . this image is centred on right ascension @xmath35 , declination @xmath36 ( j2000 ) and has a field of view of @xmath37 arcmin . sources that are common to all images in the time series are indicated by a cross @xmath32 . these sources are tested for their suitability to be used as calibration sources . circles @xmath33 indicate those sources that have been selected as calibration sources . the position of the target , asteroid 939 isberga is indicated by a square ( @xmath34 ) . ] on a single - cpu pentium - class system the correlation stage for a time series of 50 images takes only about 1 min . heidi has been designed to facilitate the production of photometric light curves for solar system objects ( target sources ) . following the selection of calibration sources as described in [ section : correlstar ] , the target sources are identified and selected . heidi uses the institut de mcanique cleste et de calcul des phmrides ( imcce ) sky body tracker ( skybot ) cone - search and resolver services @xcite to quickly identify the known solar system objects in the time series , our target sources . heidi processes the time series very quickly : for a set of 50 images it takes heidi only approximately 15 s to identify one target source , to test for its presence in the image set , to compile the list of the corresponding flux values and to produce the photometric light curve for this object , i.e. @xmath38 s per object per image . as a consequence we have implemented heidi to produce light curves for all of the objects found in the time series rather than just the primary target . the imcce skybot cone - search service identifies which solar system objects lie within a specified field of view at a given epoch : heidi calls the cone - search service and obtains a list of the objects that may be found in the time series by providing the centre coordinates in right ascension and declination and the epoch of observation . for each object returned by the cone - search service , heidi calls the imcce skybot resolver service to obtain accurate coordinates : heidi provides the name of the object and the epoch for the first image in the time series to the resolver service . if known , heidi also provides the international astronomical union ( iau ) observatory code for the site at which the images were recorded . the resolver service then returns accurate coordinates for the specified object . if the iau observatory code is provided then the resolver service returns coordinates in the topocentric frame , otherwise the coordinates are given for the geocentric frame . to determine the object coordinates in the remaining images of the time series , heidi finds the record for each object in the iau minor planet center orbit ( mpcorb ) database ( http://www.minorplanetcenter.net ) . the coordinates for each object are computed from the orbital elements in the mpcorb database on a per - image basis across the time series . these computations are much faster than obtaining coordinates from the resolver service . it takes only a fraction of a second to calculate the coordinates for an object for all images in the time series , compared with several seconds per image when retrieved from the resolver service . however , due to rounding errors in the computation , the coordinates provided by the resolver service may be more accurate than the ones computed by heidi . heidi will thus determine the difference between the coordinates provided by the resolver service and the ones computed locally and apply this offset as a correction to all computed coordinates across all images . the respective coordinates in each image of the time series are then tested to determine the presence of the target sources . to minimise false detections without introducing false non - detections , this is done in a two - step process : in a first step , an area of 10 x 10 pixels around the coordinates of the target is tested for the presence of a source ; if no source is found , the area is expanded to 15 x 15 pixels and the test repeated . it is not possible to determine whether a failure to detect a target source is due to the faintness of the object or an error in the position calculation . we thus assume that the primary target for the measurement and production of a light curve has a well - known orbit and also a good signal - to - noise ratio in the time series . if the object is detected in the images of the time series , the log flux of the object is recorded for each image . plotting the log flux of the object and the log flux of the calibration sources without correcting for the variations between images in the time series does not give a clear indication of the light curve of an object . figure [ fig : figure3 ] shows the uncorrected log flux measurements from calibration sources and the target in a time series for asteroid 939 isberga . uncorrected log flux measurements from calibration sources @xmath39 and target @xmath40 in the time series for asteroid 939 isberga . the calibration sources can clearly be seen to vary in accordance with each other . the target , asteroid 939 isberga , follows a similar trend . the light curve is not readily apparent . ] the log flux mean of the calibration sources identified in [ section : correlstar ] is calculated for each image ( similar to equation [ eqn : mean_log_flux ] but this time over the entire candidate source set ) : @xmath41 the global log flux mean is also determined : @xmath42 this is used for reference and correction of variations across the time series . for each image @xmath3 in the time series , the log flux values of the calibration sources are adjusted by adding the difference between the global log flux mean and the log flux mean for that image . for all @xmath3 , @xmath4 the difference between the global log flux mean @xmath43 and the log flux mean @xmath44 is calculated : @xmath45 for each image @xmath3 in the time series , the log flux value of the target object @xmath46 , @xmath3 ( where @xmath47 denotes the target source ) is adjusted by the same correction factor @xmath48 . to smooth out the background variations across the image set , the log flux values @xmath46 , @xmath3 for the target object @xmath47 are adjusted : @xmath49 applying this correction to the time series for asteroid 939 isberga shows a light curve that is readily apparent ( figure [ fig : figure4 ] ) . this light curve for asteroid 939 isberga produced by heidi clearly shows a primary period consistent with the expected rotation period of 2.9173 h ( 0.12155 d ) @xcite . corrected log flux measurements for calibration sources @xmath39 and target @xmath40 in the time series for asteroid 939 isberga . the variation in the calibration sources is minimal @xmath50 . the light curve of the target is clearly apparent . ] the consistency of the results achieved by heidi is further demonstrated by extending the analysis to multiple time series for asteroid 939 isberga that were obtained on three separate dates . the light curves determined by heidi for these three time series are shown superimposed in figure [ fig : figure5 ] . it can be easily seen that the three curves follow a very similar pattern . the light curves produced by heidi for asteroid 939 isberga using time series that were obtained on three separate dates show the consistency of the results . it can be easily seen that the three curves follow a very similar pattern . a primary periodicity consistent with the expected rotation period of 2.9173 h is evident . ] these curves can also be overlaid on one another to show that both the period and the amplitude are consistent ( figure [ fig : figure6 ] ) . light curves produced by heidi for asteroid 939 isberga from time series obtained on three separate dates ( 22 , 23 and 25 november 2011 ) overlaid on one another to show that both the period and the amplitude of each curve is consistent with the period and amplitude of the other curves . ] the current version ( as of november 2013 ) has good astrometric accuracy but performs only relative photometry , i.e. does not produce calibrated magnitudes . we plan to add the capability for the production of light curves calibrated to a standard photometric system . the preprocessing tasks of astrometric calibration and image alignment described in [ section : fitsalign ] are independent of the type of object being studied . the candidate sources that are rejected by the filtering process described in [ section : correlstar ] warrant further study to determine whether they were excluded due to some intrinsic variability . these excluded sources can include non - moving sources that have characteristic light curves , e.g. variable stars , supernovae and gamma - ray burst optical afterglows . in the next version of heidi , we will include the ability to also produce photometric light curves for all the excluded objects as a standard function . uncertainties on the produced light curves can be inferred from the dispersion of the magnitudes of the calibration sources and will be used to include error bars in the output . if the target source is near , i.e. within 2 to 3 arcsec , a star of similar or greater brightness , it is possible that the star is selected instead of the target source . the next version of heidi will include a step to automatically exclude detections in which the position information falls outside a predetermined tolerance . in addition , an algorithm to detect moving objects will be implemented for targets with less accurate position information or unknown targets . the high efficiency image detection & identification ( heidi ) pipeline software rapidly processes sets of astronomical images . it is very fast and reliable . heidi takes a set of images from a night of observing and produces a light curve in a matter of minutes . heidi is implemented in the programming language python . it has been tested on various linux systems and linux virtual machines . it has minimal hardware requirements and the installation process is relatively straightforward . heidi runs in virtual machines without impacting performance and can thus be installed on non - linux systems ( if a linux virtual machine is available ) . the high efficiency image detection & identification ( heidi ) pipeline software we developed has a small footprint yet is very powerful and extremely versatile . given programmes like the prompt follow - up of gamma - ray burst alerts for the detection of transient optical emissions and the gaia follow - up network for solar system objects , which require prompt follow - up and rapid analysis of time series data , heidi will be very useful and provide a service and a speed that have not been available . testing heidi on multiple fairly large time series for asteroid 939 isberga yielded extremely satisfying results . the light curves produced using time series obtained on three separate dates showed consistency with the three curves following a very similar pattern . heidi will be installed at the zadko telescope in gingin , western australia . this will allow us to analyse very rapidly all the images in the data archive of the zadko telescope and search for interesting variable sources . it will also allow prompt analysis and rapid issuing of results for time critical events such as gamma - ray burst optical afterglow emissions . the heidi package is available from the first author on request . mt thanks frdric vachier for assistance with photometry . mt thanks andrew williams for assistance with processing fits image files . the work reported on in this publication has been supported by the european science foundation ( esf ) , in the framework of the great research networking programme . dmc is supported by an australian research council future fellowship . berthier , j. , vachier , f. , thuillot , w. , fernique , p. , ochsenbein , f. , genova , f. , lainey , v. , arlot , j .- e . , 2006 , astronomical data analysis software and systems xv , 351 , 367 bertin , e. , arnouts , s. , 1996 , a&as , 117 , 393 bertin , e. , mellier , y. , radovich , m. , missonnier , g. , didelon , p. , morin , b. , 2002 , astronomical data analysis software and systems xi , 281 , 228 bertin , e. , 2006 , astronomical data analysis software and systems xv , 351 , 112 coward , d. m. , et al . , 2010 , pasa , 27 , 331 klotz , a. , bor , m. , eysseric , j. , damerdji , y. , laas - bourez , m. , pollas , c. , vachier , f. , 2008 , pasp , 120 , 1298 lang , d. , hogg , d. w. , mierle , k. , blanton , m. , roweis , s. , 2010 , aj , 139 , 1782 molnar , l. a. , et al . , 2008 , minor planet bulletin , 35 , 9 todd , m. , coward , d. m. , tanga , p. , thuillot , w. , 2013 , pasa , 30 , 14

the production of photometric light curves from astronomical images is a very time - consuming task , taking several hours or even days . larger data sets improve the resolution of the light curve , however , the time requirement scales with data volume . the data analysis is often made more difficult by factors such as a lack of suitable calibration sources and the need to correct for variations in observing conditions from one image to another . often these variations are unpredictable and corrections are based on experience and intuition . the high efficiency image detection & identification ( heidi ) pipeline software rapidly processes sets of astronomical images , taking only a few minutes . heidi automatically selects multiple sources for calibrating the images using a selection algorithm that provides a reliable means of correcting for variations between images in a time series . the algorithm takes into account that some sources may intrinsically vary on short time scales and excludes these from being used as calibration sources . heidi processes a set of images from an entire night of observation , analyses the variations in brightness of the target objects and produces a light curve all in a matter of minutes . heidi has been tested on three different time series of 50 images each of asteroid 939 isberga and has produced consistent high quality photometric light curves in a fraction of the usual processing time . the software can also be used for other transient sources , e.g. gamma - ray burst optical afterglows , gaia transient candidates . heidi is implemented in the programming language python and processes time series astronomical images in fits format with minimal user interaction . heidi processes up to 1000 images per run in the standard configuration . this limit can be easily increased , with the only real limit being system capacity , e.g. disk space , memory . heidi is not telescope - dependent and will process images even in the case that no telescope specifications are provided . heidi has been tested on various linux systems and linux virtual machines . heidi is very portable and extremely versatile with minimal hardware requirements . [ firstpage ] methods : analytical methods : data analysis methods : numerical techniques : image processing techniques : photometric astrometry

You are an expert at summarizing long articles. Proceed to summarize the following text: as is well known , the greisen - zatsepin - kuzmin ( gzk ) cutoff constrains detected uhecrs to have been produced in or near the galaxy . specifically , detected proton primaries with energies exceeding @xmath0 ev must have been produced within 50 mpc of earth , and nuclei propagation distances are constrained even further @xcite . the near - isotropy of detected uhecr arrival directions , however , suggests that galactic source locations are not easily associated with the observed arrival directions . we attempt to reconcile these observations by examining the possibility that uhecrs are galactic in origin , but consist of iron nuclei primaries with trajectories influenced by the galactic magnetic field . it has been shown by @xcite that mhd winds from young neutron stars are capable of galactic production of iron primaries that fit the observed uhecr energy spectrum . such a production mechanism implies that most source locations should be found within the galactic disk . with this in mind , we propagate iron nuclei uhecrs through a realistic model of the galactic magnetic field to investigate the possibility that galactic sources and isotropic arrival directions are not mutually exclusive phenomena . following @xcite and @xcite , we adopt a large - scale regular galactic magnetic field associated with the spiral arms of the galaxy . specifically , we choose a bisymmetric even - parity field model ( bss - s ) in which the field reverses direction between different spiral arms , but is symmetric with respect to the galactic plane . the field strength in the plane , directed along the spiral arms , at a point @xmath1 in galactocentric coordinates is given by @xmath2 with @xmath3 kpc as the galactocentric distance to maximum field strength at @xmath4 with @xmath5 , where the pitch angle is @xmath6 . the radial dependence of the field strength is given by @xmath7 where @xmath8 kpc is the galactocentric distance to the sun and @xmath9 kpc is a smoothing factor that allows for @xmath10 behavior beyond 4 kpc from the galactic center . the field equation in the galactic plane is given by @xmath11\ ] ] for this even - parity model , we introduce a @xmath12 dependence of the following form @xmath13 where the values @xmath14 kpc and @xmath15 kpc reflect the scale heights of the field in the galactic disk and halo , respectively . figure ( 1 ) gives a graphical representation of this regular field model , clearly illustrating the field association with the spiral arms . note that this model generates field values ranging approximately from @xmath16 g , with a value of @xmath17 g in the solar neighborhood . we develop two distinct approaches to the simulation of galactic uhecr trajectories . in our first method , we model a distribution of galactic sources and emit uhecrs according to an assumed @xmath18 emission energy spectrum in order to study the arrival energy spectrum at a detector . this allows for comparison between the injected and observed energy spectrum . our second approach consists of modeling the antiparticle trajectories as they depart earth , given the detected uhecr arrival spectrum . this is equivalent to plotting particle trajectories that are guaranteed to intersect earth , and this method has the advantage of allowing us to investigate potential source locations corresponding to real observed arrival directions . our simulation of uhecr particle trajectories proceeds from a reasonable distribution of neutron stars within the galactic disk to serve as injection regions for uhecrs . for simplicity , we assume that all sources are located in the galactic disk of radius 25 kpc and thickness .65 kpc ( taken from the scale height of the thin disk ) . we can derive a reasonable number of distinct sources from @xmath19 where @xmath20 myr is the program integration time and @xmath21 is the neutron star birth rate . these values produce @xmath22 distinct uhecr sources . given this set of sources , we assign a random time @xmath23 at which each source emits randomly directed uhecrs according to the @xmath24 energy spectrum , as is expected from neutron star sources @xcite . the particles then propagate through the galaxy with paths influenced by the lorentz force as they traverse the galactic magnetic field . since an earth - sized detector is too small to detect a significant number of events with computationally reasonable numbers of injections , we develop a series of larger detectors . the first is a 2d galactocentric cylindrical detector of radius @xmath8 kpc , spanning 20 - 40 pc above the galactic plane . the second detector is a 2d galactocentric ring located 30 pc above the galactic plane with an inner radius of 8.49 kpc and an outer radius of 8.51 kpc . these 2d detectors , placed at the solar distance from the galactic center , possess a number of advantages over local 3d detectors by generating a much higher detection flux while simultaneously reducing the bias for detection of local sources . with these detectors defined , we sample uhecr energy spectra to compare the detected energy spectrum with the assigned @xmath24 injection energy spectrum . to accomplish this task , we inject millions of particles from our 300,000 sources and plot the detected energy spectrum . figure ( 2 ) shows a typical detection energy spectrum from such a simulation . since the energy emission spectrum is continuous , we bin the detected data before fitting it to a function of the form @xmath25 . for the data shown , we calculate @xmath26 . thus , we find that the emission and arrival spectra are not significantly different for uhecrs . this is an important result since it is unknown _ a priori _ if the detection energy spectrum should reflect the emission spectrum when particles have the potential to be trapped in the galactic field . , matching the emission spectrum.,width=321,height=321 ] this simulation addresses the other end of the uhecr problem , propagating anti - iron nuclei from a distribution of arrival directions at earth . in this approach , we assign energies to the emitted antiparticles and propagate them through the galaxy to determine possible source locations for the corresponding uhecr particles . figure ( 3 ) shows one such sample set of trajectories for the case of isotropically distributed 50 eev iron nuclei . this method guarantees that our trajectories intersect earth , but we are now left to evaluate source location without an initial neutron star distribution . keeping in mind the need for galactic neutron stars as sources , we define a potential uhecr source location as any point along an antiparticle trajectory that intersects the galactic disk . furthermore , we closely examine those antiparticle trajectories for which the path first leaves and then reenters the disk . these particular paths are the least likely to be easily identified with their galactic sources , and their abundance would lend support to galactic uhecr origins . we proceed by first emitting a known distribution of anti - iron nuclei from earth . specifically , we choose to trace back agasa events with detected energies that exceed 100 eev . then , using the agasa energies and arrival directions , we send the particles back through the galactic field . since most of the detected high - energy agasa events are observed to have arrived away from the plane of the disk , we do nt have to worry about the exact difference between arrival direction and source location . any path that manages to reintersect the disk some time after emission will have deviated enough to be of interest to us . figure ( 4 ) shows the set of trajectories modeled from those agasa events above 100 eev . it is clear that there is little turning back towards the galactic disk once the particles have exited . the only path that remains in the galactic disk for an extended period of time corresponds to the agasa particle detected nearest the disk . with our choice of field parameters , the agasa events do not clearly point back to sources within the galactic disk . galactic field.,width=321,height=321 ] needless to say , the results of both types of simulation exhibit great sensitivity to certain parameters of the regular magnetic field model . specifically , we find that variations in field strength strongly affect possible source locations for detected uhecrs . figure ( 5 ) shows a contrast between the source location distributions for local field values of 3 and 6 @xmath27 . the 3 @xmath27 field results in an abundance of detections involving local sources , while the 6 @xmath27 case exhibits pronounced non - local galactic features in the detection spectrum . furthermore , differences in the field strength do greatly alter our agasa trajectories . figure ( 6 ) shows the trajectories produced from agasa information with a 6 @xmath27 galactic field . clearly , our choice of field parameters is important . in addition to the regular galactic field used in our models , there are a number of additional magnetic field components that may be present in the galaxy . in particular , a galactic wind similar in nature to the solar wind can significantly alter uhecr trajectories if the wind field strength is comparable to the regular field strength . such a wind model with an azimuthal field strength of 7 @xmath28 has been proposed to redirect the highest energy particles such that a single source in virgo can be detected as an isotropic distribution on earth @xcite . this strong wind effectively funnels isotropic arrival directions to the north galactic pole for a narrow range of energies . for iron nuclei , this strong version of the galactic wind guarantees diffusive behavior up to the highest energies detected thus far . local high - field regions within the galaxy also have the potential to contribute to the diffusion of uhecrs . molecular clouds , for instance , are typically associated with mg fields that could serve as scattering regions for cosmic rays . our preliminary research has indicated that the known distribution of molecular clouds has a small enough filling factor to cause little change in the paths of most uhecrs , but these and other magnetic field inhomogeneities must be better understood before the picture of uhecr propagation can be made complete .

we consider the effects of the galactic magnetic field on the propagation of ultra high energy cosmic rays ( uhecrs ) . by employing two methods of trajectory simulation , we investigate the possibility that uhecrs are produced within the galaxy with paths strongly influenced by the galactic magnetic field . such trajectories have the potential to reconcile the existing conflict between proposed local sources and isotropic uhecr arrival directions .

You are an expert at summarizing long articles. Proceed to summarize the following text: the theory of error - correcting codes for classical information has been extensively studied for almost fifty years . a fundamental question in coding theory concerns what _ capacity _ of information can be successfully transmitted through a noisy channel . call a code that maps @xmath3-bit inputs into @xmath4-bit codewords an @xmath5 code , and define its capacity to be @xmath6 . for any specific @xmath4-bit channel , let its _ capacity _ be the maximum capacity of all @xmath5 codes that successfully transmit information through it . this capacity is often taken as an asymptotic limit as @xmath4 tends towards infinity . we consider two basic kinds of @xmath4-bit noisy channels . the first is one that flips any subset of up to @xmath7 bits of each codeword that passes through it . in this model , transmission through the channel is considered successful if the @xmath3 bits of data can always be perfectly recovered . call a code that achieves this a _ @xmath7-error - correcting code_. asymptotically , it is natural to take @xmath7 as some fixed fraction @xmath0 of @xmath4 , written as @xmath8 and understood to mean @xmath9 . a second model of a noisy channel is one where each bit of each codeword that passes through it is flipped independently with probability @xmath0 . in this model , commonly referred to as the _ binary symmetric channel _ , there is no absolute bound on the number of possible errors that occur . therefore , a probabilistic definition of successful transmission is required . call a code for which the probability of successful recovery for any @xmath3 bits of data is at least @xmath10 an _ @xmath11-error - correcting code_. the subject of error - correcting codes for _ quantum _ information is much younger , developing within the past couple of years , though it has received considerable attention during this time @xcite . much of the above terminology extends naturally to quantum information by considering qubits instead of bits . call a quantum code that maps @xmath3-qubit data to @xmath4-qubit codewords an @xmath12 code . we need to specify the behavior of noisy quantum channels . a natural quantum analogue of the first model is to allow any @xmath7 qubits of each codeword that passes through it to be altered . we can take this to mean : apply an arbitrary unitary transformation to all the qubits selected for alteration . an apparently stronger definition allows the unitary transformation to also involve another set of qubits , representing an `` external environment '' , thereby simulating the effect of `` decoherence '' . an apparently weaker definition limits the unitary operations to being among : @xmath13 , @xmath14 , and @xmath15 , the standard pauli spin matrices , and @xmath16 , the unit matrix . it turns out that , by reasoning similar to that in @xcite , these three definitions of `` alter '' can be shown to be equivalent , in the sense that a code that is _ @xmath7-error - correcting _ with respect to the apparently weaker one will automatically be @xmath7-error - correcting with respect to the apparently stronger one . a quantum analog of the binary symmetric channel is the _ depolarizing channel _ , where each bit of each codeword that passes through it is independently subjected to : @xmath16 with probability @xmath17 , and @xmath13 , @xmath14 , @xmath15 each with probability @xmath18 . call a code that achieves a fidelity of at least @xmath10 on such a channel an _ @xmath11-error - correcting code_. it has previously been shown @xcite how to take some special classical linear codes ( binary and over @xmath19 ) with certain properties and transform them into quantum codes with error - correcting capabilities . in the present paper , we show how to transform quantum codes with certain properties into classical codes with error - correcting capabilities . we do not propose this as a means for constructing new classical codes ; rather , this is a means for translating existing proofs of the _ non_existence of certain classical codes into new proofs of the nonexistence of certain quantum codes . our specific results , which apply to the class of _ stabilizer _ quantum codes ( defined in the next section ) , are : by these results , we can immediately assert that , when we restrict our attention to stabilizer codes , the classical upper bounds in @xcite apply . in particular , when a quantum channel is subject to @xmath8 errors , the asymptotic capacity is bounded above by @xmath20 , where @xmath21 is the binary entropy function defined as @xmath22 . in fact , a slightly stronger but more complicated upper bound is proven in @xcite . this stronger bound is plotted in fig . 1 . this upper bound is stronger than the previously established @xmath23 bound in @xcite , though the latter bound has the advantage that it applies to nonstabilizer codes as well . it is noteworthy that all of the quantum codes proposed to date for the channels described above are stabilizer codes . other upper bounds exist for _ nondegenerate _ quantum codes ( see @xcite for a definition of nondegenerate ) . one is @xmath24 , and is based on an analogue of the classical `` sphere packing bound '' @xcite , and another asserts that the asymptotic capacity is zero if @xmath25 @xcite . it remains an open question whether the @xmath26 bound also applies to degenerate codes . very recently , it has been announced that the @xmath25 threshold bound _ does _ extend to nondegenerate codes , and this will be appear in a forthcoming paper @xcite . it is interesting to note that there exist some degenerate stabilizer codes that outperform all known nondegenerate codes on the depolarizing channel , for some values of @xmath0 @xcite . the best lower bound for this channel that we are aware of is @xmath27 @xcite . for the depolarizing channel with error probability @xmath0 , our results imply that , for stabilizer codes , the capacity is upper bounded by @xmath2 , the bound for the classical binary symmetric channel @xcite . for some values of @xmath0 , this is stronger than the previously established upper bound of @xmath23 @xcite , though the latter bound applies to nonstabilizer codes as well . the best lower bound that we are aware of for this channel is @xmath26 @xcite , and a slightly larger value for some values of @xmath0 @xcite . should any improvements to the upper bounds in @xcite for classical coding occur , they will automatically apply to quantum stabilizer codes . our results demonstrate interesting connections between quantum stabilizer codes and classical linear codes , and , for some instances of channels , yield stronger upper bounds than those that have appeared to date . in sections ii and iii , we provide a brief overview of quantum stabilizer codes and classical linear codes . in section iv , we describe how to construct a binary linear code from a quantum stabilizer code , and , in section v , we show that this construction yields the error - correcting properties required for theorems 1 and 2 . in @xcite it has been shown that many quantum codes can be described in terms of _ stabilizers_. define a stabilizer as a set of @xmath4-qubit unitary operators such that : each operator is a tensor product of @xmath4 matrices of the form @xmath13 , @xmath14 , @xmath15 , and @xmath16 , with a global phase factor of @xmath28 ; and , the set of operators is an abelian group . the code that is defined by a stabilizer is the set of all @xmath4-qubit quantum states that are fixed points of each element of the stabilizer . a stabilizer can be most easily described by a set of operators that generate it . if one negates the phase factor of some of the generators , the resulting code will change , but will have identical characteristics to the original code . thus , one can always take the phase of each generator to be @xmath29 without any loss of generality . it is convenient to denote the generators of a stabilizer in the language of binary vector spaces , as in @xcite . denote the generator @xmath30 as the @xmath31 bit vector @xmath32 , where , for @xmath33 , @xmath34 and @xmath35 for example , @xmath36 is denoted as latexmath:[$(\ , 1 \ 0 \ 1 \ 0 \ 0 \ 1 \ 0 \ 1 \ , of any two generators @xmath32 and @xmath38 is equivalent ( modulo a phase factor of @xmath39 ) to @xmath40 , where @xmath41 denotes the bit - wise sum in modulo two arithmetic . also , @xmath32 and @xmath38 commute if and only if @xmath42 where @xmath43 denotes the inner product in modulo two arithmetic . a stabilizer can then be written as an @xmath44 matrix whose rows represent the generators . for example , @xmath45 represents the eight generators of the stabilizer of a specific @xmath46 code that is 1-error - correcting ( see @xcite for a detailed analysis of this code ) . in general , if there are @xmath4 qubits and @xmath47 generators , we can encode @xmath48 data qubits ( in the above example , @xmath49 ) . the error correcting capabilities of the code are related to commutativity relationships between the error operators and the generators @xcite . an @xmath5 binary linear code is a @xmath3 dimensional subspace of @xmath50 over modulo two arithmetic . it is sufficient to specify a basis @xmath51 for such a code . then the codeword for the @xmath3-bit string @xmath52 can be taken as the linear combination @xmath53 ( this mapping is a bijection between @xmath54 and the code ) . a natural way of specifying such a code is by an @xmath55 _ generator matrix _ , whose rows are @xmath51 . an example of such a code is @xmath56 which is a @xmath57 code that is 1-error - correcting . consider a quantum stabilizer code specified by a @xmath58 matrix @xmath59 . we shall show how to construct the generator matrix of a classical binary linear code with similar error - correcting capabilities . our construction involves a transformation of the generator matrix into a useful standard form along the lines of that in @xcite . this conversion is accomplished by applying a series of basic transformations of the following two types , each of which leaves the error - correcting characteristics of the code unchanged . the first is a _ row addition _ , where the @xmath60row is added to the @xmath61row , where @xmath62 . this corresponds to replacing the @xmath61generator with the product of the @xmath61and @xmath60generator , and setting its phase to @xmath29 . the second is a _ column transposition _ , where @xmath61column is transposed with the @xmath60column in the submatrices @xmath63 and @xmath64 simultaneously . this corresponds to transposing the @xmath61qubit position with the @xmath60qubit position . we begin by applying transformations of the above types to the matrix @xmath59 in order to obtain @xmath65 where @xmath66 is the rank of the submatrix @xmath63 , and @xmath67 . this is like performing gaussian elimination on the @xmath63 submatrix . next , by performing row additions among the last @xmath68 generators and column transpositions among the last @xmath69 qubit positions , the matrix can be further converted to the form @xmath70 where @xmath71 is the rank of @xmath72 , @xmath73 , and @xmath74 . this is like performing gaussian elimination on the submatrix @xmath72 of ( [ stand1 ] ) . note that if @xmath75 and @xmath76 and then one of the last @xmath77 generators would not commute with one of the first @xmath66 generators . therefore , we can set @xmath78 , @xmath79 , and @xmath80 . thus , the form ( [ stand2 ] ) becomes @xmath81 where @xmath82 . call any set of generators in this form ( [ standard ] ) _ in standard form_. for a generator matrix in standard form , consider the classical binary linear code generated by the @xmath83 matrix @xmath84 we claim that this classical code has similar characteristics to the original quantum code . before stating this precisely , consider as an example the aforementioned @xmath46 code , that corrects one error , whose stabilizer was given by ( [ eight ] ) . converting it to standard form yields @xmath85 which is an equivalent @xmath46 code . the resulting binary linear code is @xmath86 which is a well - known @xmath87 code that corrects one error . thus , in this example , we obtain a classical code with slightly better capacity . the reader may recall that previous constructions of @xmath88 quantum codes have been based on the same classical code ( [ seven ] ) @xcite . it should be noted that the present connection is quite different , as it involves an @xmath46 quantum code . in the next section we shall show that , for any @xmath7-error - correcting @xmath12 quantum stabilizer code that is in standard form ( [ standard ] ) , the matrix ( [ classical ] ) generates a @xmath7-error - correcting @xmath89 binary linear code . also , for the case of the depolarizing channel , we shall show that , for any @xmath11-error - correcting @xmath12 quantum stabilizer code in standard form ( [ standard ] ) , the matrix ( [ classical ] ) generates an @xmath11-error - correcting @xmath89 binary linear code . furthermore , by applying operations of the types below , which also do not affect the error - correcting capabilities of the code @xcite , we can guarantee the additional property that @xmath90 , which slightly sharpens the result . the first operation is the _ column switch _ , in which the @xmath61column of submatrix @xmath63 is transposed with the @xmath61column of submatrix @xmath64 . this corresponds to : in the @xmath61qubit position , changing each instance of a @xmath13 in each generator to a @xmath15 , and each instance of a @xmath15 to a @xmath13 ( while leaving each @xmath16 and @xmath14 intact ) . the second operation is the _ column addition _ , in which the @xmath61 column of submatrix @xmath64 is added to the @xmath61column of submatrix @xmath63 . this corresponds to : in the @xmath61qubit position , changing each instance of a @xmath14 in each generator to a @xmath15 , and each instance of a @xmath15 to a @xmath14 ( while leaving each @xmath16 and @xmath13 intact ) . see @xcite for an explanation of why these two operations do not affect the characteristics of the code . in this section , we show that the constructions of the previous section satisfy the claimed error - correcting properties . the classical code whose generator matrix is given by ( [ classical ] ) consists of @xmath91 codewords in the space @xmath92 . we shall construct an _ isomorphism _ between this code and a restricted version of the quantum code specified by ( [ standard ] ) . the restricted version of the quantum code consists of @xmath91 codewords that are contained in a special set @xmath93 of @xmath94 distinct @xmath4-qubit states . this set @xmath93 has the property that it is closed with respect to @xmath15 errors among the first @xmath95 qubit positions . intuitively , the effect of bit errors on classical codewords within the space @xmath92 is equivalent to the effect of @xmath15 errors in the first @xmath95 qubit positions on quantum codewords within the space @xmath93 . formally , the isomorphism that we shall construct is a mapping @xmath96 , such that : 1 . @xmath97 is bijective . 2 . for each @xmath98 that is a codeword of the classical code , @xmath99 is a codeword of the quantum code . 3 . for each codeword @xmath98 of the classical code , and each error vector @xmath100 , @xmath101 the existence of such an isomorphism means that an error in the @xmath61bit of the classical code ( for any @xmath102 ) corresponds to a @xmath15 error in the @xmath61qubit of the restricted version of the quantum code . more precisely , if the quantum code can correct any @xmath7 errors then it can correct any @xmath7 @xmath15 errors among the first @xmath95 qubit positions , and then the following procedure for correcting any @xmath7 errors in the classical code exists . given a codeword @xmath103 subjected to an error vector @xmath104 of weight bounded by @xmath7 , first apply the mapping @xmath97 to it . by the second and third properties of @xmath97 , the result is @xmath99 subjected to at most @xmath7 @xmath15 errors among the first @xmath95 qubit positions , which can therefore be corrected . by the first property , @xmath105 can be applied to this corrected quantum codeword , yielding the correction of the original codeword . therefore , if we establish that there exists a @xmath97 that satisfies the above three properties then the classical code specified by ( [ classical ] ) must correct at least as many errors as the quantum code specified by ( [ standard ] ) . for the case of the depolarizing channel with parameter @xmath0 , if the quantum code attains fidelity @xmath10 then it attains fidelity @xmath10 for a channel that applies @xmath15 in each of the first @xmath95 qubit positions independently with probability @xmath0 ( in fact we may need to slightly modify the code by applying some column switch and column addition operations defined in section iv along the lines of the `` twirling '' techniques explained in @xcite ) . therefore , the corresponding classical code is correcting with probability at least @xmath10 on a binary symmetric channel with parameter @xmath0 . thus , the existence of the above @xmath97 also suffices for this noisy channel model . in order to construct a bijection @xmath97 with the above properties , we shall construct a useful basis for the quantum code . we begin with the stabilizer specified by the matrix in standard form ( [ standard ] ) . call the operators corresponding to the respective rows of this matrix @xmath106 . define the additional operators @xmath107 in terms of the matrix @xmath108 where @xmath109 , and @xmath110 in terms of the matrix @xmath111 by considering ( [ standard ] ) , ( [ phase ] ) , ( [ bit ] ) , and recalling the criterion for commutativity ( [ commute ] ) , it is straightforward to verify that : * @xmath112 is a set of @xmath4 independent commuting operators . * @xmath113 is a set of @xmath4 independent commuting operators . * each @xmath114 and @xmath115 commute if @xmath116 and anticommute if @xmath117 . using these properties , we can construct a basis @xmath118 for the code with some useful structural features . first , set @xmath119 to be the quantum state stabilized by @xmath112 ( this state is unique up to a global phase factor ) . in fact , @xmath120 is a quantum state with this property . next , for each @xmath121 , set @xmath122 . since @xmath114 commutes with @xmath115 if and only if @xmath117 , @xmath123 is in the + 1-eigenspace of each @xmath106 and the @xmath124 eigenspaces of @xmath107 , respectively . therefore , these states are an orthogonal basis for the quantum code . now , define the function @xmath96 as @xmath125 for each @xmath98 . we shall show that @xmath97 satisfies the three required properties . by considering ( [ standard ] ) and ( [ phase ] ) , the operator that applies @xmath15 to the @xmath61qubit ( and @xmath16 to all other qubits ) anticommutes with the @xmath61generator in the sequence @xmath126 and commutes with all others . therefore , @xmath99 is in the @xmath127 eigenspaces of @xmath126 , respectively ( recall that @xmath128 ) . thus , @xmath99 is orthogonal for each distinct @xmath103 . this proves the first property , that @xmath97 is a bijection . also , due to the close similarity between ( [ classical ] ) and ( [ bit ] ) , @xmath129 so the second property for @xmath93 holds . finally , the third property for @xmath93 holds because , using ( [ phi ] ) and ( [ similar ] ) , @xmath130 thus , @xmath97 satisfies the three required properties . i am very grateful to hans - benjamin braun for help in analyzing and plotting the functions in @xcite , david divincenzo for several interesting discussions about quantum coding theory and comments about an earlier draft of this paper , emanuel knill for providing references to existing bounds for classical codes , and juan paz for interesting discussions about stabilizer representations of codes . i am also grateful for the hospitality of the program on quantum computers and quantum coherence at the institute for theoretical physics , university of california at santa barbara , where this work was completed . this research was supported in part by nserc of canada and the u.s . national science foundation under grant no . phy94 - 07194 .

we show that within any quantum stabilizer code there lurks a classical binary linear code with similar error - correcting capabilities , thereby demonstrating new connections between quantum codes and classical codes . using this result which applies to degenerate as well as nondegenerate codes previously established necessary conditions for classical linear codes can be easily translated into necessary conditions for quantum stabilizer codes . examples of specific consequences are : for a quantum channel subject to a @xmath0-fraction of errors , the best asymptotic capacity attainable by any stabilizer code can not exceed @xmath1 ; and , for the depolarizing channel with fidelity parameter @xmath0 , the best asymptotic capacity attainable by any stabilizer code can not exceed @xmath2 . # 1 # 1*#1 * 1\{0,1 }

You are an expert at summarizing long articles. Proceed to summarize the following text: it was proposed by efimov in 1970 that if two spinless neutral bosons interact resonantly then the addition of a third identical particle leads to the appearance of an infinite number of bound three - body energy levels @xcite . this occurs simultaneously with the divergence of the @xmath1-wave scattering length @xmath2 , associated with appearance of an additional zero - energy two - body bound state . hence highly exotic efimov states appear when there is a zero or near - zero energy two - body bound state . for a long time there was no clear signature of efimov states in any naturally occuring trimer system . efimov states are not possible in atomic systems due to the long range coulomb interaction , however it may exist in the system of spinless neutral atoms . even though the efimov effect was predicted four decades ago @xcite , evidence of its existence in ultracold caesium and potassium trimers has been experimentally established only very recently @xcite . however , these trimers are obtained by manipulating two - body forces through feshbach resonances and are not naturally occuring . therefore , it is of great interest to search for the efimov effect in a naturally occuring trimer , like @xmath3he trimer . so far no experimental confirmation has been reported . the near - zero energy ( @xmath4 1 mk ) bound state ( which is also the only bound state ) of @xmath0he dimer opens the possibility of the existence of an efimov - like state in @xmath0he trimer . several authors remarked that the @xmath0he trimer may be the most promising candidate . earlier theoretical calculations show that the trimer has a @xmath5 = 0 ground state at 126 mk and a near - zero energy excited state ( @xmath6 2mk ) @xcite . the excited state has been claimed to be an efimov state . a controversy arises from the fact that the number of efimov states is highly sensitive to the binding energy of the dimer and even a very small decrease of the strength of two - body interaction makes the system unbound . strikingly , it also disappears when the two - body interaction strength is _ increased_. however in contrast with theoretical investigations , no evidence of efimov trimer has been found experimentally @xcite . in the experiments , @xmath0he trimer has been observed in its ground state only . no experimental evidence of the excited state has been reported so far . + in principle @xmath0he trimer may be considered as a very simple three - body system consisting of three identical bosons . but its theoretical treatment is quite difficult . first , the he - dimer potential is not uniquely known . very small uncertainities in the dimer potential may lead to different conclusions . secondly , the strong short - range interatomic repulsion in the he - he interaction causes large numerical errors . as @xmath0he systems are strongly correlated due to large @xmath0he@xmath0he repulsion at short separation , the effect of interatomic correlation must be taken properly into account . + in the present communication , we revisit the problem using a correlated basis function known as potential harmonics ( ph ) basis which takes care of two - body correlations @xcite . in order to include the effect of highly repulsive he - he core , we multiply the ph basis with a suitable short - range correlation function which reproduces the correct short - range behavior of the dimer wavefunction . although this correlated ph basis ( cph basis ) correctly reproduces the dimer and trimer properties , we could not find any efimov like state in trimer with the actual dimer interaction @xcite . we point out that the calculation of such a near - zero energy excited state in the shallow and extended trimer potential may involve severe numerical errors and we may miss it . thus an alternative accurate procedure is desirable . here , we apply the supersymmetric isospectral formalism for an accurate treatment . for any given potential , families of strictly isospectral potentials , _ with very different shape _ but having desirable and adjustable features are generated by supersymmetric isospectral formalism @xcite . the near - zero energy bound state will be more effectively bound in the deep narrow well of the isospectral potential and will facilitate an easier and more accurate calculation of the near - zero energy excited state . following the steps of supersymmetric quantum mechanics @xcite , for any given potential @xmath7 , one can construct a class of potentials @xmath8 , where @xmath9 represents a set of one or more continuously variable real parameters . the potential @xmath10 is isospectral in the sense that @xmath11 and @xmath10 have identical spectrum , reflection and tranmission coefficients . for simplicity we consider only one parameter @xmath12 family of isospectral potentials . we will see later that @xmath13 can take real values @xmath14 and @xmath15 . for @xmath16 , one gets back the original potential . although the set of isospectral potentials are strictly isospectral with the original potential , they have different shapes depending on the parameter @xmath13 @xcite . + fig . 1 ( color online ) the effective potential @xmath7 ( red solid curve ) and isospectral potentials @xmath17 corresponding to two values of @xmath13 : @xmath18 ( green dashed curve ) and @xmath19 ( blue dotted curve ) for the @xmath3he trimer . all energies are in mk and @xmath20 in a.u . the horizontal line indicates the energy of the first excited state in @xmath7 . in fig . 1 , we demonstrate how an original potential , @xmath7 shown by the solid ( red ) curve , changes in the isospectral potential @xmath17 for two values of the parameter @xmath13 , _ viz . _ @xmath18 ( green dashed curve ) and @xmath19 ( blue dotted curve ) . we introduce this figure here for a qualitative understanding of the features of the isospectral potentials . a complete discussion of how such isospectral potentials are calculated will be presented in sections iic and iii . although all three potentials produce identical energy spectrum , their shapes are seen to be very different . the original potential [ @xmath7 ] has a shallow and wide well with a short range repulsion . by contrast , both the isospectral potentials have a deep and narrow _ attractive _ well ( naw ) at smaller @xmath20 , while the long range part does not differ much from @xmath7 . one can also notice that as @xmath13 decreases , the narrow attractive well becomes deeper , while the intermediate barrier becomes higher . hence , the near - zero energy ( indicated in fig . 1 by a horizontal line ) excited state in @xmath7 will be at the _ same energy value _ in the isospectral potentials , but now that state will lie _ deep within the naw_. thus , while this state is weakly bound and spatially extended in the original potential , it will be strongly bound and well localized in the naw of the isospectral potential , at the same energy . clearly , computation of the wave function and its energy ( equal to the energy of the first excited state in the original potential ) will be easier . furthermore , this state becomes more strongly bound , as @xmath13 decreases . in general , as @xmath13 decreases continuously from @xmath22 to @xmath23 , @xmath10 starts developing a local minimum which shifts towards @xmath20=0 and becomes deeper and narrower . consequently , a shallow potential transforms into one having a narrow and deep potential well near the origin in the isospectral potential . the surface barrier also becomes high . such interesting properties of @xmath10 can be useful to solve near - zero energy states . such a state lies near the top of the original potential well and its wave function is spatially very extended , while the isospectral state lies well within the naw . hence the latter is strongly bound and well localized within the narrow well of @xmath24 . the parameter @xmath13 controls these features and a suitable optimum value can be chosen ( see later ) . + thus our approach consists of two steps . first , to apply a correlated quantum many - body theory for a highly correlated system like @xmath0he - trimer and second , to use the isospectral formalism for the accurate determination of the first excited state , whose energy is just a few mk . + the paper is organized as follows . in section ii , we present a brief review of the correlated potential harmonics expansion method , choice of potential and the isospectral formalism . section iii presents the results of our numerical calculation . finally we draw our conclusions in section iv . for a realistic calculation , one needs an accurate he - he interaction potential . several sophisticated he - he potentials have been proposed @xcite . among these , the commonly used ones are : tang , tonnies and yiu ( tty ) @xcite , lm2m2 @xcite , and hfd - he2 @xcite potentials . these potentials reproduce all known two - body he - he data . in the present work , we select the more popular and sophisticated tty potential . this potential has the form @xcite @xmath71,\ ] ] where @xmath72 represents the interparticle distance . the part @xmath73 has the form @xmath74 = @xmath75 wih @xmath76 = @xmath77 . the other part @xmath78 is given as @xmath79 = @xmath80 . the coefficients @xmath81 are calculated using the recurrence relation @xmath81 = @xmath82 ; @xmath83 = 1.461 , @xmath84 = 14.11 , @xmath85 = 183.5 , @xmath86 = 315766.2067 @xmath65 , d= 7.449 and @xmath87 = 1.3443 @xmath88 . the function @xmath89 is given by @xmath90 = @xmath91 with @xmath92= @xmath93 - @xmath94 . + for our numerical solution , the set of cdes [ eq . ( 8) ] is solved by hyperspherical adiabatic approximation ( haa ) @xcite . in haa , one assumes that the hyperradial motion is slow compared to the hyperangular motion . the effective potential for the hyperradial motion ( obtained by diagonalizing the potential matrix together with the diagonal hypercentrifugal repulsion for a fixed value of @xmath20 ) is obtained as a parametric function of @xmath20 . we choose the lowest eigenpotential ( @xmath7 ) as the effective potential . thus in haa , energy and wavefunction are obtained approximately by solving a single uncoupled differential equation @xmath95 \zeta_{0}(r)=0,\ ] ] subject to appropriate boundary conditions on @xmath96 . the principal advantage of the present method is two - fold . firstly , the cph basis set correctly takes care of the effect of strong short range correlation produced by the he - he interaction . secondly , the use of haa basically reduces the multidimensional problem to an effective one dimensional problem introducing the effective potential . the effective potential @xmath97 gives a clear qualitative as well as quantitative picture . for our numerical calculation we investigate the @xmath98 state and truncate the cph basis to a maximum value @xmath65 = @xmath99 , requiring proper convergence . + the ground state properties of @xmath0he dimer is obtained as a numerical solution of two - body schrdinger equation by runga - kutta algorithm . the dimer energy @xmath100 using tty potential , as well as the results from other references are presented in table i. @xmath101 & present method & -1.254 + mk & dmc [ 23 ] & -1.243 + [ 27 ] & other [ 11 ] & -1.309 + & other [ 8 ] & -1.313 + although in our earlier calculation @xcite we reported the dimer and trimer ground state properties , we include them here for completeness of the discussion . calculated rms value of @xmath55 is @xmath102 a.u . the extremely small binding energy of the dimer and the large spatial extension of the ground state wave function imply that the ground state of helium dimer is very loosely bound . the trimer ground state energy , as well as the results obtained in earlier investigations by other authors are presented in table ii for different potentials . the r.m.s . value of hyperradius is 21.389 @xmath103 thus our correlated ph basis successfully reproduces the energy values which are in very close agreement with other sophisticated calculations @xcite . + dmc + tty & 125.51 & 126.40[24 ] & 126.4 [ 7 ] & - & 125.46 [ 25 ] + lm2m2 & 126.37 & 126.40 [ 25 ] & 126.4 [ 7 ] & 125.2 [ 6 ] & + hfdhe2 & 120.28 & & 117.1 [ 7 ] & 98.1 [ 5 ] & + although we have done detailed calculation of @xmath0he trimer ground state energy , we failed to obtain any trimer excited state with this choice of two - body interaction . very recently we have analyzed in details the behavior of @xmath0he trimer excited states as a function of pairwise interaction @xmath104 where @xmath105 controls the strength of two - body interaction . @xmath105 = 1 is the physical value of the dimer interaction and @xmath105=0 corresponds to free particle limit when neither two - body nor three - body bound states appear . we found that by increasing @xmath105 from a small value , the trimer starts to support an excited state at @xmath105 = 0.978 @xcite . binding energy of the excited state gradually increases with increase in @xmath105 , attains its maximum value at @xmath105 = 0.984 , then it decreases gradually and disappears which indicates dissociation to trimer fragments as a dimer and monomer . thus the disappearance of the first excited state due to both increasing and decreasing @xmath105 clearly show that the state is an efimov state . but the value of @xmath105 for which this happens , does not correspond to the actual physical dimer interaction ( @xmath106 ) . the efimov property of other excited states are also discussed in our earlier study @xcite . however we did not claim that the trimer excited state does not exist at all , as there is a large body of work in this direction @xcite . so this is still an ongoing issue as there is considerable controversy in the earlier discussions found in refs @xcite . thus our earlier work could not resolve the problem . it may be the limitation of our basis set which is unable to produce such an elusive state . so it needs further study . + for an accurate determination of the elusive near - zero energy excited state , we use the supersymmetric isospectral formalism mentioned earlier . note that this state lies near the top of the potential well @xmath7 and is very extended spatially . consequently , a convergent calculation requires a very large number of hyperspherical partial waves ( corresponding to very large @xmath99 ) . on the other hand , the trimer ground state lies near the bottom of this well , is well bound and localized ; it therefore converges relatively easily . now the isospectral potential @xmath107 with sufficiently small positive @xmath13 will have a deep well near the origin , followed by a high barrier ( see below ) . the near - zero energy state sought after is now strongly trapped within this narrow well , _ i.e. _ it is sharply localized . hence its convergence is achieved easily enough . the isospectral potential is obtained in terms of the trimer ground state wave function ( see below ) , which by the above argument is determined with a fair degree of confidence and it offers easier calculation of the excited state . + since isospectral formalism is not a common topic , we briefly explain how an isospectral potential is obtained for a given potential @xcite . for the given potential @xmath7 having a normalized ground state @xmath108 with energy @xmath109 , [ see eq . ( 11 ) ] , a superpotential @xmath110 is defined as @xmath111 then it can easily be seen that @xmath112 and one can define a supersymmetric partner potential @xmath113 through @xmath114 such that @xmath7 and its partner have the same energy spectra , except that the ground state of @xmath7 is absent in the spectrum of its partner @xcite . + now , for the given partner potential @xmath113 , eq . ( 15 ) can be considered as a non - linear differential equation satisfied by the unknown function @xmath110 ( called riccati equation ) . with this @xmath110 , we can get back @xmath7 , using eq . ( 14 ) . but solution of the non - linear eq . ( 15 ) is not unique . for simplicity , we use the units in which @xmath115 . then the most general solution is @xcite @xmath116 where @xmath13 is an integration constant and @xmath117 is a function of @xmath20 , parametrically dependent on @xmath13 . @xmath118 is given by @xmath119}^{2 } dr^{\prime}.\ ] ] then the family of potentials @xmath120 given by @xmath121 for all allowed values of @xmath13 ( see below ) , has the _ same partner _ @xmath113 . hence the family of potentials given by eq . ( [ isp ] ) all have identical spectrum . since @xmath108 is normalized , @xmath122 . hence from eq . ( [ mgsp ] ) , one notices that the interval @xmath123 is not allowed . for all other values of @xmath13 , @xmath17 is strictly isospectral with @xmath7 @xcite . for @xmath124 , @xmath17 becomes @xmath7 . for small positive values of @xmath13 , the isospectral potential @xmath17 develops an attractive well near the origin . as @xmath125 , this well becomes deeper followed by a high barrier , before the shallower part . note that the ground state of @xmath7 , _ viz . _ @xmath108 , is necessary to calculate the family of isospectral potentials . in the present study the ground state of @xmath7 is fairly accurately calculated . we will see in the next section that different eigen states are transformed differently . therefore this transformation can not be considered as a generalized rotation in the hilbert space . thus the transformation from the original hamiltonian to the isospectral hamiltonian is different from a standard unitary transformation , even though the entire energy spectrum is preserved . first we calculate the lowest effective potential @xmath7 in the hyperradial space by hyperspherical adiabatic approximation ( haa ) as stated in sec . iib . to calculate the energy @xmath126 and wave function @xmath108 of the original potential we solve the single uncoupled equation , eq . ( 11 ) , with appropriate boundary conditions . as stated earlier , although our calculated ground state energy @xmath109 and wave function are in good agreement with other calculations , we fail to get the first excited state in such a shallow potential . here the isospectral formalism will be an effective technique to calculate several isospectral potentials of gradually varying shape , which will facilitate easier calculation of the very weakly bound state . we calculate @xmath118 from eq . ( 17 ) and then the isospectral potential for a specific value of @xmath13 is calculated from eq . ( 18 ) . + in fig . 1 , the effective potential @xmath7 calculated by the cph method is shown as a continuous ( red ) curve . this has a a soft repulsion at smaller @xmath20 , followed by a shallow minimum which supports the ground state at @xmath127 mk . as stated earlier , we fail to get the first excited state in this effective potential , by our numerical calculation . next we calculate the isospectral potential @xmath17 , for chosen values of @xmath13 . we checked that for large values of @xmath13 , calculated @xmath17 is practically indistinguishable from @xmath7 . for small values of @xmath13 , the isospectral potential develops a deep and narrow well near the origin ( left side well , lsw ) with a barrier ( intermediate barrier , ib ) separating it from the shallow well ( right side well , rsw ) . as @xmath13 decreases towards zero , lsw becomes deeper , narrower and closer to the origin and at the same time , ib becomes higher . & @xmath128 & @xmath129 & @xmath130 & @xmath131 & @xmath132 & @xmath133 + @xmath22 & & & & & 8.999 & -3.238 + 0.00010 & 4.607 & -1.897 & 5.327 & 8.333 & 9.087 & -2.848 + 0.00005 & 4.527 & -8.119 & 5.137 & 10.23 & 9.087 & -2.847 + 0.00002 & 4.457 & -22.133 & 4.943 & 13.874 & 9.085 & -2.846 + 0.00001 & 4.417 & -40.254 & 4.817 & 18.019 & 9.085 & -2.846 + in table iii , we present numerical values of the parameters of the original ( corresponding to @xmath124 ) and isospectral potentials @xmath17 for @xmath134 and @xmath135 . for each isospectral potential , we give values of the position ( @xmath136 ) and value ( @xmath137 ) of the minimum of lsw , position ( @xmath130 ) and value ( @xmath131 ) of the maximun of ib , and position ( @xmath138 ) and value ( @xmath139 ) of the minimum of rsw . we have checked by numerical calculation that the energy of both the ground and the first excited state remains independent of the choice of @xmath13 , as theory predicts . however , very small values of @xmath13 are not convenient for numerical calculation of the energy of the excited state , as numerical errors creep in , due to extreme narrowness of the lsw , in which the excited state resides . thus a judicious choice of the value of @xmath13 is necessary . by careful investigations for different values of @xmath13 , we find that the range @xmath140 for the value of @xmath13 is optimum for minimizing errors . as two typical cases demostrating the behavior of isospectral potentials , we plot @xmath141 for @xmath19 ( blue , dotted curve ) and @xmath18 ( green , dashed curve ) in fig . 1 . the very weakly bound first excited state in the original shallow potential @xmath7 now becomes strongly bound within the deep and narrow lsw of @xmath17 . the deep well and the adjacent high barrier strongly localizes this state in the isospectral potential . we solve eq . ( 11 ) , with @xmath7 replaced by @xmath17 , subject to appropriate boundary conditions to calculate the energy of the first excited state . following this isospectral technique , we do indeed get a bound first excited state . we also verified that its energy is the same , within estimated numerical errors , for values of @xmath13 lying within the chosen range . the binding energy of the first excited state of trimer is thus found to be @xmath142 mk . we present this result , together with results of other sophisticated calculations reported so far in table iv . + ref [ 11 ] & -2.282 + ref [ 7 ] & -2.280 + ref [ 8 ] & -2.277 + present method & -2.270 + here we remark that the isospectral formalism is an efficient tool to calculate near - zero energy states in shallow potential which supports at least one bound state . in the supersymmetric isospectral formalism , one can in principle calculate the energy value as well as the wave function . in our earlier calculation of the resonances of halo nuclei , we used the wave function of the isospectral potential . we calculated the probability of the system to be trapped in the well - barrier combination which facilitates the accurate determination of resonance energy @xcite . we have observed that the resonance energy is independent of @xmath13 . however in the present application to calculate the bound state wave function in the isospectral potentials and to calculate other physical observables we should take a different approach . from fig . 1 , we see that decreasing @xmath13 gradually keeps the long - range part of the isospectral potentials almost unchanged whereas the short - range part has a drastic change . naturally the wave functions of the isospectral family are now @xmath13 dependent and obviously the calculated physical observables [ e.g. average radius of the system ] will be @xmath13 dependent . in the susy isospectral formalism @xcite , it is possible to calculate the wave function of the original potential @xmath7 from the wave functions of the isospectral potential @xmath17 . the ground state @xmath143 and the first excited state @xmath144 of the isospectral potential @xmath17 corresponding to the energies @xmath109 and @xmath145 respectively are related to the ground state @xmath108 and the first excited state @xmath146 of the original potential @xmath7 by @xcite @xmath147 and @xmath148 where @xmath149 is the wronskian and @xmath108 and @xmath146 are well - behaved functions . we can calculate the ground state wave function @xmath143 of the isospectral potential @xmath17 from the original gound state wave function @xmath108 ( which is known as we calculated it to construct the family of isospectral potentials ) using eq . alternatively we can obtain @xmath143 directly by solving for the isospectral potential @xmath17 . also we can obtain @xmath150 by solving for the isospectral potential @xmath17 with energy @xmath145 . thus eq . ( 20 ) is a first order differential equation in @xmath146 and we can solve it subject to proper boundary conditions @xmath151 and @xmath152 to obtain @xmath146 . with these wave functions we can calculate any physical properties of the system and this time they will be independent of the @xmath13 . however usually in the susy isospectral formalism , people are interested only in the bound state energy and in this paper we are also interested only in the energy of the elusive first excited state . + next we discuss in detail about the choice of @xmath13 parameter . as in our earlier works @xcite , we observe that the energy is independent of the parameter @xmath13 . but making @xmath13 too small , may create large numerical error in the numerically calculated wave function @xmath150 , as the well becomes extremely narrow and very deep . then the wave function changes very rapidly over a very small interval . clearly its derivatives are inaccurate , which in turn affects the accuracy of the wave function at the next mesh point . the errors accumulate . as we solve the isospectral potential numerically , large numerical error will result for very small @xmath13 . this may mask the overall accuracy . thus the accuracy of the results are very crucially dependent on the choice of @xmath13 . there is no prescribed rule to choose @xmath13 . it is in general chosen by trial . one can choose finer mesh interval within the lsw for a small @xmath13 . but the cumulative error at successive mesh point , increases as the total number of mesh points increases . using the error estimate of the integration procedure ( we use runga - kutta algorithm ) we optimize the values of @xmath13 and mesh interval to minimize the error . in our earlier calculations @xcite for @xmath153 state of @xmath154he , the shallowness of the effective potential is removed for @xmath13 = 10 , whereas @xmath13 = 0.1 was required for @xmath155 state of the same system to get the expected behavior of the isospectral potential . thus the optimum choice of @xmath13 depends on the choice of the system and its state . + now in principle the ground state energy can also be recalculated using the isospectral potentials , in which the ground state of the original potential now lies in the very deep well of the isospectrals . but , when we solve the isospectral potential numerically , large error may creep due to the extreme narrowness of the well , as discussed above . thus when we use the isospectral formalism for the real many - body problems , ground state energy is more accurate when determined by the original potential and the excited states and resonance states are more accurate when we solve the isospectral potentials . in conclusion , we remark that the isospectral potential with a judiciously chosen value of @xmath13 can be very useful for an accurate calculation of near - zero energy states in a shallow potential and also the resonance states of halo and highly unstable systems . the present application to the very weakly bound and highly elusive first excited state in @xmath0he trimer demonstrates the novelty and practical utility of this technique . + this work has been partially supported by fapesp ( brazil ) , cnpq ( brazil ) , department of science and technology ( dst , india ) and department of atomic energy ( dae , india ) . b.c . wishes to thank fapesp ( brazil ) for providing financial assistance for her visit to the universidade de so paulo , brazil , where part of this work was done . skh wishes to thank the council of scientific and industrial research ( csir ) , india for a junior research fellowship . t.k.d . acknowledges the university grants commission ( ugc , india ) for the emeritus fellowship . + references v. efimov , phys . b * 33 * , 563 ( 1970 ) . v. efimov , nucl . a * 210 * , 157 ( 1973 ) . t. kraemar _ et al _ , nature * 440 * , 315 ( 2006 ) ; s. knoop _ et al _ , nature phys . lett . * 5 * , 227 ( 2009 ) ; m. zaccanti _ et al _ , nature phys . * 5 * , 586 ( 2009 ) . t. cornelius and w. glckle , j. chem . phys . * 85 * , 3906 ( 1986 ) . b. d. esry , c. d. lin , and c. h. greene , phys . a * 54 * , 394 ( 1996 ) . e. nielsen , d. v. fedorov and a. s. jensen , j. phys . b * 31 * , 4085 ( 1998 ) v. roudnev and s. yakovlev , chem . phys . lett . * 328 * , 97 ( 2000 ) . p. barletta and a. kievsky , phys . a * 64 * , 042514 ( 2001 ) . w. sandhas , e. a. kolganova , y. k. ho and a. k. motovilov , few - body syst . * 34 * , 137 ( 2004 ) . t. gonzlez - lezana _ et al . * 82 * , 1648 ( 1999 ) . a. k. motovilov , w. sandhas , s. a. sofianos and e. a. kolganova , eur . j. d * 13 * , 33 ( 2001 ) . h. suno and b. d. esry , phys . rev . a * 78 * , 062701 ( 2008 ) . e. braaten , h. w. hammer , d. kang , and l. platter , phys . a * 78 * , 043605 ( 2008 ) . w. schllkopf and j. p. toennies , science * 266 * , 1345 ( 1994 ) . r. brhl _ et al . * 95 * , 063002 ( 2005 ) . m. fabre de la ripelle , ann . 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we propose a novel mathematical approach for the calculation of near - zero energy states by solving potentials which are isospectral with the original one . for any potential , families of strictly isospectral potentials ( with very different shape ) having desirable and adjustable features are generated by supersymmetric isospectral formalism . the near - zero energy efimov state in the original potential is effectively trapped in the deep well of the isospectral family and facilitates more accurate calculation of the efimov state . application to the first excited state in @xmath0he trimer is presented .

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Proceed to summarize the following text: cassiopeia a is the brightest shell - type galactic supernova remnant ( snr ) in x - rays and radio , and the youngest snr observed in our galaxy . the radius of the approximately spherical shell is @xmath15 , which corresponds to @xmath16 pc for the distance @xmath17 kpc ( reed et al . the supernova which gave rise to cas a was probably first observed in 1680 ( ashworth 1980 ) . it is thought to be a type ii supernova caused by explosion of a very massive wolf - rayet star ( fesen , becker & blair 1987 ) . optical observations of cas a show numerous oxygen - rich fast - moving knots ( fmk ) , with velocities of about 5000 km s@xmath4 , and slow - moving quasi - stationary flocculi , with typical velocities of about 200 km s@xmath4 , which emit h@xmath18 and strong lines of nitrogen . x - ray observations of cas a show numerous clumps of hot matter emitting strong si , s , fe , ar , ne , mg and ca lines ( holt et al . 1994 , and references therein ) . because this snr lies at the far side of the perseus arm , with its patchy distribution of the interstellar gas , the interstellar absorption varies considerably across the cas a image ( e.g. , keohane , rudnick & anderson 1996 ) . numerous radio , optical and x - ray measurements of the hydrogen column density ( e.g. , schwarz , goss & kalberla 1997 ; hufford & fesen 1996 ; jansen et al . 1988 ; favata et al . 1997 ) show a strong scatter within a range @xmath19 , where @xmath20 . based on recent results , we consider @xmath21 as plausible values for the central region of the cas a image . in spite of considerable efforts to detect a compact remnant of the supernova explosion only upper limits on its flux had been established at different wavelengths until a pointlike x - ray source was discovered close to the cas a center ( tananbaum et al . 1999 ) in the first light observation with the x - ray observatory ( see weisskopf et al . 1996 for a description ) . after this discovery , the same source was found in the hri image of 199596 ( aschenbach 1999 ) and hri images of 1979 and 1981 ( pavlov & zavlin 1999 ) . in this letter we present the first analysis on the central source spectrum observed with ( 2 ) , together with the analysis of the , , and _ asca _ observations ( 3 ) . various interpretations of these observations are discussed in 4 . the snr cas a was observed several times during the orbital activation and calibration phase . for our analysis , we chose four observations of 1999 august 2023 with the s array of the advanced ccd imaging spectrometer ( acis ; garmire 1997 ) . in these observations cas a was imaged on the backside - illuminated chip s3 . the spectral response of this chip is presently known better that those of the frontside - illuminated chips used in a few other acis observations of cas a. we used the processed data products available from the public data archive . the observations were performed in the timed exposure mode , with a frame integration time of 3.24 s. the durations of the observations were 5.03 , 2.04 , 1.76 , and 1.77 ks . because of telemetry saturation , the effective exposures were 2.81 , 1.22 , 1.06 , and 1.05 ks , respectively . since the available acis response matrices were generated for the set of grades g02346 , we selected events with these grades . events with pulse height amplitudes exceeding 4095 adu ( @xmath22 of the total number ) were discarded as generated by cosmic rays . the images of the pointlike source look slightly elongated , but this elongation is likely caused by errors in the aspect solution , and the overall shapes of the images is consistent with the assumption that this is a point source . its positions in the four observations are consistent with that reported by tananbaum et al . ( 1999 ) : @xmath23 , @xmath24 . for each of the images , we extracted the source+background counts from a @xmath25 radius circle around the point source center , and the background from an elliptical region around the circle , with an area of about 10 times that of the circle . after subtracting the background , we obtained the source countrates @xmath26 , @xmath27 , @xmath28 , and @xmath29 ks@xmath4 ( counts per kilosecond ) . the countrate values and the light curves are consistent with the assumption that the source flux remained constant during the 4 days , with the countrate of @xmath30 ks@xmath4 . for the analysis of the point source spectrum , we chose the longest of the acis - s3 observations . we grouped the pulse - height spectrum for 306 source counts into 14 bins in the 0.85.0 kev range ( fig . 1 ) . each bin has more than 20 counts ( except for the highest - energy bin with 8 counts ) . the spectral fits were performed with the xspec package . if the source is an active pulsar , we can expect that its x - ray radiation is emitted by relativistic particles and has a power - law spectrum . the power - law fit ( upper panel of fig . 2 ) yields a photon index @xmath31 ( all uncertainties are given at a @xmath32-@xmath33 confidence level ) that is considerably larger than @xmath342.1 observed for x - ray radiation from youngest pulsars ( becker & trmper 1997 ) . the hydrogen column density , @xmath35 , inferred from the power - law fit somewhat exceeds estimates obtained from independent measurements ( see 1 ) . the ( unabsorbed ) x - ray luminosity in the 0.15.0 kev range , @xmath36 erg s@xmath4 , where @xmath37 , is lower than those observed from very young pulsars ( e.g. , @xmath38 and @xmath39 erg s@xmath4 for the crab pulsar and psr b054069 , in the same energy range ) . if the source is a neutron star ( ns ) , but not an active pulsar , thermal radiation from the ns surface can be observed . the blackbody fit ( middle panel of fig . 2 ) yields a temperature @xmath40 mk and a sphere radius @xmath41 km , which correspond to a bolometric luminosity @xmath42 erg s@xmath4 . ( we use the superscript @xmath43 to denote the observed quantities , distinguishing them from those at the ns surface : @xmath44 , @xmath45 , @xmath46 , where @xmath47^{1/2}= [ 1 - 0.41 m_{1.4}r_6^{-1}]^{1/2}$ ] is the gravitational redshift factor , @xmath48 and @xmath49 cm are the ns mass and radius ) . the temperature is too high , and the radius is too small , to interpret the detected x - rays as emitted from the whole surface of a cooling ns with a uniform temperature distribution . the inferred hydrogen column density , @xmath50 , is on a lower side of the plausible @xmath51 range . since fitting observed x - ray spectra with light - element ns atmosphere models yields lower effective temperatures and larger emitting areas ( e. g. , zavlin , pavlov & trmper 1998 ) , we fit the spectrum with a number of hydrogen and helium ns atmosphere models ( pavlov et al . 1995 ; zavlin , pavlov & shibanov 1996 ) , for several values of ns magnetic field . these fits show that the assumption that the observed radiation is emitted from the whole surface of a 10-km radius ns with a uniform temperature still leads to unrealistically large distances , @xmath5250 kpc . thus , both the blackbody fit and h / he atmosphere fits hint that , if the object is a ns , the observed radiation emerges from hot spots on its surface ( see 4 ) . an example of such a fit , for polar caps covered with a hydrogen atmosphere with @xmath53 g , is shown in the bottom panel of figure 2 . the model spectra used in this fit were obtained assuming the ns to be an orthogonal rotator ( the angles @xmath18 , between the magnetic and rotation axes , and @xmath54 , between the rotation axis and line of sight , equal @xmath55 ) . the inferred effective temperature of the caps is @xmath56 mk ( which corresponds to @xmath57 mk ) , the polar cap radius @xmath58 km , and @xmath59 . the bolometric luminosity of two polar caps is @xmath60 erg s@xmath4 . the temperature @xmath61 can be lowered , and the polar cap radius increased , if we see the spot face - on during the most part of the period extreme values , @xmath62 mk ( @xmath63 mk ) and @xmath64 km , at @xmath65 , correspond to @xmath66 . the fits with the one - component thermal models implicitly assume that the temperature of the rest of the ns surface is so low that its radiation is not seen by acis . on the other hand , according to the ns cooling models ( e.g. , tsuruta 1998 ) , one should expect that , at the age of 320 yr , the ( redshifted ) surface temperature can be as high as 2 mk for the so - called standard cooling ( and much lower , down to 0.3 mk , for accelerated cooling ) . to constrain the temperature outside the polar caps , we repeated the polar cap fits with the second thermal component added , at a fixed ns radius and different ( fixed ) values of surface temperature @xmath67 . with this approach we estimated upper limits on the lower temperature , @xmath682.3 mk ( at a 99% confidence level ) , depending on the low - temperature model chosen . these fits show that the model parameters are strongly correlated the increase of @xmath67 shifts the best - fit @xmath61 downward , and @xmath69 upward . for example , using an iron atmosphere model for the low - temperature component and assuming a hydrogen polar cap , we obtain an acceptable fit ( see fig . 1 ) for @xmath70 mk , @xmath71 km , @xmath72 mk , @xmath73 km , @xmath74 . note that this @xmath67 is consistent with the predictions of the standard cooling models , and @xmath51 is close to most plausible values adopted for the central region of the snr . we reanalyzed the archival data on cas a obtained during a long hri observation , between 1995 december 23 and 1996 february 1 ( dead - time corrected exposure 175.6 ks ) . the image shows a pointlike central source at the position @xmath75 , @xmath76 ( coordinates of the center of the brightest @xmath77 pixel ) , consistent with that reported by aschenbach ( 1999 ) . its separation from the point source position , @xmath78 , is smaller than the absolute pointing uncertainty ( about @xmath79 ; briel et al . measuring the source countrate is complicated by the spatially nonuniform background . another complication is that the 40-day - long exposure actually consists of many single exposures of very different durations . because of the absolute pointing errors , combining many single images in one leads to additional broadening of the point source function ( psf ) . to account for these complications , we used several apertures ( with radii from @xmath25 to @xmath80 ) for source+background extraction , measured background in several regions with visually the same intensity as around the source , discarded short single exposures , and used various combinations of long single exposures for countrate calculations . this analysis yields a source countrate of @xmath81 ks@xmath4 ( corrected for the finite apertures ) . we also re - investigated the archival data on cas a obtained with the hri in observations of 1979 february 8 ( 42.5 ks exposure ) and 1981 january 2223 ( 25.6 ks exposure ) . in each of the data sets there is a pointlike source at the positions @xmath82 , @xmath83 , and @xmath84 , @xmath85 , respectively , consistent with those reported by pavlov & zavlin ( 1999 ) . the separations from the position , @xmath86 and @xmath87 , and from the position , @xmath88 and @xmath89 , are smaller than the nominal absolute position uncertainty ( @xmath90 for ) . since the observations were short , estimating the source countrates is less complicated than for the hri observation . the source+background counts were selected from @xmath91-radius circles , and the background was measured from annuli of @xmath92 outer radii surrounding the circles . the source countrates , calculated with account for the hri psf ( harris et al . 1984 ) , are @xmath93 ( feb 1979 ) , @xmath94 ks@xmath4 ( jan 1981 ) , and @xmath95 ks@xmath4 ( for the combined data ) . the countrate is consistent with the upper limit of @xmath96 ks@xmath4 , derived by murray et al . ( 1979 ) from the longer of the two observations . to check whether the source radiation varied during the two decades , we plotted the lines of constant and hri countrates in figure 2 . for all the three one - component models , the domains of model parameters corresponding to the hri countrates within a @xmath97 range are broader than the 99% confidence domains obtained from the spectra . the 1-@xmath33 domains corresponding to the hri countrate overlap with the 1-@xmath33 confidence regions obtained from the spectral data . thus , the source countrates detected with the three instruments do not show statistically significant variability of the source . we also examined numerous archival _ asca _ observations of cas a ( 19931999 ) and failed to detect the central point source on the high background produced by bright snr structures smeared by poor angular resolution of the _ asca _ telescopes . in the longest of the _ asca _ sis observations ( 1994 july 29 ; 15.1 ks exposure ) the point source would be detected at a 3-@xmath33 level if its flux were a factor of 8 higher than that observed with , , and . the _ asca _ observations show that there were no strong outbursts of the central source . the observed x - ray energy flux , @xmath98 , of the compact central object ( cco ) is 3.6 , 6.5 , and @xmath99 erg @xmath100 s@xmath4 in 0.32.4 , 0.34.0 and 0.36.0 kev ranges , respectively . upper limits on its optical - ir fluxes , @xmath101 erg @xmath100 s@xmath4 and @xmath102 erg @xmath100 s@xmath4 , can be estimated from the magnitude limits , @xmath103 and @xmath104 , found by van den bergh & pritchet ( 1986 ) . this gives , e.g. , @xmath105 for the energy range , and @xmath106 for the and ranges . the flux ratios are high enough to exclude coronal emission from a noncompact star as the source of the observed x - ray radiation . a hypothesis that cco is a background agn or a cataclismic variable can not be completely rejected , but its probability looks extremely low , given the high x - ray - to - optical flux ratio , the softness of the spectrum , and the lack of indications on variability . the strong argument for cco to be a compact remnant of the cas a explosion is its proximity to the cas a center . in particular , this source lies @xmath80@xmath107 south of the snr geometrical center determined from the radio image of cas a ( see reed et al . 1995 , and references therein ) . the source separation , @xmath108@xmath91 , from the snr expansion center , found by van den bergh & kamper ( 1983 ) from the analysis of proper motions of fmks , corresponds to a transverse velocity of 50250 km s@xmath4 ( for @xmath5 kpc , @xmath109 yr ) . much higher transverse velocities , 8001000 km s@xmath4 , correspond to the separation , @xmath110@xmath111 , from the position of the apparent center of expanding snr shell derived by reed et al . ( 1995 ) from the radial velocities of fmks . thus , if cco is the compact remnant of the sn explosion , it is moving south ( or sse ) from the cas a center with a transverse velocity of a few hundred km s@xmath4 , common for radio pulsars . if cco is an isolated ( nonaccreting ) object , it might be an active pulsar with an unfavorable orientation of the radio beam ( a limit on the pulsed flux of 80 mjy at 408 mhz was reported by woan & duffett - smith 1993 ) . however , a lack of a plerion or a resolved synchrotron nebula , together with the steep x - ray spectrum and low luminosity ( see 2 ) do not support this hypothesis . the lack of the pulsar activity has been found in several x - ray sources associated with young compact remnants of sn explosions ( e.g. , gotthelf , vasisht & dotani 1999 ) ; it may be tentatively explained by superstrong ( @xmath112 g ) magnetic fields which may suppress the one - photon pair creation in the pulsar s acceleration gaps ( baring & harding 1998 ) . if cco is an isolated ns without pulsar activity , one may assume that the observed x - rays are emitted from the ns surface . in this case , we also have to assume an intrinsically nonuniform surface temperature distribution to explain the small size and high temperature of the emission region . slight nonuniformity of the surface temperature can be caused by anisotropy of heat conduction in the strongly magnetized ns crust ( greenstein & hartke 1983 ) . however , this nonuniformity is not strong enough to explain the small apparent areas of the emitting regions . some nonuniformity might be expected in magnetars , if they are indeed powered by decay of their superstrong magnetic fields ( thompson & duncan 1996 ; heyl & kulkarni 1998 ) and a substantial fraction of the thermal energy is produced in the outer ns crust . in this case , the hotter regions of the ns surface would be those with stronger magnetic fields . if additional investigations will demonstrate quantitatively that the observed luminosity of @xmath113 erg s@xmath4 can be emitted from a small fraction , @xmath114 , of the magnetar s surface , we should expect that the radiation is pulsed , with a probable period of a few seconds typical for magnetars . we can also speculate that cco is a predecessor of a soft gamma - repeator ( such a hypothesis has been proposed by gotthelf et al . 1999 for the central source of the kes 73 , which shows a spectrum similar to cco , albeit emitted from a larger area ) . higher temperatures of polar caps can be explained by different chemical compositions of the caps and the rest of the ns surface . light - element polar caps could form just after the sn explosion via fallback of a fraction of the ejected matter onto the magnetic poles . due to fast stratification in the strong gravitational field , the upper layers of the polar caps will be comprised of the lightest element present . the thermal conductivity in the liquid portion of thin degenerate ns envelopes , which is responsible for the temperature drop from the nearly isothermal interior to the surface , is proportional to @xmath115 , where @xmath116 is the ion charge ( yakovlev & urpin 1980 ) . this means that low-@xmath116 envelopes are more efficient heat conductors than high-@xmath116 ones , so that a light - element ( h , he ) surface has a higher effective temperature for a given temperature @xmath117 at the outer boundary of the internal isothermal region . approximately , the effective surface temperature is proportional to @xmath118 if the chemical composition of the envelope does not vary with depth , so that the surface of a hydrogen envelope can be @xmath119 times hotter than that of an iron envelope . numerical calculations of chabrier , potekhin & yakovlev ( 1997 ) give a smaller factor ( 1.61.7 for temperatures of interest ) , with account for burning of light elements into heavier ones in the hot bottom layers of the envelope , but neglecting the effects of strong magnetic fields which can somewhat increase this factor ( heyl & hernquist 1997 ) . hence , the light - element cap should be hotter than the rest of the ns surface . for instance , for @xmath120 mk , the effective temperatures of the h cap and fe surface are @xmath121 mk and @xmath122 mk , for @xmath123 , @xmath71 km . as we have shown in 2 , a two - component model spectrum with such temperatures is consistent with the observed cco spectrum , for @xmath124 km . the thickness of the hydrogen cap , @xmath125 g @xmath100 , needed to provide such a temperature difference , corresponds to the total cap mass @xmath126 . for lower @xmath127 , the temperature difference will be smaller , but still appreciable for @xmath128 . such an explanation of the cco radiation is compatible only with the standard cooling scenario the difference of chemical compositions could not account for a large ratio , @xmath129 , of the cap and surface temperatures required by the accelerated cooling . let us consider the hypothesis that the observed x - ray radiation is due to accretion onto a ns or a black hole ( bh ) . to provide a luminosity @xmath130 erg s@xmath4 , the accretion rate should be @xmath131 g s@xmath4 , where @xmath132 is the accretion efficiency ( @xmath133 for accretion onto the surface of a ns ) . although the luminosity and the accretion rate are very small compared to typical values observed in accreting binaries , they are too high to be explained by accretion from circumstellar matter ( csm ) very high csm densities and/or low object velocities relative to the accreting medium are required . for instance , the bondi formula , @xmath134 , gives the following relation between the csm baryon density @xmath135 and velocity @xmath136 km s@xmath4 : @xmath137 @xmath138 . even at @xmath139 , which is lower than a typical pulsar velocity , the required density exceeds that expected in the cas a interiors by about 34 orders of magnitude , unless the ns or bh moves within a much denser ( and sufficiently cold ) csm concentration . this estimate for @xmath135 can be considered as a lower limit because accretion onto a bh , or onto a ns in the propeller regime , is much less efficient . we can not , however , exclude that cco is accreting from a secondary component in a close binary or from a fossil disk which remained after the sn explosion . we can rule out a massive secondary component from the above - mentioned @xmath140 and @xmath141 limits , we estimate @xmath142 , @xmath143 . we can also exclude a persistent low - mass x - ray binary ( lmxb ) or a transient lmxb in outburst the object would have a much higher x - ray luminosity than observed , and the accretion disk would be much brighter in the ir - optical range ( van paradijs & mcclintock 1995 ) . however , cco might be a compact object with a fossil disk , or an lmxb with a dwarf secondary component , in a long - lasting quiescent state ( e.g. , an m5 dwarf with @xmath144 would have @xmath145 , for the adopted distance and extinction ) . an indirect indication that cco could be a compact accreting object is that its luminosity and spectrum resemble those of lmxbs in quiescence , although we have not seen variability inherent to such objects . if the accreting object were a young ns , it would be hard to explain how the matter accretes onto the ns surface a very low magnetic field and/or long rotation period , @xmath146 s , would be required for the accreting matter to penetrate the centrifugal barrier . the criterion suggested by rutledge et al . ( 1999 ) to distinguish between the ns and bh lmxbs in quiescence , based on fitting the quiescent spectra with the light - element ns atmosphere models , favors the bh interpretation , although the applicability of this criterion to a system much younger than classical lmxbs may be questioned . on the other hand , in at least some of bh binaries optical radiation emitted by the accretion flow was detected in quiescence ( e.g. , narayan , barret & mcclintock 1997 ) at a level exceeding the upper limit on the cco optical flux . finally , one could speculate that the cco progenitor was a binary with an old ns , and this old ns has sufficiently slow rotation and low magnetic field to permit accretion onto the ns surface from a disk of matter captured in the aftermath of the sn explosion . in this case , cco could have properties of an accreting x - ray pulsar with a low accretion rate . ( a similar model was proposed by popov 1998 for the central source of rcw 103 , although he assumed accretion from the ism . ) to conclude , we can not firmly establish the nature of cco based on the data available it can be either an isolated ns with hot spots or a compact object ( more likely , a bh ) accreting from a fossil disk or from a dwarf binary companion . although the cco spectrum and luminosity strongly resemble those of other radio - quiet compact sources in snrs , these sources may not necessarily represent a homogeneous group e.g. , the central source of kes 73 shows 11.7 s pulsations and remarkable stability , and was proposed to be a magnetar ( gotthelf et al . 1999 ) , whereas the central source of rcw 103 shows long - term variability and no pulsations ( gotthelf , petre & vasisht 1999 ) . we favor the isolated ns interpretation of cco because it has not displayed any variability . critical observations to elucidate its nature include searching for periodic and aperiodic variabilities , deep ir imaging , and longer acis observations which would provide more source quanta for the spectral analysis . we are grateful to norbert schulz for providing the acis response matrices , to gordon garmire , leisa townsley and george chartas for their advices on the acis data reduction , and to niel brandt , sergei popov , and jeremy heyl for useful discussions . the and data were obtained through the high energy astrophysics science archive research center online service , provided by the nasa s goddard space flight center . the work was partially supported through nasa grants nag5 - 6907 and nag5 - 7017 .

the central pointlike x - ray source of the cas a supernova remnant was discovered in the first light observation and found later in the archival and images . the analysis of these data does not show statistically significant variability of the source . because of the small number of photons detected , different spectral models can fit the observed spectrum . the power - law fit yields the photon index @xmath04.1 , and luminosity @xmath1@xmath2@xmath3 erg s@xmath4 , for @xmath5 kpc . the power - law index is higher , and the luminosity lower , than those observed from very young pulsars . one can fit the spectrum equally well with a blackbody model with @xmath68 mk , @xmath7@xmath8 km , @xmath9@xmath10 erg s@xmath4 . the inferred radii are too small , and the temperatures too high , for the radiation could be interpreted as emitted from the whole surface of a uniformly heated neutron star . fits with the neutron star atmosphere models increase the radius and reduce the temperature , but these parameters are still substantially different from those expected for a young neutron star . one can not exclude , however , that the observed emission originates from hot spots on a cooler neutron star surface . because of strong interstellar absorption , the possible low - temperature component gives a small contribution to the observed spectrum ; an upper limit on the ( gravitationally redshifted ) surface temperature is @xmath112.3 mk , depending on chemical composition of the surface and star s radius . amongst several possible interpretations , we favor a model of a strongly magnetized neutron star with magnetically confined hydrogen or helium polar caps ( @xmath12 mk , @xmath13 km ) on a cooler iron surface ( @xmath14 mk ) . such temperatures are consistent with the standard models of neutron star cooling . alternatively , the observed radiation may be interpreted as emitted by a compact object ( more likely , a black hole ) accreting from a fossil disk or from a late - type dwarf in a close binary . submitted to _ the astrohysical journal _

You are an expert at summarizing long articles. Proceed to summarize the following text: savannas ecosystems are characterized by the long - term coexistence between a continuous grass layer and scattered or clustered trees @xcite . occurring in many regions of the world , in areas with very different climatic and ecological conditions , the spatial structure , persistence , and resilience of savannas have long intrigued ecologists @xcite . however , despite substantial research , the origin and nature of savannas have not yet been fully resolved and much remains to be learned . savanna tree populations often exhibit pronounced , non - random spatial structures @xcite . much research has therefore focused on explaining how spatial patterning in savannas arises @xcite . in most natural plant systems both facilitative and competitive processes are simultaneously present @xcite and hard to disentangle @xcite . some savanna studies have pointed toward the existence of short - distance facilitation @xcite , while others have demonstrated evidence of competition @xcite , with conflicting reports sometimes arriving from the same regions . different classes of savannas , which can be characterized by how much rainfall they typically receive , should be affected by different sets of processes . for example , in semiarid savannas water is extremely limited ( low mean annual precipitation ) and competition among trees is expected to be strong , but fire plays little role because there is typically not enough grass biomass to serve as fuel . in contrast , humid savannas should be characterized by weaker competition among trees , but also by frequent and intense fires . in - between these extremes , in mesic savannas , trees likely have to contend with intermediate levels of both competition for water and fire @xcite . competition among trees is mediated by roots that typically extend well beyond the crown @xcite . additionally , fire can lead to local facilitation due to a protection effect , whereby vulnerable juvenile trees placed near adults are protected from fire by them @xcite . we are particularly interested in how the interplay between these mechanisms governs the spatial arrangement of trees in mesic savannas , where both mechanisms may operate . on the other side , it has frequently been claimed that pattern formation in arid systems can be explained by a combination of long - distance competition and short - distance facilitation @xcite . this combination of mechanisms is also known to produce spatial structures in many other natural systems @xcite . although mesic savannas do not display the same range of highly regular spatial patterns that arise in arid systems ( e.g. , tigerbush ) , similar mechanisms might be at work . specifically , the interaction between long - range competition and short - range facilitation might still play a role in pattern formation in savanna tree populations , but only for a limited range of parameter values and possibly modified by demographic stochasticity . although the facilitation component has often been thought to be a key component in previous vegetation models @xcite , rietkerk and van de koppel @xcite , speculated , but did not show , that pattern formation could occur without short - range facilitation in the particular example of tidal freshwater marsh . in the case of savannas , as stated before , the presence of adult trees favor the establishment of new trees in the area , protecting the juveniles against fires . considering this effect , we take the facilitation component to its infinitesimally short spatial limit , and study its effect in the emergence of spatially periodic structures of trees . to our knowledge , this explanation , and the interrelation between long - range competition and local facilitation , has not been explored for a vegetation system . one of our main results is that when considering the limit of local facilitation and nonlocal competition , clustering of trees appears . here we develop a minimalistic model of savannas that considers two of the factors , as already mentioned , thought to be crucial to structure mesic savannas : tree - tree competition and fire , with a primary focus on spatially nonlocal competition . employing standard tools used in the study of pattern formation phenomena in physics ( stability analysis and the structure function ) @xcite , we explore the conditions under which the model can produce non - homogeneous spatial distributions . a key strength of our approach is that we are able to provide a complete and rigorous analysis of the patterns the model is capable of producing , and we identify which among these correspond to situations that are relevant for mesic savannas . we further examine the role of demographic stochasticity in modifying both spatial patterns and the conditions under which trees persist in the system in the presence of fire , and discuss the implications of these results for the debate on whether the balance of processes affecting savanna trees is positive , negative , or is variable among systems . this is the framework of our study : the role of long - range competition , facilitation and demographic fluctuations ( in the second part of the paper ) in the spatial structures of mesic savannas . to complete our work we include an appendix where we study the effect of external fluctuations ( mimicking for e.g. rainfall ) on savanna dynamics . our model is inspired by the one presented by calabrese et al . in @xcite . it complements theirs by providing further analytical results that clearly demonstrate that this simple system , where we focus on the local limit of facilitation , can produce the full spectrum of spatial patterns reported from models employing both short - range facilitation and long - range inhibition ( competition ) . in this section we derive the deterministic equation for the local density of trees , such that dynamics is of the logistic type and we only consider tree - tree competition and fire . we study the formation of patterns via stability analysis and provide numerical simulations of our model , showing the emergence of spatial structures . calabrese et al . @xcite introduced a simple discrete - particle lattice savanna model that considers the birth - death dynamics of trees , and where tree - tree competition and fire are the principal ingredients . these mechanisms act on the probability of establishment of a tree once a seed lands at a particular point on the lattice . in the discrete model , seeds land in the neighborhood of a parent tree with a rate @xmath0 , and establish as adult trees if they are able to survive both competition neighboring trees and fire . as these two phenomena are independent , the probability of establishment is @xmath1 , where @xmath2 is the probability of surviving the competition , and @xmath3 is the probability of surviving a fire event . from this dynamics , we write a deterministic differential equation describing the time evolution of the global density of trees ( mean field ) , @xmath4 , where the population has logistic growth at rate @xmath0 , and an exponential death term at rate @xmath5 . it reads : @xmath6 generalizing eq . ( [ eq : mf ] ) , we propose an evolution equation for the space - dependent ( local ) density of trees , @xmath7 : @xmath8 we allow the probability of overcoming competition to depend on tree crowding in a local neighborhood , decaying exponentially with the density of surrounding trees as @xmath9 where @xmath10 is a parameter that modulates the strength of the competition , and @xmath11 is a positive kernel function that introduces a finite range of influence . this model is related to earlier models of pattern formation in arid systems @xcite , and subsequent works @xcite , but it differs from standard kernel - based models in that the kernel function accounts for the interaction neighborhood , and not for the type of interaction with the distance . note also that the nonlocal term enters nonlinearly in the equation . following @xcite , @xmath12 is assumed to be a saturating function of grass biomass , @xmath13 , similar to the implementation of fire of jeltsch _ et al . _ in @xcite @xmath14 where @xmath15 governs the resistance to fire , so @xmath16 means no resistance to fires . notice how our model is close to the one in @xcite through the definitions of @xmath2 and @xmath3 , although we consider the probability of surviving a fire depending on the local density of trees , and in @xcite it depends on the global density . the deterministic differential equation that considers tree - tree competition and fire for the spatial tree density is @xmath17 where @xmath18 thus , we have a logistic - type equation with an effective growth rate that depends nonlocally on the density itself , and which is a combination of long - range competition and local facilitation mechanisms ( fire ) . the probability of surviving a fire is higher when the local density of trees is higher , as can be seen from the definition in equation ( [ probfire ] ) . in figure [ transition ] we show numerical solutions for the mean field equation ( [ eq : mf ] ) ( lines ) and the spatially explicit model ( equation [ sav1 ] ) ( dots ) in the stationary state @xmath19 using different values of the competition . we have used a top - hat function as the competition kernel , @xmath11 ( see section [ lsa ] for more details on the kernel choice ) . we observe a very good agreement of both descriptions which becomes worse when we get closer to the critical point @xmath20 , where the model presents a phase transition from a tree - grass coexistence to a grassland state . this disagreement appears because while the mean field equation describes an infinite system , the eq . ( [ sav1 ] ) description forces us to choose a size for the system . the model reproduces the long - term coexistence between grass and trees that is characteristic of savannas . to explore this coexistence , we study the long - time behavior of the system and analyze the homogeneous stationary solutions of eq . ( [ sav1 ] ) , which has two fixed points . the first one is the absorbing state representing the absence of trees , @xmath21 , and the other can be obtained , in the general case , by numerically solving @xmath22 in the regime where @xmath23 is small ( near the critical point ) , if competition intensity , @xmath10 , is also small , it is possible to obtain an analytical expression for the critical value of the probability of surviving a fire , @xmath20 , @xmath24 outside of the limit where @xmath25 , we can solve eq . ( [ steq ] ) numerically in @xmath23 to show that the critical value of the fire resistance parameter , @xmath20 , does not depend on competition . a steady state with trees is stable for higher fire survival probability ( fig . [ transition ] ) . the model , then , shows a transition from a state where grass is the only form of vegetation to another state where trees and grass coexist at @xmath20 . in what follows , we fix @xmath26 , so we choose our temporal scale in such a way that time is measured in units of @xmath5 . this choice does not qualitatively affect our results . , as a function of the resistance to fires parameter , @xmath15 . the lines come from the mean field solution , eq . ( [ steq ] ) , and the dots from the numerical integration of eq . ( [ sav1 ] ) over a square region of @xmath27 . we have chosen @xmath26 , and @xmath28 . in the case of the spatial model , @xmath23 involves an average of the density of trees over the studied patch of savanna.,scaledwidth=40.0% ] the spatial patterns appearing in the nonlocal savanna model can be studied by performing a linear stability analysis @xcite of the stationary homogeneous solutions of equation ( [ sav1 ] ) , @xmath29 . the stability analysis is performed by considering small harmonic perturbations around @xmath30 , @xmath31 , @xmath32 . after some calculations ( [ appa ] ) , one arrives at the dispersion relation @xmath33}{(\sigma-\rho_{0}+1)}-1,&\end{aligned}\ ] ] where @xmath34 , @xmath35 , is the fourier transform of the kernel , @xmath36 the critical values of the parameters of the transition to pattern , @xmath37 and @xmath38 , and the fastest growing wavenumber @xmath39 are obtained from the simultaneous solution of @xmath40 note that @xmath39 represents the most unstable mode of the system , which means that it grows faster than the others and eventually dominates the state of the system . therefore , it determines the length scale of the spatial pattern . these two equations yield the values of the parameters @xmath10 and @xmath15 at which the maximum of the curve @xmath41 , right at @xmath39 , starts becoming positive . this signals the formation of patterns in the solutions of eq . ( [ sav1 ] ) . as eq . ( [ eq : sec ] ) is explicitly written as @xmath42 the most unstable wavenumber @xmath39 can be obtained by evaluating the zeros of the derivative of the fourier transform of the kernel . equation ( [ reldisper ] ) shows that competition , through the kernel function , fully determines the formation of patterns in the system . the local facilitation appears in @xmath43 and it is not relevant in the formation of spatial structures . if the fourier transform of @xmath44 never takes positive values , then @xmath45 is always negative and only the homogeneous solution is stable . however , when @xmath46 can take negative solutions then patterns may appear in the system . what does this mean in biological terms ? imagine that we have a family of kernels described by a parameter @xmath47 : @xmath48 ( @xmath49 gives the range of competition ) . the kernels are more peaked around @xmath50 for @xmath51 and more box - like when @xmath52 . it turns out that this family of functions has non - negative fourier transform for @xmath53 , so that no patterns appear in this case . a lengthy discussion of this property in the context of competition of species can be found in @xcite . thus , the shape of the competition kernel dictates whether or not patterns will appear in the system . if pattern formation is possible , then the values of the fire and competition parameters govern the type of solution ( see below ) . our central result for nonlocal competition is that , contrary to conventional wisdom , it can , in the limit of infinitesimally short ( purely local ) facilitation , promote the clustering of trees . whether or not this occurs depends entirely on the shape of the competition kernel . for large @xmath47 we have a box - like shape , and in these cases trees compete strongly with other trees , roughly within a distance @xmath49 from their position . the mechanism behind this counterintuitive result is that trees farther than @xmath49 away from a resident tree area are not able to _ invade _ the zone defined by the radius r around the established tree ( their seeds do not establish there ) , so that an exclusion zone develops around it . for smaller @xmath47 there is less competition and the exclusion zones disappear . for a more detailed analysis , one must choose an explicit form for the kernel function . our choice is determined by the original @xmath54 taken in @xcite , so that it decays exponentially with the number of trees in a neighborhood of radius @xmath49 around a given tree . thus , for @xmath44 we take the step function ( limit @xmath55 ) @xmath56 as noticed before , the idea behind the nonlocal competition is to capture the effect of the long roots of a tree . the kernel function defines the area of influence of the roots , and it can be modeled at first order with the constant function of equation ( [ kerneldd ] ) . thus the parameter r , which fixes the _ nonlocal _ interaction scale , must be of the order of the length of the roots @xcite . since the roots are the responsible for the adsorption of resources ( water and soil nutrients ) , a strong long - range competition term implies strong resource depletion . for this kernel the fourier transform is @xcite @xmath57 and its derivative is @xmath58 , where @xmath59 , and @xmath60 is the @xmath61-order bessel function . since @xmath34 can take positive and negative values , pattern solutions may arise in the system , that will in turn depend on the values of @xmath10 and @xmath15 . the most unstable mode is numerically obtained as the first zero of @xmath62 , eq . ( [ derlam ] ) , which means the first zero of the bessel function @xmath63 . this value only depends on @xmath49 , being independent of the resistance to fires and competition , and it is @xmath64 . because a pattern of @xmath65 cells is characterized by a wavenumber @xmath66 , where @xmath67 is the system size , the typical distance between clusters , @xmath68 , using the definition of the critical wavenumber is given by @xmath69 . in other words , it is approximately the range of interaction @xmath49 . this result is also independent of the other parameters of the system . since we are interested in the effect of competition and fire on the distribution of savanna trees , we will try to fix all the parameters but @xmath15 and @xmath10 . we will explore the effect of different values of these parameters on the results . first , we have chosen ( as in @xcite ) the death rate @xmath26 , and solving eq . ( [ steq ] ) we will roughly estimate the birth rate , @xmath0 . we will work in the limit of intermediate to high mean annual precipitation , so water is non - limiting and thus we can neglect the effects of competition ( @xmath70 ) . at this intermediate to high mean annual precipitation the empirically observed upper limit of savanna tree cover is approximately @xmath71 @xcite . to reach this upper limit in the tree cover , disturbances must also be absent , implying no fire ( @xmath72 ) . in this limit , the mean field equation ( [ eq : mf ] ) is quantitatively accurate , as it is shown in figure [ transition ] , and the stationary mean field solution of the model depends only on the birth rate @xmath73 it can be solved for @xmath0 for a fixed @xmath71 , and it yields @xmath28 @xcite . in the following we just consider the dependence of our results on @xmath10 and @xmath15 . in particular , @xmath74 . the phase diagram of the model , computed numerically , is shown in fig . [ phspace ] , where we plot the spatial character of the steady solution ( homogeneous or inhomogeneous ) as a function of @xmath10 and @xmath15 . note that increasing competition enhances the inhomogeneous or pattern solution . this is because , as we are now in the case of a kernel giving rise to clusters , increasing @xmath10 makes it more difficult to enter the exclusion zones in - between the clusters . for very strong competition ( high , unrealistic , @xmath10 ) , fire has no influence on the pattern . the critical line separating these two solutions ( pattern and homogeneous)can be analytically computed as a function of the parameters @xmath10 , @xmath15 , @xmath30 and @xmath75 ( see eq . ( [ criline ] ) in [ appb ] ) . in figure [ phspace ] we have plotted ( with crosses ) this critical line separating homogeneous and pattern solutions for the step kernel . note that the stationary density of trees , @xmath30 , must be computed numerically from eq . ( [ steq ] ) . ) for @xmath76 , @xmath77 , and a step kernel . the absorbing - active transition is shown at @xmath20 with circles ( o ) . the homogeneous - pattern transition ( eq . ( [ criline ] ) ) is indicated with crosses ( x ) . the diamond , the square , and the up - triangle show the value of the parameters @xmath15 and @xmath10 taken in figures [ patterns](a)-(c ) respectively . the stars point out the transition to inhomogeneous solutions in the stochastic model as described in section [ stochastic ] , with @xmath78.,scaledwidth=40.0% ] with @xmath28 , in the absence of fire ( @xmath72 ) , and for weak competition , we can take the limits @xmath79 and @xmath72 of the dispersion relation eq . ( [ reldisper ] ) , leading to @xmath80 in fig . [ transition ] , for large @xmath15 , it can be seen that typically @xmath81 , so eq . ( [ limitceroinfinity ] ) becomes negative . this result means that in this limit , trees are uniformly distributed in the system as there is no competition , and space does not play a relevant role in the establishment of new trees . such situation could be interpreted as favorable to forest leading to a fairly homogeneous density of trees . this result agrees with the phase plane plotted in figure [ phspace ] . in biological terms , there are no exclusion zones in the system because there is no competition . the previous analysis provides information , depending on the competition and fire parameters , about when the solution is spatially homogenous and when trees arrange in clusters . however , the different shapes of the patterns have to be studied via numerical simulations @xcite of the whole equation of the model . we have taken a finite square region of savanna with an area of @xmath82 ha . , allowed competition to occur in a circular area of radius @xmath83 , and employed periodic boundary conditions and a finite differences algorithm to obtain the numerical solution . similarly to what has been observed in studies of semiarid water limited systems @xcite , different structures , including gaps , stripes , and tree spots , are obtained in the stationary state as we increase the strength of competition for a fixed value of the fire parameter or , on the other hand , as we decrease the resistance to fires for a given competition intensity . in both equivalent cases , we observe this spectrum of patterns as far as we go to a more dry state of the system , where resources ( mainly water ) are more limited ( see figs . [ patterns](a)-[patterns](c ) ) and competition is consequently stronger . this same sequence of appearance of patterns has been already observed in the presence of different short - range facilitation mechanisms @xcite . it indicates that , when @xmath10 is increased ( i.e. the probability of surviving competition is decreased ) , new trees can not establish in the exclusion areas so clustering is enhanced . on the other hand , in the case of fire - prones savannas , previous works had only shown either tree spot @xcite or grass spots @xcite . therefore , at some values of the parameter space ( see fig . [ patterns]b ) , the patterns in our deterministic approach are not observed in mesic savannas , and should correspond to semiarid systems . however , we will show in the following sections that under the parameter constraints of a mesic savanna , and considering the stochastic nature of the tree growth dynamics in the system ( i.e. demographic noise ) , our model shows realistic spatial structures . ) , ( b ) striped grass vs. tree ( @xmath84 ) , and ( c ) tree spots ( @xmath85 ) patterns in the deterministic model in a square patch of savanna of @xmath27 . @xmath86 , @xmath87 , @xmath76 and @xmath77 in all the plots.,scaledwidth=48.0% ] a much more quantitative analysis of the periodicity in the patterns can be performed via the structure function . this will be helpful to check the previous results and , especially , for the analysis of the data of the stochastic model of the next section , for which we will not present analytical results . the structure function is defined as the modulus of the spatial fourier transform of the density of trees in the stationary state , @xmath88 where the average is a spherical average over the wavevectors with modulus @xmath89 . the structure function is helpful to study spatial periodicities in the system , similar to the power spectrum of a temporal signal . its maximum identifies dominant periodicities , which in our case are the distances between tree clusters . note that the geometry of the different patterns can not be uncovered with the structure function , since it involves a spherical average . in fig . [ maxstructure ] , we show the transition to patterns using the maximum of the structure function as a function of the competition parameter . a peak appears when there are spatial structures in the system , so @xmath90\neq0 $ ] . however , we do not have information about the values where the shapes of the patterns change . taking @xmath91 , the peak is always at @xmath92 for our deterministic savanna model , independently of the competition and fire resistance parameters , provided that they take values that ensure the emergence of patterns in the system ( see the line labeled by @xmath93 in fig . [ strucboth ] ; for the definition of @xmath94 see next section ) . this result is in good agreement with the theoretical result provided for the wavelength by the linear stability analysis @xmath95 , which is also independent of competition and resistance to fires . at long times . the fire parameter is fixed at @xmath86 . black circles refers to the deterministic model and red squares to the stochastic model , @xmath96.,scaledwidth=33.0% ] ) for different values of the demographic noise intensity . @xmath97 , @xmath86 , @xmath91 , @xmath77 , @xmath76.,scaledwidth=33.0% ] the perfectly periodic patterns emerging in fig . [ patterns ] from the deterministic model seem to be far from the disordered ones usually observed in aerial photographs of mesic savannas and shown by individual based models @xcite . we have so far described a savanna system in terms of the density of trees with a deterministic dynamics . the interpretation of the field @xmath98 is the density of tree ( active ) sites in a small volume , @xmath99 . if we think of trees as reacting particles which are born and die probabilistically , then to provide a reasonable description of the underlying individual - based birth and death dynamics , we have to add a noise term to the standard deterministic equation . it will take into account the _ intrinsic _ stochasticity present at the individual level in the system . if we take a small volume , @xmath99 , the number of reactions taking place is proportional to the number of particles therein , @xmath100 , with small deviations . if @xmath100 is large enough , the central limit theorem applies to the sum of @xmath100 independent random variables and predicts that the amplitude of the deviation is of the order of @xmath101 @xcite . this stochasticity referred to as demographic noise . the macroscopic equation is now stochastic , @xmath102- \nonumber \\ & -&\alpha\rho(\mathbf{x } , t)+\gamma\sqrt{\rho(\mathbf{x } , t)}\eta(\mathbf{x},t),\end{aligned}\ ] ] where @xmath103 ( but we take it as a constant , @xcite ) modulates the intensity of @xmath104 , a gaussian white noise term with zero mean and correlations given by dirac delta distributions @xmath105 the complete description of the dynamics in eq.([savsto ] ) should have the potential to describe more realistic patterns . we first investigate the effect of demographic noise on the persistence of trees in the system . we show in ( fig . [ noisy - transition ] ) that the critical point , @xmath20 , depends on the value of the competition parameter @xmath10 . this effect is rather small , so that when @xmath10 increases the transition to the grassland state appears only for a slightly larger @xmath15 ( i.e , less frequent fire ) . the reason seems to be that fire frequency and intensity depend on grass biomass . seasonally wet savannas support much more grass biomass that serves as fuel for fires during the dry season @xcite . dry savannas have much lower grass biomass , so they do not burn as often or as intensely . the shift of the critical value of @xmath15 when competition is stronger is consistent with the one showed in @xcite , as can be seen comparing figure 2 in @xcite with figure [ noisy - transition ] here . besides , the values obtained for @xmath20 are larger when we consider the demographic stochasticity @xcite neglected in the deterministic field approach . ) with @xmath78 and average the density of trees in the steady state.,scaledwidth=33.0% ] we explore numerically the stochastic savanna model using an algorithm developed in @xcite ( see [ sec : numerical ] ) . note that the noise makes the transition to pattern smoother so the change from homogeneous to inhomogeneous spatial patterns is not as clear as it is in the limit where the demographic noise vanishes ( see fig . [ maxstructure ] ) . the presence of demographic noise in the model , as shown in fig . [ phspace ] ( red stars ) , also decreases the value of the competition strength at which patterns appear in the system , as has been observed in other systems . mathematically , these new patterns appear since demographic noise maintains fourier modes of the solution which , due to the value of the parameters , would decay in a deterministic approach @xcite . biologically , exclusion zones are promoted by demographic noise , since it does not affect regions where there are not trees . on the other hand in vegetated areas fluctuations may enhance tree density , leading to stronger competition . the presence of demographic noise in the model allows the existence of patterns under more humid conditions . this result is highly relevant for mesic savannas , as we expect competition to be of low to intermediate strength in such systems . we show two examples of these irregular patterns in fig . [ stopatterns](a ) and fig . [ stopatterns](b ) . unrealistic stripe - like patterns no longer appear in the stochastic model . we have studied the dynamics of the system for some values of the fire and competition parameters . demographic noise influences the spatial structures shown by the model . the deterministic approach shows a full spectrum of patterns which are not visually realistic for mesic savannas ( but for arid systems ) . the role of the noise is to transform this spectrum of regular , unrealistic patterns into more irregular ones ( figures [ stopatterns](a)-[stopatterns](d ) ) that remind the observed in aerial photographs of real mesic savannas . on the other hand , these patterns are statistically equivalent to the deterministic ones , as it is shown with the structure function in fig . [ strucboth ] . the dominant scale in the solution is given by the interaction radio , @xmath49 , and it is independent of the amplitude of the noise ( see the structure function in figure [ strucboth ] , peaked around @xmath106 independently of the noise ) . besides , over a certain treshold in the amplitude , demographic noise destroys the population of trees . therefore , the model presents an active - absorbing transition with the noise strength , @xmath94 , being the control parameter . . @xmath86 , @xmath87 , @xmath76 and @xmath77 in all the plots . ( a ) @xmath78 , @xmath107 . ( b ) @xmath78 , @xmath108 . ( c ) @xmath109 , @xmath110 . ( b ) @xmath78 , @xmath110.,scaledwidth=48.0% ] understanding the mechanisms that produce spatial patterns in savanna tree populations has long been an area of interest among savanna ecologists @xcite . a key step in such an analysis is defining the most parsimonious combination of mechanisms that will produce the pattern in question . in this paper the combination of long - range competition for resources and the facilitation induced by fire are considered the responsible of the spatial structures , in the line of studies of vegetation pattern formation in arid systems , where also a combination of long - range inhibition and short - range facilitation is introduced @xcite . the main difference is that the facilitation provided by the protection effect of adult trees against fires in our savanna model takes the short - range facilitation to its infinitesimally short limit ( i.e , local limit ) . under this assumption we have studied the conditions under which our model could account for patterns . we have shown that nonlocal competition combined with local facilitation induces the full range of observed spatial patterns , provided the competition term enters nonlinearly in the equation for the density of trees , and that competition is strong enough . the key technical requirement for this effect to occur is that the competition kernel must be an almost constant function in a given competition region , and decay abruptly out of the region . we verify this condition working with supergaussian kernel functions . in practice , this means that competition kernels whose fourier transform takes negative values for some wavenumber values , will lead to competition driven clustering . the other mechanism we have considered for a minimalistic but realistic savanna model , fire , has been shown to be relevant for the coexistence of trees and grass and for the shape of the patterns . however , competition is the main ingredient allowing pattern solutions to exist in the model . if the shape of the kernel allows these types of solutions , then the specific values of fire and competition parameters determine the kind of spatial structure that develops . it is also worth mentioning that one can observe the full spectrum of patterns in the limit where fires vanish ( @xmath72 ) , so there is no facilitation at all , provided competition is strong enough . however , when there is no competition , @xmath70 , no patterns develop regardless of the value of the fire term . therefore , we conclude that the nonlocal competition term is responsible for the emergence of clustered distributions of trees in the model , with the fire term playing a relevant role only to fix the value of the competition parameter at which patterns appear . in other words , for a given competition strength , patterns appear more readily when fire is combined with competition . a similar mechanism of competitive interactions between species has been shown to give rise to clusters of species in the context of classical ecological niche theory . scheffer and van nes @xcite showed that species distribution in niche space was clustered , and pigolotti et al . @xcite showed that this arises as an instability of the nonlocal nonlinear equation describing the competition of species . long - distance competition for resources in combination with the local facilitation due to the protection effect of adult trees in the establishment of juvenile ones can explain the emergence of realistic structures of trees in mesic savannas . in these environmental conditions , competition is limited , so we should restrict to small to intermediate values of the parameter @xmath10 , and the effect of fires is also worth to be taken into account . however , these two ingredients give a full range of patterns observed in vegetated systems , but not in the particular case of savannas . it is necessary to consider the role of demographic noise , which is present in the system through the stochastic nature of the birth and death processes of individual trees . in this complete framework our model shows irregular patterns of trees similar to the observed in real savannas . the other important feature of savannas , the characteristic long - time coexistence of trees and grass is well captured with our model ( figures [ transition ] and [ noisy - transition ] ) . besides , the presence of demographic noise , as it is shown in figure [ noisy - transition ] , makes our approach much more realistic , since the persistence of trees in the face of fires is related to the water in the system . on the other hand , demographic stochasticity causes tree extinction at lower fire frequencies ( larger @xmath15 ) than in the deterministic case . this is because random fluctuations in tree density are of sufficient magnitude that this can hit zero even if the deterministic stationary tree density ( for a given fire frequency ) is greater than zero . this effect vanishes if we increase the system size . the demographic noise is proportional to the density of trees ( proportional to @xmath111 ) , so fluctuations are smaller if we study bigger patches of savannas . as usually happens in the study of critical phenomena in statistical mechanics , the extinction times due to demographic noise increase exponentially with the size of the system for those intensities of competition and fire that allow the presence of trees in the stationary state . over the critical line , this time will follow a power law scaling , and a logarithmic one when the stationary state of the deterministic model is already absorbing ( without trees ) @xcite . we have shown the formation of patterns in a minimal savanna model , that considers the combination of long - range competition and local facilitation mechanisms as well as the transition from trees - grass coexistence to a grass only state . the salient feature of the model is that it only considers nonlocal ( and nonlinear ) competition through a kernel function which defines the length of the interaction , while the facilitation is considered to have an infinitesimally short influence range . our model thus differs from standard kernel - based savanna models that feature both short - range facilitation and long - range competition . the same sequence of spatial patterns appears in both approaches , confirming rietkerk and van de koppel s @xcite suggestion that short - range facilitation does not induce spatial pattern formation by itself , and long - distance competition is also needed . it also suggests that long - range competition could be not only a necessary , but also a sufficient condition to the appearance of spatial structures of trees . inspired by @xcite , we have proposed a nonlocal deterministic macroscopic equation for the evolution of the local density of trees where fire and tree - tree competition are the dominant mechanisms . if the kernel function falls off with distance very quickly ( the fourier transform is always positive ) the system only has homogenous solutions . in the opposite case , patterns may appear depending on the value of the parameters ( @xmath10 and @xmath15 ) , and in a sequence similar to the spatial structures appearing in standard kernel - based models . under less favorable environmental conditions , trees tend to arrange in more robust structures to survive ( fig . [ patterns](d ) ) . biologically , trees are lumped in dense groups , separated by empty regions . entrance of new trees in these _ exclusion zones _ is impossible due to the intense competition they experience there . a great strength of our approach is that our deterministic analysis is formal , and we have shown the different spatial distributions of the trees that occur as competition becomes more intense , concluding that self organization of trees is a good mechanism to promote tree survival under adverse conditions @xcite . trees tend to cluster in the high competition ( low resources ) limit ( fig . [ patterns](d ) ) , due to the formation of exclusion zones caused by non - local competition , and not as a result of facilitation . however , because we are dealing with a deterministic model , the patterns are too regular and the transition between the grass - only and a tree - populated states is independent of tree competition . we therefore considered stochasticity coming from the stochastic nature of individual birth and death events , to provide a more realistic description of savanna dynamics . calabrese et al . @xcite also noted that savanna - to - grassland transition was independent of competition intensity in the mean field approach , but not when demographic noise was included . in the present model , both the grassland to savanna transition and the spatial structures that develop are influenced by demographic stochasticity . in the case of spatial structures , demographic noise is specially relevant , since it turns much of the unrealistic patterns of the deterministic model into more realistic ones , that remind the observed in real savannas . it also allows the existence of periodic arrangements of trees in more humid systems , which means environmental conditions closer to mesic savannas . we have quantified the characteristic spacing of spatial patterns through the structure function . the irregular patterns produced by the stochastic model still have a dominant wavelength whose value is the same as in the deterministic model and depends only on the value of the range of the interaction , @xmath49 , in the kernel function . the match between the typical spatial scale of the patterns and the characteristic distance over which nonlocal competition acts indicates that competition is responsible for the presence of clustered spatial structures . r.m - g is supported by the jaepredoc program of csic . r.m - g . and c.l . acknowledge support from micinn ( spain ) and feder ( eu ) through grant no . fis2007- 60327 fisicos . we acknowledge federico vzquez and emilio hernndez - garca for their comments and discussion . we also acknowledge the detailed reading and insightful comments of three anonymous referees which greatly helped to improve this manuscript . this appendix shows the details of the linear stability analysis , in particular how it is obtained the dispersion relation in eq . ( [ reldisper ] ) . we consider the stationary solution @xmath23 plus a small harmonic perturbation , @xmath112 where @xmath32 . substituting eq . ( [ ansatz ] ) into the original equation ( [ sav1 ] ) , and retaining only linear terms in @xmath113,we arrive to the relation dispersion @xmath114 & \nonumber \\ & + bc\sigma\frac{1 - 2\rho_{0}}{\sigma+1-\rho_{0}}-1 , & \nonumber \\\end{aligned}\ ] ] where @xmath115 is the fourier transform of the kernel , @xmath116 , and @xmath117 , provided that we deal with normalized kernels . equation ( [ rd ] ) can be written as eq . ( [ reldisper ] ) using the definition of @xmath43 . we show here the analytical expression for the critical line in the transition from homogeneous to inhomogeneous solutions . starting from eq . ( [ eq : first ] ) it is possible to write an expression for the value of the resistance to fires parameter , @xmath15 , at which the macroscopic equation ( [ sav1 ] ) starts showing pattern solutions , as a function of the competition parameter , @xmath10 , and the most unstable mode @xmath118 . considering the value of the parameters taken in our study , @xmath28 and @xmath26 , it is @xmath119}{10\left[1 - 2\rho_{0}+ \delta \hat{g}(k_{c})\rho_{0}(1+\rho_{0})-{\rm e}^{\delta\pi r^{2}\rho_{0}}/5\right ] } & \nonumber \\ & + \frac{(\rho_{0}-1)\sqrt{5\left[5(\rho_{0}-1)^{2 } ( \delta \hat{g}(k_{c})\rho_{0}-1)^{2}-4{\rm e}^{\delta\pi r^{2}\rho_{0}}\rho_{0}\right]}}{10\left[1 - 2\rho_{0}+ \delta \hat{g}(k_{c})\rho_{0}(1+\rho_{0})-{\rm e}^{\delta\pi r^{2}\rho_{0}}/5\right]}.&\end{aligned}\ ] ] this complicated expression must be evaluated numerically together with the solution of eq . ( [ steq ] ) for the stationary density of trees , which is also a function of the competition and fire parameters . we show the results in figure [ phspace ] , where the curve , represented with the black crosses , fits perfectly with the numerical results from the linear stability analysis . the integration of stochastic equations where the noise amplitude depends on the square root of the variable , @xmath120 , and there are absorbing states ( i.e , states where the system stays indefinitely ) , has awaken a great interest , specially in the study of critical phenomena ( i.e , properties of the system that appear when it is close to the critical point , often the absorbing state ) . the amplitude of the fluctuations tends to zero there , and thus numerical instabilities may appear . recently @xcite a very efficient method has been developed , but we have used in this work an older one , presented in @xcite , since its implementation is easier and it gives precise results working far from the transition point . it consists on discretizing the langevin equation , taking a step size @xmath121 in the variable . to apply the method to equation ( [ savsto ] ) , first of all we discretize the space . particularly , we compute the integral in the exponential term approximating it by a sum of the field evaluated in the nodes of the discrete space @xmath122 then , we integrate the temporal dependence . the key of the algorithm is to prevent @xmath123 to take negative values . from a general equation @xmath124 where @xmath125 is a gaussian white noise with zero mean and delta correlated , it is @xmath126 where @xmath127 . @xmath128 is a gaussian number with zero mean and unit variance . at this point , to prevent @xmath123 to take negative values , the author in @xcite proposes to dicretize the density setting @xmath129 and to truncate the gaussian distribution from where @xmath128 is obtained simetrically so that @xmath130 . the negatives values are avoided requiring @xmath131 . it can be done in many ways but following @xcite we use @xmath132 finally , rescaling the equation , we can achieve a discretized version in which positive and zero - mean noise are ensured at the cost of a `` quantized '' density . one of the key ingredients for the long coexistence between grass and trees is the largely inhomogeneous temporal distribution of precipitations over time @xcite . we have studied this environmental variability following the idea in @xcite , considering the switching between unstressed vegetation growth , given by the first term in ( [ sav1 ] ) , and drought - induced vegetation decay , represented with the second term in eq . ( [ sav1 ] ) . these processes take place each time step with probability @xmath133 and @xmath134 , respectively . from now on , we call @xmath135&=&b_{eff}(\rho)\left[\rho(\mathbf{x},t)-\rho^{2}(\mathbf{x},t)\right ] , \nonumber \\ f_{d}[\rho(\mathbf{x},t)]&=&-\alpha\rho(\mathbf{x},t),\end{aligned}\ ] ] and @xmath136=\frac{1}{2}\left[f_{b}[\rho(\mathbf{x},t)]\pm f_{d}[\rho(\mathbf{x},t)]\right].\ ] ] the random dynamics of the system is written in terms of a stochastic partial differential equation , @xmath137+f_{-}[\rho(\mathbf{x},t)]\xi_{dn}(t),\ ] ] where @xmath138 is a dichotomous noise ( dmn ) , assuming values @xmath139 ( wet season ) and @xmath140 ( dry season ) with probability @xmath133 and @xmath134 , respectively . if the rate of random switching , taken as the inverse of the integration time step , is relatively fast respect to the rate of convergence to equilibrium in each of the two states , we can replace the noise term in eq . ( [ dichsde ] ) with its average value , @xmath141 . it is meaningful since the rainfall seasons are much shorter than the time needed to reach one of the equlibrium stationary states of death and birth processes , @xmath142 , respectively . this substitution leads to a deterministic equation @xmath143+f_{-}[\rho(\mathbf{x},t)](1 - 2p),\ ] ] where we will be able to perform linear stability analysis as usual . the new dispersion relation is easily obtained , @xmath144}{(\sigma-\rho_{0}+1)},&\end{aligned}\ ] ] which means that the main effect of the dichotomous noise is to renormalize the rates @xmath5 and @xmath0 . the patterns observed now are the same as the ones in the deterministic case , though the regions where they emerge change in accordance with this renormalization . thus , the effect of stochastic precipitation , as modeled with this random switching mechanism , is a change of the parameter values for the different transitions observed in the deterministic continuum model eq . ( [ sav1 ] ) . according to the value of @xmath133 , an absorbing - active phase transition is observed , @xmath145 . small values of p , meaning long dry season , lead to an absorbing state while increasing the probability of raining implies the appearence of trees in the system . in this latter case , the solution can be either homogeneous or showing spatial patterns , depending on fire and competition . this attempt to model rainfall has not been very succesful and does not give a lot of new information . much effort of future research should be put on this point , trying to get much more realistic modelling of external environmental variability , according to empirical observations , with long runs of dry years and rare wet years . calabrese , f. vzquez , c. lpez , m. san miguel , and v. grimm . the individual and interactive effects of tree - tree establishment competition and fire on savanna structure and dynamics . the american naturalist , 175 , 3 , ( 2010 ) . r. lefever , n. barbier , p. couteron , and o. lejeune . deeply gapped vegetation patterns : on crown / root allometry , criticality and desertification . journal of theoretical biology , 2 , 261 , ( 2009 ) , 194 - 209 . i. dornic , h. chat , and m.a . muoz , integration of langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions . physical review letters , 94 , ( 2005 ) , 100601 .

we propose a model equation for the dynamics of tree density in mesic savannas . it considers long - range competition among trees and the effect of fire acting as a local facilitation mechanism . despite short - range facilitation is taken to the local - range limit , the standard full spectrum of spatial structures obtained in general vegetation models is recovered . long - range competition is thus the key ingredient for the development of patterns . the long time coexistence between trees and grass , and how fires affect the survival of trees as well as the maintenance of the patterns is studied . the influence of demographic noise is analyzed . the stochastic system , under the parameter constraints typical of mesic savannas , shows irregular patterns characteristics of realistic situations . the coexistence of trees and grass still remains at reasonable noise intensities .

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Proceed to summarize the following text: the star formation ( sf ) is a fundamental process in the evolution of galaxies and is far from being well understood . the sf is usually characterized by the initial mass function ( imf ) and the total sf rate ( sfr ) , which depends on many factors such as the density of the interstellar gas , its morphology , its metallicity , _ etc . _ generally , four major factors drive star formation in galaxies : large scale gravitational instabilities , cloud compression by density waves , compression in a rotating galactic disk due to shear forces , and random gas cloud collisions . in galaxies with previous stellar generations additional sf triggers exist , such as shock waves from stellar winds and supernova explosions . in dense environments , such as clusters of galaxies and compact groups , tidal interactions , collisions with other galaxies , ism stripping , and cooling flow accretion probably play some role in triggering the sf process . the triggering mechanisms were reviewed recently by elmegreen ( 1998 ) . while `` global '' phenomena play a large part in grand design spirals , random collisions of interstellar clouds have been proposed as one explanation for dwarf galaxies with bursts of sf . due to their small size , lack of strong spiral pattern , and sometimes solid - body rotation ( _ e.g. , _ martimbeau 1994 , blok & mcgaugh 1997 ) , the star formation in dwarf galaxies can not be triggered by compression from gravitational density waves or by disk shear . therefore , understanding sf in dwarf galaxies should be simpler than in other types of galaxies , because the number of possible trigger mechanisms is reduced . the h@xmath0 emission from a galaxy measures its ongoing sfr ( kennicutt 1983 , kennicutt 1994 ) . the blue luminosity of a galaxy measures its sf integrated over the last @xmath1 yrs ( gallagher 1984 ) . the red continuum radiation originates both from relatively young stars which already evolved into red giants and super - giants , and from a large population of aged low - mass stars , if previous sf episodes took place . hunter ( 1998 ) tested a set of sf predictors on two small samples of dwarf galaxies , one observed by them and another derived from de blok ( 1997 ) . they found that the ratio of hi surface density to the critical density for the appearance of ring instabilities did not correlate with the star formation , but that the stellar surface brightness did . from this , they concluded that possibly some stellar energy input provides the feedback mechanism for star formation . brosch ( 1998 ) confirmed that the strongest correlation among a number of parameters tested on a sample of virgo cluster dwarf irregular galaxies was between the average h@xmath0 surface brightness and the mean blue surface brightness . this is similar to the findings of phillipps & disney ( 1985 ) for spiral galaxies , where in a sample of 77 spiral galaxies from kennicutt & kent ( 1983 ) a correlation was found between the total h@xmath0 emission ( expressed as specific sfr or as h@xmath0 equivalent width ) and the average blue surface brightness . on the level of individual hii regions in dwarf irregular galaxies , heller ( 1998 : hab98 ) showed that a correlation exists also between the h@xmath0 line flux and the red continuum flux underneath the region , measured with the same aperture as the line flux . we emphasize that in this case the correlation is between local quantities , not for overall galactic properties . these correlations indicate that the regulation of sf in irr s is local and by the existing stellar population . the self - regulated evolution of dwarf galaxies has recently been modeled by andersen & burkert ( 1997 ) . we concentrate here on samples of late - type dwarf irregular galaxies ( digs ) in the virgo cluster ( vc ) and elsewhere in the nearby ( within 100 mpc ) universe . the reason for selecting digs is to limit the number of possible sf trigger mechanisms ; digs are devoid of large - scale sf triggers , as explained above . we take advantage of the availability of net h@xmath0 line images to determine the general pattern of the distribution of hii regions over the irregular galaxies . as far as we could ascertain , such a study of a large sample of irregular galaxies was never published . previous attempts to classify digs were _ e.g. , _ by sandage & binggeli ( 1984 ) for low surface brightness ( lsb ) objects , by loose & thuan ( 1985 ) for bcds , and by patterson & thuan ( 1996 ) for digs . all used broad - band images to perform the classification . we do not study here the morphology of the galaxies , as reflected by their light distribution on broad - band images , but rather the morphology of their ensemble of hii regions . our goal is to gain some insight on the star - forming properties of this class of galaxies . the question of the spatial distribution of hii regions in irregular galaxies has been studied previously by hodge ( 1969 ) for seven nearby objects , and by hunter & gallagher ( 1986 ) for a larger sample of galaxies . the primary sample consists of 52 digs in the vc with hi measurements from hoffman _ ( 1987 , 1989 ) . the sample was constructed in order to enable the detection of weak dependencies of the star formation properties on the hydrogen content and on the surface brightness . we selected two sub - samples by surface brightness ; one represents a high surface brightness ( hsb ) group and is classified as either bcd or anything+bcd , and another represents a low surface brightness ( lsb ) sample and includes only imiv or imv galaxies . the uniform morphological classification , which bins the digs in the hsb or lsb groups , is exclusively from binggeli ( 1985 , vcc ) . the galaxies were observed at the wise observatory ( wo ) from 1990 to 1997 , with ccd imaging through broad bands and narrow h@xmath0 bandpasses in the rest frame of each galaxy . the discussion of all observations and their interpretation is the subject of other papers ( ab98 , hab98 ) . we restrict the discussion here to the localization of the star - forming regions on the broad - band , or continuum light images of the digs . in some virgo galaxies we did not detect h@xmath0 emission ; these objects have been omitted in table 1 leaving 30 digs from our combined virgo sample . the two virgo cluster samples are augmented here by 83 irregular galaxies for which data were collected from the literature . not all these objects are dwarfs but all appear to be , or are classified as , irregular galaxies ; we call them here digs and are not strict in qualifying an object as `` dwarf '' . three objects of the extended sample appear in two references ; these objects have been classified independently and have two entries in table 1 . the total number of classifications is thus 124 , but only 110 different objects have been considered . we inspected h@xmath0 and broad - band or red continuum images from strobel ( 1991 ) , miller & hodge ( 1994 ) , mcgaugh ( 1995 ) , van zee ( 1996 ) , marlowe ( 1997 ) , martin ( 1997 ) , hilker ( 1998 ) , and gavazzi ( 1998 ) . the images from van zee include objects analyzed in her phd thesis ( van zee 1996 ) and some galaxies from an unpublished comparison sample . the objects were selected to be lsb digs and were checked not to show obvious signs of interaction on the palomar sky survey plates . the objects from van zee contribute 27 galaxies to the extended sample . the objects studied by gavazzi ( 1998 ) and included here consist of eight galaxies classified as irr , most in the a1367 cluster . the hilker ( 1997 ) object is an lmc - like galaxy in the fornax cluster . ho ii was studied in detail by puche ( 1992 ) ; we used the published images for the present classification . the 12 galaxies studied by marlowe ( 1997 ) are classified as either amorphous or blue compact , are intrinsically faint ( _ i.e. , _ dwarf ) , and are nearby . the sample of martin ( 1997 ) contains a heterogeneous assemblage of star - forming dwarf galaxies . three objects with spiral morphology ( n2537 , vii zw403 , and n4861 ) from her list were excluded from the present analysis , leaving 12 galaxies to be considered here . additionally , four objects were added from the study of dwarf galaxies in the m81 system ( miller & hodge 1994 ) and four other from the study of digs by strobel ( 1991 ) . finally , we included 13 objects from the morphological study of lsb disk galaxies of mcgaugh ( 1995 ) which did not show strong spiral patterns on the published images . the only restrictions to the inclusion of a galaxy in the extended sample were that the object should be classified as an irregular galaxy in the original publication and that it would have a net h@xmath0 and an off - line image . this resulted in a very heterogeneos collection of irregular galaxies ; those from martin ( 1997 ) and marlowe ( 1997 ) are mainly low - luminosity , nearby objects , while those from gavazzi ( 1998 ) , being at @xmath270 mpc , are of high luminosity and would not be strictly classifiable as `` dwarfs '' . most of the galaxies in gavazzi were rejected because they were not irr galaxies and so were all the objects analyzed by hunter ( 1998 ) , which do not have published images but only azimuthal averages of line and continuum emission . we could not use the large h@xmath0 and continuum image set in koopman ( 1997 ) because it contains only spiral and lenticular galaxies . the entire selection of 110 galaxies classified here is listed in table 1 , where the objects from our virgo sample are identified by their number in the virgo cluster catalog ( binggeli 1985 ) . the analysis reported here is based primarily on the visual inspection of the net h@xmath0-line and off - line images of each galaxy . these are usually presented side - by - side in the original publications , at the same scale and with enough `` gray - scale stretch '' to allow easy perception of the hii regions on the net - line image , and of the general outline of the galaxy in the off - line image . this facilitates the comparison and the determination of whether the hii regions are distributed mostly at the edges or near the center of an object . these two cases have been noted in table 1 as e for edge and c for center , and are the primary morphological index used for this classification . there are a few cases of mixed morphology , which have been so noted in table 1 . one such example is ugc 7178 , from the primary sample of van zee ( 1996 ) , which has four hii regions , two near the center and two at the edge and is classified here as c+e . other galaxies do not show preferential sf at either the center or the edge , but display a @xmath2linear distribution of hii regions showing up as a `` spine '' on the galaxy image . these objects are noted as l = linear , and/or sp = spine types and are probably related to the `` cometary '' galaxies noted by loose & thuan ( 1985 ) . a few objects have a number of hii regions arranged on the ( partial ) circumference of an ellipse ; these are marked el = ellipse in table 1 . finally , some irr s show a scattering of hii regions and are accordingly marked d = diffuse . ugc11820 in van zee s primary sample ( van zee 1996 ) has its hii regions arranged on @xmath2a spiral arm . it is possible that this is a case of mistaken classification and the object is probably a spiral , as listed in ugc and in ned . a similar case may be object 127037 in gavazzi ( 1998 ) . our classification , by the distribution of the visible hii regions , should be compared with that of patterson & thuan ( 1996 ) , where six classes of digs were distinguished on the basis of broad - band b and i images . the classifications are an extension of the loose & thuan ( 1986 ) scheme and bin the digs into dwarf spirals ( ds ) , nucleated dwarf irregulars ( dine ) , dwarf ellipticals with a central ridge ( dire ) , dwarfs with `` asymmetric star formation '' ( dia ) , objects with randomly scattered star formation similar to the gr8 galaxy ( gr8 ) , and dwarfs which show a bar with whispy extensions ( dib ) . the two ugc objects in common , u300 and u2162 , have both been classified by patterson & thuan as dia , while we classify them as e and a , confirming the asymmetry mention and adding the qualifier that the hii regions tend to be at the edges of the galaxies . we used a second morphological index to flag a symmetric or asymmetric distribution ( s / a ) of hii regions . a galaxy is labelled asymmetric ( a ) in the distribution of its hii regions if these are located predominantly on one side of the galaxy . in other words , the label a is assigned if it is possible to draw on the continuum or broad - band image of a galaxy a diameter which bisects it so that most of the hii regions are on one side of this diameter . if no such diameter seems to exist , the galaxy is classified as symmetrical ( s ) in the distribution of its hii regions . this asymmetry criterion is similar to that used by hodge ( 1969 ) , with the exception that hodge used the `` reference frame '' of the hii region distribution while we used that of the red continuum light . note that the secondary classification of the asymmetry is independent of the primary classification of edge / center / spine described above . this secondary classifying index is also listed in table 1 . in some cases the images were too poor to allow a classification . these objects have a question mark in the table . the references for the images are listed in table 2 . we show in figure 1 examples of the primary and secondary classifications using objects from ab98 and hab98 . the top row shows an object with edge distribution of hii regions , which is asymmetric . the middle row shows a galaxy where the hii regions are arranged in a linear , spine - like configuration . the bottom row shows an object with a centrally located , symmetric distribution of hii regions . obviously , a consistent classification requires similar types of display , contrast of images , _ etc . _ this is not always the case , as some sources did not provide two images ( line and continuum ) for a galaxy , but only one with overlaid contours for the missing information ( _ e.g. , _ martin 1997 or marlowe 1997 ) and we also did not have control over the display mode . nevertheless , the few galaxies appearing in two references allow some measure of confidence in the classification : n1800 , n5253 , and ii zw40 are common to the samples from marlowe ( 1997 ) and martin ( 1997 ) ; their classification , performed completely independently on different images , is virtually identical for all three cases . to put the classification on a more `` objective '' and numerical basis , and to aviod possible biases caused by the tendency to detect structures when visually inspecting random distributions of dots , we formed two indices which quantify the degree of concentration of hii regions ( `` concentration index''=ci ) and the amount of asymmetry in the hii region distribution ( `` asymmetry index''=ai ) . these indices are calculated from counts of hii identified in various regions of the galaxies net - line images we inspected and are listed in table 1 . the definition of these two parameters is fairly intuitive and is explained below . ci is defined as the ratio of the number of hii regions in the inner part of the galaxy to that in the outer part . we count the hii regions within half the semi - major axis from the center , and divide this by one - quarter of the number of hii regions external to this region . the one - quarter factor brings the comparison to a number per equal - area basis and is , in fact , a ratio of the number surface density of hii regions . ci can have values between zero and infinity , as the outer part of the galaxy may be devoid of hii regions . a value of unity represents a uniform distribution of hii regions , while galaxies with no emission near the center have ci=0 . galaxies with emission localized in their centers and no emission detected in their outer parts have ci=@xmath3 ; this is represented in the table as ci=100 . the asymmetry index ai is defined as the ratio of the number of hii regions counted in the `` hii poor '' half of the galaxy to that in the `` hii rich '' area , where the divider is the bisecting diameter selected visually to show the largest contrast between the two galaxy halfs . ai ranges from 0 to 1 , with unity representing a symmetric distribution of hii regions . the smaller the value of ai , the more asymmetric is the distribution of hii regions . we explain the two - dimensional morphological classification with the example of vcc 17 , shown in the top row of images in fig . vcc 17 has two central hii regions and four regions in its outer part . its ci is therefore @xmath4 . a diameter may be drawn on the figure which puts two hii regions on one side of it and four on the other side . this appears to be the most extreme asymmetry , making ai=@xmath5=0.5 . our choice of irregular galaxies in which to study the patterns of star formation was quite deliberate . as mentioned in the introduction , these objects should be devoid of large - scale sf triggering mechanisms such as density waves or various disk instabilities . therefore the triggering mechanisms should be simpler to disentangle in this kind of objects . the star formation indicator used here was the h@xmath0 line emission and the distribution of the hii regions was checked against the light distribution of either the red continuum , or any broad - band images supplied by the authors of a specific paper . the summary statistic for the distribution among types and the symmetric or asymmetric morphologies is presented in table 3 . the galaxies with mixed morphologies have been counted once in each bin , thus the total number of cases listed in the table is larger than the number of actual galaxies inspected . galaxies with incomplete or dubious classification , which have only a question mark in the s / a column of table 1 , have been excluded from the statistic . table 3 shows that most galaxies have an e - type distribution of hii regions . specifically , we find that about half of all classifications are e and a. the only other bin populated by a significant amount of irregular galaxies is c and s ( @xmath2one quarter of all classifications ) and the other bins are essentially empty . most galaxies , which were selected only to be irregulars based on their appearance on broad - band images , form preferrentially their hii regions in their outer regions ( type a has @xmath22/3 of the cases ) . the objects in which the distribution of hii regions appears symmetrical are mostly those where it is also central . this is very similar to what loose & thuan ( 1985 ) found for bcds . we show the distribution of the two morphological indices in figures 2 and 3 ( the latter shown to emphasize the behavior of the morphological indices for ci@xmath60 ) . it is clear that there is a strong dychotomy , because of the objects with essentially central h@xmath0 emission which make up the rightmost part of the plot ( ai@xmath7 ) . the distribution of the objects with inner * and * outer hii regions is shown in figure 4 . the figure shows the dominance of the highly asymmetric coverage of digs by hii regions , with most cases concentrating at [email protected] . this implies that in most digs one half of the galaxy has twice or more the number of hii regions than the other half . gerola & seiden ( 1978 ) proposed that the mechanism regulating the sf in irr s is the stochastic self - propagating sf ( sspsf ) . their simulations , as well as the more recent ones by jungwiert & palous ( 1994 ) , produce preferentially flocculent or grand - design spirals . if such a mechanism operates in digs it should produce an expanding sf wave which would engulf the entire galaxy , or at least those regions with suitable ism density . unfortunately , the 3d simulations of sspsf relevant to digs ( comins 1983 , 1984 ) do not show `` snapshots '' of the sf proceeding with time through the galaxy . such plots could have been analyzed in a similar manner to the galaxy images and possibly some constraint could have been derived . the few papers which do show such plots ( _ e.g. , _ gerola 1980 ) have too few figures to make their analysis statistically significant . note also that in large , slowly - rotating disks , like the lmc , the sspsf tends to produce mostly long filaments of young stars which show no special preference of galactic location ( feitzinger 1987 ) . if the galaxy is small , and a number of sns explode off its center , it is possible in principle to have a compression wave travel through the gas and form stars in suitable location while escaping from the galaxy in places where the ism is thin or altogether absent . it is difficult to estimate the likelihood of such a mechanism but it is possible to examine it in a well - resolved object . the nearby dig ho ii was studied intensively by puche ( 1992 ) . a comparison of their h@xmath0 and off - line images shows that ho ii forms stars near its center . the hi synthesis map indicates that the h@xmath0 emission originates either at the interfaces between large holes in the hi distribution or in the small hi holes . thus this case also does not argue in favour of the sspsf forming stars asymmetrically , or at the edges of a galaxy . if the ism in an irregular galaxy is preferentially aligned with its long dimension , a spine of hii regions could form by the sspsf mechanism . similarly , if the galaxy is in a symmetrical gravitational potential well , its ism could concentrate at the bottom of the potential and begin forming stars there in a c configuration . is there a mechanism of sf which would produce predominantly lop - sided regions of sf at the edge of a galaxy ? the question was posed recently for the lmc by de boer ( 1998 ) , who proposed a mechanism to explain the observed distribution of giant sf structures lining the edge of the lmc . they postulate that the interaction between the lmc gas and gas in the halo of the milky way causes gas compression followed by star formation . the sf takes place at the interface between the two gas distributions , where the interaction beweeen the lmc gas and the mw gas occurs . the rotation of the lmc moves the regions with newly formed stars away from the place of formation , the star formation ceases , and the newly formed stars simply age . we tested this possibility with the objects in our virgo samples which have measurements of h@xmath0 and underlying red continuum emission for individual hii regions . the test is reported in detail in hab98 . if the mechanism of de boer ( 1998 ) is at work , we expect a decrease of the h@xmath0 line intensity simultaneously with an enhancement of the red stellar continuum under the hii region as it ages . the test we performed involved a comparison of the ratios of h@xmath0 intensities and of the red continua for the brightest and the faintest hii regions identified in the same galaxy . these ratios were compared for 13 objects with multiple hii regions and a trend in the opposite direction from that expected was found ; the more intense the underlying red continuum , the stronger the line emission is . this argues against the de boer ( 1998 ) proposition and in favour of a mechanism regulating the star formation through the existing local stellar population . mcgaugh ( 1995 ) mention that the m=1 density wave mode may be possible in lsb galaxies , giving rise to one - armed spirals . in principle , it is possible that the excess of irregulars with asymmetric , edge - concentrated star formation is due to this phenomenon , but we deem it unlikely . the reason is that if the sf process would be driven by a density wave , the one - armed spiral pattern should be visible not only in the distribution of hii regions but also in the red continuum image . a possibly related phenomenon , of displaced light centers with respect to the outermost isophotes in a sample of extremely late - type spirals was recently reported by matthews & gallagher ( 1997 ) . the phenomenon was explained as a consequence of the disk , which is the luminous galaxy , orbiting in an off - center position within an extended dark halo ( levine & sparke 1998 ) . the question of dark matter ( dm ) halos in the context of irregular galaxies has also been discussed by hunter ( 1998 ) , where the conclusion was that the dm may affect the star formation by enhancing the gravitational potential . it is possible that such a model could explain also the asymmetric patterns of star formation which we reported above , but its exploration , with special emphasis on local density enhancements by the dm , is beyond the scope of this paper . we analysed the distribution of regions where star formation takes place at present in a sample of 110 irregular galaxies . our results can be summarized as follows : 1 . star formation takes place predominantly at the edges of dwarf irregular galaxies , mostly to one side of a galaxy . existing models of star formation in such objects do not predict such a distribution of star forming regions over a galaxy . the proposals of de boer ( 1998 ) , of an interaction between the galaxy and some surrounding medium which compresses the ism and thus enhances the star formation , and of mcgaugh ( 1995 ) , of strong single - arm spiral patterns in dwarf galaxies which could give rise to asymmetric distribution of star - forming regions , can probably be rejected . no good explanation was identified for the peculiar location of the hii regions in irregular galaxies . nb is grateful for continued support of the austrian friends of tel aviv university and for the hospitality of the space telescope science institute , where most of this paper was written . ea is supported by a special grant from the ministry of science to develop tauvex , a uv imaging experiment . ah acknowledges support from the us - israel binational science foundation and travel grants from the sackler institute for astronomy . astronomical research at tel aviv university is partly supported by a grant from the israel science foundation . discussions on this subject with mario livio appreciated . we are grateful for constructive remarks on this subject from lyle hoffman , crystal martin , and an anonymous referee . liese van zee and giuseppe gavazzi kindly supplied electronic copies of images from their samples of galaxies , used for some of the comparisons presented here . almoznino , e. & brosch , n. 1998 , mnras , in press ( ab98 ) . andersen , r .- p . & burkert , a. 1997 , apj , preprint . brosch , n. , heller , a. & almoznino , e. 1998 , apj , 504 ( september 10 ) , in press . binggeli , b. , sandage , a. & tamman , g.a . 1985 , aj , 90 , 1681 . comins , n.f . 1983 , apj , 266 , 543 . comins , n.f . 1984 , apj , 284 , 90 . de blok , e. & mcgaugh , s.s . 1997 , mnras 290 , 533 . de boer , k.s . , braun , j.m . , vallenari , a. & mebold , u. 1998 , a&a , 329 , l49 . gerola , h. & seiden , p.e . 1978 , apj , 223 , 129 . gerola , h. , seiden , p.e . & schulman , l.s . 1980 , apj , 242 , 517 . gallagher , j.s . , hunter , d.a . & tutukov , a.v . 1984 , apj , 284 , 544 . gavazzi , g. , catinella , b. , carraso , l. boselli , a. & contursi , a. 1998 , aj , 115 , 1745 . elmegreen , b.g . 1998 , in _ origins of galaxies , stars , planets and life _ woodward , h.a . thronson , & m. shull , eds . ) , asp series , in press . feitzinger , j.v . , haynes , r.f . , klein , u. , wielebinski , r. & perschke , m. 1987 , vistas in astr . , 30 , 243 . heller , a. , almoznino , e. & brosch , n. 1998 , mnras , submitted ( hab98 ) . hilker , m. , bomans , d.j . , infante , l. & kissler - patig , m. 1997 , a&a , 327 , 562 . hodge , p. 1969 , apj , 156 , 847 . hoffman , g.l . , helou , g. , salpeter , e.e . , glosson , j. & sandage , a. 1987 , apjs , 63 , 247 . hoffman , g.l . , lewis , b.m . , helou , g. , salpeter , e.e . & williams , h.l . 1989 , apjs , 69 , 65 . hunter , d.a . & gallagher , j.s . 1986 , pasp , 98 , 5 . hunter , d.a . , elmegreen , b.g . & baker , a.l . 1998 , apj , 493 , 595 . jungwiert , b. & palous , j. 1994 , a&a , 287 , 55 . kennicutt , r.c . 1998 , apj , 498 , 541 . kennicutt , r.c . 1983 , apj , 272 , 54 . kennicutt , r.c . , tamblyn , p. & congdon , c.w . 1994 , apj , 435 , 22 . levine , s.e . & sparke , l.s . 1998 , apjl , in press ( astro - ph/9803146 ) . loose , h .- h . & thuan , t.x . 1985 in _ star - forming dwarf galaxies _ ( d. kunth , t.x . thuan & j. tran than van , eds . ) , gif sur yvette : editions frontieres , p. 73 . marlowe , a.t . , meurer , g.r . & heckman , t.m . 1997 , apjs , 112 , 285 . martin , c. 1997 , apj , 491 , 561 . martimbeau , n. , carignian , c. & ray , j .- 1994 , aj , 107 , 543 . matthews , l.d . & gallagher , j.s . 1997 , aj , 114 , 1899 . mcgaugh , s.s . , schombert , j.m . & bothun , g.d . 1995 , aj , 109 , 2019 . miller , b.w . & hodge , p. 1994 , apj , 427 , 656 . patterson , r.j . & thuan , t.x . 1996 , apjs , 107 , 103 . phillipps , s. & disney , m. 1985 , mnras , 217 , 435 . puche , d. , westphal , d. , brinks , e. & roy , j .- r . 1992 , aj , 103 , 1841 . schmidt , m. 1959 , apj , 129 , 243 . strobel , n.v . , hodge , p. & kennicutt , r.c . 1991 , apj , 383 , 148 . tresse , l. & maddox , s.j . 1998 , apj , 495 , 691 . van zee , l. 1996 , phd thesis , cornell university . * figure 1 : examples of galaxies of different types and morphologies . each row shows the galaxy as imaged through the h@xmath0 filter at left , through the continuum filter at the center , and in the net - h@xmath0 line at right . the galaxies are vcc 17 ( e & a ) , vcc 1374 ( l , sp & s ) , and vcc 10 ( c & s ) . a 10 arcsec bar at the upper right corner of each row sets the scale . * figure 2 : distribution of classification indices for digs . each independent classification is represented by a circle . the points in regions of high concentration have been `` jiggled '' slightly in an attempt to separate points which are very close together . concentration index values of 100 represent galaxies with ci=@xmath3 ( _ i.e. , _ all hii regions in central locations ) . * figure 3 : distribution of classification indices for digs . this is an expansion of figure 2 near ci@xmath60 . the points have * not * been jiggled in this presentation . * figure 4 : distribution of the asymmetry index among galaxies which have ci@xmath9 . most of the objects concentrate in the low ai bins , indicating a preference for asymmetric distribution of hii regions . u300 & e & 0 & a & 0.5 & 1 & u191 & e & 0 & a & 0.58&2 u521 & c & 100 & s & 1 & 1 & u634 & e & 0 & a & 0 & 2 u2684 & e+c & 100 & a & 1 & 1 & u891 & l , sp & 100 & s&1&2 u2984 & e , el & 0 & a & 0.38 & 1 & u1175 & e&0 & a & 0&2 u3174 & e & 1.33 & a & 0.4 & 1 & u2162 & e & 0 & a&0.5 & 2 u3672 & e & 0 & a & 0 & 1 & u3050 & c+e&1.33 & a&0.5&2 u4660 & d & 0 & a & 0.33 & 1 & u4762 & e & 0&a & 0.5&2 u5716 & e , el & 0&a & 0 & 1 & u5764 & c+e & 0&a & 0&2 u7178 & c+e & 0.67 & a&0.17 & 1 & u5829 & e & 0&a & 0&2 ugca357 & e & 0 & a&0 & 1 & u7300 & e & 0&a & 0.33 & 2 haro43 & l , sp & 2 & s&1 & 1 & u8024 & e & 0 & s?&0.29&2 u9762 & d ? & - & ? & - & 1 & u9128 & e & 0 & s&0 & 2 u10281 & e & 0&a & 0 & 1 & ddo210 & c & 100&s&0 & 2 u11820 & e & 0.87 & a&0.58 & 1 & n1427a & e & 0.2 & a&0.11&4 97073 & e , el & 0.31 & a & 0 & 3 & v0017 & e & 2&a & 0.5&7 97079 & e & 0 & a&0 & 3 & v0169 & e & 0&a & 0.5 & 7 97087 & l , sp & 1&a&1 & 3 & v0217 & e&0 & a & 0.5&7 97138 & e&0.66 & a&0.25 & 3&v0328 & e & 0&a & 0.5 & 7 108085 & e & 0&a & 0 & 3&v0350 & c ? & 0 & s & 0&7 127037 & e , el&0.57 & a&0.20 & 3&v0477 & e & 0&a & 0&7 160086 & c & 0.57&s & 0.75 & 3 & v0530 & c & 100&a&0&7 160139 & e & 0&a & 0.22 & 3&v0826 & e&0 & a & 0.5&7 haro 14 & c & 100&a&1 & 5 & v0963 & c & 100&s&0&7 n625 & c & 100&s & 1 & 5 & v1455 & c & 100 & a&0 & 7 n1510 & c & 100&s&1 & 5&v1465 & c+e & 2.5 & a & 0&7 n1705 & c & 100&s&1 & 5&v1468 & e & 0&a & 1&7 n1800 & c+e & 1 & s&0 & 5 & v1585 & e & 0&a & 0&7 n2101 & c+e & 0.33&s & 0.23 & 5 & v1753 & e & 0&a & 0&7 ii zw40 & c & 1 & s & 1 & 5&v1952 & e & 0&a&0 & 7 n2915 & c+e & 100&a & 0.66 & 5&v1992 & e & 0&a & 0&7 n3125 & c+e & 1&a & 1 & 5&v2034 & d ? & - & ? & - & 7 n3955 & c & 100&a?&1 & 5 & v0010 & c&100&s & 1 & 8 n4670 & c & 100&s & 1 & 5 & v0144 & c&100&s&1 & 8 n5253 & c & 100&s & 1 & 5&v0172 & c+e&0&a&0 & 8 n1569 & c & 1&s&0.33 & 6&v0324 & c&100&s&1 & 8 n1800 & e & 100&a & 1 & 6 & v0410 & c&100&s&1 & 8 ii zw 40 & c & 1&s & 0.25 & 6 & v0459 & e&0&a & 1&8 n2363 & c & 100&s & 1 & 6 & v0513 & c&100&s&1 & 8 i zw 18 & c & 4&s & 0 & 6&v0562 & c & 100 & s & 1 & 8 m82 & c & 100&s & 1 & 6 & v0985 & e&100&a&1 & 8 n3077 & c & 100&s & 1 & 6 & v1179 & c&100&s&1 & 8 sex a & e & ? & a & ? & 6&v1374 & l , sp & 2&a&1 & 8 n3738 & c & 100&a & 1 & 6&v1725 & l , sp & 4&a&0 & 8 n4214 & l , sp & 0.5&s ? & 0.33 & 6&v1791 & e&0&a & 0.5&8 n4449 & l , sp & ? & a & ? & 6&ho ii & c&1 & s&1 & 9 n5253 & c & 100 & s & 1 & 6&ddo 47 & d&1.28 & s & 0.89&10 ho i & e & 0.13&a & 0.35 & 11 & sex b & e & 0.36 & a & 0.5&10 ho ix & e & 0&a & 0.20 & 11 & ddo 167 & c & 25&a & 0.66 & 10 ic 2547 & e & 0&a & @xmath20.5&11 & ddo 168 & l , sp & 1 & s & 0.71&10m81db & l , sp & 0.25&s & 0.33 & 11 & ddo 187 & e & 0 & a & 0.5&10f415 - 3 & c & 2 & a & 0 & 12 & f469 - 2 & c+e & 2 & a & 0 & 12 f561 - 1 & e & 1.14 & a & 0.75 & 12 & f563-v1 & d & 100 & s & 1 & 12 f611 - 1 & e & 0 & a & 0 & 12 & u12695 & e & 0.5 & a & 0.14 & 12 f562-v2 & c+e & 0 & a & 0.5 & 12 & f746 - 1 & e & 0.36 & a & 0.87 & 12 u5709 & c & 100 & s & 1 & 12 & f568 - 1 & c+e & 0.28 & s & 1 & 12 f583 - 5 & l , sp & 0 & s & 0.75 & 12 & f585 - 3 & c & 0.8 & s & 0.66 & 12 u5675 & c & 0 & s & 1 & 12 1 & van zee 1996 primary & 142 & van zee 1996 secondary & 13 3 & gavazzi 1998 & 84 & hilker 1998 & 1 5 & marlowe 1997 & 12 6 & martin 1997 & 12 7 & heller 1998 & 17 8 & almoznino & brosch 1998 & 13 9 & puche 1992 & 1 10 & strobel 1991 & 4 11 & miller & hodge 1994 & 4 12 & mcgaugh 1995 & 13 & total number of images & 113

the location of hii regions , which indicates the locus of present star formation in galaxies , is analyzed for a large collection of 110 irregular galaxies ( irr ) imaged in h@xmath0 and nearby continuum . the analysis is primarily by visual inspection , although a two - dimensional quantitative measure is also employed . the two different analyses yield essentially identical results . hii regions appear preferentially at the edges of the light distribution , predominantly on one side of the galaxy , contrary to what is expected from stochastic self - propagating star formation scenarios . this peculiar distribution of star forming regions can not be explained by a scenario of star formation triggered by an interaction with extragalactic gas , or by a strong one - armed spiral pattern . # 1#2#3#4#5#6#7 to#2 ' '' ''

You are an expert at summarizing long articles. Proceed to summarize the following text: in this section we describe the main results of the paper . ( in case any of the terminology appears to be new to the reader , the corresponding definitions are given in the next section . ) if a contact structure on a @xmath0-manifold is cooriented , then every legendrian knot ( i.e. a knot that is everywhere tangent to the contact distribution ) has a natural framing ( a continuous normal vector field ) . hence when studying legendrian knots in such contact manifolds the main question is to distinguish those of them that realize isotopic framed knots . similarly if the contact structure is parallelized , then every transverse knot ( i.e. a knot that is everywhere transverse to the contact distribution ) also has a natural framing , and when studying transverse knots in such contact manifolds again the main question is to distinguish those of them that realize isotopic framed knots . vassiliev invariants proved to be an extremely useful tool in the study of framed knots , and the conjecture is that they are sufficient to distinguish all the isotopy classes of framed knots . vassiliev invariants can also be easily defined in the categories of legendrian and of transverse knots . in this paper we study the relationship between the groups of vassiliev invariants of these three categories of knots , and explore when these invariants can be used to distinguish legendrian knots that realize isotopic framed knots . consider a contact manifold @xmath4 with a cooriented contact structure . fix an abelian group @xmath5 , a connected component @xmath6 of the space of framed immersions of @xmath7 into @xmath4 , and a connected component @xmath8 of the space of legendrian immersions of @xmath7 into @xmath4 . we study the relation between the groups of @xmath5-valued vassiliev invariants of framed knots from @xmath6 and of @xmath5-valued vassiliev invariants of legendrian knots from @xmath9 . the main results obtained in this paper are described below . [ first ] the groups of @xmath5-valued vassiliev invariants of legendrian knots from @xmath9 and of framed knots from @xmath6 are canonically isomorphic , provided that the euler class of the contact bundle vanishes on every @xmath10 realizable by a mapping @xmath11 . ( see theorem [ isomorphismobtained ] and proposition [ interpretationconditionii ] . ) using theorem [ first ] we show that : the groups of @xmath5-valued vassiliev invariants of legendrian knots from @xmath9 and of framed knots from @xmath6 are canonically isomorphic , provided that one of the following conditions holds : 1 : : the contact structure is tight ; 2 : : the euler class of the contact bundle is in the torsion of @xmath12 ( in particular if the euler class is zero ) . 3 : : the contact manifold is closed and admits a metric of negative sectional curvature . ( see sections [ homologysphere ] and [ negativecurvature ] and theorem [ tight ] . ) as a corollary , we get that for any surface @xmath2 the group of finite order arnold s @xmath1-type invariants of wave fronts on @xmath2 is isomorphic to the group of vassiliev invariants of framed knots in the spherical cotangent bundle @xmath3 of @xmath2 . previously the isomorphism of the groups of vassiliev invariants of legendrian and of framed knots was known only in the case where @xmath13 and @xmath4 is the standard contact @xmath14 ( result of d. fuchs and s. tabachnikov @xcite ) or the standard contact solid - torus ( result of j. hill @xcite ) . the proofs of these isomorphisms were based on the fact that for the @xmath15-valued vassiliev invariants of framed knots in these manifolds there exists a universal vassiliev invariant also known as the kontsevich integral . ( currently the existence of the kontsevich integral is known only for a total space of an @xmath16-bundle over a compact oriented surface with boundary , see the paper @xcite of andersen , mattes , and reshetikhin . ) thus the approach used in @xcite and @xcite to show the isomorphism of the groups of vassiliev invariants is not applicable for almost all contact @xmath0-manifolds and abelian groups @xmath5 and our results appear to be a strong generalization of the results of fuchs , tabachnikov and hill . we also construct the first examples where vassiliev invariants can be used to distinguish legendrian knots that realize isotopic framed knots and are homotopic as legendrian immersions . these are also the first examples where the groups of vassiliev invariants of legendrian and of framed knots from the corresponding components of the spaces of legendrian and of framed immersions are not canonically isomorphic . the manifold @xmath17 admits infinitely many cooriented contact structures for which there exist legendrian knots that can be distinguished by @xmath18-valued vassiliev invariants even though they realize isotopic framed knots and are homotopic as legendrian immersions . ( see theorem [ example1 ] and theorem [ example2 ] in which the similar result is proved for any orientable total space of an @xmath7-bundle over a nonorientable surface of a sufficiently high genus . ) for transverse knots we obtain the following result ( see theorem [ isomorphismobtainedtransverse ] ) : let @xmath19 be a contact manifold with a parallelized contact structure , then the groups of @xmath5-valued vassiliev invariants of transverse and of framed knots ( from the corresponding components of the spaces of transverse and of framed immersions ) are canonically isomorphic . in this paper @xmath5 is an abelian group ( not necessarily torsion free ) , and @xmath4 is a connected oriented @xmath0-dimensional riemannian manifold ( not necessarily compact ) . a _ contact structure _ on a @xmath0-dimensional manifold @xmath4 is a smooth field @xmath20 of tangent @xmath21-dimensional planes , locally defined as a kernel of a differential @xmath22-form @xmath23 with non - vanishing @xmath24 . a manifold with a contact structure possesses the canonical orientation determined by the volume form @xmath25 . the standard contact structure in @xmath14 is the kernel of the @xmath22-form @xmath26 . a _ contact element _ on a manifold is a hyperplane in the tangent space to the manifold at a point . for a surface @xmath2 we denote by @xmath3 the space of all cooriented ( transversally oriented ) contact elements of @xmath2 . this space is the spherical cotangent bundle of @xmath2 . its natural contact structure is the distribution of tangent hyperplanes given by the condition that the velocity vector of the incidence point of a contact element belongs to the element . a contact structure is _ cooriented _ if the @xmath21-dimensional planes defining the contact structure are continuously cooriented ( transversally oriented ) . a contact structure is _ oriented _ if the @xmath21-dimensional planes defining the contact structure are continuously oriented . since every contact manifold has a natural orientation we see that every cooriented contact structure is naturally oriented and every oriented contact structure is naturally cooriented . a contact structure is _ parallelizable _ ( _ parallelized _ ) if the @xmath21-dimensional vector bundle @xmath27 over @xmath4 is trivializable ( trivialized ) . since every contact manifold has a canonical orientation , one can see that every parallelized contact structure is naturally cooriented . a contact structure @xmath28 on a manifold @xmath4 is said to be _ overtwisted _ if there exists a @xmath21-disk @xmath29 embedded into @xmath4 such that the boundary @xmath30 is tangent to @xmath28 while the disk @xmath29 is transverse to @xmath28 along @xmath30 . not overtwisted contact structures are called _ tight_. a _ curve _ in @xmath4 is an immersion of @xmath7 into @xmath4 . ( all curves have the natural orientation induced by the orientation of @xmath7 . ) a _ framed curve _ in @xmath4 is a curve in @xmath4 equipped with a continuous unit normal vector field . a _ legendrian curve _ in a contact manifold @xmath19 is a curve in @xmath4 that is everywhere tangent to @xmath28 . if the contact structure on @xmath4 is cooriented , then every legendrian curve has a natural framing given by the unit normals to the planes of the contact structure that point in the direction specified by the coorientation . to a legendrian curve @xmath31 in a contact manifold with a parallelized contact structure one can associate an integer that is the number of revolutions of the direction of the velocity vector of @xmath31 ( with respect to the chosen frames in @xmath28 ) under traversing @xmath31 according to the orientation . this integer is called the _ maslov number _ of @xmath31 . the set of maslov numbers enumerates the set of the connected components of the space of legendrian curves in @xmath14 ( cf . [ h - principlelegendrian ] ) . a _ transverse _ curve in a contact manifold @xmath19 is a curve in @xmath4 that is everywhere transverse to @xmath28 . if the contact structure on @xmath4 is parallelized , then a transverse curve has a natural framing given by the unit normals corresponding to the projections of the first of the two coordinate vectors of the contact planes on the @xmath21-planes orthogonal to the velocity vectors of the curve . a transverse curve in a contact manifold with a cooriented contact structures is said to be _ positive _ if at every point the velocity vector of the curve points into the coorienting half - plane , and it is said to be _ negative _ otherwise . there are two connected components of the space of transverse curves in @xmath14 , they consist of positive and negative transverse curves respectively . in general if @xmath19 is a contact manifold with a cooriented contact structure , then every connected component of the space of unframed curves contains two connected components of the space of transverse curves . they consist of positive and negative transverse curves respectively . a _ knot ( framed knot ) _ in @xmath4 is an embedding ( framed embedding ) of @xmath7 into @xmath4 . in a similar way we define legendrian and transverse knots in @xmath4 . a _ singular ( framed ) _ knot with @xmath32 double points is a curve ( framed curve ) in @xmath4 whose only singularities are @xmath32 transverse double points . an _ isotopy _ of a singular ( framed ) knot with @xmath32 double points is a path in the space of singular ( framed ) knots with @xmath32 double points under which the preimages of the double points on @xmath7 change continuously . an @xmath5-valued framed ( resp . legendrian , resp . transverse ) knot invariant is an @xmath5-valued function on the set of the isotopy classes of framed ( resp . legendrian , resp . transverse ) knots . a transverse double point @xmath33 of a singular knot can be resolved in two essentially different ways . we say that a resolution of a double point is positive ( resp . negative ) if the tangent vector to the first strand , the tangent vector to the second strand , and the vector from the second strand to the first form the positive @xmath0-frame . ( this does not depend on the order of the strands ) . if the singular knot is legendrian ( resp . transverse ) , then these resolution can be made in the category of legendrian ( resp . transverse ) knots . a singular framed ( resp . legendrian , resp . transverse ) knot @xmath34 with @xmath35 transverse double points admits @xmath36 possible resolutions of the double points . the sign of the resolution is put to be @xmath37 if the number of negatively resolved double points is even , and it is put to be @xmath38 otherwise . let @xmath39 be an @xmath5-valued invariant of framed ( resp . legendrian , resp . transverse ) knots . the invariant @xmath39 is said to be of _ finite order _ ( or _ vassiliev invariant _ ) if there exists a nonnegative integer @xmath32 such that for any singular knot @xmath40 with @xmath35 transverse double points the sum ( with appropriate signs ) of the values of @xmath39 on the nonsingular knots obtained by the @xmath36 resolutions of the double points is zero . an invariant is said to be of order not greater than @xmath32 ( of order @xmath41 ) if @xmath32 can be chosen as the integer in the definition above . the group of @xmath42-valued finite order invariants has an increasing filtration by the subgroups of the invariants of order @xmath41 . [ h - principlelegendrian ] _ @xmath43-principle for legendrian curves . _ for @xmath19 a contact manifold with a cooriented contact structure , we put @xmath44 to be the total space of the fiberwise spherization of the contact bundle , and we put @xmath45 to be the corresponding locally trivial @xmath7-fibration . the @xmath43-principle proved for the legendrian curves by m. gromov ( @xcite , pp.338 - 339 ) says that the space of legendrian curves in @xmath19 is weak homotopy equivalent to the space of free loops @xmath46 in @xmath44 . the equivalence is given by mapping a point of a legendrian curve to the point of @xmath44 corresponding to the direction of the velocity vector of the curve at this point . in particular the @xmath43-principle implies that the set of the connected components of the space of legendrian curves can be naturally identified with the set of the conjugacy classes of elements of @xmath47 . [ description]_description of legendrian and of transverse knots in @xmath14 . _ the contact darboux theorem says that every contact @xmath0-manifold @xmath19 is locally contactomorphic to @xmath14 with the standard contact structure that is the kernel of the @xmath22-form @xmath26 . a chart in which @xmath19 is contactomorphic to the standard contact @xmath14 is called _ a darboux chart . _ transverse and legendrian knots in the standard contact @xmath14 are conveniently presented by the projections into the plane @xmath48 . identify a point @xmath49 with the point @xmath50 furnished with the fixed direction of an unoriented straight line through @xmath48 with the slope @xmath51 . then the curve in @xmath14 is a one parameter family of points with ( non - vertical ) directions in @xmath52 . a curve in @xmath14 is transverse if and only if the corresponding curve in @xmath52 is never tangent to the chosen directions along itself . while a generic regular curve has a regular projection into the @xmath48-plane , the projection of a generic legendrian curve into the @xmath48-plane has isolated critical points ( since all the planes of the contact structure are parallel to the @xmath51-axis ) . hence the projection of a generic legendrian curve may have cusps . a curve in @xmath14 is legendrian if and only if the corresponding planar curve with cusps is everywhere tangent to the field of directions . in particular this field is determined by the curve with cusps . let @xmath19 be a contact manifold with a cooriented contact structure . let @xmath9 be a connected component of the space of legendrian curves in @xmath4 , and let @xmath6 be the connected component of the space of framed curves that contains @xmath9 . ( such a component exists because a legendrian curve in a manifold with a cooriented contact structure is naturally framed , and a path in the space of legendrian curves corresponds to a path in the space of framed curves . ) let @xmath53 ( resp . @xmath54 ) be the group of @xmath5-valued order @xmath41 invariants of legendrian ( resp . framed ) knots from @xmath9 ( resp . from @xmath6 ) . clearly every invariant @xmath55 restricted to the category of legendrian knots in @xmath9 is an element @xmath56 . this gives a homomorphism @xmath57 . [ isomorphism ] let @xmath58 be a contact manifold with a cooriented contact structure . let @xmath9 be a connected component of the space of legendrian curves in @xmath4 , and let @xmath6 be the connected component of the space of framed curves that contains @xmath9 . then the following two statements * a * and * b * are equivalent . a : : @xmath59 for any @xmath60 and any knots @xmath61 representing isotopic framed knots . b : : @xmath62 is a canonical isomorphism . if the mapping from the isotopy classes of legendrian knots in @xmath9 to the isotopy classes of framed knots in @xmath6 is surjective , then the proof of theorem [ isomorphism ] is obvious . however in general this mapping is not surjective and the proof of theorem [ isomorphism ] is given in section [ proofisomorphism ] . the famous bennequin inequality shows that this mapping is not surjective even in the case where @xmath4 is the standard contact @xmath14 . theorem [ isomorphism ] implies that to obtain the isomorphism between the groups @xmath54 and @xmath53 it suffices to show that statement @xmath63 of theorem [ isomorphism ] is true for the connected components @xmath9 and @xmath6 of the spaces of legendrian and of framed curves . [ conditions]_condition @xmath64 . _ in @xcite d. fuchs and s. tabachnikov showed that statement @xmath63 holds for all the connected components of the space of legendrian curves when the ambient manifold is the standard contact @xmath14 and the group @xmath5 is @xmath15 . ( one can verify that the proof of this theorem of fuchs and tabachnikov goes through for @xmath5 being any abelian group . ) they later observed @xcite that since their proof of this fact is mostly local , the similar fact should be true for a big class of contact manifolds . however in fact the proof of their theorem is not completely local and is also based on the existence of well - defined bennequin invariant and maslov number for a legendrian knot in @xmath14 . in general the bennequin invariant is not well - defined unless the knot is zero - homologous and the maslov number is not well - defined unless either the knot is zero - homologous or the contact structure is parallelizable . thus the generalization of this theorem to the case of manifolds other than @xmath14 meets certain difficulties . ( and in fact the corresponding result does not hold for a big class of contact manifolds , see section [ nonisomorphicgroups ] . ) by analyzing the proof of the theorem of fuchs and tabachnikov ( see section [ reasonstoappear ] ) we get that it can be generalized to the case of an arbitrary contact @xmath0-manifold with a cooriented contact structure , provided that the connected component @xmath6 ( containing @xmath9 ) satisfies the following * condition @xmath64 : * the connected component @xmath6 of the space of framed curves contains infinitely many components of the space of legendrian curves . ( see proposition [ interpretationconditionii ] for the homological interpretation of condition @xmath64 . ) this generalization of the result of fuchs and tabachnikov and theorem [ isomorphism ] imply the following theorem . [ isomorphismobtained ] let @xmath58 be a contact manifold with a cooriented contact structure , and let @xmath9 be a connected component of the space of legendrian curves in @xmath4 . let @xmath6 be the connected component of the space of framed curves that contains @xmath9 . let @xmath53 ( resp . @xmath54 ) be the group of @xmath5-valued order @xmath41 invariants of legendrian ( resp . framed ) knots from @xmath9 ( resp . from @xmath6 ) . then the groups @xmath53 and @xmath54 are canonically isomorphic , provided that @xmath6 satisfies condition @xmath64 . now we give a homological interpretation of condition @xmath64 . [ interpretationconditionii ] let @xmath19 be a contact manifold with a cooriented contact structure , let @xmath65 be the euler class of the contact bundle , and let @xmath6 be a component of the space of framed curves in @xmath4 . then @xmath6 does not satisfy condition @xmath64 if and only if there exists @xmath66 such that @xmath67 and @xmath23 is realizable by a mapping @xmath11 with the property that @xmath68 is a loop free homotopic to loops realized by curves from @xmath6 . for the proof of proposition [ interpretationconditionii ] see subsection [ proofinterpretationconditionii ] . [ homologysphere]_some immediate corollaries of theorem [ isomorphismobtained ] and the generalization of the theorem of fuchs and tabachnikov about the isomorphism of the groups of the @xmath15-valued vassiliev invariants in the case of @xmath69 . _ proposition [ interpretationconditionii ] implies that if the contact structure is parallelizable ( and hence the euler class of the contact bundle is zero ) then all the connected components of the space of framed curves satisfy condition @xmath64 . applying theorem [ isomorphismobtained ] we conclude that for any abelian group @xmath5 and for every connected component of the space of legendrian curves @xmath9 and for the containing it component of the space of framed curves @xmath6 the groups @xmath53 and @xmath54 of @xmath5-valued vassiliev invariants are canonically isomorphic . clearly the value of the euler class of the contact bundle is zero if @xmath4 is an integer homology sphere . hence for any abelian group @xmath5 we obtain the isomorphism of the groups @xmath53 and @xmath54 of @xmath5-valued vassiliev invariants . this generalizes the theorem of d. fuchs and s. tabachnikov @xcite saying that for the standard contact @xmath14 and for @xmath13 the quotient groups @xmath70 and @xmath71 are canonically isomorphic . the proof of this theorem of fuchs and tabachnikov was based on the fact that for the @xmath15-valued vassiliev invariants of framed knots in @xmath14 there exists the universal vassiliev invariant constructed by t. q. t. le and j. murakami @xcite . ( for unframed knots in @xmath14 the construction of the universal vassiliev invariant is the classical result of m. kontsevich @xcite , and the invariant itself is the famous kontsevich integral . ) the existence of the universal vassiliev invariant is currently known only for a very limited collection of @xmath0-manifolds , and only for @xmath5 being @xmath15 , @xmath72 , or @xmath73 . ( andersen , mattes , reshetikhin @xcite proved its existence in the case where @xmath13 and @xmath4 is the total space of an @xmath16-bundle over a compact oriented surface @xmath2 with @xmath74 . ) thus the approach used in @xcite to show the isomorphism of the quotient groups is not applicable for almost all contact @xmath0-manifolds and abelian groups @xmath5 , and theorem [ isomorphismobtained ] appears to be a strong generalization of the result of fuchs and tabachnikov . let @xmath19 be a contact manifold with a cooriented contact structure , and let @xmath6 be a connected component of the space of framed curves in @xmath4 . theorem [ isomorphismobtained ] implies that the group of @xmath5-valued order @xmath41 invariants of legendrian knots from a connected component @xmath75 of the space of legendrian curves does not depend on the choice of a cooriented contact structure , provided that for this choice @xmath6 satisfies condition @xmath64 . and hence in these cases the group can not be used to distinguish cooriented contact structures on @xmath4 . ( see remark [ homologysphere ] and theorems [ atoroidal ] and [ tight ] for the list of cases when the connected components of the space of framed curves are known to satisfy condition @xmath64 . ) [ arnoldsj+]*finite order arnold s @xmath1-type invariants of wave fronts on surfaces . * a very interesting class of contact manifolds satisfying the conditions of theorem [ isomorphismobtained ] is formed by the spherical cotangent bundles @xmath3 of surfaces @xmath2 with the natural contact structure on @xmath3 ( see [ definitions ] ) . the theory of the invariants of legendrian knots in @xmath3 is often referred to as the theory of arnold s @xcite @xmath1-type invariants of fronts on a surface @xmath2 . the natural contact structure on @xmath3 is cooriented . ( the coorientation is induced from the coorientation of the contact elements of @xmath2 . ) one can verify that for orientable @xmath2 the standard contact structure on @xmath3 is parallelizable , and hence all the components of the space of framed curves satisfy condition @xmath64 . if @xmath2 is not orientable , then the standard cooriented contact structure on @xmath3 is not parallelizable , but one can still verify ( cf . proposition 8.2.4 @xcite ) that every connected component of the space of framed curves satisfies condition @xmath64 . hence for any abelian group @xmath5 and for any surface @xmath2 we obtain the canonical isomorphism of the groups of @xmath5-valued order @xmath41 invariants of legendrian and of framed knots ( from the corresponding components of the spaces of legendrian and of framed curves in @xmath3 with the standard contact structure ) . or equivalently we get that the groups of @xmath5-valued order @xmath41 @xmath1-type invariants of fronts on @xmath2 and of @xmath5-valued order @xmath41 invariants of framed knots in @xmath3 ( from the corresponding components of the two spaces ) are canonically isomorphic . previously it was known that for @xmath76 and @xmath77 the quotient groups @xmath70 and @xmath71 are canonically isomorphic . the proof of this result of j. w. hill @xcite was based on the fact that for the @xmath15-valued vassiliev invariants of framed knots in @xmath78 there exists the universal vassiliev invariant constructed by v. goryunov @xcite . ( for unframed knots in @xmath14 the existence of the universal vassiliev invariant is the classical result of m. kontsevich @xcite , and the invariant itself is the famous kontsevich integral . ) our results generalize the result of j. w. hill ( even in the case of @xmath79 . the following theorem describes another big class of contact manifolds for which the groups of vassiliev invariants of legendrian and of framed knots ( from the corresponding components of the two spaces of curves ) are canonically isomorphic . [ atoroidal ] let @xmath19 be a contact manifold ( with a cooriented contact structure ) such that @xmath80 , and for every mapping @xmath81 of the two - torus the homomorphism @xmath82 is not injective . then all the components of the space of framed curves in @xmath4 satisfy condition @xmath64 , and hence the groups of @xmath42-valued order @xmath41 invariants of legendrian and of framed knots ( from the corresponding components of the spaces of legendrian and framed curves ) are canonically isomorphic . for the proof of theorem [ atoroidal ] see subsection [ proofatoridal ] . [ negativecurvature ] _ the isomorphism of the groups of vassiliev invariants in the case of closed manifolds admitting a metric of negative sectional curvature and other corollaries of theorem [ atoroidal ] . _ let @xmath4 be a closed manifold admitting a metric of negative sectional curvature . a well - known theorem by a. preissman ( see @xcite pp . 258 - 265 ) says that every nontrivial commutative subgroup of the fundamental group of a closed @xmath0-dimensional manifold of negative sectional curvature is infinite cyclic . hence for every mapping @xmath81 the kernel of @xmath83 is nontrivial . it is also known that the universal covering of such @xmath4 is diffeomorphic to @xmath14 , and hence @xmath80 . thus every closed manifold @xmath4 admitting a metric of negative sectional curvature satisfies all the conditions of theorem [ atoroidal ] and for an arbitrary cooriented contact structure on such @xmath4 we obtain the isomorphism of the groups of @xmath5-valued order @xmath41 invariants of legendrian and of framed knots from the corresponding components of the spaces of legendrian and of framed curves . another important class of contact manifolds for which every connected component of the space of framed curves satisfies condition @xmath64 is formed by contact manifolds with a tight contact structure . the following theorem appeared as a result of discussions of stefan nemirovski and the author . [ tight ] let @xmath19 be a contact manifold with a tight cooriented contact structure . then all the components of the space of framed curves in @xmath4 satisfy condition @xmath64 , and hence the groups of @xmath42-valued order @xmath41 invariants of legendrian and of framed knots ( from the corresponding components of the spaces of legendrian and framed curves ) are canonically isomorphic . for the proof of theorem [ tight ] see subsection [ prooftight ] . let @xmath4 be a contact manifold with a parallelized contact structure @xmath28 . let @xmath84 be a connected component of the space of transverse curves in @xmath19 , and let @xmath6 be the connected component of the space of framed curves that contains @xmath84 . ( such a component exists because a transverse curve in a manifold with a parallelized contact structure is naturally framed , and a path in the space of transverse curves corresponds to a path in the space of framed curves . ) let @xmath85 ( resp . @xmath54 ) be the group of @xmath5-valued order @xmath41 invariants of transverse ( resp . framed ) knots from @xmath84 ( resp . from @xmath6 ) . clearly every invariant @xmath55 restricted to the category of transverse knots in @xmath84 is an element @xmath86 . this gives a homomorphism @xmath87 . [ isomorphismtransverse ] let @xmath58 be a contact manifold with a parallelized contact structure . let @xmath84 be a connected component of the space of transverse curves in @xmath19 , and let @xmath6 be the component of the space of framed curves that contains @xmath84 . then the following two statements * a * and * b * are equivalent . a : : @xmath59 for any @xmath88 and any knots @xmath89 representing isotopic framed knots . b : : @xmath90 is a canonical isomorphism . the proof of theorem [ isomorphismtransverse ] is analogous to the proof of theorem [ isomorphism ] . similar to the case of theorem [ isomorphism ] the proof of theorem [ isomorphismtransverse ] becomes obvious if the mapping from the isotopy classes of transverse knots in @xmath84 to the isotopy classes of framed knots in @xmath6 is surjective . however in general this mapping is not surjective and to obtain the proof of theorem [ isomorphismtransverse ] one follows the ideas of the proof of theorem [ isomorphism ] ( the famous bennequin inequality shows that this mapping is not surjective even for the standard contact @xmath14 . ) thus to obtain the isomorphism between the groups @xmath54 and @xmath85 it suffices to show that statement @xmath63 of theorem [ isomorphismtransverse ] is true for the connected components @xmath84 and @xmath6 of the spaces of transverse and of framed curves . in @xcite d. fuchs and s. tabachnikov showed that statement @xmath63 holds for all the connected components of the space of transverse curves in the case where @xmath4 is the standard contact @xmath14 and @xmath13 . ( one can verify that the proof of this theorem of fuchs and tabachnikov goes through for @xmath5 being any abelian group . ) they later observed @xcite that since their proof of this fact is mostly local , the similar fact should be true for a big class of contact manifolds . however in fact the proof of their theorem is not completely local and is based on the existence of a well - defined bennequin invariant for a transverse knot in @xmath14 . unfortunately the bennequin invariant is not well - defined unless the knot is zero homologous . and the generalization of this theorem to the case of manifolds other than @xmath14 meets certain difficulties that are similar to the ones we meet when we generalize the analogous theorem of fuchs and tabachnikov for legendrian knots , see [ reasonstoappear ] . we imitate the arguments we use in [ reasonstoappear ] and obtain that statement * a * of theorem [ isomorphismtransverse ] is true for any contact @xmath0-manifold with a parallelized contact structure . ( observe that in the case of transverse knots , on the contrary to the case of legendrian knots , no extra conditions on the contact manifold appear to be needed for the statement * a * to be true . ) thus we get the following theorem . [ isomorphismobtainedtransverse ] let @xmath58 be a contact manifold with a parallelized contact structure , and let @xmath84 be a connected component of the space of transverse curves in @xmath4 . let @xmath6 be the connected component of the space of framed curves that contains @xmath84 . let @xmath85 ( resp . @xmath54 ) be the group of @xmath5-valued order @xmath41 invariants of transverse ( resp . framed ) knots from @xmath84 ( resp . from @xmath6 ) . then the groups @xmath85 and @xmath54 are canonically isomorphic . this generalizes the theorem of d. fuchs and s. tabachnikov @xcite saying that for the standard contact @xmath14 and for @xmath13 the quotient groups @xmath91 and @xmath71 are canonically isomorphic . the proof of this theorem of fuchs and tabachnikov was based on the fact that for the @xmath15-valued vassiliev invariants of framed knots in @xmath14 there exists the universal vassiliev invariant constructed by t. q. t. le and j. murakami @xcite . ( for unframed knots in @xmath14 the construction of the universal vassiliev invariant is the classical result of m. kontsevich @xcite , and the invariant itself is the well - known kontsevich integral . ) the existence of the universal vassiliev invariant is currently known only for a very limited collection of @xmath0-manifolds , and only for @xmath5 being @xmath15 , @xmath72 or @xmath73 . ( andersen , mattes , reshetikhin @xcite proved its existence in the case where @xmath13 and @xmath4 is the total space of an @xmath16-bundle over a compact oriented surface @xmath2 with @xmath74 . ) thus the approach used in @xcite to show the isomorphism of the quotient groups is not applicable for almost all contact @xmath0-manifolds and abelian groups @xmath5 , and theorem [ isomorphismobtainedtransverse ] appears to be a strong generalization of the result of fuchs and tabachnikov . let @xmath19 be a contact manifold with a parallelized contact structure , and let @xmath6 be a connected component of the space of framed curves in @xmath4 . theorem [ isomorphismobtainedtransverse ] implies that for any @xmath92 the group of @xmath5-valued order @xmath41 invariants of transverse knots from a connected component of the space of transverse curves contained in @xmath6 does not depend on the choice of a parallelized contact structure . hence this group can not be used to distinguish parallelized contact structures on @xmath4 . in this section we construct a big class of examples when vassiliev invariants distinguish legendrian knots that realize isotopic framed knots and are homotopic as legendrian curves . theorem [ isomorphism ] says that in these examples the groups of vassiliev invariants of legendrian and of framed knots are not canonically isomorphic , and we obtain the first known examples when these groups are not canonically isomorphic . theorem of r. lutz @xcite says that for an arbitrary orientable @xmath0-manifold @xmath4 every homotopy class of distributions of @xmath21-planes tangent to @xmath4 contains a contact structure . ( the theorem of ya . eliashberg @xcite says even more that every homotopy class of the distributions of @xmath21-planes tangent to @xmath4 contains a positive overtwisted contact structure . ) however in our constructions we will use only the euler classes of contact bundles . for this reason we start with the following proposition . [ existcontact ] let @xmath4 be an oriented @xmath0-manifold and let @xmath93 be an element of @xmath12 . then @xmath94 can be realized as the euler class of a cooriented contact structure on @xmath4 if and only if @xmath95 , for some @xmath96 . for the proof of proposition [ existcontact ] see subsection [ proofexistcontact ] . let @xmath28 be a cooriented contact structure on @xmath97 such that the euler class of the contact bundle is nonzero . ( the euler class takes values in @xmath98 , and proposition [ existcontact ] says that for any even @xmath99 there exists a cooriented contact structure on @xmath17 with the euler class @xmath100 . ) let @xmath34 be a knot in @xmath17 that crosses exactly once one of the spheres @xmath101 . the theorem of chow @xcite and rashevskii @xcite says that there exists a legendrian knot @xmath102 that is @xmath103-small isotopic to @xmath34 as an unframed knot . let @xmath104 be the legendrian knot that is the same as @xmath102 everywhere except of a small piece located in a chart contactomorphic to the standard contact @xmath14 where it is changed as it is shown in figure [ change.fig ] ( see [ description ] ) . [ example1 ] a : : legendrian knots @xmath102 and @xmath104 belong to the same component of the space of legendrian curves and realize isotopic framed knots . b : : there exists a @xmath18-valued order one invariant @xmath105 of legendrian knots , such that @xmath106 . for the proof of theorem [ example1 ] see subsection [ proofexample1 ] . let @xmath107 , @xmath108 , be the knot that is the same as @xmath102 everywhere except of a small piece located in a chart contactomorphic to the standard contact @xmath14 where it is changed in the way described by the addition of @xmath100 zigzags shown in figure [ change.fig ] . the proof of theorem [ example1 ] implies that all @xmath107 s are homotopic as legendrian curves and realize isotopic framed knots , but for all @xmath109 legendrian knots @xmath110 and @xmath111 are not legendrian isotopic . the order one invariant of legendrian knots @xmath105 constructed in the proof of theorem [ example1 ] has the property that @xmath112 . hence this @xmath105 distinguishes all the @xmath107 s . [ overtwisted]*examples of nonisotopic legendrian knots with overtwisted complements that realize isotopic framed knots and are homotopic as legendrian immersions . * let @xmath113 be an embedded into @xmath4 disk centered at a point @xmath114 . the theorem of eliashberg @xcite says that every homotopy class of distributions of @xmath21-planes tangent to @xmath4 contains an overtwisted contact structure that has @xmath113 as the standard overtwisted disk . in the example of theorem [ example1 ] we can start with an overtwisted contact structure that has @xmath113 as an overtwisted disk and with an unframed knot @xmath34 that is far away from @xmath113 . then since both @xmath102 and @xmath104 were constructed using a @xmath103-small approximation of @xmath34 , we can assume that they are also far away from @xmath113 . and we have constructed examples of nonisotopic legendrian knots with overtwisted complements that realize isotopic framed knots and are homotopic as legendrian immersions . previously such examples were unknown and the theorem of ya . eliashberg and m. fraser @xcite says that such examples are impossible if the ambient manifold is @xmath115 . [ otherexamples ] _ below we describe another big family of examples where finite order invariants distinguish legendrian knots that realize isotopic framed knots and are homotopic as legendrian immersions . _ let @xmath2 be a nonorientable surface that can be decomposed as a connected sum of the klein bottle @xmath34 and a surface @xmath116 . let @xmath4 be an orientable manifold that admits a structure of a locally trivial @xmath7-fibration @xmath117 . ( for example one can take @xmath4 to be the spherical tangent bundle @xmath118 of @xmath2 . ) consider an @xmath7-fibration @xmath119 induced from @xmath120 by the mapping @xmath121 that corresponds to the solid loop in figure [ example2.fig ] . ( in this figure the enumeration of the end points of the arcs indicates which pairs of points should be identified to obtain the loop . ) since the solid loop is an orientation preserving loop in @xmath2 , we get that @xmath122 ( torus ) . put @xmath123 to be the natural mapping of the total space of the induced fibration @xmath119 into the total space of @xmath117 . a homology class in @xmath124 projecting to the dashed loop in figure [ example2.fig ] has intersection @xmath22 with the class @xmath125\in h_2(m , \z)$ ] realized by @xmath126 . thus there exists @xmath127 such that @xmath128)=1 $ ] . proposition [ existcontact ] says that for every @xmath129 the class @xmath130 is realizable as the euler class of a cooriented contact structure on @xmath4 . thus for every @xmath129 there exists a cooriented contact structure on @xmath4 such that the value of the euler class of the contact bundle on @xmath125 $ ] is equal to @xmath131 . let @xmath28 be a cooriented contact structure on @xmath4 such that the euler class @xmath132 of the contact bundle satisfies @xmath133)=2r$ ] , for some nonzero @xmath129 . let @xmath34 be an arbitrary legendrian knot such that its projection to @xmath2 ( considered as a loop ) is free homotopic to the solid loop in figure [ example2.fig ] . let @xmath134 be legendrian knots that are the same as @xmath34 everywhere except of a chart ( contactomorphic to the standard contact @xmath14 ) where @xmath104 and @xmath135 are different from @xmath34 as it is described in figure [ example4.fig ] , see [ description ] . ( the number of cusps in figure [ example4.fig ] is @xmath133)=2r\neq 0 $ ] . ) [ example2 ] the knots @xmath104 and @xmath135 described above belong to the same component @xmath9 of the space of legendrian curves and realize isotopic framed knots . there exists a @xmath18-valued order one invariant @xmath105 of legendrian knots from @xmath9 such that @xmath136 . for the proof of theorem [ example2 ] see subsection [ proofexample2 ] . similarly to [ overtwisted ] one verifies that the contact structure and the knots @xmath104 and @xmath135 in the statement of theorem [ example2 ] can be chosen so that the restrictions of the contact structure to the complements of @xmath104 and of @xmath135 are overtwisted . using the ideas of the proof of theorem [ example2 ] one can construct many other examples of legendrian knots that can be distinguished by vassiliev invariants of legendrian knots even though they realize isotopic framed knots and are homotopic as legendrian immersions . for example as a solid loop in figure [ example2.fig ] we could take any loop @xmath137 such that the number of double points that separate @xmath137 into two orientation reversing loops is odd , and the value of the euler class of the contact bundle on @xmath125\in h_2(m,\z)$ ] is nonzero . [ commute ] let @xmath138 be a locally trivial @xmath7-fibration of an oriented manifold @xmath139 over a ( not necessarily orientable ) manifold @xmath140 . let @xmath141 be the class of an oriented @xmath7-fiber of @xmath120 , and let @xmath23 be an element of @xmath142 . then : _ proof of proposition [ commute ] . _ if we move an oriented fiber along the loop @xmath146 , then in the end it comes to itself either with the same or with the opposite orientation . it is easy to see that it comes to itself with the opposite orientation if and only if @xmath144 is an orientation reversing loop in @xmath140 . _ proof of proposition [ preissman ] . _ it is well - known that any closed @xmath2 , other than @xmath150 admits a hyperbolic metric . ( it is induced from the universal covering of @xmath2 by the hyperbolic plane . ) the theorem of a. preissman ( see @xcite pp . 258 - 265 ) says that if @xmath4 is a closed riemannian manifold of negative sectional curvature , then any nontrivial abelian subgroup @xmath151 is isomorphic to @xmath18 . thus if @xmath152 is closed , then any nontrivial commutative @xmath153 is infinite cyclic . if @xmath2 is not closed , then the statement of the proposition is also true because in this case @xmath2 is homotopy equivalent to a bouquet of circles . [ toughandtechnical ] let @xmath154 ( klein bottle ) be a surface not necessarily closed or orientable . let @xmath4 be an orientable @xmath0-manifold , and let @xmath117 be a locally trivial @xmath7-fibration . let @xmath155 be the class of an oriented @xmath7-fiber of @xmath120 , and let @xmath156 be an element with @xmath157 . let @xmath137 be an element of the centralizer @xmath158 of @xmath23 . then there exist @xmath159 and nonzero @xmath160 such that @xmath161 . [ prooftoughandtechnical ] _ proof of proposition [ toughandtechnical ] . _ since @xmath23 and @xmath137 commute in @xmath162 we get that @xmath163 and @xmath164 commute in @xmath149 . proposition [ preissman ] and the fact that @xmath157 imply that there exist @xmath165 with @xmath166 , @xmath99 , and nonzero @xmath160 such that @xmath167 and @xmath168 . _ an important homomorphism._[hansen ] let @xmath139 be a manifold , let @xmath171 be the space of free loops in @xmath139 , and let @xmath172 be a loop . an element @xmath173 is realizable by a mapping @xmath174 with @xmath175 . let @xmath176 be the element corresponding to the trace of the point @xmath177 under the homotopy of @xmath178 described by @xmath23 . let @xmath179 be the homomorphism that maps @xmath180 to @xmath181 . since the @xmath21-cell of @xmath182 is glued to the @xmath22-skeleton along the commutation relation of the meridian and of the longitude of @xmath182 , we get that @xmath179 is a surjective homomorphism of @xmath183 onto the centralizer @xmath184 of @xmath185 . if @xmath186 for @xmath187 , then the mappings @xmath188 and @xmath189 of @xmath182 corresponding to these loops can be deformed to be identical on the @xmath22-skeleton of @xmath182 . clearly the obstruction for @xmath188 and @xmath189 to be homotopic as mappings of @xmath182 ( with the mapping of the @xmath22-skeleton of @xmath182 fixed under homotopy ) is an element of @xmath190 obtained by gluing together the boundaries of the @xmath21-cells of the two tori . in particular we get the proposition of v. l. hansen @xcite saying that @xmath191 is an isomorphism , provided that @xmath192 . [ h - principleforcurves ] _ @xmath43-principle for curves in @xmath4 . _ for a @xmath0-dimensional manifold @xmath4 we put @xmath193 to be the manifold obtained by the fiberwise spherization of the tangent bundle of @xmath4 , and we put @xmath194 to be the corresponding locally trivial @xmath195-fibration . the @xmath43-principle ( that can be found in @xcite ) says that the space of curves in @xmath4 is weak homotopy equivalent to @xmath196 ( the space of free loops in @xmath193 ) . the weak homotopy equivalence is given by mapping a curve @xmath34 to a loop @xmath197 that sends a point @xmath198 to the point of @xmath193 corresponding to the direction of the velocity vector of @xmath34 at @xmath199 . [ obstruction ] let @xmath104 and @xmath135 be two framed knots that coincide pointwise as embeddings of @xmath7 . then there is an integer obstruction @xmath200 for them to be isotopic as framed knots with the embeddings of @xmath7 fixed under the isotopy . this obstruction is calculated as follows . let @xmath201 be the knot obtained by shifting @xmath104 along the framing and reversing the orientation on the shifted copy . together @xmath104 and @xmath201 bound a thin strip . we put @xmath202 to be the intersection number of the strip with a very small shift of @xmath135 along its framing . for two singular framed knots @xmath207 and @xmath208 with @xmath32 transverse double points that coincide pointwise as immersions of @xmath7 , we put @xmath209 to be the value of @xmath210 on the nonsingular framed knots @xmath104 and @xmath135 that coincide pointwise as embeddings of @xmath7 and are obtained from @xmath207 and @xmath208 by resolving each pair of the corresponding double points of @xmath207 and of @xmath208 in the same way . ( the value @xmath211 does not depend on the resolution as soon as the corresponding double points of the two knots are resolved in exactly the same way . ) as before @xmath211 is the integer valued obstruction for @xmath207 and @xmath208 to be isotopic as singular framed knots with the immersion of @xmath7 corresponding to the two knots fixed under isotopy . for a singular framed knot @xmath212 with @xmath32 transverse double points we denote by @xmath213 , @xmath99 , the isotopy class of a singular framed knot with @xmath32 transverse double points that coincides with @xmath212 as an immersion of @xmath7 and has @xmath214 . [ homotopy ] let @xmath104 and @xmath135 be framed knots ( resp . singular framed knots with @xmath32 transverse double points ) that coincide pointwise as embeddings ( resp . immersions ) of @xmath7 . then @xmath104 and @xmath135 are homotopic as framed knots ( resp . singular framed knots with @xmath32 transverse double points ) if and only if @xmath202 is even . [ proofhomotopy]_proof of proposition [ homotopy ] . _ clearly if @xmath202 is even , then @xmath104 and @xmath135 are framed homotopic . ( we can change the obstruction by two by creating a small kink and passing through a double point at its vertex . ) every oriented @xmath0-dimensional manifold @xmath4 is parallelizable , and hence it admits a @xmath215-structure . a framed curve @xmath34 in @xmath4 represents a loop in the principal @xmath216-bundle of @xmath217 . ( the @xmath0-frame corresponding to a point of @xmath34 is the velocity vector , the framing vector , and the unique third vector of unit length such that the @xmath0-frame defines the positive orientation of @xmath4 . ) one observes that the values of the @xmath215-structure on the loops in the principal @xmath216-bundle of @xmath217 realized by @xmath104 and @xmath135 are different provided that @xmath202 is odd . but these values do not change under homotopy of framed curves . hence if @xmath202 is odd , then @xmath104 and @xmath135 are not framed homotopic . [ strange ] using the self - linking invariant of framed knots one can easily show that if @xmath104 and @xmath135 in [ obstruction ] are pointwise coinciding zero - homologous framed knots and @xmath218 , then @xmath104 is not isotopic to @xmath135 in the category of framed knots . however for knots that are not zero - homologous this is not generally true , see [ proofaexample1 ] . for this reason we introduce the following definitions . if for an unframed knot @xmath34 there exist isotopic framed knots @xmath104 and @xmath135 that coincide with @xmath34 pointwise and have @xmath218 , then we say that @xmath34 _ admits finitely many framings_. for @xmath34 that admits finitely many framings we put _ the number of framings @xmath219 of _ @xmath34 to be the minimal positive integer @xmath220 such that there exist isotopic framed knots @xmath104 and @xmath135 that coincide with @xmath34 pointwise and have @xmath221 . one can easily show that if @xmath34 admits finitely many framings , then there are exactly @xmath219 isotopy classes of framed knots realizing the isotopy class of the unframed knot @xmath34 . proposition [ homotopy ] implies that @xmath219 is even . [ decrease ] let @xmath19 be a contact @xmath0-manifold with a cooriented contact structure , let @xmath6 be a connected component of the space of framed curves , and let @xmath8 be a connected component of the space of legendrian curves in @xmath58 . a : : let @xmath34 be an unframed knot obtained by forgetting the framing on a knot from @xmath6 . then there exists a legendrian knot from @xmath222 realizing the isotopy class of @xmath34 . b : : if @xmath203 is an isotopy class of framed knots in @xmath6 that is realizable by a legendrian knot from @xmath9 , then the isotopy class of @xmath223 ( see [ obstruction ] ) is also realizable by a legendrian knot from @xmath9 . c : : let @xmath40 be an unframed singular knot with @xmath32 double points obtained by forgetting the framing on a singular knot from @xmath6 . then there exists a singular legendrian knot from @xmath224 realizing the isotopy class of @xmath40 . d : : if @xmath225 is an isotopy class of singular framed knots in @xmath6 that is realizable by a singular legendrian knot from @xmath9 , then the isotopy class of @xmath226 is also realizable by a singular legendrian knot from @xmath9 . [ proofdecrease ] _ proof of statement * a * of proposition [ decrease ] . _ let @xmath44 be the fiberwise spherization of the @xmath21-dimensional contact vector bundle , and let @xmath45 be the corresponding locally trivial @xmath7-fibration . we denote by @xmath227 the class of an oriented @xmath7-fiber of @xmath228 . for a legendrian curve @xmath229 denote by @xmath230 the loop in @xmath44 obtained by mapping a point @xmath198 to the point of @xmath44 corresponding to the direction of the velocity vector of @xmath31 at @xmath231 . the @xmath43-principle [ h - principlelegendrian ] says that legendrian curves @xmath104 and @xmath135 in @xmath4 belong to the same component of the space of legendrian curves in @xmath4 if and only if @xmath232 and @xmath233 are free homotopic loops in @xmath44 . w. l. chow @xcite and p. k. rashevskii @xcite showed that every unframed knot @xmath34 is isotopic to a legendrian knot @xmath31 ( and this isotopy can be made @xmath103-small ) . deforming @xmath34 we can assume ( see [ h - principleforcurves ] ) that : * 1 : * @xmath34 and @xmath31 coincide in the neighborhood of @xmath234 , * 2 : * @xmath34 and @xmath31 realize the same element @xmath235\in \pi_1(m , k_l(1))$ ] , and * 3 : * that liftings to @xmath44 of legendrian curves from @xmath9 are free homotopic to a loop @xmath23 in @xmath44 such that @xmath236 and @xmath237 \in \pi_1(m , k_l(1))$ ] . take a chart of @xmath4 ( that is contactomorphic to the standard contact @xmath14 ) containing a piece of the legendrian knot . from the formula for the maslov number deduced in @xcite it is easy to see that the modifications of the legendrian knot corresponding to the insertions of two cusps shown in figure [ twocusp.fig ] ( see [ description ] ) induce multiplication by @xmath241 of the lifting of @xmath31 to an element of @xmath242 . ( here the sign depends on the choice of an orientation of the fiber used to define @xmath238 . ) performing this operation sufficiently many times we obtain the legendrian knot from @xmath9 realizing the isotopy class of the unframed knot @xmath34 . _ proof of statement * b * of proposition [ decrease ] . _ take a chart of @xmath4 ( that is contactomorphic to the standard contact @xmath14 ) containing a piece of the knot @xmath203 and perform the homotopy in @xmath9 shown in figure [ kink.fig ] , see [ description ] . ( observe that a self - tangency point of the projection of a legendrian curve in @xmath14 to the @xmath48-plane corresponds to a double point of the legendrian curve . ) straightforward verification ( cf . the formula for the bennequin invariant deduced in @xcite ) shows that the legendrian knot we obtain in the end of the homotopy realizes @xmath223 . the fact that statement * b * of theorem [ isomorphism ] implies statement * a * is clear . thus we have to show that statement * a * implies statement . this is done by showing that there exists a homomorphism @xmath243 such that @xmath244 and @xmath245 . [ definitionofpsi]_definition of @xmath249 . _ if the isotopy class of the knot @xmath250 is realizable by a legendrian knot @xmath251 , then put @xmath252 . the value @xmath253 is well - defined because if @xmath254 is another knot realizing @xmath34 , then @xmath255 by statement * a * of theorem [ isomorphism ] . let @xmath256 be the component of the space of unframed curves that corresponds to forgetting framings on the curves from @xmath6 . propositions [ decrease ] and [ homotopy ] imply that if an unframed knot @xmath257 admits finitely many framings ( see [ strange ] ) , then all the isotopy classes of framed knots from @xmath6 realizing the isotopy class of the unframed knot @xmath258 are realizable by legendrian knots from @xmath9 . thus we have defined the value of @xmath249 on all the framed knots from @xmath6 that realize unframed knots admitting finitely many framings . if @xmath257 admits infinitely many framings , then either * 1 ) * all the isotopy classes of framed knots from @xmath6 realizing the isotopy class of @xmath258 are realizable by legendrian knots from @xmath9 or * 2 ) * there exists a knot @xmath259 realizing the isotopy class of @xmath258 such that @xmath203 is realizable by a legendrian knot from @xmath9 and @xmath260 ( see [ obstruction ] ) is not realizable by a legendrian knot from @xmath9 . ( in this case @xmath261 etc . also are not realizable by legendrian knots from @xmath9 , see [ decrease ] . ) in the case * 1 ) * the value of @xmath249 is already defined on all the framed knots from @xmath6 realizing @xmath258 . in the case * 2 ) * put @xmath262 ( proposition [ decrease ] implies that the sum on the right hand side is well - defined . ) similarly put @xmath263 now we have defined @xmath249 on all the framed knots ( from @xmath6 ) realizing @xmath258 . doing this for all @xmath258 for which case * 2 ) * holds we define the value of @xmath249 on all the knots from @xmath6 . if @xmath260 is realizable by a legendrian knot @xmath265 , then consider a singular legendrian knot @xmath266 with @xmath35 double points that are vertices of @xmath35 small kinks such that we get @xmath31 if we resolve all the double points positively staying in the class of the legendrian knots . ( to create @xmath266 we perform the first half of the homotopy shown in figure [ kink.fig ] in @xmath267 places on @xmath265 . ) let @xmath268 be the set of the @xmath36 possible resolutions of the double points of @xmath266 . for @xmath269 put @xmath270 to be the sign of the resolution , and put @xmath271 to be the nonsingular legendrian knot obtained via the resolution @xmath272 . since @xmath39 is an order @xmath41 invariant of legendrian knots we get that @xmath273 ( observe that if we resolve @xmath100 double points of @xmath266 negatively , then we get the isotopy class of @xmath274 . ) this finishes the proof of the proposition . let @xmath275 be a singular framed knot with @xmath35 double points . let @xmath268 be the set of the @xmath36 possible resolutions of the double points of @xmath40 . for @xmath276 put @xmath270 to be the sign of the resolution , and put @xmath277 to be the isotopy class of the knot obtained via the resolution @xmath272 . if the isotopy class of @xmath40 is realizable by a singular legendrian knot from @xmath9 , then identity holds for @xmath40 , since @xmath39 is an order @xmath41 invariant of legendrian knots ( and the value of @xmath249 on a framed knot @xmath248 realizable by a legendrian knot @xmath251 was put to be @xmath280 ) . if @xmath281 admits finitely many framings , then all the isotopy classes of singular framed knots from @xmath6 realizing the isotopy class of @xmath281 are realizable by singular legendrian knots from @xmath9 , and we get that identity holds for @xmath40 . if @xmath281 admits infinitely many framings and all the isotopy classes of singular framed knots from @xmath6 realizing @xmath281 are realizable by singular legendrian knots from @xmath9 , then automatically holds for @xmath40 . if @xmath281 admits infinitely many framings but not all the isotopy classes of singular framed knots from @xmath6 realizing @xmath281 are realizable by singular legendrian knots from @xmath9 then put @xmath282 to be the framed knot realizing @xmath281 that is realizable by a singular legendrian knot from @xmath9 and such that @xmath283 , @xmath284 , are not realizable by singular legendrian knots from @xmath9 . proposition [ decrease ] says that @xmath285 , @xmath284 , are realizable by singular legendrian knots from @xmath9 and hence identity holds for @xmath285 , @xmath286 . using proposition [ mainidentity ] and the fact that identity holds for @xmath285 , @xmath286 , we show that holds for @xmath287 . namely , @xmath288 considering the values of @xmath55 on the @xmath36 possible resolutions of a singular framed knot with @xmath267 singular small kinks we get that @xmath51 should satisfy identity . hence @xmath245 and this finishes the proof of theorem [ isomorphism ] . the proof of the theorem of fuchs and tabachnikov that says that statement @xmath63 of theorem [ isomorphism ] is true for all the connected components of the space of legendrian curves when the ambient contact manifold is the standard contact @xmath14 is based on the following three observations : 1 : : there are two types of cusps arising under the projection of the part of a legendrian knot that is contained in a darboux chart to the @xmath48-plane ( see [ description ] ) . they are formed by cusps for which the branch of the projection of the knot going away from the cusp is locally located respectively above or below the tangent line at the cusp point , see figure [ twocusp.fig ] . for a legendrian knot @xmath34 and @xmath290 we denote by @xmath291 the legendrian knot obtained from @xmath34 by the modification corresponding to an addition of @xmath100 cusp pairs of the first type and @xmath292 cusp pairs of the second type to the projection of the part of @xmath34 located in a darboux chart . + let @xmath104 and @xmath135 be legendrian knots in the standard contact @xmath14 that realize isotopic unframed knots . then for any @xmath293 and @xmath294 large enough there exist @xmath295 such that the legendrian knot @xmath296 is legendrian isotopic to @xmath297 . 2 : : if there exists @xmath92 such that legendrian knots @xmath298 and @xmath299 are legendrian isotopic , then every vassiliev invariant of legendrian knots takes equal values on @xmath104 and on @xmath135 . 3 : : the number @xmath32 from the previous observation exists if the ambient contact manifold is @xmath14 and the legendrian knots @xmath104 and @xmath135 belong to the same component of the space of legendrian curves and realize isotopic framed knots . the first two observation are true for any contact @xmath0-manifold ( since the proof of the corresponding facts is local ) . but the number @xmath32 from the statement of the third observation does not exist in general . in the case of the ambient manifold being @xmath14 fuchs and tabachnikov showed the existence of such @xmath32 using the explicit calculation involving the maslov classes and bennequin invariants of legendrian knots . however in order for the bennequin invariant to be well - defined the knots have to be zero - homologous , and in order for the maslov class to be well - defined the knots have to be zero - homologous or the contact structure has to be parallelizable . below we show that such @xmath32 exists for any @xmath104 and @xmath135 that realize isotopic framed knots and belong to the same component of the space of legendrian curves , provided that the connected component @xmath6 of the space of framed curves that contains @xmath104 and @xmath135 satisfies condition @xmath64 . ( we assume that the contact structure on @xmath4 is cooriented . ) _ proof of the fact that @xmath302 can be chosen so that @xmath303 . _ let @xmath305\rightarrow m$ ] be the isotopy changing @xmath104 to @xmath135 in the category of framed knots . analyzing the proof of fuchs and tabachnikov one verifies that for @xmath306 large enough the legendrian isotopy @xmath307 changing @xmath301 to @xmath308 can be chosen so that for every @xmath309 $ ] the legendrian knot @xmath310 is contained in a thin tubular neighborhood @xmath311 of @xmath312 and is isotopic ( as an unframed knot ) to @xmath313 inside @xmath311 . for two framed knots @xmath313 and @xmath314 realizing unframed knots that are isotopic inside @xmath311 there is a well - defined @xmath18-valued obstruction to be isotopic inside @xmath311 in the category of framed knots . this obstruction is the difference of the self - linking numbers of the inclusions of @xmath313 and @xmath315 into @xmath14 induced by an identification of @xmath311 with the standard solid torus in @xmath14 . ( one verifies that for @xmath313 and @xmath315 that are isotopic as unframed knots inside @xmath311 this difference does not depend on the choice of the identification of @xmath311 with the standard solid torus in @xmath14 . ) from the formula for the bennequin invariant stated in @xcite one gets that the value of the obstruction for @xmath301 to be isotopic as a framed knot to @xmath104 inside @xmath316 is equal to @xmath317 . similarly the value of the obstruction for @xmath308 to be isotopic as a framed knot to @xmath135 inside @xmath318 is equal to @xmath319 . clearly the value of the obstruction for @xmath320 to be isotopic to @xmath321 inside @xmath311 does not depend on @xmath33 ( for the isotopy @xmath322 changing @xmath104 to @xmath135 in the category of framed knots ) , and we get that @xmath303 . _ proof of the fact that if @xmath6 satisfies condition @xmath64 , then @xmath304 . _ let @xmath323 be the class of an oriented @xmath7-fiber of @xmath45 . from the @xmath43-principles for legendrian and for unframed curves ( see [ h - principlelegendrian ] and [ h - principleforcurves ] ) one obtains that every component of the space of legendrian curves contained in @xmath6 corresponds to the conjugacy class of @xmath324 , for some @xmath325 . ( connected components of the space of free loops in @xmath44 are naturally identified with the conjugacy classes of the elements of @xmath47 . ) from the formula for the maslov number deduced in @xcite and the @xmath43-principle for legendrian curves one gets that @xmath296 is contained in the component of the space of legendrian curves that corresponds to the conjugacy class of @xmath327 . using the fact that @xmath232 and @xmath233 are conjugate in @xmath47 ( since @xmath104 and @xmath135 are legendrian homotopic ) and the fact that since the contact structure is cooriented @xmath238 is in the center of @xmath47 ( see [ commute ] ) , we get that @xmath297 is contained in the component that corresponds to the conjugacy class of @xmath328 . since @xmath296 and @xmath297 are legendrian isotopic ( and hence legendrian homotopic ) we get that @xmath329 and @xmath330 are conjugate in @xmath47 , and using [ commute ] we get that @xmath232 is conjugate to @xmath331 . but since @xmath6 satisfies condition @xmath64 we have @xmath332 , and hence @xmath304 . from the identities @xmath303 and @xmath304 one gets that @xmath333 and @xmath334 . assume that @xmath335 . ( the case where @xmath336 is treated similarly . ) put @xmath337 . it is easy to show that since @xmath301 and @xmath297 are legendrian isotopic , then @xmath338 and @xmath339 are also legendrian isotopic . ( basically one can keep the @xmath340 extra cusp pairs close together on a small piece of the projection of the part of the knot contained in a darboux chart during the whole isotopy process . ) but @xmath338 and @xmath339 are obtained from @xmath104 and @xmath135 by the modification corresponding to the addition of @xmath341 pairs of cusps of each of the two types , and we can take @xmath32 from the observation * 2 * to be @xmath341 . this shows that @xmath104 and @xmath135 can not be distinguished by the vassiliev invariants of legendrian knots provided that @xmath6 satisfies condition @xmath64 , and that @xmath104 and @xmath135 realize isotopic framed knots and are homotopic as legendrian immersions . hence statement * a * of theorem [ isomorphism ] is true provided that @xmath6 satisfies condition @xmath64 . the @xmath43-principle for curves [ h - principleforcurves ] says that the set @xmath256 of the connected components of the space of curves in @xmath4 is naturally identified with the set of the connected components of the space of free loops in the spherical tangent bundle @xmath193 of @xmath4 . hence it is also naturally identified with the set of conjugacy classes of the elements of @xmath342 . ( from the long homotopy sequence of the fibration @xmath194 we see that it is also naturally identified with the set of conjugacy classes of the elements of @xmath162 . ) choose a @xmath215-structure on @xmath4 . it is easy to see ( cf . [ homotopy ] and [ proofhomotopy ] ) that the set @xmath343 of the connected components of the space of framed curves in @xmath4 is identified with the product @xmath344 . here the @xmath345-factor is the value of the @xmath215-structure on the loop in the principal @xmath216-bundle of @xmath217 that corresponds to a framed curve from the connected component , see [ proofhomotopy ] . ( this value does not depend on the choice of the framed curve in the component . ) the @xmath43-principle for the legendrian curves says that the set of the connected components of the space of legendrian curves is naturally identified with the set of homotopy classes of free loops in @xmath44 ( the spherical contact bundle of @xmath4 ) . hence it is also naturally identified with the set of conjugacy classes of the elements of @xmath47 . since every contact manifold is oriented and the contact structure was assumed to be cooriented , we get that the planes of the contact structure are naturally oriented . this orientation induces the orientation of the @xmath7-fibers of @xmath45 . put @xmath323 to be the class of the oriented @xmath7-fiber of @xmath45 . the theorem of chow @xcite and rashevskii @xcite says that every connected component of the space of curves contains a legendrian curve . straightforward verification shows that the insertion of the zig - zag into the legendrian curve @xmath34 ( see figure [ twocusp.fig ] ) changes the value of the @xmath215-structure on the corresponding framed curve . it is easy to verify ( see @xcite ) that the two connected components of the space of legendrian curves that contain respectively @xmath34 and @xmath34 with the extra zigzag correspond to the conjugacy classes of @xmath346 and of @xmath347 ( or of @xmath348 ) in @xmath47 . ( we obtain @xmath347 or @xmath348 depending on which of the two possible zig - zags we insert . ) let @xmath8 be a connected component of the space of legendrian curves in @xmath58 that corresponds to the conjugacy class of @xmath349 . then every connected component @xmath350 of the space of legendrian curves corresponds to the conjugacy class of @xmath351 , for some @xmath160 . put @xmath93 to be the euler class of the contact bundle . consider the locally - trivial @xmath7-fibration @xmath362 induced by @xmath322 from the @xmath7-fibration @xmath363 . one can verify that @xmath364 is the euler class of @xmath120 . on the other hand the euler class of @xmath120 is @xmath365 and it is naturally identified with the value of @xmath93 on the homology class realized by @xmath126 . this implies that if @xmath6 does not satisfy condition @xmath64 , then there exists a homology class @xmath23 from the statement of the proposition . on the other hand the existence of the class @xmath23 from the statement of the proposition implies that there exists a legendrian curve @xmath366 such that @xmath346 is conjugate to @xmath367 , for @xmath32 being the value of @xmath93 ( the euler class of the contact bundle ) on the homology class realized by @xmath126 . ( proposition [ existcontact ] says that @xmath95 , for some @xmath368 , and hence @xmath32 is even . ) this means that @xmath6 does not satisfy condition @xmath64 and we have proved proposition [ interpretationconditionii ] . similar to [ proofinterpretationconditionii ] we get that to prove that all the components of the space of framed curves satisfy condition @xmath64 , it suffices to show that @xmath346 and @xmath353 are not conjugate in @xmath47 , for all @xmath369 and @xmath370 . let @xmath371 and @xmath160 be such that @xmath372 we have to show that @xmath373 . identity implies that @xmath356 and @xmath376 commute in @xmath162 . hence there exists a mapping of the two - torus @xmath377 such that @xmath378 and @xmath379 . by the assumption of the theorem @xmath380 has a nontrivial kernel . thus there exist @xmath159 with at least one of @xmath100 and @xmath292 being nonzero such that @xmath381 , and hence @xmath382 thus @xmath384 . applying to the last identity we get that @xmath385 . since @xmath80 we see that @xmath238 has infinite order in @xmath47 , and hence @xmath386 . if @xmath32 is zero , then we are done . hence we have to look at the case of @xmath387 . ( we assumed that @xmath383 . ) from we get that @xmath388 , and hence by proposition [ commute ] @xmath389 is in the center of @xmath47 . thus @xmath390 . on the other hand using we get that @xmath391 . since @xmath238 has infinite order in @xmath47 we get that @xmath392 . by our assumptions @xmath383 and we have @xmath373 . this finishes the proof of theorem [ atoroidal ] . let @xmath94 be the euler class of the contact bundle of @xmath19 . proposition [ interpretationconditionii ] implies that it suffices to show that @xmath393 , for every homology class @xmath394 realizable by a mapping @xmath81 . the result of d. gabai ( see corollary 6.18 @xcite ) implies that every @xmath395 realizable by a mapping of @xmath182 can be realized by a collection of spheres and of a torus that are embedded into @xmath4 . finally the result of ya . eliashberg ( see @xcite theorem @xmath396 ) says that for a tight contact structure the value of @xmath93 on any embedded torus or sphere is zero . hence @xmath393 . this finishes the proof of theorem [ tight ] . since the contact structure is cooriented we get that the tangent bundle @xmath217 is isomorphic to the sum @xmath398 of the oriented contact bundle @xmath28 and the trivial oriented line bundle @xmath399 . the tangent bundle of every orientable @xmath0-manifold is trivializable and we get that the second stiefel - whitney class of the contact bundle is zero . but the second stiefel - whitney of @xmath28 is the projection of the euler class of @xmath28 under the natural mapping @xmath400 , and we get that @xmath95 for some @xmath397 . consider an oriented @xmath21-dimensional vector bundle @xmath401 over @xmath4 with the euler class @xmath402 . the second stiefel - whitney class @xmath403 of @xmath401 is zero , since it is the projection of @xmath404 . since @xmath401 is an oriented vector bundle we have @xmath405 . consider the sum @xmath406 of @xmath401 with the trivial oriented @xmath22-dimensional vector bundle @xmath399 . clearly the total stiefel - whitney class of the @xmath0-dimensional oriented vector bundle @xmath406 is equal to @xmath22 , and the euler class of @xmath406 is equal to @xmath407 . using the interpretation of the stiefel - whitney and the euler classes of @xmath406 as obstructions for the trivialization of @xmath406 , we get that @xmath406 is trivializable . since the tangent bundle of an oriented @xmath0-dimensional manifold is trivializable , we see that @xmath401 is isomorphic to an oriented sub - bundle of @xmath217 . since @xmath4 is oriented this sub - bundle of @xmath217 is also cooriented now the theorem of lutz @xcite , that says that every homotopy class of distributions of @xmath21-planes tangent to @xmath4 contains a contact structure , implies the existence of a cooriented contact structure with the euler class @xmath93 . [ proofaexample1 ] _ proof of statement * a * of theorem [ example1 ] . _ clearly ( see figure [ homotopy.fig ] ) the two legendrian knots @xmath102 and @xmath104 belong to the same component of the space of legendrian curves . it is easy to see that if @xmath102 realizes the isotopy class of a framed knot @xmath408 , then @xmath104 realizes the isotopy class of @xmath409 ( see [ obstruction ] for the definition of @xmath409 ) . below we show that @xmath408 and @xmath409 are isotopic framed knots . let @xmath410 be the sphere that crosses @xmath408 at exactly one point , and let @xmath411\times s^2 $ ] be a thin tubular neighborhood of @xmath412 . fix @xmath413 ( below called the north pole ) and the direction in @xmath414 ( below called the zero meridian ) . we can assume that the knot @xmath408 inside @xmath411\times s^2 $ ] looks as follows : it intersects each @xmath415\times s^2 $ ] at the north pole of the corresponding sphere , and the framing of the knot is parallel to the zero meridian . consider an automorphism @xmath416 that is identical outside of @xmath411\times s^2 $ ] such that it rotates each @xmath417\times s^2 $ ] by @xmath418 around the north pole in the clockwise direction . clearly under this automorphism @xmath408 gets two extra negative twists of the framing and @xmath419 . on the other hand it is easy to see that @xmath420 is diffeotopic to the identity , since it corresponds to the contractible loop in @xmath421 . hence we see that @xmath408 and @xmath409 are isotopic framed knots . this finishes the proof of statement * a * of theorem [ example1 ] . [ props1xs2 ] let @xmath28 be a cooriented contact structure on @xmath97 with a nonzero euler class @xmath93 of the contact bundle . let @xmath44 be the spherical contact bundle , let @xmath45 be the corresponding locally trivial @xmath7-fibration , and let @xmath323 be the class of an oriented @xmath7-fiber of @xmath228 . then @xmath238 is of finite order in @xmath47 and @xmath422 . _ proof of proposition [ props1xs2 ] . _ consider the oriented @xmath21-plane bundle @xmath423 that is the restriction of the contact bundle over @xmath4 to the sphere @xmath424 . the euler class of @xmath120 is the value of @xmath93 on the homology class realized by @xmath425 , and hence is nonzero . let @xmath426 be the manifold obtained by the fiberwise spherization of @xmath120 , and let @xmath427 be the corresponding locally trivial @xmath7-fibration . since the euler class of @xmath120 is nonzero we get that a certain multiple of the class of the fiber of @xmath428 is homologous to zero . but @xmath429 is generated by the class of the fiber , and hence the class of the fiber of @xmath428 is of finite order in @xmath429 . this implies that @xmath323 is of finite order . _ proof of statement * b * of theorem [ example1 ] . _ let @xmath9 be the connected component of the space of legendrian curves that contains @xmath102 and @xmath104 . figure [ homotopy.fig ] shows that @xmath102 can be changed to @xmath104 ( in the space of legendrian curves ) by a sequence of isotopies and one passage through a transverse double point . hence if there exists a @xmath18-valued invariant @xmath105 of legendrian knots from @xmath9 that increases by one under every positive passage through a transverse double point of a legendrian knot , then it distinguishes @xmath102 and @xmath104 . ( clearly if such @xmath105 does exist , then it is an order one invariant of legendrian knots . ) below we show the existence of such @xmath105 in the connected component @xmath9 . put @xmath430 . let @xmath431 be a legendrian knot , and let @xmath432 be a generic path in @xmath9 connecting @xmath102 and @xmath433 . let @xmath434 be the set of moments when @xmath432 crosses the discriminant ( i.e. the subspace of singular knots ) in @xmath9 , and let @xmath435 , @xmath436 , be the signs of these crossings . for a generic path @xmath437 put @xmath438 . it is clear that if @xmath105 ( with @xmath430 ) does exist , then @xmath439 . to show that @xmath105 does exist we have to verify that for every legendrian knot @xmath431 and for a generic path @xmath432 connecting @xmath433 to @xmath102 the value of @xmath440 does not depend on the choice of a generic path @xmath432 connecting @xmath102 and @xmath433 , or equivalently we have to show that @xmath441 for every generic closed loop @xmath432 connecting @xmath102 to itself . there are two codimension two strata of the discriminant of @xmath9 . they are formed respectively by singular legendrian knots with two transverse double points , and by legendrian knots with one double point at which the two intersecting branches are tangent of order one . straightforward verification shows that @xmath442 , for every small closed loop @xmath137 going around a codimension two stratum of @xmath9 . this implies that for every generic loop @xmath432 connecting @xmath102 to itself the value of @xmath440 depends only on the element of @xmath443 realized by @xmath432 . hence to prove the existence of @xmath105 it suffices to show that @xmath441 for every @xmath444 . the @xmath43-principle says that the space of legendrian curves in @xmath19 is weak homotopy equivalent to the space of free loops @xmath46 in the spherical contact bundle @xmath44 of @xmath97 . ( the mapping giving the equivalence lifts a legendrian curve @xmath34 to a loop @xmath346 in @xmath44 by sending @xmath198 to the point in @xmath44 that corresponds to the direction of the velocity vector of @xmath34 at @xmath199 . ) thus @xmath443 is naturally isomorphic to @xmath448 . proposition [ props1xs2 ] says that @xmath422 and from [ hansen ] we get that @xmath449 is isomorphic to the centralizer @xmath450 of @xmath451 . using propositions [ commute ] and [ props1xs2 ] we see that either @xmath452 or @xmath453 , for some nonzero @xmath454 . hence there exists @xmath160 and nonzero @xmath455 such that @xmath456 . ( one should take @xmath32 and @xmath210 to be divisible by @xmath120 if @xmath453 . ) but the loop @xmath23 in @xmath443 corresponding to @xmath457 is just the sliding of @xmath102 @xmath32 times along itself according to the orientation . ( this deformation is induced by the rotation of the parameterizing circle . ) this loop does not intersect the discriminant , and hence @xmath458 . this finishes the proof of statement * b * of theorem [ example1 ] . [ part1]_@xmath104 and @xmath135 are homotopic legendrian curves and they realize isotopic framed knots . _ let @xmath459 be the class of the @xmath7-fiber of the fibration @xmath45 . the @xmath43-principle says that the connected component of the space of legendrian curves that contains @xmath34 corresponds to the conjugacy class of @xmath370 . from the formula for the maslov number deduced in @xcite it is easy to see that the connected components containing @xmath104 and @xmath135 correspond to the conjugacy classes of @xmath460 and of @xmath461 . let @xmath462 be an element projecting to the class @xmath155 of the @xmath7-fiber of @xmath117 . the value of the euler class of the contact bundle on the homology class realized by @xmath126 is equal to @xmath463 . ( here @xmath322 is the mapping from the description of the euler class of the contact bundle . ) and because of the reasons explained in the proof of proposition [ interpretationconditionii ] we get that @xmath464 , for the legendrian knot @xmath34 used to construct @xmath104 and @xmath135 . now proposition [ commute ] implies that @xmath232 and @xmath233 are conjugate in @xmath47 and hence @xmath104 and @xmath135 are in the same component of the space of legendrian curves . the fact that @xmath104 and @xmath135 realize isotopic framed knots is clear , because as unframed knots they are the same , and as it is shown in @xcite every pair of extra cusps corresponds to the negative extra twist of the framing . [ ideaexample2]_the idea of the proof of the fact that @xmath104 and @xmath135 can be distinguished by an order one invariant of legendrian knots . _ let @xmath465 be a point in @xmath4 . let @xmath40 be a singular unframed knot with one double point . the double point separates @xmath40 into two oriented loops . deform @xmath40 preserving the double point , so that the double point is located at @xmath465 . choosing one of the two loops of @xmath40 we obtain an ordered set of two elements @xmath466 , or which is the same an element @xmath467 . clearly there is a unique element of the set @xmath468 that corresponds to the original singular unframed knot @xmath40 , where @xmath468 is is the quotient set of @xmath469 modulo the consequent actions of the following groups : 1 : : @xmath162 whose element @xmath401 acts on @xmath470 by sending it to @xmath471 . ( this corresponds to the ambiguity in deforming @xmath40 , so that the double point is located at @xmath465 . ) 2 : : @xmath345 that acts via the cyclic permutation of the two summands . ( this corresponds to the ambiguity in the choice of one of the two loops of @xmath40 . ) assume that @xmath478 is an invariant of legendrian knots from @xmath9 such that under every ( generic transverse ) positive passage through a discriminant in @xmath9 it increases by @xmath479 , where @xmath40 is the unframed singular knot corresponding to the crossing of the discriminant . clearly such @xmath478 is an order one invariant of framed knots from @xmath9 . to prove the theorem we show the existence of such @xmath478 , and then we show that it distinguishes @xmath104 and @xmath135 . [ existence]_the existence of @xmath478 . _ let @xmath432 be a generic path in @xmath9 that starts with @xmath104 . let @xmath434 be the set of instances when @xmath432 crosses the discriminant ( i.e. the subspace of singular knots ) in @xmath9 , and let @xmath435 , @xmath436 , be the signs of these crossings . let @xmath480 be those instances for which the value of @xmath481 on the corresponding singular unframed knots is @xmath22 . for a generic path @xmath437 put @xmath482 . let @xmath256 be the connected component of the space of unframed curves obtained by forgetting the framings on curves from @xmath6 , and let @xmath201 be the unframed knot obtained by forgetting the framing on @xmath484 . similarly to the above for a generic path @xmath432 in @xmath256 starting with @xmath201 we put @xmath485 . ( as above @xmath486 is the set of instances when the value of @xmath481 on the singular unframed knots obtained under @xmath432 is equal to @xmath22 , and @xmath487 , @xmath488 , are the signs of the corresponding crossings of the discriminant . ) the codimension two stratum of the discriminant of @xmath256 consists of singular curves whose only singularities are two distinct transverse double points . straightforward verification shows that @xmath489 for any small loop @xmath137 going around the codimension two stratum . hence @xmath490 is a homomorphism . there are two codimension two strata in @xmath9 . they consist of respectively singular legendrian curves whose only singularities are two distinct transverse double points and of singular legendrian curves whose only singularity is one double point at which the two branches are tangent . considerations similar to the ones above show that @xmath491 is a homomorphism . the @xmath43-principle says that the space of legendrian curves in @xmath19 is weak homotopy equivalent to the space of free loops in the spherical contact bundle @xmath44 of @xmath4 . ( the mapping that gives the equivalence lifts a legendrian curve @xmath34 in @xmath58 to a loop @xmath346 in @xmath44 by mapping @xmath495 to the point of @xmath44 that corresponds to the velocity vector of @xmath34 at @xmath199 . ) since @xmath422 for @xmath4 from the statement of the theorem , we obtain ( see [ hansen ] ) the natural isomorphism @xmath496 . since @xmath497 and @xmath18 is torsion free , we get that to show the existence of @xmath478 it suffices to show that for every @xmath498 there exist @xmath369 and @xmath499 such that @xmath500 and @xmath501 . clearly @xmath517 and since @xmath518 is surjective there exists a loop @xmath519 such that @xmath520 . let @xmath521 be the loop corresponding to the deformation under which @xmath104 slides once around itself according to the orientation of @xmath104 . ( this deformation is induced by the rotation of the circle parameterizing @xmath104 . ) clearly @xmath522 does not cross the discriminant and hence @xmath523 . to prove the existence of @xmath478 it suffices to show that @xmath524 , for @xmath499 such that @xmath525 . but this @xmath432 is @xmath526 . thus it suffices to show that @xmath527 . since @xmath523 we get that @xmath528 and thus it suffices to show that @xmath529 . in [ explain ] we show that @xmath530 can be realized as a power of the loop described by the deformation shown in figure [ obstruction.fig ] . then since one of the loops of the only singular knot arising under this deformation is contractible and the value of @xmath533 on such a singular knot is zero we get that @xmath534 . this finishes the proof of the existence of @xmath478 modulo the explanations given in [ explain ] . [ explain ] _ now we show that @xmath530 can be realized as a sequence of loops described by the deformation shown in figure [ obstruction.fig ] . _ the @xmath43-principle for curves [ h - principleforcurves ] says that @xmath535 is weak homotopy equivalent to the space of free loops @xmath196 in the spherical tangent bundle @xmath193 of @xmath4 . in particular @xmath536 . in subsubsection [ hansen ] we introduced a surjective homomorphism @xmath33 from @xmath537 onto @xmath538 . let @xmath539 be loops such that @xmath540 . as it was explained in [ hansen ] the obstruction for @xmath23 and @xmath137 to be homotopic is an element of @xmath541 . since every orientable @xmath0-manifold is parallelizable we get that @xmath542 . clearly @xmath80 for @xmath4 from the statement of the theorem and hence @xmath543 . consider the loop @xmath544 that looks the same as @xmath23 except for a small period of time when we perform the deformation shown in figure [ obstruction.fig ] . clearly @xmath545 , and straightforward verification show that the @xmath18-valued obstruction for @xmath544 and @xmath137 to be homotopic differs by one from the obstruction for @xmath23 and @xmath137 to be homotopic . hence performing this operation ( or its inverse ) sufficiently many times we can change @xmath23 to be homotopic to @xmath137 . [ distinguish ] _ let us show that @xmath478 distinguishes @xmath104 and @xmath135 . _ let @xmath547\rightarrow \mathcal l$ ] be a generic path connecting @xmath104 and @xmath135 . to prove the theorem we have to show that @xmath548 . let @xmath201 ( resp @xmath549 ) be the unframed knot obtained by forgetting the framing on @xmath104 ( resp @xmath135 ) . let @xmath550\rightarrow \mathcal c$ ] be the isotopy that deforms @xmath201 into @xmath549 in the category of unframed curves under which @xmath201 all the time stays in a thin tubular neighborhood of @xmath549 . consider a homotopy @xmath551 that corresponds to a product of paths @xmath552 . ( @xmath553 connects @xmath201 to itself . ) clearly @xmath554 . for a loop @xmath555 ( that connects @xmath201 to itself ) put @xmath556 to be a mapping such that for every @xmath495 the mappings @xmath557 and @xmath558 are the same . the value of the euler class of the contact bundle on the homology class realized by @xmath559 is equal to @xmath560 . _ loop @xmath564 . _ since @xmath565 is an orientation preserving loop and @xmath4 is orientable , we get that the @xmath7-fibration over @xmath7 ( parameterizing the knots ) induced from @xmath117 by @xmath566 is trivializable . hence we can coherently orient the fibers of this fibration . the orientation of the @xmath7-fiber over @xmath198 induces the orientation of the @xmath7-fiber of @xmath120 that contains @xmath567 . the loop @xmath564 is the deformation of @xmath201 under which every point of @xmath201 slides once around the fiber of @xmath120 that contains this point ( staying inside the fiber ) in the direction specified by the orientation of the fiber corresponding to this point . the @xmath43-principle for curves says that @xmath256 is weak homotopy equivalent to the space of free loops @xmath196 in the spherical tangent bundle @xmath193 of @xmath4 . ( the mapping that gives the equivalence lifts a curve @xmath34 in @xmath4 to a loop @xmath346 in @xmath193 by mapping @xmath495 to the point of @xmath193 that corresponds to the velocity vector of @xmath34 at @xmath199 . ) let @xmath568 be the surjective homomorphism described in [ hansen ] , and let @xmath569 be the element that projects to the class @xmath155 of the @xmath7-fiber of @xmath117 . using proposition [ toughandtechnical ] one verifies that for every @xmath561 there exist @xmath369 such @xmath570 , for some @xmath159 . let @xmath571 be the loop described in figure [ obstruction.fig ] . ( it is easy to see that @xmath571 is in the center of @xmath572 . ) similar to [ explain ] we get that @xmath573 for some @xmath574 . [ bholdsforgamma ] _ if @xmath576 then @xmath577 and * b * holds for @xmath432 . _ the loop @xmath564 crosses the discriminant twice , both crossings occur with the same sign and the values of @xmath533 on the corresponding singular knots are equal to one . these crossings occur in the fiber over the double point of @xmath565 . ( since the double point of @xmath565 separates it into two orientation reversing loops , the two points of @xmath201 contained in this fiber induce opposite orientations of it , and the two branches of @xmath201 that intersect the fiber slide in the opposite directions under @xmath564 . ) hence @xmath578 . ( the sign depends on the orientation of the @xmath7-fibers of @xmath579 used to induce the orientations of the fibers containing the points of @xmath201 . ) * acknowledgments . * i am very grateful to stefan nemirovski , serge tabachnikov and oleg viro for the valuable discussions and suggestions . i am deeply thankful to h. geiges and a. stoimenow for the valuable suggestions , and to o. baues , m. bhupal , n. a campo , a. cattaneo , j. frhlich , j. latschev , a. shumakovich , and v. turaev for many valuable discussions . this paper was written during my stay at the max - planck - institut fr mathematik ( mpim ) , bonn , and it is a continuation of the research conducted at the eth zurich @xcite . i would like to thank the directors and the staff of the mpim and the staff of the eth for hospitality and for providing the excellent working conditions . 99999 v.i . invarianty i perestroiki ploskih frontov , _ trudy mat . inst . steklov . * 209 * ( 1995 ) ; english translation : _ invariants and perestroikas of wave fronts on the plane , _ singularities of smooth mappings with additional structures , _ proc . steklov inst . * 209 * ( 1995 ) , pp . j. e. andersen , j. mattes , n. reshetikhin : _ quantization of the algebra of chord diagrams , _ math . cambridge philos . , vol . * 124 * no . 3 ( 1998 ) , 451467 w. l. chow : _ ber systemevon linearen partiellen differentialgleichungen erster ordnung , _ math . * 117 * ( 1939 ) , pp . 98105 m. do carmo : _ riemannian geometry , _ birkhuser , boston ( 1992 ) ya . eliashberg : _ contact @xmath0-manifolds twenty years since j. martinet s work , _ ann . fourier ( grenoble ) * 42 * no . 1 - 2 ( 1992 ) , pp . 165192 ya . eliashberg : _ classification of overtwisted contact structures on @xmath0-manifolds , _ invent . * 98 * ( 1989 ) , pp . 623637 ya . eliashberg , m. fraser : _ classification of topologically trivial legendrian knots , _ crm proceedings and lecture notes , vol . * 15 * ( 1998 ) , pp . 1751 d. fuchs and s. tabachnikov : _ invariants of legendrian and transverse knots in the standard contact space , _ topology vol . * 36 * , no . 5 ( 1997 ) , 10251053 d. fuchs and s. tabachnikov : joint results , private communication with s. tabachnikov ( 1999 ) d. gabai : _ foliations and topology of @xmath0-manifolds , _ j. differential geom . * 18 * no . 3 ( 1983 ) , pp . 445503 v. goryunov : _ vassiliev type invariants in arnold s @xmath583-theory of plane curves without direct self - tangencies , _ topology vol . * 37 * no . 3 ( 1998 ) , m. gromov : _ partial differential relations , _ springer - verlag , berlin heidelberg ( 1986 ) v. l. hansen : _ on the fundamental group of the mapping space , _ compositio math . * 28 * ( 1974 ) , pp . 3336 j. w. hill : _ vassiliev - type invariants in @xmath1-theory of planar fronts without dangerous self - tangencies , _ c. r. acad . paris sr . , vol . * 324 * no . 5 ( 1997 ) , 537542 m. kontsevich : _ vassiliev s knot invariants , _ gelfand seminar , pp . 137150 , adv . soviet math . , 16 , part 2 , amer . soc . , providence , ri ( 1993 ) . r. lutz : _ structures de contact sur les fibrs principaux en cercles de dimension @xmath0 , _ ann . fourier , vol . * 3 * ( 1977 ) , pp . 115 t. q. t. le and j. murakami : _ the universal vassiliev - kontsevich invariant for framed oriented links , _ compositio math . , vol . * 102 * no . 1 ( 1996 ) pp . k. rashevskii : _ about the possibility to connect any two points of a completely nonholonomic space by an admissible curve , _ uchen . libknecht ped . ser . mat . * 2 * ( 1938 ) , pp . 8394 v. tchernov : _ arnold - type invariants of wave fronts on surfaces _ , to appear in topology ; preprint http://xxx.lanl.gov/ math.gt/9901133 ( 1999 ) v. tchernov : _ finite order invariants of legendrian , transverse and framed knots in contact @xmath0-manifolds , _ preprint http://xxx.lanl.gov math.sg/9907118 ( 1999 )

we show that for a large class of contact @xmath0-manifolds the groups of vassiliev invariants of legendrian and of framed knots are canonically isomorphic . as a corollary , we obtain that the group of finite order arnold s @xmath1-type invariants of wave fronts on a surface @xmath2 is isomorphic to the group of vassiliev invariants of framed knots in the spherical cotangent bundle @xmath3 of @xmath2 . on the other hand we construct the first examples of contact manifolds for which vassiliev invariants of legendrian knots can distinguish legendrian knots that realize isotopic framed knots and are homotopic as legendrian immersions .

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Proceed to summarize the following text: there are two types of supernova remnants ( snrs ) in the x - ray region from a morphological point of view : a shell structure and a filled - center structure . some snrs in each type contain a point source at its center . therefore , the point source at its center does not always affect the morphology of the snr . the shell structure is generated as the result of a blast wave propagating inside the interstellar matter ( ism ) , while the origin of the filled - center structure has not yet been established . a cloud evaporation model can explain the filled - center structure ( white , long 1991 ) . the clouds in the interstellar space will gradually evaporate after passage of the blast wave , which enhances the brightness of the central part of the snr , resulting in the filled - center structure . many of them show evidence of the interactions with molecular clouds , and have been studied from a theoretical point of view ( chevalier 1999 ) . another explanation is a radiative phase model ( cox 1972 ) . when the density at the shell region increases , the radiative instability proceeds and reduces the temperature of the shell region so that it becomes x - ray dim in the shell region . rho and petre ( 1998 ) proposed a new group of snr , `` mixed - morphology ( mm ) snr '' , among the filled - center snrs based on a rosat observation . they are characterized as having 1 ) a shell structure in the radio region , 2 ) a filled - center structure in the x - ray region , 3)an absence of a compact source in its center , and 4 ) thin thermal emission in the x - ray region showing a solar or sub - solar abundance . the forth criterion is considered to mean that the x - ray emitting plasma is not contaminated by the ejecta . they selected 7snrs belonging to this group as proto - typical mm snrs : w28 , w44 , 3c400.2 , kes27 , msh11 - 61a , 3c391 and ctb1 . they also claimed 7 other snrs as probably belonging to this group . in young snrs , like cassiopeia - a , there is a large inhomogeneity in metal abundance ( hwang et al . 2000 ) . even in a middle - aged snr , like the cygnus loop , the abundance in the shell region is quite different from that in its center ( miyata et al . 1998 , 1999 ) , whereas the absolute intensity in the center is quite weak . although these two snrs belong to the shell structure , it will take long time before the convection of the ejecta with the ism is completed . therefore , the uniform abundance in the snr may become an important clue to form a new class of snrs . among the mm snrs , yoshita et al . ( 2001 ) reported on the x - ray structure of 3c400.2 using the asca satellite , which has better energy resolution than that of rosat . they found no spectral variation across the snr , with a possible exception of the abundance of fe . due to an elongation in the radio image , they studied a possible interaction of two snrs . they concluded that it was generated from a single snr , rather than two . the snr , kes27 ( g327.4 + 0.4 ) , is classified as a proto - typical mm type snr . in the radio wavelength , it shows a shell structure with a diameter of @xmath7 with a slight complexity , a typical shell with an arm in the northwest . it has a spectral index of @xmath8 where @xmath9 ( milne et al.1989 ) , which is a typical value of shell - type snrs in the radio region . no optical emission has been detected from kes 27 ( van den bergh 1978 ; kirshner , winkler 1979 ) . the x - ray observation of kes27 was initially motivated by the fact that the remnant was located within the error circle of the cosb unidentified @xmath10-ray source , cg3270 ( hermsen et al . lamb and markert ( 1981 ) observed kes27 with the einstein ipc , and found that the x - ray emission was centrally peaked . an ipc image was also given by seward ( 1990 ) , which clearly showed clumpy x - ray emission . seward et al . ( 1996 ) reported using rosat data that there were several unresolved point - like sources as well as diffuse emission . it showed not only emission from inside , but also that from the bright eastern shell , which coincides with the bright radio emission . spectral studies of the diffuse emission showed little difference between the central region and the eastern rim . we report here on an observation of kes27 using the asca satellite to study its x - ray structure , particularly its spectral variation . the asca observation of kes27 was performed on 1994 august 21@xmath1122 . we retrieved these data from darts astrophysical database at the isas plain center . the siss ( yamashita et al . 1997 ) were operated in a combination of the 2-ccd bright mode and of the 4-ccd faint mode . some data were obtained in the 2-ccd mode , while others were obtained in the 4-ccd mode . since the radio shell of kes27 has the diameter of @xmath12 , most of the remnant can be covered by the field of view ( fov ) of the sis in the 4-ccd mode , which is a square of @xmath13 . since the major part of the observation was done in the 4-ccd mode , we only selected the 4-ccd faint mode data . we excluded all of the data taken at an elevation angle from the earth rim below @xmath14 from the night earth rim and @xmath15 ( sis 0 ) or @xmath16 ( sis 1 ) from the day earth rim , a geomagnetic cutoff rigidity lower than 6gv , and the region of the south atlantic anomaly . after screening the above criteria , we further removed the time region of a sudden change of the corner pixels of x - ray events . we then removed the hot and flickering pixels and corrected cti , dfe and echo effects ( dotani et al . 1995 ) from our data set . the exposure times after the screening were 9ks for sis 0 , and 11ks for sis 1 . the giss were operated in the ph mode with the standard bit assignments ( makishima et al . the gis data were also screened in a different way . we excluded all of the data taken at an elevation angle from the earth rim below @xmath14 , a geomagnetic cutoff rigidity lower than 6gv , and the region of the south atlantic anomaly . the exposure times after the screening were 13ks for gis 2 and gis 3 , respectively . we subtracted the non - x - ray background and the cosmic x - ray background from the data . we then made a correction for vignetting . figure [ fig : kes27_gismost ] shows the gis image of kes27 in the 0.710kev energy band . the overlaid contours are the radio map reproduced from the molonglo observatory synthesis telescope ( most ) observation ( milne et al . 1985 , 1989 ) . it clearly shows the centrally peaked x - ray emission confined by the radio shell . in the radio map , there are two bright regions drawn in red contour : one is the east rim and the other is the center . therefore , the radio - bright rim in the east is relatively dim in x - rays , while the radio - bright center is bright in x - rays . figure [ fig : kes27_gisimage ] shows gis images for two energy bands : the low - energy ( 0.72kev ) band and the high - energy ( 210kev ) band . comparing the low - energy band image with that obtained by the pspc ( seward et al . 1996 ) , we noticed that the asca image shows no rim brightening in the eastern rim . the eastern rim was detected with the einstein ipc with relatively weak intensity ( seward 1990 ) . considering the difference in the energy bands of these detectors , the emission from the eastern rim must be soft in the spectrum . seward et al . ( 1996 ) reported several point - like x - ray sources detected by the pspc inside the remnant . comparing the image obtained by the hri , they found that these sources show soft emission . due to the spatial resolution of asca , we found no point - like feature corresponding to them . the asca image shows a centrally peaked structure that does not coincide with the point source reported . we generated the gis band ratio map by dividing the 210kev image by the 0.72kev image , as shown in figure [ fig : kes27_gishard ] . it shows that the central part shows a harder spectrum than that of the outer region . the hardest region statistically coincides with the x - ray brightest region , which might suggest the existence of a point source in the center . we extracted the gis and sis spectra from the circular region with a diameter of 18@xmath17 , which is shown by the outermost circle in the thick solid line in figure [ fig : kes27_gis0710 ] , which surely covers the entire x - ray emission observed . since kes27 is located near the galactic plane , ( @xmath18 , @xmath19 ) = ( , ) , the galactic ridge emission should be taken into account . therefore , we extracted the annular region around the source as the background spectra . we selected the annular region of inner and outer diameters of 24@xmath17 and 30@xmath17 as the background for the gis data . similarly , we selected the region of the sis fov outside a circle of 22@xmath17 diameter as background for the sis data . in the subtraction process , we took into account the vignetting effect of the x - ray optics . after background subtraction , we found that there is no emission above 4kev . we , therefore , employed the spectra up to 4kev for a spectral analysis . we performed a simultaneous fit of the model using the gis and sis data . from the spectra in figure [ fig : kes27_spec_entire ] , the emission lines of si and s are clearly seen , which indicates that the x - ray emission comes from a thin thermal plasma in origin . we employed the vmekal model in ftool 5.1 where the free parameters were the interstellar absorption feature , @xmath20 , electron temperature , @xmath21 , abundances of mg , si , s , ar , and fe . the other metal abundances were fixed to the solar values . we obtained a statistically good fit , the results of which are listed in table [ tbl : kes27_fitpar ] , of the entire region . the best - fit curves are also shown in figure [ fig : kes27_spec_entire ] . we should note that the obtained abundances are statistically consistent with those of the solar values . in the spectral fitting , we need not require to employ a model with the non - equilibrium ionization ( nei ) condition . the ionization parameter , @xmath22 , the product of the electron density and the elapsed time after the shock heating , is longer than 10@xmath23@xmath2s , which indicates that the plasma almost reaches the collisional ionization equilibrium ( cie ) condition . using the plasma density , the value of @xmath22 gives us an age estimate of longer than @xmath4 yr , suggesting that the snr is in the radiative stage . we then tried to search for a spectral variation across the remnant . since the x - ray band ratio map shows a point symmetric structure , we divided the image into two regions : an inner region and an outer region . the inner region has a diameter of 6@xmath17 centered on the intensity peak in figure [ fig : kes27_gis0710 ] . the outer region has an 18@xmath17 diameter . figure [ fig : kes27_speceach ] shows the spectra obtained from the two regions . we also performed a model fitting by employing the same model to that employed for the entire region . the best - fit parameters are given in table [ tbl : kes27_fitpar ] as well as the best - fit curves in figure [ fig : kes27_speceach ] . we can notice that most of them show statistically the same value , with the exception of @xmath21 . we can say that there is a temperature decrease from the inner region towards the outer region that causes the harder spectra in the center . as we mentioned before , the x - ray brightest region does not coincide with the point source detected by rosat . if there is a point source inside the snr , it usually shows a hard spectrum and a possible pulsation . seward et al . ( 1996 ) noticed the possibility that enhanced central emission could be due to a synchrotron nebula . our results do not require a power - law type component , like that expected for a possible point source inside the snr . we picked up photons in the 210kev range of the gis from the inner region . the total number of photons was 381 with a time resolution of 62.5ms . a fourier analysis showed no prominent pulsation component between 0.1252000s . due to the poor statistics , we could not obtain a meaningful upper limit for the pulsation . the origin of the central emission is still unknown . so far , there is no reliable distance estimate to kes27 . seward et al . ( 1996 ) assigned a distance of 6.5kpc by comparing the column density with that of the neighboring snr , rcw103 ( 5@xmath24 away from kes27 ) , whose distance was determined by a h measurement . case and bhattacharya ( 1998 ) used a refined @xmath25_d _ relation to derive a distance of 4.0kpc . here , we adopt a distance of 5kpc , an intermediate value of two estimates . in order to determine the extent of the x - ray emitting plasma , we measured the solid angle subtended by a half - maximum intensity contour . in figure [ fig : kes27_gis0710 ] , the half - maximum intensity contour is drawn by the curved red line . the solid angle subtended by this contour corresponds to a circle with a diameter of . since kes27 shows a filled - center structure with a point - symmetric shape , we assume that the x - ray emitting plasma occupies a sphere with diameter @xmath26 and a uniform density . if this is the case , the half - maximum intensity contour becomes a circle with a diameter of @xmath27 . therefore , we calculated various parameters while assuming the diameter of the plasma to be 13@xmath17 with a uniform density . we also assumed the electron density , @xmath28 , to be equal to the proton density , @xmath29 , for simplicity . these are summarized in table [ tbl : kes27_prop ] . vwe performed a similar image analysis using the rosat pspc . we found that the solid angle subtended by the half - maximum intensity contour is equal to a circle with a diameter of 12@xmath17 , which is almost equal to that by using the asca gis . this can be understood because kes27 has such a high @xmath20 that the emission is confined to only the asca energy range . the apparent size in the radio region is @xmath30 ( whiteoak , green 1996 ) , which is bigger by 50% in size than that we measured in the x - ray region . the density given in table [ tbl : kes27_prop ] is that of the x - ray emitting plasma . the total mass in the x - ray emitting region is a part of the mass affected by the sn explosion . the radio - bright region outside the x - ray emitting region must have a higher density than that inside , since it cools down and becomes x - ray dim . judging from the apparent sizes in the radio and x - ray regions , the total mass occupied outside the x - ray bright region must be by at least one order magnitude bigger than that in the x - ray bright region . therefore , if the ejecta contain plenty of metal , it must be dissolved into the ism , resulting in the solar abundance . this shows that kes27 is a typical mm snr . in the radio band , kes27 shows a bright structure along the eastern rim . the bright radio feature on one side of the remnant resembles the radio morphology of snrs that are known to show an interaction with the molecular cloud , such as 3c391 and w44 . in general , a detection of shock - excited oh maser emission is a strong evidence for an interaction with the molecular cloud . the detection of oh maser emission was reported for 3c391 ( frail et al . 1996 ) and w44 ( claussen et al . 1997 and references therein ) , while no oh emission has been detected toward kes27 ( frail et al . therefore , rho and petre ( 1998 ) noted that there is no evidence of an actual interaction with the molecular cloud in kes27 . the asca observation verified the centrally peaked and thin thermal x - ray nature for kes27 . in addition , we first found a spectral variation from the inner region towards the outer region . it should be noted that the harder spectrum in the inner region comes not from the heavier interstellar absorption but from the higher temperature . the value of @xmath21 in the inner region is higher by 40% than that in the outer region , while there is no variation of the abundances of heavy elements from a statistical point of view . the temperature gradient suggests that thermal conduction does not play an important role . however , our analysis may be an oversimplification due to showing an one - temperature model . other snrs having both a radio shell and centrally peaked x - ray emission with a thermal nature , like kes27 , have a uniform temperature distribution . for example , no significant @xmath21 variation has been found in 3c400.2 ( yoshita et al . 2001 ) , g69.4 + 1.2 ( yoshita et al . 2000 ) . as mentioned in subsection 3.1 , there are two radio - bright regions in kes27 : one is in the east rim and the other is in the center . the bright - radio features are generally seen in the rim of the snr where the matter is compressed by a blast wave . when the matter is compressed , the polarization becomes prominent . the polarization map given by milne et al . ( 1989 ) shows strong polarization in the east rim as well as in the center where both regions are radio - bright . however , the east rim is x - ray dim and the center is x - ray bright . therefore , it must be a projection effect that the bright radio feature in the center coincides with the x - ray bright region . future x - ray observations with a higher spatial resolution , such as by using chandra or xmm - newton , will reveal the relation between the x - ray intensity peak and the radio emission . we observed kes27 , a proto - typical mm snr , using asca gis and sis data . we confirmed a filled - center structure in the x - ray region with an angular diameter of 13@xmath17 , which is about 50% smaller than that of the radio bright region . the x - ray spectrum can be well fitted by a cie ( vmekal ) model with solar abundances . comparing the spectrum from the inner region with that from the outer region , all of the spectral parameters are consistent with each other , with an exception of @xmath21 . the inner region clearly shows a higher value of @xmath21 by 40% than that in the outer region . this suggests that the thermal conduction does not play an important role . our analysis using asca supports that kes27 is a proto - typical mm snr . there are two radio - bright regions in kes27 : one is in the east rim and the other is in the center . the east rim is x - ray dim while the center is x - ray bright . by taking into account the radio polarization , the coincidence between the x - ray and the radio in the center is found to be a projection effect . a finer observation in the future will reveal the detailed structure in the x - ray region as well as any coincidence between the x - ray and radio regions . the authors would like to express their special thanks to the asca team . this research was partially supported by the grant - in - aid for scientific research by the ministry of education , culture , sports , science and technology of japan ( 13440062 , 13874032 ) . case , g. l. , & bhattacharya , d. 1998 , apj , 504 , 761 chevalier , r. a. 1999 , apj , 511 , 798 claussen , m. j. , frail , d. a. , goss , w. m. , & gaume , r. a. 1997 , apj , 489 , 143 cox , d. p. 1972 , apj , 178 , 159 dotani , t. , yamashita , a. , rasmussen , a. , & the sis team 1995 , asca news , 3 , 25 frail , d. a. , goss , w. m. , reynoso , e. m. , giacani , e. b. , green , a. j. , & otrupcek , r. 1996 , aj , 111 , 1651 hermsen , w. swanenburg , b. n. , bignami , g. f. , boella , g. , buccheri , r. , scarsi , l. , kanbach , g. , mayer - hasselwander , h. a. , masnou , j. l. , & paul , j. a. 1977 , nature , 269 , 494 hwang , u. , holt , s. s. , & petre , r. 2000 , apj , 537 , l119 kirshner , r. p. , & winkler , p. f.,jr . 1979 , apj , 227 , 853 lamb , r. c. , & market , t. h. 1981 , apj , 244 , 94 makishima , k. , tashiro , m. , ebisawa , k. , ezawa , h. , fukazawa , y. , gunji , s. , hirayama , m. , idesawa , e. , 1996 , pasj , 48 , 171 milne , d. k. , caswell , j. l. , haynes , r. f. , kesteven , m. j. , wellington , k. j. , roger , r. s. , & bunton , j. d. 1985 , proc . astron . soc . australia , 6 , 78 milne , d. k. , caswell , j. l. , kesteven , m. j. , haynes , r. f. , & roger , r. s. 1989 , proc . australia , 8 , 187 miyata , e. , & tsunemi , h. 1999 apj , 525 , 305 miyata , e. , tsunemi , h. , kohmura , t. , suzuki , s. , & kumagai , s. 1998 , pasj , 50 , 257 rho , j. , & petre , r. 1998 , apj , 503 , l167 seward , f. d. 1990 , apjs , 73 , 781 seward , f. d. , kearns , k. e. , & rhode , k. l. 1996 , apj , 471 , 887 van den bergh , s. 1978 , apjs , 38 , 119 white , r. l. , & long , k. s. 1991 , apj , 373 , 543 whiteoak , j. b. z , & green , a. j. 1996 , a&as , 118 , 329 yamashita , a. , dotani , t. , bautz , m. , crew , g. , ezuka , h. , gendreau , k. , kotani , t. , mitsuda , k. , 1997 , ieee trans . nucl . sci . , 44 , 847 yoshita , k. , miyata , e. , & tsunemi , h. 2000 , pasj , 52 , 867 yoshita , k. , tsunemi , h. , miyata , e. , & mori , k. 2001 , pasj , 53 , 93 lccc parameter & entire & inner & outer + @xmath31 & 2.4@xmath32 & [email protected] & 2.6@xmath33 + @xmath21[kev ] & 0.71@xmath6 & [email protected] & 0.59@xmath6 + mg & 1.1@xmath34 & @xmath35 & 1.1@xmath36 + si & [email protected] & 0.9@xmath34 & [email protected] + s & [email protected] & 1.2@xmath37 & [email protected] + ar & @xmath38 & @xmath39 & @xmath40 + fe & 1.2@xmath41 & 1.5@xmath42 & 0.9@xmath43 + @xmath44/d.o.f . & 333/342 & 102/90 & 339/280 + + & & & + + lc parameter & value + diameter [ pc ] & 18.9@xmath45 + temperature [ kev ] & 0.71@xmath6 + density [ @xmath2 ] & 0.39@xmath47 @xmath48 + thermal energy [ erg ] & @xmath49 @xmath50 + total mass [ @xmath51 & 34@xmath52 @xmath50 + + + +

we report here on the observation of kes27 , a proto - typical mixed - morphology snr , using asca . it clearly shows a filled - center structure in the x - ray region while a shell structure in the radio region . there are two radio bright regions : one is in the center , while the other is in the east rim . the x - ray intensity peak coincides well with the radio bright region at the center . the x - ray spectrum was well - fitted by a collisional ionization equilibrium model with solar abundances . taking into account the ionization parameter ( @xmath010@xmath1@xmath2s ) and the plasma density ( 0.39@xmath3@xmath2 ) , we found that the age of the snr is longer than @xmath4 yr . the hardness ratio map indicates that the inner region shows a harder spectrum than that in the outer region , which does not come from the heavier interstellar absorption feature , but from the higher temperature . there is a temperature gradient from the innner region ( [email protected] kev ) toward outer region ( 0.59@xmath6 kev ) , indicating that the thermal conduction does not play an important role .

You are an expert at summarizing long articles. Proceed to summarize the following text: early studied in the 1930 s @xcite , the counter - rotating machines arouse a greater interest in the turbomachinery field , particularly for their potential improvement of the efficiency with respect to conventional machines by recovering kinetic energy from the front rotor exit - flow and by adding energy to the flow . the first counter - rotating machines have appeared in aeronautic @xcite and marine applications @xcite in open configuration . conventional designs of high speed counter - rotating fans are based on quite expensive methods and require a systematic coming and going between theoretical methods such as the lifting line theory or the strip - analysis approach @xcite and cfd analysis @xcite . moreover , the axial spacing , which has a major role on the rotors interaction and consequently on the noise @xcite , is a key parameter to find a compromise between high aerodynamic and good acoustic performance for high speed fans @xcite . in order to reduce this interaction , the axial spacing of high speed fans has to be relatively large , resulting in a decrease in the aerodynamic performance @xcite . for the same reason , the rear rotor ( rr ) diameter has to be smaller ( about 10@xmath2 according to @xcite ) than the front rotor ( fr ) diameter to reduce interaction between the fr tip vortex and the rr blade tip . contrary to that , in the case of low speed fans axial spacing could be shortened using the benefit of a relatively low rotor interaction . therefore these machines see a revival of interest in several distinct configurations open and ducted flows , shrouded or not shrouded rotors in various subsonic regime applications @xcite . recent research work dealt with the effects of global parameters like rotation speed ratio @xcite , local phenomena such as tip vortex flows @xcite and improvement of cavitation performance for pumps @xcite . all previous studies have shown the benefit of rr in improving the global efficiency and in increasing the operating flow - rate range while maintaining high efficiency . the counter - rotating systems ( crs ) moreover allow to reduce the fans diameter and/or to reduce the rotation rate . more axial spacing is needed compared to one simple fan , but not much more than a rotor - stator stage . however , it requires a more complex shaft system . another interesting feature of crs is that it makes it possible to design axial - flow fans with very low angular specific speed @xmath3 with @xmath4 the mean angular velocity , @xmath5 the flow rate , @xmath6 the total pressure rise , and @xmath7 the fluid density . with such advantages , the crs becomes a very interesting solution and the interaction between the rotors needs to be better understood in order to design highly efficient crs . however , only a few studies have been concerned with , on the one hand , the effect of the axial spacing , and , on the other hand , the design method @xcite , particularly with rotors load distribution for a specified design point . this paper focuses on two major parameters of ducted counter - rotating axial - flow fans in subsonic regime : the rotation rate ratio , @xmath8 and the relative axial spacing , @xmath9 . in some cases , these systems are studied by using two identical rotors or the rr is not specifically designed to operate with the fr . in this study , the fr is designed as conventional rotor and the rr is designed on purpose to work with the fr at very small axial spacing . in this first design , the total work to perform by the crs was arbitrarily set up approximately to two halves one half respectively for the fr and rr . in [ sec : design ] the method that has been used to design the front and the rear rotors is firstly described . the experimental set - up is presented in [ sec : setup ] . then the overall performances of the system in its default configuration and the effects of varying the rotation ratio and the relative axial spacing between the rotors are discussed in [ sec : results ] . the design of the rotors is based on the use of the software mft ( mixed flow turbomachinery ) , a 1d code developed by the dynfluid laboratory @xcite based on the inverse method with simplified radial equilibrium to which an original method has been added specifically for the design of the rr of the counter - rotating system . from the specified total pressure rise , volume flow - rate and rotating speed , optimal values of the radii @xmath10 and @xmath11 are first proposed . in a second step , the tip and the hub radii as well as the radial distribution of the circumferential component of the velocity at the rotor outlet , @xmath12 , could be changed by the user . the available vortex models are the free vortex ( @xmath13 ) , the constant vortex ( @xmath14 ) and the forced vortex ( @xmath15 ) . the velocity triangles are then computed for @xmath16 radial sections , based on the euler equation for perfect fluid with a rough estimate of the efficiency of @xmath17 and on the equation of simplified radial equilibrium ( radial momentum conservation ) . the blades can then be defined by the local resolution of an inverse problem considering a 2d flow and searching for the best suited cascade to the proposed velocity triangles by the following parameters : @xmath18 the stagger angle , computed from the incidence angle , @xmath19 giving the lower pressure variation on the suction surface of the blade using equations [ eq : gamma ] and [ eq : a ] . the solidity , @xmath20 and the chord length , @xmath21 are thus computed at the hub and at the tip using equations [ eq : sigma ] and [ eq : c ] where @xmath22 denotes the lieblein s diffusion factor@xcite . the intermediate chords are obtained by linearisation . finally , the camber coefficients @xmath23 are computed using equation [ eq : coef_port ] . @xmath24 these empirical equations have been validated for naca-65 cascades @xcite , for @xmath25 and @xmath26 . velocity triangles for the crs . the fluid is flowing from left to right . , width=294 ] the behaviour of the designed machine resulting from the above method can then be analysed using a direct method in order to determine whether the design point is achieved and what are the characteristics of the machine at the neighbourhood of the design point . the effects due to real fluid are taken partially into account with in - house loss models and the introduction of an axial - velocity distribution which considers the boundary layers at the hub and casing . thus , the characteristics of the machine can be obtained in the vicinity of the design - point discharge . regarding the crs , the geometrical dimensions , the number of blades of fr and of rr and their rotation rates are imposed . in particular , the number of blades of each rotor was chosen in order to prevent to have the same blade passing frequency or harmonics for both rotors in the lower frequencies range . the system that is presented here has moreover been designed to have a pure axial exit - flow . an iterative procedure is then performed . the pressure rise of the fr is initially chosen and then designed and quickly analysed as explained . an estimate of the pressure rise that rr would made is then performed , based on this analysis . if the total pressure rise of the crs is not met , the design pressure rise of fr is varied and the calculus are made again . in this method , losses and interactions in - between the two rotors are not taken into account . any recirculation happening near the blade passage or near the blade hub or tip is not predicted by mft as it is based on simplified radial equilibrium . . .[tab : speci]design point of the counter - rotating system for air at @xmath27 [ cols= " < , < , < , < " , ] fans characteristics : ( a ) static pressure rise @xmath28 _ vs _ flow rate @xmath5 ; ( b ) static efficiency @xmath29 _ vs _ flow rate @xmath5 . the axial spacing is @xmath30 . @xmath31 : fr rotating alone at @xmath32 rpm ( rr has been removed ) , @xmath33 : rr rotating alone at @xmath34 rpm ( fr has been removed ) and @xmath35 : crs at @xmath32 rpm and @xmath36 . the @xmath37 and the dashed lines stand for the design point of the crs ] the fr rotating alone has a very flat curve ( @xmath31 in fig . [ fig : caract_r1_r2_r12_nominal ] ) . the nominal flow - rate of fr is slightly greater than the design point it is @xmath38 greater . the measured static pressure rise at the design point is @xmath39 pa , with a relatively low static efficiency of @xmath40 . this is not surprising with no shroud and a large radial gap . moreover , this is consistent with the estimated static pressure rise by mft , which is around @xmath41 pa . the rr rotating alone has a steeper curve ( @xmath33 in fig . [ fig : caract_r1_r2_r12_nominal ] ) and its nominal flow - rate @xmath42 [email protected]@xmath44 is lower than the design flow - rate of fr and crs . this is consistent with the bigger stagger angle of the blades ( see tab . [ tab : geospeci ] ) and can be explained by examining the velocity triangles in fig . [ fig : trianglevit ] and considering the case with the fr coupled to the rr : the incoming velocity @xmath45=@xmath46 has an axial component as well as a tangential component . hence , the flow angle in the relative reference frame reads : @xmath47 now the case without the fr is considered and it is assumed that the flow through the honeycomb is axial . since the tangential component does not exist any more , @xmath48 . mft estimates @xmath49 m.s@xmath44 , @xmath50 m.s@xmath44 and @xmath51 m.s@xmath44 , which leads to @xmath52 at the blade mid - span . supposing that rr rotating alone reaches its maximum efficiency for @xmath52 , equation [ eq : tanbeta ] implies that @xmath53 m.s@xmath44 , _ i.e. _ @xmath54 [email protected]@xmath44 . this is exactly the nominal flow - rate of rr rotating alone ( see fig . [ fig : caract_r1_r2_r12_nominal ] and tab . [ tab : rendmax ] ) . it is clear from the above analysis why the nominal flow - rate of rr is lower than the design flow - rate . the characteristic curve of the crs ( @xmath35 in fig . [ fig : caract_r1_r2_r12_nominal ] ) is steeper than the characteristic curve of fr . it is roughly parallel to the rr curve . the nominal flow - rate of the crs matches well with the design flow - rate , _ @xmath55 [email protected]@xmath44 . the static pressure rise at the nominal discharge ( @xmath56 pa ) is @xmath57 lower than the design point ( @xmath58 pa ) , which is not so bad in view of the rough approximations used to design the system . please notice that the static pressure rise of the crs is not equal to the addition of the static pressure rise of the fr with the pressure static rise of the rr , taken separately . the crs has a high static efficiency ( @xmath59 ) compared to a conventional axial - flow fan or to a rotor - stator stage with similar dimensions , working at such reynolds numbers @xcite . the gain in efficiency with respect to the fr is @xmath60 points , whilst an order of magnitude of the maximum gain using a stator is typically @xmath61 points@xcite . awaiting for more accurate local measurements of the flow angle at the exit of the crs , a simple test of flow visualization with threads affixed downstream of the crs was performed . it has been observed that without the rr the flow is very disorganized . when the rr is operating , at the design configuration ( @xmath36 and @xmath62 ) , the flow is less turbulent , the threads are oriented with a small angle at the exit . this small angle seems , however to decrease when @xmath8 is increased between @xmath55 and @xmath63 . this is consistent with the results in section [ subsec : influence ] where it is found that the nominal operating point is observed for a value of @xmath8 higher than the design value . the flow - rate range for which the static efficiency lays in the range @xmath64 is : @xmath65 [email protected]@xmath44 , that is from @xmath66 of the nominal flow - rate up to @xmath67 of the nominal flow - rate . one open question is to what extent the global performances of the crs are affected by the axial spacing and the speed ratio , and whether the efficient range could be extended by varying the speed ratio . crs characteristics at @xmath32 rpm , @xmath30 and @xmath68 $ ] : ( a ) static pressure rise @xmath28 _ vs _ flow rate @xmath5 ; ( b ) static efficiency @xmath29 _ vs _ flow rate @xmath5 . @xmath69 : @xmath70 , @xmath71 : @xmath72 , @xmath73 : @xmath74 , @xmath75 : @xmath76 , @xmath77 : @xmath36 , @xmath78 : @xmath79 , @xmath80 : @xmath81 , @xmath31 : @xmath82 , @xmath83 : @xmath84 , @xmath35 : @xmath85 and @xmath86 : @xmath87 . the blue @xmath37 and the dashed lines stand for the design point of the crs ] maximal static efficiency @xmath29 _ vs _ @xmath8 for the crs with @xmath32 rpm and @xmath30 . ] in this paragraph , the rotation rate of fr is kept constant at @xmath32 rpm , and the rotation rate of rr is varied from @xmath88 to @xmath89 rpm . the corresponding @xmath8 are @xmath90 . the axial spacing is @xmath30 . the overall performances of the crs in these conditions are plotted in fig . [ fig : caractrend_n1_2000 ] . as expected , the more the rotation rate of rr increases , the more the static pressure rise of the crs increases and the nominal flow - rate of the crs increases . the maximal efficiency as a function of @xmath8 is plotted in fig . [ fig : etadetheta ] . for very low rotation rates of rr , _ i.e. _ for @xmath70 ( @xmath69 in fig . [ fig : caractrend_n1_2000 ] ) and @xmath72 ( @xmath71 in fig . [ fig : caractrend_n1_2000 ] ) , the system is very inefficient : in the first case when the rr is at rest the maximum efficiency hardly reaches @xmath91 which is below the maximal efficiencies of both fr and rr alone . the maximum flow - rate that can be reached is moreover very low in both cases compared to the discharge goal of @xmath92 [email protected]@xmath44 . in the range @xmath93 $ ] , _ i.e. _ @xmath94 $ ] rpm , the system is highly efficient . the maximum efficiency increases with @xmath8 to reach a maximum value of @xmath95 for @xmath82 and is then quasi - constant ( @xmath96 for @xmath97 ) . this is a very interesting feature of the counter - rotating system . one could imagine , simply by varying the rr rotation rate , to work at a constant pressure rise with an efficiency greater than @xmath98 for a large flow - rate range . for instance in the present case , the system could give a constant static pressure rise of @xmath99 pa with @xmath100 for @xmath101 [email protected]@xmath44 with @xmath32 rpm , @xmath30 and @xmath102 $ ] . one could also imagine to work at a constant flow - rate with high static efficiency . for instance in the present case , the system could give a constant flow - rate of @xmath92 [email protected]@xmath44 with @xmath100 for @xmath103 pa with @xmath32 rpm , @xmath30 and @xmath93 $ ] . crs characteristics at various axial spacing : ( a ) static pressure rise @xmath28 _ vs _ flow rate @xmath5 ; ( b ) static efficiency @xmath29 _ vs _ flow rate @xmath5 . the rotation ratio of fr is @xmath104 and @xmath36 . @xmath35 : @xmath30 , @xmath77 : @xmath105 , @xmath106 : @xmath107 , @xmath69 : @xmath108 , @xmath109 : @xmath110 and @xmath83 : @xmath111 . the blue @xmath37 and the dashed lines stand for the design point of the crs ] figure [ fig : caract_distance ] shows the characteristics curves at the design rotation rates , i.e. , @xmath32 rpm and @xmath36 . regarding @xmath112 $ ] , the overall performances do not change significantly and the variation is in the uncertainty range . the efficiency does not vary significantly either . in other studies@xcite it was reported that the axial spacing had a more significant influence on the overall performances . this was noticed as well in this study . for a=@xmath113 and a=@xmath114 , the global performances are decreased by @xmath115 pa ( @xmath116 ) comparing to the other spacings . however , even for a=@xmath114 , the crs still shows good performances with high efficiency compared to the conventional fan systems . a counter - rotating axial - flow fan has been designed according to an iterative method that is relatively fast . it is based on semi - empirical modelization that partly takes into account the losses , boundary layers at hub and casing , and the effects of low reynolds numbers ( below @xmath117 ) . the overall performances at the nominal design point are slightly lower than predicted , with a static pressure rise @xmath57 lower . the static efficiency is however remarkably high ( @xmath118 ) and corresponds to a @xmath60 points gain in efficiency with respect to the fr maximal efficiency and to a @xmath61 points gain with respect to the rr . the overall measurements give first clues that allow to validate the design method . the counter - rotating system has a very flexible use that allows to work at constant flow - rate on a wide range of static pressure rises or to work at constant pressure rise on a wide range of flow - rates , with static efficiency bigger than @xmath98 , simply by varying the rr rotation rate . one could thus imagine an efficient closed - loop - controlled axial - flow fan . the overall performances moreover do not significantly vary with the axial spacing in the range @xmath119 $ ] . however , for @xmath110 and @xmath120 the overall performances slightly decrease . 10 bechet , s. , negulescu , c. , chapin , v. , and simon , f. , 2011 . `` integration of cfd tools in aerodynamic design of contra - rotating propellers blades '' . in 3rd ceas conference ( council of european aerospace societies ) . blandeau , v. p. , joseph , p. f. , and tester , b. j. , 2009 . `` broadband noise prediction from rotor - wake interaction in contra - rotating propfans '' . in 15th aiaa / ceas aeroacoustics conference -30th aiaa aeroacoustics conference , aiaa 20093137 . shigemitsu , t. , furukawa , a. , watanabe , s. , and okuma , k. , 2005 . `` air / water two - phase flow performance of contra - rotating axial flow pump and rotational speed control of rear rotor '' . in asme 2005 fluids engineering division summer meeting june 19 - 23 , 2005 , houston , texas , usa , pp . 10691074 . cho , l. , choi , h. , lee , s. , and cho , j. , 2009 . `` numerical and experimental analyses for the aerodynamic design of high performance counter - rotating axial flow fans '' . in proceedings of the asme 2009 fluids engineering division summer meeting - fedsm2009 , colorado usa , pp . fedsm200978507 . noguera , r. , rey , r. , massouh , f. , bakir , f. , and kouidri , s. , 1993 . `` design and analysis of axial pumps '' . in asme fluids engineering , second pumping machinery symposium , washington , usa . , pp . 95111 . lieblein , s. , schwenk , f. c. , and broderick , r. l. , 1953 . diffusion factor for estimating losses and limiting blade loading in axial - flow - compressor blade elements . tech . rep . tm e53d01 , national advisory committee for aeronautics . sarraf , c. , nouri , h. , ravelet , f. , and bakir , f. , 2011 . `` experimental study of blade thickness effects on the global and local performances of a controlled vortex designed axial - flow fan '' . , p. 684

_ an experimental study on the design of counter - rotating axial - flow fans was carried out . the fans were designed using an inverse method . in particular , the system is designed to have a pure axial discharge flow . the counter - rotating fans operate in a ducted - flow configuration and the overall performances are measured in a normalized test bench . the rotation rate of each fan is independently controlled . the relative axial spacing between fans can vary from @xmath0 to @xmath1 . the results show that the efficiency is strongly increased compared to a conventional rotor or to a rotor - stator stage . the effects of varying the rotation rates ratio on the overall performances are studied and show that the system has a very flexible use , with a large patch of high efficient operating points in the parameter space . the increase of axial spacing causes only a small decrease of the efficiency _

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Proceed to summarize the following text: hamiltonian structures play a fundamental role in mathematical physics . it s enough to recall a few examples : classical mechanics , electrodynamics , quantum mechanics , hydrodynamics and general relativity . however , when applying the classical methods and technics of symplectic geometry to pdes , one faces significant difficulties , both analytical and conceptual . part of the problem is that symplectic forms that arise in many applications are weak symplectic forms on infinite dimensional manifolds . more importantly , often integral curves of pdes are not differentiable in time in the function spaces one would normally use ; in the linear case , this corresponds to the fact that the operators involved are unbounded . stock examples include the euler and klein - gordon equations . when dealing with such systems one has to pay careful attention to domains of definitions as many standard formulas become only formal relationships . their justification is often cumbersome and requires some ad hoc methods . the goal of this paper is to contribute to the development of techniques that are useful for the treatment of nonlinear pdes with non - differentiable ( in time ) solutions and build a framework that allows a systematic and rigorous study of such systems and is applicable to the broad range of physical phenomena . previous work in this vein is @xcite . specifically , this article is devoted to the study of the euler equations for an ideal fluid on the compact manifold , the example that provides the main inspiration and motivation . the goal is to understand in what exact sense ( if any ) the flow generated by euler s equation consists of poisson maps . since the classic work of arnold @xcite , it has been known that _ formally _ the euler equation could be viewed as a hamiltonian system . ( expositions of this may be found in @xcite and @xcite ) . the work of @xcite showed the remarkable fact that in appropriate function spaces , the flow of the euler equations in lagrangian representation ( in sobolev function spaces @xmath6 for @xmath7 ) is given by a smooth vector field and hence all the difficulties are resolved in that context . this work also shows that one can perform a reduction ( euler - poincar reduction ) to eulerian representation to rigorously derive that the solutions obtained this way satisfy the euler equations ( taking into account one derivative loss due to the reduction procedure ) . from the work of @xcite , the reduced flow of the euler equations in @xmath6 are known to form a continuous flow in @xmath6 ( both in time and in the initial velocity field ) , and regarded as maps from @xmath6 to @xmath8 , they are @xmath3 . another remarkable property of the solutions also follows from this same work namely that the individual particle trajectories are @xmath9 in time , a fact not so easy to see directly in eulerian representation ( see @xcite ) . while a version of the _ symplectic nature _ of the flow of the euler equations follows directly from the results in @xcite ( taking into account the loss of one derivative ) , it is not so clear that there is a well defined poisson sense for the results . in fact , the work of @xcite ( and many subsequent papers by other authors ) shows that in the poisson context , this derivative loss is a nontrivial issue in defining a good sense in which one has a poisson manifold and in which the euler equations then define a hamiltonian system in the poisson sense . _ the main purposes of this paper is to fill this gap by means of a nonsmooth lie - poisson reduction procedure on appropriate classes of functions . _ this article has the following structure . in [ s : eulersol ] we give important background information on euler equation and manifolds of diffeomorphisms . then , we recall the basic ideas of poisson reduction in [ s : poisson ] . our results are presented in next two sections . in [ s : weakpoisson ] we prove that tangent bundle of a weak riemannian manifold carries a poisson structure in an appropriate sense , provided that the manifold possesses a _ smooth _ riemannian connection . the later requirement is fulfilled on the groups of diffeomorphisms according to the work of @xcite . in [ s : eulerflow ] we utilize this result to show that the flow of euler equation is poisson in an appropriate sense . we conclude with short discussion of presented results in [ s : future ] . in this section we present some classical results concerning the euler equation that motivated our study . the notation and exposition follows @xcite . the euler equations on compact manifold are traditionally formulated in the following way . let @xmath10 be a compact riemannian @xmath2-manifold possibly with boundary @xmath11 . find a time dependent vector field @xmath12 , ( which has an associated flow denoted @xmath13 ) such that 1 . @xmath14 is a given initial condition with @xmath15 2 . the euler equations hold : @xmath16 for some scalar function @xmath17 ( the pressure ) , 3 . @xmath18 , and 4 . @xmath19 is parallel to @xmath11 . it is standard that above equation can be formally rewritten as an ode on the space of divergence free vector fields with a derivative loss . but it was discovered by @xcite that this is literally true with no derivative loss in lagrangian representation . we recall how this proceeds . let @xmath20 be a volume form on the manifold @xmath10 . let @xmath21 denote the space of mappings of sobolev class @xmath22 from an @xmath2-manifold @xmath10 to a manifold @xmath23 . for @xmath24 , let @xmath25 @xmath26 then both @xmath27 are smooth infinite dimensional manifolds and topological groups , moreover @xmath28 is a closed submanifold and a subgroup of @xmath29 . let @xmath30 and @xmath31 be the canonical projections and let @xmath32 be the identity element of the groups @xmath33 . then @xmath34 @xmath35 where @xmath36 denotes the space of @xmath4 divergence free vector fields on @xmath10 that are parallel to the boundary . a given riemannian metric on @xmath10 induces a right invariant weak riemannian metric on @xmath28 given by @xmath37 for @xmath38 where scalar product under the integral sign is taken in @xmath10 . as was shown in @xcite , @xmath28 possesses a smooth riemannian connection and , as a consequence , a smooth spray , which we will denote @xmath39 . [ t : flow ] for @xmath5 , the weak riemannian metric has a @xmath9 spray @xmath40 . let @xmath41 be the ( local , @xmath9 ) flow of @xmath39 . let @xmath42 ( the material velocity field ) and @xmath43 ( the particle position field ) . then the solution of the euler equation with initial condition @xmath44 is given by @xmath45 from the properties of the diffeomorphism group , one sees that this result shows that the euler equations ( [ euler_equations ] ) are well - posed in @xmath6 in eulerian representation . first , recall the following basic and simple result about poisson reduction ( see , for example , @xcite ) . suppose that @xmath46 is a lie group that acts on a poisson manifold @xmath47 and that for each @xmath48 the action map @xmath49 is a poisson map . suppose that the quotient @xmath50 is a smooth manifold and the projection @xmath51 is a submersion . then , there is a unique poisson structure @xmath52 on @xmath50 such that @xmath53 is a poisson map . it is given by @xmath54 where @xmath55 is a poisson bracket in @xmath47 and @xmath56 is a set of smooth functions on @xmath50 . if @xmath57 is a hamiltonian vector field for a g - invariant hamiltonian @xmath58 , then @xmath53 also induces reduction of dynamics . there is a function @xmath59 such that @xmath60 . since @xmath53 is a poisson map it transforms @xmath57 on @xmath47 to @xmath61 on @xmath50 , that is , @xmath62 . denoting the flow of @xmath57 by @xmath63 and the flow of @xmath61 by @xmath64 we obtain commutative diagram @xmath65 our strategy is to apply the above procedure to the context of fluids . to do so , define the map @xmath66 via @xmath67 where @xmath68 . let @xmath69 be given by @xmath70 for @xmath71 by proposition [ t : flow ] , @xmath64 is the flow of euler equation on @xmath72 , i.e. @xmath73 satisfies the euler equations ( [ euler_equations ] ) . it is clear from the preceding developments that @xmath63 ( as a flow of a spray ) is a flow of hamiltonian vector field on @xmath74 . the following commutative diagram @xmath75 suggests that the flow of euler equation itself , which is obtained from @xmath63 via poisson reduction , should be a hamiltonian flow in the sense of poisson manifolds and this is certainly formally true ( see , for instance @xcite for both the case considered here as well as the case of free boundary problems ) . however , as noted in this reference and elsewhere , there are difficulties in finding the right class of functions so that one gets a poisson structure in a precise sense . to justify the formal insight in precise function spaces , one has to overcome two hurdles . the first hurdle is that @xmath76 is only a weak symplectic manifold , and therefore does not necessary carry a poisson bracket in any obvious way without special ad hoc hypotheses such as `` the needed functional derivatives exist '' which have long been recognized as awkward at best . the second hurdle is that @xmath76 is not a lie group in the usual sense ( left multiplication is not smooth ) , and @xmath53 is not a smooth map ( inversion in @xmath28 is not smooth ) . therefore , the well developed theory of poisson and lie - poisson reduction is not directly applicable in this case , even though the loss of derivatives one suffers from these transformations is well understood . the main point of this paper is to resolve these difficulties in what we believe is a satisfactory way . we do this in the following sections . let @xmath77 be a weak riemannian manifold modelled on banach space @xmath78 with metric @xmath79 . then @xmath80 possesses a canonical _ weak _ symplectic form that is given in charts by the following standard formula ( see , e.g. , @xcite ) : @xmath81 where @xmath82 , @xmath83 . for a smooth function @xmath84 on a ( strong ) symplectic manifold @xmath85 , let @xmath86 denote its hamiltonian vector field . then @xmath87 makes @xmath88 into a poisson manifold . since @xmath89 is weak , formula [ f : symppoisson ] does not automatically define poisson bracket @xmath90 for arbitrary functions @xmath91 since @xmath92 may fail to exist and even if they do , one has to make additional hypotheses to obtain the jacobi identity . however , under the two additional hypothesis : 1 . @xmath77 has smooth riemannian connection ; 2 . the inclusion @xmath93 ( the literal dual space ) via @xmath94 is dense , it will be shown that one can define a poisson bracket on the subalgebra @xmath95 of @xmath96 . here @xmath97 are covariant partial derivatives on @xmath80 , the definition of which will be given below . this newly defined bracket _ makes @xmath98 into a lie algebra and retains essential dynamical properties of a `` true '' poisson bracket , including the jacobi identity and the fact that flows of hamiltonian vector fields are poisson maps and , of course , energy is conserved . _ moreover , we will show that the bracket indeed is related to the canonical weak symplectic form in the way that one would expect . in the following we assume that conditions ( 1 ) and ( 2 ) are satisfied . [ [ covariant - partial - derivatives . ] ] covariant partial derivatives . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + first , we introduce covariant partial derivatives on @xmath80 . let @xmath99 and @xmath100 be natural projections , @xmath101 be a christoffel map and @xmath102 be a connector map . in local representation , @xmath103 define @xmath104 by @xmath105 it is standard that @xmath106 is a diffeomorphism ( see @xcite ) . for @xmath107 we set @xmath108 @xmath109 in local representation , this reads @xmath110 and @xmath111 similarly , for @xmath112 we define @xmath113 ( here @xmath114 is the space of linear maps @xmath115 ) by @xmath116 @xmath117 the following lemmas are readily verified . [ t : prop1 ] let @xmath118 be a vector field on @xmath80 , @xmath119 be a vector field on @xmath120 , @xmath121 . then @xmath122 [ t : prop2 ] for @xmath123 , we have @xmath124 where @xmath125 is the parallel translation of @xmath126 along the curve @xmath13 with @xmath127 . let @xmath128 now we can define the bracket @xmath129 via @xmath130 [ [ preliminaries - on - the - poisson - structure . ] ] preliminaries on the poisson structure . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the following is the first main result . [ t : prop4 ] the bracket maps @xmath131 into @xmath132 and also maps @xmath133 into @xmath134 . [ [ remark . ] ] remark . + + + + + + + by definition of the covariant partial derivatives , @xmath135 for @xmath136 . the theorem asserts that if @xmath137 then , in fact , @xmath138 , i.e. there are @xmath139 such that @xmath140 and the maps @xmath141 have appropriate smoothness . define operator @xmath142 . this definition extends the usual notion of covariant derivative from vector fields along curves on @xmath77 to arbitrary curves on @xmath80 . let @xmath143 and @xmath137 . choosing @xmath125 as in lemma [ t : prop2 ] , we obtain @xmath144 to proceed further , we need to calculate the quantity @xmath145 where @xmath146 is an arbitrary element of @xmath147 . let @xmath148 be a parametric surface in @xmath80 with the following properties : 1 . @xmath149 ; 2 . @xmath150 is a parallel translation of @xmath126 ; 3 . @xmath151 is a parallel translation of @xmath152 ; 4 . @xmath153 for all @xmath22 ; 5 . @xmath148 is a parallel translation of @xmath150 for all @xmath22 . then , keeping in mind lemmas [ t : prop1 ] , [ t : prop2 ] and symmetry of riemannian connection , one checks the following : @xmath154 @xmath155 @xmath156\ ] ] @xmath157\ ] ] @xmath158 @xmath159 [ t : lemma1,2 ] ) . let @xmath160 denote the ricci curvature tensor . then @xmath161 @xmath162 by construction of @xmath148 , we have @xmath163 . applying lemma [ t : lemma1,2 ] we obtain @xmath164 @xmath165 thus , @xmath166 by bianchi s identity . similar calculations yield @xmath167 @xmath168 substituting this into the formulas for @xmath169 and using bianchi s identity once again , we get @xmath170 similarly , @xmath171 @xmath172 as @xmath173 is smooth , the statement of the theorem follows . [ [ hamiltonian - vector - fields . ] ] hamiltonian vector fields . + + + + + + + + + + + + + + + + + + + + + + + + + + the smoothness structure of hamiltonian vector fields is given as follows . [ t : prop5,6 ] the vector field @xmath57 is a @xmath174 hamiltonian vector field ( with respect to canonical weak symplectic form ) on @xmath80 of class @xmath174 if and only if @xmath175 . moreover , @xmath176 in local representation , we have @xmath177 indeed , @xmath178 substituting this expression into the formula for @xmath89 and using the symmetry of @xmath179 we obtain the desired result . let @xmath180 be a hamiltonian vector field , @xmath181 be arbitrary . then @xmath182 on the other hand , by lemma [ t : prop1 ] @xmath183 setting @xmath184 and comparing the above expressions we see that @xmath185 similarly , setting @xmath186 yields @xmath187 thus , @xmath188 . conversely , let @xmath189 . defining a vector field @xmath57 by formula [ e : ham ] and substituting into formula [ e : omega ] one obtains for arbitrary vector @xmath190 @xmath191 [ t : prop7 ] let @xmath192 be arbitrary . then @xmath193 by proposition [ t : prop5,6 ] , the vector fields @xmath92 are defined whenever @xmath194 is . then @xmath195 [ t : pbrack ] the bracket @xmath196 is antisymmetric , bilinear , derivation on each factor and makes @xmath134 into a lie - algebra . antisymmetry , linearity and property of being derivation follows directly from the definition of the bracket . by theorem [ t : prop4 ] @xmath196 leaves @xmath134 invariant . then , jacobi identity follows from proposition [ t : prop7 ] in the usual way , for example as in @xcite . now , @xmath80 has both symplectic and poisson structures , and therefore two generally different definitions of hamiltonian vector fields . we need to check that in our case these coincide . to do so , let @xmath197 temporarily denote the hamiltonian vector field with respect to poisson structure @xmath196 and @xmath86 denotes the hamiltonian vector field with respect to canonical symplectic form corresponding to function the @xmath198 . recall , that @xmath197 is defined as a vector field such that @xmath199= { \left \{h , f \right \ } } \quad \forall h\in \mathcal{k}.\ ] ] thus , for all @xmath200 , @xmath201 & = { \frac{\partial h}{\partial \eta } } \cdot \frac{\partial f}{\partial v } - { \frac{\partial h}{\partial v } } \cdot \frac{\partial f}{\partial \eta}\\ & = dh \cdot \mathbf{x}_{f}^p = { \frac{\partial h}{\partial \eta } } \cdot t \tau \mathbf{x}_{f}^p+{\frac{\partial h}{\partial v } } \cdot k \mathbf{x}_{f}^p\end{aligned}\ ] ] and therefore , @xmath202 and @xmath203 . comparing this with formula [ e : ham ] , we see that @xmath204 . finally , from the coordinate expression , it is easy to see that @xmath86 is a well defined @xmath174 vector field for any @xmath205 . previously we established that classes @xmath206 are preserved under bracketing . unfortunately , for @xmath205 and a diffeomorphism @xmath207 the composition @xmath208 does not have to be in any class @xmath209 . one can , however , compose with _ symplectic diffeomorphisms . _ [ t : compdif ] let @xmath210 be a symplectic @xmath174 diffeomorphism , @xmath205 . then @xmath211 . we have @xmath212 and so by proposition [ t : prop5,6 ] , @xmath213 . [ t : flowpoisson ] let @xmath63 be a flow of a smooth hamiltonian vector field on @xmath214 . then @xmath63 is a poisson , i.e. for all @xmath215 @xmath216 @xmath63 is symplectic with respect to the weak riemannian form . since @xmath63 preserves class @xmath134 , the statement follows from jacobi identity by the usual argument . as we stated earlier , in @xcite it is shown that @xmath28 carries a smooth riemannian connection , and therefore the results of the previous section apply . therefore , by those results , the space @xmath74 carries a poisson structure ( in the precise sense given there ) which we denote @xmath196 . let @xmath217 , @xmath218 , @xmath219 stand for the corresponding connector maps on the underlying manifold @xmath10 , on @xmath29 and @xmath28 respectively , while @xmath220 , @xmath221 , @xmath222 are the corresponding connections and @xmath179 , @xmath223 , @xmath224 are the corresponding christoffel maps . in the following @xmath79 denotes the riemannian metric on @xmath10 , @xmath29 , @xmath28 and an induced scalar product on @xmath225 depending on the context . the relationship between these metrics is given by [ e : scal ] . recall the notation from [ s : poisson ] . namely , let @xmath63 be the flow of the spray on @xmath76 , @xmath64 denote the flow of euler equation on @xmath72 and @xmath226 , @xmath227 . recall also that we have the commutative diagram [ t : commdiag ] the following diagram is commutative : @xmath228 now we prepare and recall from @xcite some useful lemmas . let @xmath229 . define @xmath230 via @xmath231 . then @xmath232 indeed , notice that @xmath233 by right invariance of the spray . thus , @xmath234 is an integral curve of @xmath39 . since @xmath235 , the statement of the lemma follows from uniqueness of integral curves . recall that by definition , @xmath236 for all @xmath237 . let @xmath238 . then , using the preceding lemma , we obtain @xmath239 notice , that @xmath240 for any @xmath229 . indeed , @xmath241 thus @xmath242 and the proposition is proved . [ [ a - poisson - structure - on - the - lie - algebra . ] ] a poisson structure on the lie algebra . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + now , we construct a poisson bracket @xmath243 on @xmath72 so that @xmath53 is a poisson map . for @xmath244 such that @xmath245 define @xmath246 as in [ s : weakpoisson ] , define @xmath247 and @xmath248 let @xmath249 . then @xmath243 is a bilinear map @xmath250 and a derivation on each factor . moreover , it satisfies jacobi identity on @xmath251 , that is for all @xmath252 , and @xmath253 , @xmath254 let @xmath255 . recall , that for @xmath256 , @xmath257 is an algebra . thus , @xmath258 is a bilinear bounded map @xmath259 ( and @xmath260 ) , hence smooth . this implies that @xmath261 bilinearity and derivation property of @xmath243 trivially follows from properties of @xmath262 and @xmath79 . now we calculate @xmath263 . let @xmath264 . since @xmath265 , the frchet derivative of @xmath266 exists and coincides with its gateaux derivative . thus , by bilinearity of scalar product and @xmath220 , @xmath267 [ t : lem5 ] let @xmath268 , @xmath249 , and let @xmath269 be @xmath4 vector fields on m. then @xmath270 by the sobolev theorems , @xmath118 is a @xmath271 vector field on @xmath10 . by properties of the riemannian connection , for all @xmath272 @xmath273 thus , @xmath274 let @xmath275 be a flow of @xmath118 on @xmath10 . since @xmath118 is divergence free , @xmath20 is @xmath275 invariant , i.e. @xmath276 , where @xmath277 denotes a pullback by @xmath275 . then @xmath278 [ t : lem6 ] let @xmath279 , @xmath280 . then for all @xmath281 @xmath282 we compute as follows : @xmath283 [ t : hodge ] . let @xmath118 be an @xmath4 vector field on @xmath10 , @xmath284 . there is an @xmath285 function @xmath286 and an @xmath4 vector field @xmath119 with @xmath119 divergence free , such that @xmath287 further , the projection maps @xmath288 @xmath289 are continuous linear maps on @xmath290 . the decomposition is orthogonal in @xmath291 sense , that is for all @xmath292 @xmath293 [ t : lem7 ] there is a bilinear continuous map @xmath294 @xmath295 such that for all @xmath296 @xmath297 fix coordinate system @xmath298 on @xmath10 and let @xmath299 denote components of metric tensor , @xmath300 denote components of vector field @xmath301 in the chosen system . let @xmath302 ( as usually , the summation on repeated indexes is understood ) . then @xmath303 where @xmath304 since @xmath4 is an algebra for @xmath305 it follows that @xmath306 is an @xmath4 vector field . now we set @xmath307 and use [ e : hodge ] . by lemmata [ t : lem5]-[t : lem7 ] , we have @xmath308 thus for any @xmath309 , @xmath310 \\ & \quad + ddf(v)\cdot b(dg(v),v)-ddg(v)\cdot b(df(v),v ) \\ & \quad + ddg(v ) \cdot p_e { \nabla}_{df(v)}v - ddf(v ) \cdot p_e { \nabla}_{dg(v)}v,\end{aligned}\ ] ] and hence @xmath311 and @xmath312 . [ [ remark.-1 ] ] remark . + + + + + + + if @xmath313 , then @xmath314 , @xmath315 are @xmath174 as maps @xmath316 , hence @xmath317 . now we prove the jacobi identity . to simplify notation , we set @xmath318 moreover , since in the following argument all functions are evaluated at the same point @xmath319 , we will write @xmath320 instead of @xmath321 , etc . by lemmata [ t : lem6]-[t : lem7 ] , we obtain @xmath322,{\nabla}_{f}-b_f \right \rangle}+ { \left \langleddg \cdot ( b_h - { \nabla}_{h } ) , { \nabla}_{f}-b_f \right \rangle } \\ & \qquad + { \left \langleddh \cdot ( { \nabla}_{g } - b_g),{\nabla}_{f}-b_f \right \rangle } \\ & = { \left \langle[dh , dg],{\nabla}_{f}-b_f \right \rangle}+d_{ghf}-d_{hfg},\end{aligned}\ ] ] where @xmath323 and @xmath324 $ ] is a lie bracket of vector fields on @xmath10 . notice that lie bracket of divergence free vector fields is divergence free . for @xmath325 @xmath326,{\nabla}_{f}-b_f \right \rangle}={\left \langle\left[\left[dh(v),dg(v)\right],df(v)\right],v \right \rangle}.\ ] ] since terms of type @xmath327 cancel out in the jacobi cycle @xmath328 and so the jacobi identity for bracket @xmath243 follows from the jacobi identity for vector fields . however , for @xmath329 lie bracket of @xmath330 and @xmath331 is an @xmath332 vector field , hence merely continuous and therefore @xmath333,df(v)\right]$ ] may fail to exist . therefore , in this case more care is needed . let @xmath334 @xmath335}v \right \rangle}\ ] ] with this notation in mind , by lemma [ t : lem5 ] and the hodge decomposition @xmath326,{\nabla}_{f } \right \rangle}={\left \langle{\nabla}_{dh}dg-{\nabla}_{dg}dh,{\nabla}_{df}v \right \rangle}= - a_{ghf}+a_{hgf}.\ ] ] similarly , by definition of @xmath336 @xmath326,b_f \right \rangle}= c_{fhg}.\ ] ] by a well known formula for riemannian connection , @xmath337}z,\ ] ] for all sufficiently smooth vector fields @xmath338 . thus , @xmath339}v \right \rangle}= c_{fgh}.\ ] ] thus , @xmath340 and so @xmath341 [ [ remark.-2 ] ] remark . + + + + + + + if @xmath342 , @xmath249 , then by lemma [ t : lem5 ] @xmath343,v \right \rangle}.\ ] ] this shows that bracket @xmath243 is naturally related to lie - poisson bracket on @xmath344 . now we establish the relationship between poisson bracket @xmath243 on @xmath72 that we just introduced and poisson bracket @xmath196 on @xmath28 . for @xmath345 define @xmath346 [ t : eqdef ] define the function spaces @xmath347 and @xmath348 then @xmath349 for @xmath350 @xmath351 and for all @xmath352 @xmath353 without loss of generality @xmath354 . since @xmath53 is not even a @xmath271 function @xmath355 it is not obvious that @xmath356 is defined . however , differentiating @xmath357 and @xmath358 as functions @xmath359 one obtains the required result . [ t : dfrv ] under the assumptions of the theorem , @xmath360 it is well known ( @xcite ) that @xmath361 . notice , that for @xmath362 , @xmath363 where time derivative is taken in @xmath76 . by lemma [ t : prop1 ] @xmath364 thus , by right invariance of the metric on @xmath28 @xmath365 [ t : dfrn ] under the assumptions of the theorem @xmath366 \right \rangle}_e\ ] ] that is , @xmath367 first , we calculate @xmath368 . let @xmath369 , @xmath370 be a parallel translation of @xmath126 with @xmath371 . recall that @xmath372 then , by lemma [ t : prop2 ] , @xmath373 since connection on @xmath28 is right invariant , i.e. , @xmath374 we have @xmath375=\left [ \widetilde{k } \frac{d}{dt}_{{t=0 } } v_t \right ] \circ \eta^{-1 } = 0.\ ] ] by lemma [ t : prop1 ] @xmath376 combining above equalities together , we get @xmath377.\ ] ] @xmath378 \right \rangle}_e.\ ] ] we claim that for all @xmath379 @xmath380 \right \rangle}={\left \langlez,{\nabla}_{y}x \right \rangle}.\ ] ] recall that by construction ( see @xcite ) , @xmath381 @xmath382 @xmath383 by a well known formula of differential geometry , we have @xmath384 and hence @xmath385 = p_e[{\nabla}_{y}x].\ ] ] by the hodge decomposition @xmath386 \right \rangle } = { \left \langlez,{\nabla}_{y}x \right \rangle}={\left \langleb(z , x),y \right \rangle}.\ ] ] by the above developments and right invariance of metric on @xmath28 , we have @xmath387 calculating @xmath356 at @xmath309 by lemmata [ t : dfrv],[t : dfrn ] , we obtain @xmath388 [ t : pipoisson ] map @xmath389 is a poisson map , i.e. for all @xmath390 pointwise in @xmath391 @xmath392 @xmath393 since @xmath53 is the identity on @xmath72 , the statement follows immediately from theorem [ t : eqdef ] . [ t : fpoisson ] let @xmath394 and @xmath395 are such that @xmath396 , @xmath397 , @xmath398 , @xmath399 , @xmath400 . then @xmath401 in particular , @xmath63 preserves @xmath402 and for @xmath403 pointwise in @xmath391 @xmath404 without loss of generality @xmath405 . first , we notice that covariant partial derivatives of @xmath406 at @xmath407 are elements of @xmath76 . indeed , @xmath408 there is a function @xmath409 such that @xmath410 thus , @xmath411 however , by proposition [ t : compdif ] @xmath412 for any @xmath413 , hence there is @xmath414 such that for all @xmath19 , @xmath415 in a similar sense , one shows that @xmath416 . thus , @xmath417 is well defined and depends only on values of @xmath418 calculated at point @xmath419 . however , @xmath420 also depends only on values of covariant partial derivatives at @xmath419 . then , we choose @xmath421 such that @xmath422 the equality @xmath423 follows from proposition [ t : flowpoisson ] . by the preceding arguments , the same holds if we replace @xmath424 with @xmath425 . this concludes the first part of the proposition . the second part then follows . [ s : redpoisson ] the map @xmath426 is poisson with respect to the bracket @xmath243 . . then @xmath428 . by proposition [ t : fpoisson ] @xmath429 and we have pointwise in @xmath430 : @xmath431 in the previous sections we successfully implemented a nonsmooth lie - poisson reduction technique for the study of the euler equations of ideal fluid flow . this enabled us to find a precise sense in which the flow of euler equation on the lie algebra of divergence free vector fields ( parallel to the boundary of the fluid region ) is a hamiltonian system in the poisson sense and that the flow consists of poisson maps , despite the fact that this flow is believed ( as maps from @xmath4 to @xmath4 ) to be continuous , but not differentiable . a key part of this process was to introduce a poisson structure on the space of divergence free vector fields . as one would expect from the bracket derived via a type of lie - poisson reduction , this bracket is closely related to the formal lie - poisson bracket on the dual to the lie algebra of divergence free vector fields . even though we consider only euler s equation , the technique developed here is directly applicable to several other important systems those which can be written as an ode on groups of diffeomorphisms , such as the following : 1 . the camassa - holm ( ch ) equation on @xmath432see @xcite : @xmath433 2 . the averaged euler equations ( or the lae-@xmath434 equations)see @xcite : @xmath435 where @xmath436 and @xmath12 satisfies appropriate boundary conditions , such as the no - slip conditions @xmath437 on @xmath438 . 3 . the epdiff equation ( also called the averaged template matching equation ) on a compact manifold @xmath10see @xcite and @xcite : @xmath439 with appropriate boundary conditions , such as the no - slip conditions @xmath437 on @xmath11 . the epdiff equations reduce to the ch equations in the case @xmath440 . these equations may be derived as the right reduction to the identity of the geodesic motion on the appropriate lie group ( see , for example , @xcite and @xcite for the case of the ch equations ) , and the preceding references for the other equations . the crucial technical fact that enables our methods to work in both cases is the smoothness of the spray on the lie group . for the case of the ch equations and the lae-@xmath434 equations on regions with no boundary , this is due to @xcite and for regions with boundary to @xcite . for the case of the epdiff equations , a rather convincing plausibility argument is given @xcite . one important direction in which we would like to pursue these ideas is that of nonsmooth solutions . even for the ideal euler equations , this is interesting because of the singular solutions , such as point vortices , vortex filaments and sheets . they clearly have themselves an interesting poisson structure , as was investigated by @xcite and @xcite . there are similar interesting singular solutions for the epdiff equations , whose geometry is investigated in @xcite . it would be very interesting if , on the smaller spaces appropriate for these classes of singular solutions that are introduced in these references , the smooth spray property still holds and , if that is the case , whether or not one could then carry out the program in the present paper . another interesting direction for the present research is to the case of free boundary problems , a notoriously difficult case for infinite dimensional poisson structures , even at the formal level ( see @xcite , @xcite , @xcite and @xcite . ) holm , d.d . and j. e. marsden [ 2003 ] , momentum maps and measure valued solutions ( peakons , filaments , and sheets ) of the euler - poincar equations for the diffeomorphism group . in marsden , j. e. and t. s. ratiu , editors , _ festshrift for alan weinstein ( to appear)_. birkhuser boston .

this paper provides a precise sense in which the time @xmath0 map for the euler equations of an ideal fluid in a region in @xmath1 ( or a smooth compact @xmath2-manifold with boundary ) is a poisson map relative to the lie - poisson bracket associated with the group of volume preserving diffeomorphism group . this is interesting and nontrivial because in eulerian representation , the time @xmath0 maps need not be @xmath3 from the sobolev class @xmath4 to itself ( where @xmath5 ) . the idea of how this difficulty is overcome is to exploit the fact that one does have smoothness in the lagrangian representation and then carefully perform a lie - poisson reduction procedure .

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Proceed to summarize the following text: from the bethe - weizsacker mass formula , it is well understood that the symmetry energy from bulk matter is the difference between the energy of pure neutron matter and pure symmetric matter . mathematically , it can we written as @xmath0 where @xmath1 and @xmath2 . @xmath3 , @xmath4 , and @xmath5 are the neutron , proton , and nuclear matter densities , respectively . the symmetry energy has great importance in the dense matter existing in the neutron stars , but only indirect information can be extracted from astrophysical observations @xcite . it is also important in the quark gluon plasma ( qgp ) and hadron gas ( hg ) phase @xcite . the qgp and hg phases existed in the early stage of the evolution of universe ( about 15 billion years ago ) and are inaccessible nowadays . it is difficult to recreate these conditions , although numerous experiments are occurring at the relativistic heavy ion collider ( rhic ) and the large hadron collider ( lhc ) @xcite . heavy - ion reactions , during which matter goes through compression and expansion , are considered to be the true testing ground for the hot and dense matter phases . the nuclear equation of state ( neos ) and the density dependence of the symmetry energy can be probed by some observables in intermediate - energy heavy - ion collisions ( hics ) . the softness of the neos has been well described in the literature in the last couple of decades @xcite . however , the density dependence of the symmetry energy , from the coulomb barrier to the deconfinement of nuclear matter , is a hot topic in the present era . at sub - saturation densities , the density dependence of the symmetry energy is studied by observables such as the neutron - to - proton ratio , isotopic and isobaric scaling , isospin diffusion , isospin fractionation and/or distillation , and isospin migration . recently , the msu group @xcite claimed the softness of the symmetry energy at sub - saturation densities by using the double neutron - to - proton ratio and isospin diffusion from two isotopic systems , @xmath6sn + @xmath6sn and @xmath7sn + @xmath7sn at e = 50 mev / nucleon . in another study , again soft symmetry energy was claimed by using the isospin diffusion for the same set of reactions , but at e = 35 mev / nucleon @xcite . in a recent study , soft symmetry energy is also favored for the same set of reactions at e = 50 mev / nucleon by using the neutron - to - proton ratio @xcite . in all the studies , the problem of sub - saturation density dependence of the symmetry energy seems to be addressed to some extent ; however , the uncertainties are still large enough to justify the large amount of work that is under way in many laboratories all over the world . in contrast , the present status of supra - saturation density dependence of the symmetry energy is quite uncertain and interesting . the high - density behavior of the symmetry energy in the literature is studied by using two important parameters : one is the yield ratio parameter and second is the flow parameter . the yield ratio parameter has been studied in term of single and double ratios of neutrons to protons @xcite , single and double ratios of @xmath8 @xcite , the @xmath9 ratio @xcite , the @xmath10 ratio @xcite , and isospin fractionation @xcite , while , the flow parameter has been studied in terms of relative and differential flows ( single and double ratios ) of neutrons to protons or @xmath11h to @xmath11he @xcite , and in terms of the ratio @xcite or difference @xcite of neutron - to - proton elliptic flow . before using the @xmath11h and @xmath11he particle yield and flow ratios for the density dependence of the symmetry energy at high incident energies , one must check the production of these particles in the supra - saturation density region , which is obtained during the highly compressed stage only . however , the production of neutrons and protons occurs in large amounts and can explain the high density dependence of symmetry energy with great accuracy . favorable results with neutron and proton elliptic flow at e = 400 mev / nucleon were also observed in 2011 . in one of the studies , the softness of the symmetry energy with @xmath12 is predicted by comparing the fopi collaboration data with the neutron - to - proton elliptic flow ratio @xcite . in the same year , cozma _ et al . _ @xcite predicted the softness of symmetry energy with @xmath13 = 2 by comparing the fopi data with the neutron - to - proton elliptic flow difference . even then uncertainty lies in the results , in terms of determination of symmetry energy : in the first study , symmetry energy is momentum independent , while in later one it is from momentum - dependent interactions . moreover , the studies were limited to only 400 mev / nucleon . let us examine some interesting features from the ratio parameters at supra - saturation densities . all the ratio parameters show sensitivity to the symmetry energy . in the literature , it is also claimed that @xmath14 and @xmath9 have more sensitivity than @xmath8 @xcite . the sensitivities of all the parameters is checked in term of transverse momentum and rapidity distribution dependence @xcite , while pion and kaon ratio studies are extended with the isospin asymmetry of the system and the incident energy @xcite . in recent years , when the pion ratio has been compared with the fopi data by using the two well known models ibuu04 and imiqmd , in terms of isospin asymmetry and incident energy , the predictions for the symmetry energy are found to be totally opposite . imiqmd predicts stiff symmetry energy ( @xmath15 ) @xcite , while ibuu04 predicts soft symmetry energy ( x=1 ) @xcite . in the present era , the @xmath8 ratio is supposed to be a strong candidate for predicting the high - density behavior of symmetry energy . just as for @xmath16 and @xmath17 , neutrons and protons are also produced in large amounts up to 1 gev / nucleon . even around 400 mev / nucleon , the production of neutrons and protons is greater than the production of pions . unfortunately , the neutron - to - proton ratio parameter in most studies is restricted only with the transverse momentum and kinetic energy dependencies @xcite . to draw a fruitful conclusion in the near future , first , it is very important to study the isospin asymmetry and incident energy dependencies of the neutron - to - proton ratio , just as in the recent @xmath8 study , and then compare the sensitivity to the symmetry energy from both ratios , as the pion ratio results were recently compared with the fopi experimental findings . second , one has to avoid choosing randomly any type of fragment to study the supra - saturation density dependence of symmetry energy . for this , it is important to check whether or not a particular type of fragment is formed in the region @xmath18 , which is simple when one addresses the sub - saturation density dependence of symmetry energy . finally , with increasing incident energy , the time evolution of the density has a different trend at two extremes : one at the time of maximum compression and the second at the freeze - out time ( t = 200 fm / c ) . it also becomes interesting to see the different stiffnesses of the symmetry energy dependence of the neutron - to - proton ratio for incident energies at the time of maximum compression and at freeze - out time . in the concluding remarks of this paper , we have tried to address the following goals : * to check the sensitivities of different kind of fragments to the high - density behavior of symmetry energy . * to check the behavior of the neutron - to - proton ratio at the time of maximum compression and at freeze - out time . * to check the isospin asymmetry and incident energy dependences of single and double neutron - to - proton ratios from different neutron - rich systems to the high - density behavior of symmetry energy , and then compare the sensitivity of symmetry energy from the neutron - to - proton ratio with that from the pion ratio . this study is similar to recent studies using the pion ratio @xcite . for the present study , the isospin quantum molecular dynamics ( iqmd ) model is used to generate the phase space of nucleons , which is discussed in sec . ii . the results are discussed in sec . iii , and we summarize the results in sec . in the iqmd model @xcite , nucleons are represented by wave packets , just as in the qmd model of aichelin @xcite . these wave packets of the target and projectile interact via the full skyrme potential energy , which is represented by @xmath19 and is given as : @xmath20 here @xmath21 is the coulomb energy and @xmath22 originates from the density dependence of the nucleon optical potential , and is given as @xmath23 the first two of the three parameters of eq . [ equation2 ] ( @xmath24 and @xmath25 ) are determined by demanding that , at normal nuclear matter densities , the binding energy should be equal to 16 mev and the total energy should have a minimum at @xmath26 . the third parameter @xmath27 is usually treated as a free parameter . its value is given in term of the compressibility : @xmath28 the different values of compressibility give rise to soft and hard equations of state . the soft equation of state is employed in the present study with the parameters @xmath29 mev , @xmath30 mev , and @xmath27 = 7/6 , corresponding to an isoscaler compressibility of @xmath31 mev . in the third term @xmath32 is the potential part of the symmetry energy , which is adjusted on the basis of calculations from the microscopic or phenomenological many - body theory , having the form @xmath33 here @xmath34mev , parameterized on the basis of the experimental value of the symmetry energy , is known as the symmetry potential energy coefficient . on the basis of the @xmath35 value , symmetry energy is divided into two types with @xmath36 and @xmath37 , corresponding to soft and stiff symmetry energies , respectively . the total symmetry energy per nucleon employed in the simulation is the sum of the kinetic and potential terms and is given as @xmath38 where @xmath39 mev is known as the symmetry kinetic energy coefficient . the kinetic symmetry energy originates from the fermi - dirac distribution @xcite . finally , we get a density and isospin - single particle potential in nuclear matter as follows : @xmath40 here @xmath41 , @xmath42 , and @xmath43 . the potential also depends on the momentum - dependent interactions , which are optional in the iqmd model . note that the @xmath27 used in the determination of the equation of state and @xmath35 used in the determination of symmetry energy are different parameters . the interesting feature of symmetry energy is that its value increases with decreasing @xmath35 at sub - saturation densities , while the opposite is true at supra - saturation densities . in other words , soft symmetry energy is more pronounced at sub - saturation densities , while stiff symmetry energy is more pronounced at supra - saturation densities . in the calculations , we use the isospin - dependent in - medium cross section in the collision term and the pauli blocking effects as in the qmd model @xcite . the cluster yields are calculated by means of the coalescence model , in which particles with relative momentum smaller than @xmath44 and relative distance smaller than @xmath45 are coalesced into a one cluster . the value of @xmath45 and @xmath44 for the present work are 3.5 fm and 268 mev / c , respectively . the neutron - to - proton ratio is among the first observables that was proposed as a possible sensitive probe for symmetry energy prediction at sub - saturation densities @xcite ; however , some studies are also performed using the rapidity distribution and transverse momentum dependencies at supra - saturation densities . in this article , the sensitivities of free nucleons , light charged particles ( lcps , having charge number between 1 and 2 ) , and intermediate mass fragments ( imfs , having charge number between 3 and @xmath46 ) to the high density behavior of symmetry energy are checked , providing the results of incident energy and isospin asymmetry dependencies of single and double neutron - to - proton ratios with the high - density sensitive fragments . to perform the study , thousand of events are simulated for the isotopes of sn , namely @xmath6sn + @xmath6sn , @xmath7sn + @xmath7sn and @xmath47sn + @xmath47sn between incident energies of 50 and 600 mev / nucleon at semicentral geometry by using the soft and stiff symmetry energies of @xmath36 and 1.5 , respectively . as discussed earlier , a soft equation of state with an isospin- dependent nucleon - nucleon ( nn ) cross section of @xmath48 is employed . the incident energy and isospin asymmetry dependences of single and double neutron - to - proton ratios , just as for the @xmath8 ratio @xcite , are considered as a point of importance in the present study . the single ratio is just the ratio of neutrons to protons and is represented in the study by @xmath49 , while double ratio is the ratio of the single ratios of any two isotopes of sn . in order to study the systematics of the isospin effects , the single ratio of the isotope with a greater number of neutrons is always mentioned in the numerator when the double ratio is calculated . mathematically , the double ratio is represented by @xmath50 and is given as @xmath51 ( color online ) time evolution of the average density for the @xmath47sn + @xmath47sn reaction with the soft symmetry energy ( @xmath52 ) at semicentral geometry . the different lines represent different incident energies ranging from 50 to 600 mev / nucleon.,width=340 ] to predict the high - density behavior of symmetry energy , the very first point is to understand the time evolution of the average density at different incident energies . with increasing incident energy , the density will be expected to be greater than the normal nuclear matter density in the most compressed region . as we know , the density of the environment surrounding the nucleons of a fragment plays a crucial role in determining the physical process behind its formation . in fig . [ fig:1 ] , we display the average density @xmath53 reached in a typical reaction as a function of time at different incident energies for @xmath47sn + @xmath47sn by using the soft symmetry energy @xmath36 . the average nucleon density is calculated as @xcite @xmath54\rangle,\end{aligned}\ ] ] with @xmath55 and @xmath56 being the position coordinates of the @xmath57 and @xmath58 nucleons , respectively . as we have expected , with increasing of incident energy , the density is found to increase in the compression zone . interestingly , at lower beam energies , the maximum density reached is lower and the reaction time is longer . with increasing incident energy , the life - time of the high - density nuclear matter gets shorter due to instability . for example , at b = 2 fm the average density reaches a maximum and is close to normal nuclear matter density at @xmath59= 18 and 33 fm / c , respectively , for e = 50 mev / nucleon ; but for the case of e = 600 mev / nucleon , the respective times are 10 and 20 fm / c . this means that the difference between the two times is almost 15 fm / c at e = 50 mev / nucleon , while it is only 10 fm / c at e = 600 mev / nucleon . this clearly indicates that the matter shows high - density behavior only for a small time interval , which decreases with increasing incident energy . since we are interested in the sensitivities of different kinds of fragments and their neutron - to - proton ratios , only those fragments that lie in the high - density region ( @xmath18 ) will be sensitive to the high - density behavior of the symmetry energy . ( color online ) time evolution of free nucleons ( top ) , lcps ( middle ) , and imfs ( bottom ) at semicentral geometry for @xmath47sn + @xmath47sn using the soft symmetry energy ( @xmath60 ) . the different lines have the same meaning as in fig . [ fig:1 ] . the vertical line in each panel represents our time limit before which the system can be in the supra - saturation region ( refer to fig . 1).,width=340 ] to check the sensitivities of different kind of fragments , in fig . [ fig:2 ] we display time evolution of free nucleons ( top ) , lcps ( middle ) , and imfs ( bottom ) at semicentral geometry for incident energies ranging from 50 to 600 mev / nucleon . the behavior for all kinds of fragments is consistent with the results in the literature @xcite . the production of free nucleons increases with incident energy , and lcp production decreases after 400 mev / nucleon . in ref . @xcite , lcp production is correlated with the nuclear stopping and is also found to have a maximum at 400 mev / nucleon . imf production is found to decrease after 100 mev / nucleon . this is due to the different origin of the production of imfs as compared to free and lcps . for more details about the incident energy dependence of imfs , see ref . @xcite . our main task is to check the sensitivities of the fragments in the high - density region . for this , we apply the limit that at least one particle must be produced before the time 20 fm / c , because , in an average , after that time the density becomes lower than normal nuclear matter density for all the incident energies under consideration . the free nucleons are highly sensitive at all the energies . this is not true for lcps and imfs . lcps are produced in this region only after the incident energy reaches 200 mev / nucleon . in contrast , no imfs are produced in the supra - saturation density region . this means that imfs are not so sensitive to the high - density dependence of symmetry energy ; however , they can be used at sub - saturation and saturation densities @xcite . interestingly , the single neutron - to - proton ratio from imfs is found to change with the incident energy ( not shown here ) , but this is mainly due to coulomb interactions . here we conclude that the neutron - to - proton ratio from free nucleons as well as lcps can act as a probe of the high - density behavior of the symmetry energy . one more interesting observation is obtained from fig . [ fig:1 ] . with increasing incident energy , the time evolution of the density is exactly opposite during the compression and expansion stages . that is , in the expansion stage the average density is found to decrease with increasing incident energy , which was earlier increasing in the compressed zone . now , we have two aspects of the basis of the time evolution of density : one is the compressed - zone time and second is the freeze - out time . interestingly , if the density behavior is opposite at the two times , then it would supposedly affect the magnitude of the symmetry energy as well as its effect on the nuclear matter during the whole time evolution . ( color online ) excitation function of @xmath61 , which is proportional to @xmath62 , at semicentral geometry . the left panel is at the time of maximum compression , while the right one is at the freeze - out time . solid and open circles with a dashed line represent the contributions of the soft ( @xmath35=0.5 ) and stiff ( @xmath35=1.5 ) symmetry energies , respectively , for the system @xmath47sn + @xmath47sn.,width=340 ] to see the virtual change in the symmetry energy due to the change in the density , we display in fig . [ fig:3 ] the incident energy dependence of @xmath61 , which is proportional to the symmetry energy , at the time of the maximum compression ( left panel ) and at the freeze - out time ( right panel ) . at the time of maximum compression , the symmetry energy rises with the increasing incident energy ( increase in density ) . as the density is more than the normal nuclear matter density in this region , the stiff symmetry energy is stronger than the soft one . with increasing incident energy ( increase in density ) , the stiff symmetry energy is changing drastically , while , the soft symmetry energy shows little change . this exactly coincided with the ideal picture of density dependence of the symmetry energy . on the other hand , if we look at the energy dependence of @xmath61 at @xmath59 = 200 fm / c , the situation is totally different . the symmetry energy is found to decrease with increasing incident energy ( decrease in density ) . now the density is lower than the normal nuclear matter density , so the magnitude of the soft symmetry energy is greater than that of the stiff symmetry energy . in other words , the supra - saturation ( sub - saturation ) density region is more neutron rich with @xmath63 ( @xmath60 ) . the effect from the sub- and supra - saturation density behaviors of symmetry energy will compete and contribute in the final observables . due to the different behavior of density at different times , it is important to observe the isospin effects at the time of maximum compression and at the freeze - out time to understand the high - density behavior of symmetry energy . for this purpose , in the coming sections , the incident energy and isospin asymmetry dependencies of the single and double neutron - to - proton ratios from free nucleons and lcps are analyzed . ( color online ) excitation function of the single neutron - to - proton ratio for free nucleons ( top panel ) and lcps ( bottom panel ) at semicentral geometry . the left and right panels are the same expect they are at the time of maximum compression and at the freeze - out time , respectively . solid and open circles represent the soft and stiff symmetry energies , respectively . solid and dashed lines corresponds to the systems of @xmath6sn + @xmath6sn and @xmath7sn + @xmath7sn , respectively . , width=340 ] in order to address the sensitivity of symmetry energy at the time of maximum compression and at freeze - out time , we display the incident energy and isospin asymmetry dependencies of the single neutron - to - proton ratio at different times in figs . [ fig:4 ] , [ fig:5 ] and [ fig:6 ] . the left and right panels are at the time of maximum compression and freeze - out time , respectively . in fig . [ fig:4 ] , the ratios from free nucleons and lcps are displayed in the top and bottom panels . the many interesting facts are revealed in the figure . the incident energy dependencies of the ratios are found to be highly sensitive to symmetry energy for the two different times . as we know , the relative strength of symmetry energy is opposite at sub- and supra - saturation densities with @xmath60 and @xmath63 . in the range of 50 - 150 mev / nucleon , only the low density part up to about 1.1@xmath26 contributes . therefore , in the low - energy region , for free nucleons , we can see the high ratio with the soft symmetry energy at both the times under consideration . at and above 200 mev / nucleon , a broad range of densities up to 1.8@xmath26 is involved . of course , at about 200 mev / nucleon and above , for the behavior of the high - density symmetry energy , there is a combined effect for particles going through both low - and high - density region . at higher energies , a higher @xmath64 ratio is observed with the stiff symmetry energy for the neutron rich system @xmath47sn + @xmath47sn at the time of maximum compression , which is true with the soft symmetry energy at the freeze - out time . the result is similar for free nucleons and lcps . however , lcps are not as sensitive and the ratio is even less than the ratio of the system . this is due to the excess number of protons involved in the production of lcps compared to free nucleons . these protons will lower the ratio for lcps . it is clearly visible that the ratio at both times is almost the same with the stiff symmetry energy , but changes drastically with the soft symmetry energy . this is due to the fact that , at the time of maximum compression , the density is in the supra - saturation region and the stiff symmetry energy is much higher ( see fig.[fig:3 ] ) than the soft symmetry energy . therefore , the stiff symmetry energy is able to separate most of the neutrons near the time of maximum compression and then accelerate the neutrons toward higher kinetic energy at later times . however , the soft symmetry energy is not so high , and the separation of the neutrons takes place for a longer time . after 50 - 60 fm / c ( see fig . [ fig:1 ] ) , the density drops to the sub - saturation density region and now the soft symmetry energy has a quite high magnitude ( see fig . [ fig:3 ] ) compared to the stiff one . the soft symmetry energy in this region is still separating the neutrons as well as accelerating them high kinetic energy . that is why the ratio with the soft symmetry energy drastically changes when one goes from compression to freeze - out time , but remains almost constant with the stiff symmetry energy . mainly , the neutron - to - proton ratio is found to decrease with the incident energy for free nucleons as well as for lcps , just like the @xmath8 ratio . the decrease in the ratio may be due to two reasons : * one reason may be the role of coulomb interactions with incident energy . with increasing incident energy , chances of break - up of initial correlations among the nucleons becomes stronger , and the production of free nucleons including neutrons and protons will increase . however , at very low incident energy , the production of neutrons is more due to the symmetry energy because of its repulsive ( attractive ) nature for neutrons ( protons ) . in short , due to coulomb interactions , a shift of protons takes place from low to high incident energies . the effect of the coulomb interactions can be checked by taking the double ratio , which is discussed in fig . [ fig:7 ] . * the contribution of pions from secondary - chance nucleon - nucleon collisions might increase with the beam energy . if a first - chance nucleon - nucleon collision converts a neutron to a proton by producing a @xmath16 , then subsequent collisions of the energetic protons can convert them back to neutrons by producing a @xmath17 . therefore , at sufficiently high energy , the neutrons , which are produced due to symmetry energy , are changing into the protons and further producing @xmath65 s , which will lead to a decease in the neutron - to - proton ratio . this can be confirmed by using the double ratio concept . if the double ratio is still deceasing with incident energy , then it means that , in addition to the coulomb interactions , the phenomenon of secondary nucleon - nucleon collisions is also very important . one more point of interest is that the difference between the soft and stiff symmetry energies at freeze - out time is found to decrease with incident energy for free nucleons , while it increases for lcps . of the above two reasons , the first one is applicable for free nucleons as well as for lcps . the second one is applicable only for free nucleons , as the energy in this study is up to 600 mev / nucleon , which is quite sufficient to produce pions . to see the effect of the high - density behavior of symmetry energy on the isospin asymmetry dependence , we display the ratio from free nucleons and lcps in figs . [ fig:5 ] and [ fig:6 ] at only high energies ( 200 , 400 , and 600 mev / nucleon ) . due to the instability of the highly compressed zone , we are not able to differentiate between the results of symmetry energy obtained at the time of maximum compression ; however , we had earlier obtained some important conclusions from fig . [ fig:4 ] , where incident energy dependence was discussed . the results from figs . [ fig:5 ] and [ fig:6 ] at the freeze - out time reveal many important points . the isospin asymmetry dependence of the ratio from free nucleons is highly sensitive to the symmetry energy compared to lcps , i.e. , the ratio from free nucleons is found to be sharply increasing with the isospin asymmetry of the system compared to lcps . this is due to the fact that isospin effects on the ratio from free nucleons are strongly affected by the symmetry energy and weakly affected by coulomb interactions , while the opposite is true for the ratio from lcps . as discussed earlier , the ratio is found to decease with the incident energy , which is also true here for the isospin asymmetry dependence . the difference between the soft and stiff symmetry energy results comes from the behavior of free nucleon emission with the isospin asymmetry of the system , i.e. , the greater the isospin asymmetry of the system , the greater the contribution of neutrons in the ratio due to the symmetry energy . the soft symmetry energy is stronger at the freeze - out time , which will lead to an increase in the ratio more sharply than the stiff symmetry energy . this effect is again weakly observed in the ratio from lcp s . isospin asymmetry dependence of the single neutron - to - proton ratio for free nucleons at different incident energies . the left panel is at the time of maximum compression , while the right panel is at the freeze - out time . solid and open circles represent the soft and stiff symmetry energies , respectively . , width=340 ] same as in fig . [ fig:5 ] but for the lcps.,width=340 ] ( color online ) excitation function of the double neutron - to - proton ratio from different isotopes of sn for free nucleons ( top ) and lcps ( bottom ) . the vertical line in the bottom panel represent the energy limit above which @xmath50 of lcps becomes more or less insensitive . solid and open circles represent the soft and stiff symmetry energies , respectively . the solid , dashed , and dot - dashed line corresponds to double ratios from @xmath47sn + @xmath47sn to @xmath7sn + @xmath7sn , @xmath7sn + @xmath7sn to @xmath6sn + @xmath6sn , and @xmath47sn + @xmath47sn to @xmath6sn + @xmath6sn , respectively . , width=340 ] isospin asymmetry dependence of the double neutron - to - proton ratio from free nucleons at different incident energies . the different symbols have the same meaning as in fig . [ fig:5 ] . , width=340 ] incident energy dependence of the power - law exponent @xmath66 from fig . [ fig:8 ] . the symbols and lines are the same as in figs . 5 and 8.,width=340 ] in order to cancel the coulomb effects and to see the effect of symmetry energy , we show in fig . [ fig:7 ] , the incident energy dependence of the double ratio from different isotopes of sn with different combinations , namely , @xmath47sn + @xmath47sn and @xmath7sn + @xmath7sn , @xmath7sn + @xmath7sn and @xmath6sn + @xmath6sn , @xmath47sn + @xmath47sn and @xmath6sn + @xmath6sn , having differences of 8 , 12 , and 20 neutrons and the same number of protons . the upper and lower panels are for free nucleons and lcps . the double ratio is found to increase with the difference of the number of neutrons in the different combinations . it is similar to the results obtained at sub - saturation densities by different models @xcite . the main point to be discussed here is that the double ratio is found to decrease with increasing incident energy . as we have discussed in fig . [ fig:4 ] , there may be two reasons for the decrease of the single ratio with increasing incident energy . one was the coulomb effect , which is canceled out here . the second was the pion effect , which is still active in the double ratio and becomes more and more dominant with increasing incident energy . due to that effect , the double ratio is found to decrease with the incident energy . it indicates that the pion production effect is very important at high incident energy and is equally useful for understanding the high - density behavior of the incident energy @xcite . in contrast , this effect is valid only for the double ratio from the free nucleons and not from the lcps . the double ratio from lcps is found to be constant above 200 mev / nucleon . this indicates that the effect of the symmetry energy for the ratio from lcps can be analyzed only near sub - saturation densities close to 1.1@xmath26 . the decrease in the single ratio for the lcps was only due to the coulomb interactions at higher incident energies , which is canceled out by taking the double ratio ; the double ratio from the symmetry energy becomes independent of the incident energy after 200 mev / nucleon . this type of dependence for the single @xmath8 ratio can be observed above 1 gev / nucleon @xcite . the behavior of symmetry energy for the double ratio is exactly the same as that for the single ratio . this indicates that lcp production is also not a sensitive probe for investigating the high - density behavior of the symmetry energy . the only possible probe from the fragments is the double ratio of neutrons to protons from free nucleons . another possible probe is the @xmath8 ratio , which recently was compared with the experimental data of the fopi by the ibuu04 and imiqmd calculations @xcite . in order to strengthen our conclusion , in fig . [ fig:8 ] , we display the isospin asymmetry dependence of the double ratio from free nucleons at different incident energies . all the curves are fitted with a power law of the form @xmath67 , where @xmath68 is the double ratio from free nucleons and @xmath13 is the double ratio of the systems . the power - law exponent @xmath66 is found to vary drastically with the symmetry energy , which is to be discussed later in fig . [ fig:9 ] . after canceling the coulomb effects , the trend for the double ratio is the same as that of the single ratio in fig . [ fig:5 ] . it reflects the fact that the isospin effects for free nucleons is stronger for more neutron - rich systems and is mainly due to the symmetry energy . however , the decrease in the isospin effect with the increase of incident energy is due to the production of pions at sufficiently high energy . the difference in the double ratio obtained with the soft and stiff symmetry energies here is also found to increase from the neutron - poor to the neutron - rich system , just like the single neutron - to - proton ratio in fig . [ fig:5 ] as well as the single pion ratio in the literature @xcite . to see the clear systematics of the incident energy toward the symmetry energy , we plot the incident energy dependence of the power exponent @xmath66 in fig . [ fig:9 ] , which is extracted from the curves of fig . [ fig:8 ] . with increasing incident energy , the sensitivity of the symmetry energy goes on decreasing toward the double ratio ; however , the soft symmetry energy is more sensitive in comparison with the stiff one . in brief , when one goes from the sub - saturation to the supra - saturation density region , the soft symmetry still has a crucial role to play compared to the stiff one . this is due to the density ( fig . [ fig:1 ] ) , which undergoes a sudden change between the supra- and sub - saturation density regions with time at higher incident energies . finally , from this study , we confirm that the high - density behavior of symmetry energy can be studied by using the single and double ratios of neutrons to protons from free nucleons . in comparison , the double ratio is more accurate for this purpose , due to its greater sensitivity to the soft symmetry energy . meanwhile , the lighter and heavier fragments ratio can be considered good candidates at sub - saturation densities , and also have been used in the literature many times by different groups @xcite . in order to investigate the high - density behavior of the symmetry energy , isospin asymmetry and beam energy dependencies of neutron - to - proton ratios ( single and double ) from different kinds of fragments are studied by using the iqmd model . the single neutron - to - proton ratio from free nucleons and lcps is found to decrease ( increase ) with incident energy ( with the isospin asymmetry of the system ) . stronger isospin effects are observed with the soft symmetry energy . similar results with the @xmath8 ratio are also observed by li _ et al . _ and feng _ et al . _ , but with opposite behavior for symmetry energy . the double neutron - to - proton ratio from free nucleons is highly sensitive to the symmetry energy , incident energy , and isospin asymmetry of the system . however , the sensitivity of the neutron - to - proton double ratio from lcps to the nuclear symmetry energy is almost beam - energy independent above 200 mev / nucleon . the same trend is observed for the single @xmath8 ratio above 1 gev / nucleon . the sensitivity of the soft symmetry energy to the ratio parameter is strongly affected by the choice of times , which is not true for the stiff symmetry energy . in simple words , just like the @xmath8 ratio , the neutron - to - proton double ratio from free nucleons can act as a useful probe to constrain the high - density behavior of symmetry energy . experiments are planned at msu , gsi , riken , and frib to determine the high - density behavior of symmetry energy by using the neutron - to - proton ratio . this work is supported in part by the chinese academy of sciences support program for young international scientists under grant no . 2010y2jb02 , the national science foundation of china under contract no . 11035009 , and no . 10979074 , by the the knowledge innovation project of the chinese academy of sciences under grant no . kjcx2-ew - n01 , and by the 973-program under contract no . 2007cb815004 . j. m. lattimer and m. prakash , science * 304 * , 536 ( 2004 ) ; 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the symmetry energy at sub- and supra - saturation densities has great importance for understanding the exact nature of asymmetric nuclear matter as well as neutron stars , but it is poorly known , especially at supra - saturation densities . we will demonstrate here whether or not the neutron - to - proton ratios from different kinds of fragments can determine the supra - saturation behavior of the symmetry energy . for this purpose , a series of sn isotopes were simulated at different incident energies using the isospin quantum molecular dynamics ( iqmd ) model with either a soft or a stiff symmetry energy . it is found that the single neutron - to - proton ratio from free nucleons as well as light charged particles ( lcps ) is sensitive to the symmetry energy , incident energy , and isospin asymmetry of the system . however , with the double neutron - to - proton ratio , this is true only for the free nucleons . it is possible to study the high - density behavior of symmetry energy by using the neutron - to - proton ratio from free nucleons .

You are an expert at summarizing long articles. Proceed to summarize the following text: among the many possibilities for coherent manipulation of quantum systems , stimulated raman adiabatic passage ( stirap ) is one of the most widely used and studied @xcite . this technique transfers population adiabatically between two states @xmath0 and @xmath1 in a three - state quantum system , without populating the intermediate state @xmath2 even when the time - delayed driving fields are on exact resonance with the respective pump and stokes transitions . the technique of stirap relies on the existence of a dark state , which is a time - dependent coherent superposition of the initial and target states only , and which is an eigenstate of the hamiltonian if states @xmath0 and @xmath1 are on two - photon resonance . because stirap is an adiabatic technique , it is robust to variations in most of the experimental parameters . in the early applications of stirap in atomic and molecular physics its efficiency , most often in the range 90 - 95% , has barely been scrutinized because such an accuracy suffices for most purposes . because stirap is resilient to decoherence linked to the intermediate state ( which is often an excited state ) this technique has quickly attracted attention as a promising control tool for quantum information processing @xcite . the latter , however , demands very high fidelity of operations , with the admissible error at most @xmath3 , which is hard to achieve with the standard stirap because , due to its adiabatic nature , it approaches unit efficiency only asymptotically , as the temporal pulse areas increase . for usual pulse shapes , e.g. , gaussian , the necessary area for the @xmath3 benchmark is so large that it may break various restrictions in a real experiment . several scenarios have been proposed to optimize stirap in order to achieve such an accuracy . because the loss of efficiency in stirap derives from incomplete adiabaticity , unanyan _ @xcite , and later chen _ et al . _ @xcite , have proposed to annul the nonadiabatic coupling by adding a third pulsed field on the transition @xmath4 . however , this field must coincide in time with the nonadiabatic coupling exactly ; its pulse area , in particular , must equal @xmath5 , which makes the pump and stokes fields largely redundant . an alternative approach to improve adiabaticity is based on the dykhne - davis - pechukas formula @xcite , which dictates that nonadiabatic losses are minimized when the eigenenergies of the hamiltonian are parallel . this approach , however , prescribes a strict time dependences for the pump and stokes pulse shapes @xcite , or for both the pulse shapes and the detunings @xcite . another basic approach to robust population transfer , which is an alternative to adiabatic techniques , is the technique of composite pulses , which is widely used in nuclear magnetic resonance ( nmr ) @xcite , and more recently , in quantum optics @xcite . this technique , implemented mainly in two - state systems , replaces the single pulse used traditionally for driving a two - state transition by a sequence of pulses with appropriately chosen phases ; these phases are used as a control tool for shaping the excitation profile in a desired manner , e.g. , to make it more robust to variations in the experimental parameters intensities and frequencies . recently , we have proposed a hybrid technique composite adiabatic passage ( cap ) which combines the techniques of composite pulses and adiabatic passage via a level crossing in a two - state system @xcite . cap can deliver extremely high fidelity of population transfer , far beyond the quantum computing benchmark , and far beyond what can be achieved with a single frequency - chirped pulse . recently , the cap technique has been demonstrated experimentally in a doped solid @xcite . to state @xmath1 via a sequence of pump - stokes pulse pairs . on one - photon resonance ( top ) , the order of the pump and stokes pulses is reversed from pair to pair , while off single - photon resonance it is the same for all pulse pairs . ] in this paper , we combine the two basic techniques of composite pulses and stirap into a hybrid technique , which we name _ composite stirap_. this technique , which represents a sequence of an odd number of forward and backward ordinary stiraps , @xmath6 , adds to stirap the very high fidelity of composite pulses . each individual stirap can be very inaccurate , the affordable error being as much as 20 - 30% , but all errors interfere destructively and cancel in the end , thereby producing population transfer with a cumulative error far below the quantum computing benchmark of @xmath3 . we derive an analytical formula for the composite phases , applicable to an arbitrary odd number of pulse pairs @xmath7 ; the phases do not depend on the shape of the pulses and their mutual delay . the dynamics of a three - state @xmath8 system ( fig . [ tog ] ) is described by the schrdinger equation , [ schr ] i_t ( t ) = ( t)(t ) , where the vector @xmath9^t$ ] contains the three probability amplitudes . the hamiltonian in the rotating - wave approximation and on two - photon resonance between states @xmath0 and @xmath1 is [ h ] ( t ) = 2 , where @xmath10 and @xmath11 are the rabi frequencies of the pump and stokes fields , @xmath12 is the one - photon detuning between each laser carrier frequency and the bohr frequency of the corresponding transition , and @xmath13 is the population loss rate from state @xmath2 ; we assume @xmath14 . states @xmath0 and @xmath1 are coupled by @xmath10 , while states @xmath2 and @xmath1 are coupled by @xmath11 . the evolution of the system is described by the propagator @xmath15 , which connects the amplitudes at the initial and final times , @xmath16 and @xmath17 : @xmath18 . the mathematics is substantially different when the pump and stokes fields are on resonance or far off - resonance with the corresponding transition : therefore we consider these cases separately . first , we will consider the one - photon resonance , @xmath19 . then there is a mapping between the three - state problem and a corresponding two - state problem described by the hamiltonian ( t ) = 2 . ( in this correspondence , @xmath10 and @xmath11 are assumed real . ) in general , if the two - state propagator is parameterized in terms of the complex cayley - klein parameters @xmath20 and @xmath21 ( @xmath22 ) as = , we can write the propagator of stirap as [ u - ab ] = . if @xmath10 and @xmath11 are reflections of each other , @xmath23 [ e.g. , if @xmath24 and @xmath25 are identical symmetric functions of time ] , where @xmath26 is the pulse delay , then it is easily shown that @xmath27 . we use this property to parameterize the stirap propagator as @xmath28 in the adiabatic limit , @xmath29 ; hence @xmath30 . .pump and stokes phases for different number of pulse pairs @xmath7 for resonant composite stirap . [ cols="<,<",options="header " , ] the fidelity of the nonresonant composite stirap is illustrated in fig . [ c - stirap ] ( bottom frames ) and fig . [ contoffres ] . again , composite stirap greatly outperforms single stirap in terms of fidelity and robustness . as a function of the pulse delay and the peak rabi frequency for a single off - resonant stirap ( top ) and a sequence of five stiraps ( bottom ) , with phases given by eq . , for @xmath31 pulse shapes , eqs . . the detuning is @xmath32 . ] composite stirap may be affected by several sources of errors . in the first place , errors in the composite phases should be held low in order to keep the high fidelity . we found that an error below 1% in the phases , which is relatively easy to achieve in the lab , can be tolerated . in fig . [ c - stirap ] we have added a curve , which demonstrates the fidelity of composite stirap for @xmath33 and a standard deviation of 0.01 radians in the composite phases ; despite this error , the technique still has ultrahigh fidelity , with an error below @xmath3 . stirap owes much of its great popularity to the fact that it can operate , unlike other techniques , in the presence of population losses from the middle state @xmath2 . however , when ultrahigh fidelity is aimed the presence of such losses can reduce the fidelity and they can not be very large . ( the decay can be harmful only in the resonant case , while off - resonant composite stirap is much more resilient to them . ) we have found that in the resonant case , if the decay rate is sufficiently low , or if the pulse duration is sufficiently short ( @xmath34 ) , composite stirap still maintains high fidelity and outperforms the standard stirap , as seen in fig . [ decay ] . as @xmath13 increases above @xmath35 , stirap behaves better but the fidelities of both stirap and composite stirap drop rapidly and are inadequate for quantum computing purposes . the presence of losses can be compensated with higher rabi frequency ; as a rough estimate the scaling law @xmath36 applies . it is also important to note that in the presence of decay the pulse pairs should be as close to each other as possible , as in the inset of fig . [ decay ] . this is readily achieved with microsecond and nanosecond pulses , e.g. , produced by acoustooptic modulators , as has been demonstrated recently in a doped - solid experiment @xcite . because composite stirap involves @xmath7 pulse pairs , its duration is longer than stirap by the same factor , given that there are no gaps between the pulse pairs , as shown in the inset of fig . [ decay ] . in return , composite stirap gives a fidelity which can not be achieved with ordinary stirap , even with the much higher pulse areas . thus the main advantage of composite stirap over ordinary stirap is the ultrahigh fidelity . the main advantage of composite stirap over other variations of stirap , which provide `` shortcuts '' to adiabaticity by eliminating or reducing the nonadiabatic coupling @xcite is the simplicity of implementation , which requires just the control of the relative phases between the pulse pairs , and the preserved robustness of stirap with respect to variations in the interaction parameters . the `` shortcuts '' techniques use less pulse area , and therefore are faster than composite stirap ( although still slower than resonant techniques which use areas of just @xmath37 @xcite ) , but they give away most of the robustness of stirap by imposing strict restrictions on the pulse shapes , and some of them on the detunings too ; some of them even place considerable transient population in the intermediate state . pulse shapes with @xmath38 and @xmath19 . ] the hybrid technique proposed here combines two popular methods for manipulation of quantum systems stirap and composite pulses . it greatly outperforms the standard stirap in terms of fidelity due to cancelation of the nonadiabatic errors by destructive interference . the greatly enhanced fidelity , well beyond the quantum computing benchmark , while preserving stirap s robustness against variations in the interaction parameters , makes composite stirap a promising technique for quantum information processing . this work is supported by the bulgarian nsf grants d002 - 90/08 and dmu-03/103 , and the alexander von humboldt foundation . k. bergmann , h. theuer , and b. w. shore , rev . phys . * 70 * , 1003 ( 1998 ) ; n. v. vitanov , m. fleischhauer , b. w. shore , and k. bergmann , adv . at . , , opt . phys . * 46 * , 55 ( 2001 ) ; n. v. vitanov , t. halfmann , b. w. shore , and k. bergmann , annu . phys . chem . * 52 * , 763 ( 2001 ) . m. hennrich , t. legero , a. kuhn , and g. rempe , phys . * 85 * , 4872 ( 2000 ) ; a. kuhn , m. hennrich , and g. rempe , ibid . * 89 * , 067901 ( 2002 ) ; t. wilk , s.c . webster , h.p . specht , g. rempe , and a. kuhn , ibid . * 98 * , 063601 ( 2007 ) ; j.l . srensen , d. mller , t. iversen , j.b . thomsen , f. jensen , p. staanum , d. voigt , and m. drewsen , new j. phys . * 8 * , 261 ( 2006 ) . g. dridi , s. gurin , v. hakobyan , h.r . jauslin , and h. eleuch , phys . a * 80 * , 043408 ( 2009 ) . levitt and r. freeman , j. magn . * 33 * , 473 ( 1979 ) ; r. freeman , s. p. kempsell , and m. h. levitt , j. magn * 38 * , 453 ( 1980 ) ; m.h . levitt , prog . nmr spectrosc . * 18 * , 61 ( 1986 ) ; r. freeman , _ spin choreography _ ( spektrum , oxford , 1997 ) . h. hffner , c. f. roos , and r. blatt , phys . rep . * 469 * , 155 ( 2008 ) ; n. timoney , v. elman , s. glaser , c. weiss , m. johanning , w. neuhauser , and c. wunderlich , phys . a * 77 * , 052334 ( 2008 ) ; b.t . torosov and n.v . vitanov , phys . a * 83 * , 053420 ( 2011 ) ; s.s . ivanov and n.v . vitanov , opt . lett . * 36 * , 7 ( 2011 ) ; g.t . genov , b.t . torosov , and n.v . vitanov , phys . a * 84 * , 063413 ( 2011 ) .

we introduce a high - fidelity technique for coherent control of three - state quantum systems , which combines two popular control tools stimulated raman adiabatic passage ( stirap ) and composite pulses . by using composite sequences of pairs of partly delayed pulses with appropriate phases the nonadiabatic transitions , which prevent stirap from reaching unit fidelity , can be canceled to an arbitrary order by destructive interference , and therefore the technique can be made arbitrarily accurate . the composite phases are given by simple analytic formulas , and they are universal for they do not depend on the specific pulse shapes , the pulse delay and the pulse areas .

You are an expert at summarizing long articles. Proceed to summarize the following text: in order to be manipulated and transmitted information should be encoded into some degree of freedom of a physical system . ultimately , this means that the input alphabet should correspond to the spectrum of some observable , _ i.e. _ that information is transmitted using _ quantum signals_. at the end of the channel , to retrieve this kind of quantum information , one should measure the corresponding observable . as a matter of fact , the measurement process unavoidably introduces some disturbance , and may even destroys the signal , as it happens in many quantum optical detectors , which are mostly based on the irreversible absorption of the measured radiation . actually , even in a measurement scheme that somehow preserves the signal for further uses , one is faced by the information gain versus state disturbance trade - off , _ i.e. _ by the fact that the more information is obtained , the more the signal under investigation is being modified . actually , the most informative measurement of an observable @xmath0 on a state @xmath1 corresponds to its ideal projective measurement , which is also referred to as von neumann _ second kind _ quantum measurement @xcite . in an ideal projective measurement the outcome @xmath2 occurs with the intrinsic probability density @xmath3 , whereas the system after the measurement is left in the corresponding eigenstate @xmath4 . a projective measurement is obviously repeatable , since a second measure gives the same outcome as the first one . however , the initial state is erased , and the conditional output do not permit to obtain further information about the input signal . the opposite case corresponds to a fully non - destructive detection scheme , where the state after the measurement can be made arbitrarily close to the input signal , and which is characterized by an almost uniform output statistics , _ i.e. _ by a data sample that provides almost no information . besides fundamental interest , the realization of a projective measurement of the quadrature would have application in quantum communication based on continuous variables . in facts , it provides a reliable and controlled source of optical signals . on the other hand , a fully non - destructive measurement scheme is an example of a quantum repeater , another relevant tool for the realization of quantum network . between these two extremes we have the entire class of quantum nondemolition ( qnd ) measurements . such intermediate schemes provide only a partial information about the measured observable , and correspondingly are only partially distorting the signal under investigation . in particular , in this paper , we show how to attain an optimized qnd measurement of quadrature _ i.e. _ scheme which minimizes the information gain versus state disturbance trade - off . most of the schemes suggested for back - action evading measurements are based on nonlinear interaction between signal and probe taking place either in @xmath5 or @xmath5 media ( both fibers and crystals ) @xcite , or on optomechanical coupling @xcite . a beam - splitter based scheme has been earlier suggested to realize optical von neumann measurement @xcite . here , we focus our attention on an interferometric scheme which requires only linear elements and single - mode squeezers . a schematic diagram of the suggested setup is given in fig . [ f : setup ] . the signal under examination @xmath6 and the probe ( meter ) state @xmath7 are given by @xmath8 where @xmath9 , @xmath10 are eigenstates of the field quadratures @xmath11 , @xmath10 of the two modes , and @xmath12 and @xmath13 are the corresponding wave - functions . the two beams are linearly mixed in a mach - zehnder interferometer with internal phase - shift given by @xmath14 . there are also two @xmath15 plates , each imposing a @xmath16 phase - shift . overall , the interferometer equipped with the plates is equivalent to a beam splitter of transmittivity @xmath17 . however , the interferometric setup is preferable to a single beam splitter since it permits a fine tuning of the transmittivity . after the interferometer , one of the two output modes is revealed by homodyne detection , whereas the second mode is firstly displaced by an amount that depends on the outcome of the measurement ( feedback assisted amplitude modulation ) , and then squeezed according to the transmittivity of the interferometer ( see details below ) . as we will see , either by tuning the phase - shift of the interferometer , or by exciting the probe state @xmath7 in a squeezed vacuum , and by varying the degree of squeezing , the action of the setup ranges from a projective to a non - destructive measurement of the field quadrature as follows : 1 . the statistics of the homodyne detector ranges from a distribution arbitrarily close to the intrinsic quadrature probability density of the signal state @xmath18 to an almost uniform distribution ; 2 . the conditional output state , after registering a value @xmath19 for the quadrature of the signal mode , ranges from a state arbitrarily close to the corresponding quadrature eigenstate @xmath20 to a state that approaches the input signal @xmath6 . the two features can be summarized by saying that the present scheme realizes the whole set of qnd measurements of a field quadrature . in addition , the interferometer can be tuned in order to minimize the information gain versus state disturbance trade - off , _ i.e. _ to achieve an optimal qnd measurement of quadrature . such a kind of measurement provides the maximum information about the quadrature distribution of the signal , while keeping the conditional output state as close as possible to the incoming signal . the paper is structured as follows . in the next section we analyze the dynamics of the measurement scheme , and describe in details the action of linear feedback and tunable squeezing on the conditional output state and on the homodyne distribution . in section [ s : lim ] we analyze the limiting cases of strongly squeezed and anti - squeezed probes , which correspond to projective and non - destructive measurements respectively . in section [ s : opt ] we introduce two fidelity measures , in order to quantify how close are the conditional output and the homodyne distribution to the input signal and its quadrature distribution respectively . as a consequence , we are able to individuate an optimal set of configurations that minimize the trade - off between information gain and state disturbance . section [ s : outro ] closes the paper with some concluding remarks . let us now describe the interaction scheme in details . the evolution operator of the interferometer is given by @xmath21 $ ] , such that the input state @xmath22 after the interferometer the quadrature of one of the modes ( say mode @xmath23 ) is revealed by homodyne detection . the distribution of the outcomes is given by @xmath24 \qquad \pi(x)=|x\rangle\langle x| \label{probx}\;,\end{aligned}\ ] ] @xmath25 being the povm of the homodyne detector . since the reflectivity of the interferometer is given by @xmath26 from an outcome @xmath0 by the homodyne we infer a value @xmath27 for the quadrature of the input signal . the corresponding probability density is given by @xmath28 \right|^2 \label{probx0}\;,\end{aligned}\ ] ] and the conditional output state for the mode @xmath29 @xmath30 the amplitude of this conditional state is then modulated by a feedback mechanism , which consists in the application of a displacement @xmath31 , with @xmath32 . such displacing action can be obtained by mixing the mode with a strong coherent state of amplitude @xmath33 ( _ e.g. _ the laser beam also used as local oscillator for the homodyne detector , see fig . [ f : setup ] ) in a beam splitter of transmittivity @xmath34 close to unit , with the requirement that @xmath35 an experimental implementation using a feedforward electro - optic modulator has been presented in @xcite . the resulting state is given by @xmath36 finally , this state is subjected to a single - mode squeezing transformation @xmath37 $ ] by a degenerate parametric amplifier ( dopa ) . by tuning the squeezing parameter to a value @xmath38 and using the relation @xmath39 we arrive at the final state @xmath40\:|y\rangle \label{output}\;.\end{aligned}\ ] ] the wave - function of this conditional output state is thus given by @xmath41}{\sqrt{\int dy\ : |\psi_{\sc s}(y)|^2 \ : \left|\psi_{\sc p}\left[\tan\phi(y - x_0)\right ] \right|^2 } } \label{outwave}\;.\end{aligned}\ ] ] eq . ( [ probx0 ] ) and eqs . ( [ output],[outwave ] ) summarize the filtering effects of the probe wave - function on the output statistics and the conditional state respectively . for the probe mode in the vacuum state we have @xmath42 such that the homodyne distribution of eq . ( [ probx0 ] ) results @xmath43 where @xmath44 denotes convolution and @xmath45 a gaussian of mean @xmath19 and variance @xmath46 . the quadrature distribution of the corresponding output state is given by @xmath47 eqs ( [ vacprobx0 ] ) and ( [ vacond ] ) account for the noise introduced by vacuum fluctuations . this noise can be manipulated by suitably squeezing the probe , thus realizing the whole set of qnd measurement . squeezed or anti - squeezed vacuum probes are described by the wave - functions @xmath48 where the information about squeezing stays in the requirement @xmath49 . notice that squeezing the probe introduces additional energy in the system . the mean photon number of the states in ( [ sqvac ] ) is given by @xmath50 . using a squeezed vacuum probe eqs . ( [ vacprobx0 ] ) and ( [ vacond ] ) rewrites as @xmath51 eq . ( [ sqcond1 ] ) says that by squeezing the probe the statistics of the homodyne detectors can be made arbitrarily close to the intrinsic quadrature distribution @xmath52 , whereas eq . ( [ sqcond2 ] ) shows that , for any value of the outcome @xmath19 , the conditional output @xmath53 approaches the corresponding quadrature eigenstate @xmath20 . for @xmath54 the mean energy of the conditional output state @xmath53 increases , since it is approaching a quadrature eigenstate ( an exact eigenstate would have infinite energy ) . notice that this amount of energy is mostly provided by the probe state itself , rather than by the displacement and squeezing stages of the setup the improvement in the precision due to squeezing , compared to that of a vacuum probe , can be quantified by the ratio of variances in the filtering gaussian of eqs . ( [ vacond ] ) and ( [ sqcond2 ] ) . calling this ratio @xmath55 we have @xmath56 and thus , for squeezing not too low , @xmath57 . for an anti - squeezed vacuum probe eqs . ( [ vacprobx0 ] ) and ( [ vacond ] ) rewrites as @xmath58 eqs . ( [ sqcond3 ] ) and ( [ sqcond4 ] ) says that by anti - squeezing the probe the statistics of the homodyne detectors is approaching a flat distribution over the real axis , and correspondingly that the conditional output can be made arbitrarily close to the incoming signal , independently on the actual value of @xmath19 . notice that , in principle , both projective and non - destructive measurements could be obtained with vacuum probe , simply by varying the internal phase - shift of the interferometer according to eqs . ( [ vacprobx0 ] ) and ( [ vacond ] ) . however , this would affect also the _ rate _ of the events at the output ( since @xmath14 governs the transmittivity of the interferometer ) , and therefore may be not convenient from practical point of view . on the other hand , when a fine tuning of the variances in eqs . ( [ sqcond1]-[sqcond4 ] ) is needed ( as for example in the optimization of the scheme , see next section ) it can be conveniently obtained by varying @xmath14 , without the need of varying the degree of squeezing of the probe . so far we considered the two extreme cases of infinitely squeezed or antisqueezed probes . now we proceed to quanti - fy the trade - off between the state disturbance and the gain of information for the whole set of intermediate cases . there are two relevant parameters : i ) how close the output signal is to the input state , and ii ) how close the homodyne distribution is to the intrinsic quadrature probability . according to eq . ( [ output ] ) , after the outcome @xmath2 is being registered the conditional output state is given by @xmath59 . since the outcome @xmath2 occurs with the probability @xmath60 of eq . ( [ probx0 ] ) , the density matrix describing the output ensemble after a large number of measurements is given by @xmath61 indeed , this is the state that can be subsequently manipulated , or used to gain further information on the system . the resemblance between input and output can be quantified by the _ average state fidelity _ @xmath62 inserting eq . ( [ output ] ) in eq . ( [ avf ] ) we obtain @xmath63 where , for the squeezed vacuum probes we are taking into account , the transfer function is given by @xmath64 @xmath65 being the variance of the probe wave - function , _ i.e. _ @xmath66 for squeezed probe and @xmath67 for anti - squeezed probes . f take values from zero to unit , and it is a decreasing function of the probe squeezing . if also the initial signal is gaussian the fidelity results @xmath68 @xmath69 being the variance of the signal wave - function . ( [ gf ] ) interpolates between the two extreme cases of the previous section . in order to check this behavior we evaluate ( [ gf ] ) for strong squeezing or anti - squeezing . we have @xmath70 in order to quantify how close the quadrature probability of the input signal is to the homodyne distribution at the output we introduce the _ average distribution fidelity _ @xmath71 which also ranges from zero to one and it is an increasing function of the probe squeezing . for both gaussian signal and probe we obtain @xmath72 and therefore @xmath73 notice that @xmath74 and @xmath75 are _ global _ figures of fidelity @xcite , _ i.e. _ compare the input and the output on the basis of the whole quantum state or probability distributions rather than by their first moments , as it happens by considering customary qnd parameters ( see for example @xcite , for a more general approach in the case of two - dimensional hilbert space see @xcite ) . as a matter of fact the quantity @xmath76 is not constant , and this means that by varying the squeezing of the probe we obtain different trade - off between information gain and state disturbance . an optimal choice of the probe , corresponding to maximum information and minimum disturbance , maximizes @xmath76 . the maximum is achieved for @xmath77 , corresponding to fidelities @xmath78\simeq 86 \%$ ] and @xmath79\simeq 91\%$ ] . notice that for a chosen signal , the optimization of the qnd measurement can be achieved by tuning the internal phase - shift of the interferometer , without the need of varying the squeezing of the probe . for a nearly balanced interferometer we have @xmath80 : in this case the optimal choice for the probe is a state slightly anti - squeezed with respect to the signal , _ i.e. _ @xmath81 . finally , the fidelities are equal for @xmath82 , corresponding to @xmath83=g[x_{\sc e}]\simeq 88 \%$ ] for non gaussian signals the behavior is similar though no simple analytical form can be obtained for the fidelities . in this case , in order to find the optimal qnd measurement , one should resort to numerical means @xcite . in conclusions , we have suggested an interferometric scheme assisted by squeezing and linear feedback to realize an arbitrary qnd measurement of a field quadrature . compared to previous proposals the main features of our setup can be summarized as follows : i ) it involves only linear coupling between signal and probe , ii ) only single mode transformations on the conditional output are needed , and iii ) the whole class of qnd measurements may be obtained with same setup either by tuning the internal phase - shift of the interferometer , or by varying the squeezing of the probe . the present setup permits , in principle , to achieve both a projective and a fully non - destructive quantum measurement of a field quadrature . in practice , however , the physical constraints on the maximum amount of energy that can be impinged into the optical channels pose limitations to the precision of the measurements . this agrees with the facts that both an exact repeatable measurement and a perfect state preparation can not be realized for observables with continuous spectrum @xcite . of course , other limitations are imposed by the imperfect photodetection and by the finite resolution of detectors @xcite . compared to a vacuum probe , the squeezed / anti - squeezed meters suggested in this paper provide a consistent noise reduction in the desired fidelity figure already for moderate input probe energy . in addition , by varying the squeezing of the probe an optimal qnd measure can be achieved , which provides the maximum information about the quadrature distribution of the signal , while keeping the conditional output state as close as possible to the incoming signal . 99 j. von neumann , _ mathematical foundations of quantum mechanics _ ( princeton univ . press , princeton , nj , 1955 ) , pp . 442 - 445 . m. d. levenson et al , phys . * 57 * , 2743 ( 1986 ) . a la porta et al , phys . 62 * , 28 ( 1989 ) . s. pereira et al , phys . lett . * 72 * , 214 ( 1994 ) . j. poizat , p. grangier , phys . lett . * 70 * , 271 ( 1993 ) . s. r. friberg , phys . lett . * 69 * , 3165 ( 1992 ) . r.bruckmeier et al , appl . b * 64 * , 203 ( 1997 ) . f. x. kartner , h. a. haus , phys . a * 47 * , 4585 ( 1993 ) ; appl . b * 64 * , 219 ( 1997 ) . k. jacobs et al , phys . a * 49 * , 1961 ( 1994 ) . m. pinard et al , phys . a * 51 * , 2443 ( 1995 ) . g. m. dariano , m. f. sacchi , phys . a 231 , 325 ( 1997 ) . m. g. a. paris , phys . lett . a * 217 * , 78 ( 1996 ) . p. k. lam et al , phys . lett . * 79 * , 1471 ( 1997 ) . k. banaszek , phys . lett . * 86 * , 1366 ( 2001 ) m. g. a. paris , unpublished . d. walls and g. milburn , _ quantum optics _ springer verlag ( berlin , 1994 ) . c. fuchs , a. peres , phys . a * 53 * , 2038 ( 1996 ) . m. ozawa , publ . rims kyoto * 21 * , 279 ( 1985 ) . h. f. hofmann , phys . a * 61 * , 033815 ( 2000 ) .

we suggest an interferometric scheme assisted by squeezing and linear feedback to realize the whole class of field - quadrature quantum nondemolition measurements , from von neumann projective measurement to fully non - destructive non - informative one . in our setup , the signal under investigation is mixed with a squeezed probe in an interferometer and , at the output , one of the two modes is revealed through homodyne detection . the second beam is then amplitude - modulated according to the outcome of the measurement , and finally squeezed according to the transmittivity of the interferometer . using strongly squeezed or anti - squeezed probes respectively , one achieves either a projective measurement , _ i.e. _ homodyne statistics arbitrarily close to the intrinsic quadrature distribution of the signal , and conditional outputs approaching the corresponding eigenstates , or fully non - destructive one , characterized by an almost uniform homodyne statistics , and by an output state arbitrarily close to the input signal . by varying the squeezing between these two extremes , or simply by tuning the internal phase - shift of the interferometer , the whole set of intermediate cases can also be obtained . in particular , an optimal quantum nondemolition measurement of quadrature can be achieved , which minimizes the information gain versus state disturbance trade - off .

You are an expert at summarizing long articles. Proceed to summarize the following text: it has been recently pointed out @xcite that the radiative corrections to the masses of the higgs bosons in the framework of the mssm @xcite , @xcite , @xcite can be relatively large . the leading correction to the effective potential comes from the top quark and squark loops , being proportional to the top yukawa coupling , which is considered to be big due to the heaviness of the top quark . the other corrections happen to be smaller , though in some cases their effect is not negligible as well . the net effect of the radiative corrections is to increase the masses of the higgs bosons . this increase may be very significant for the future searches , since it can achieve several dozen gev , implying that the higgs mass could exceed the @xmath0 boson mass . considering the tree - level higgs potential one finds out that the value of the lightest higgs boson mass is restricted by the inequality @xmath1 this strict limit is , however , violated by the radiative corrections . the radiative corrections to the supersymmetric boson masses proceed > from the one - loop effective potential @xmath2 where @xmath3 denotes the conventional supertrace and @xmath4 is the scale at which all the couplings in the tree - level potential are renormalized . @xmath5 are the field dependent masses of all the possible particles running through the loops . in what follows we limit ourselves with the top and stop contributions as the main ones . as far as the radiative corrections appear to be large achieving 30 % one can wonder about the values of the higher order contributions . since according to eq.([3 ] ) they have the @xmath6 form one is expecting to have @xmath7 at the second loop , @xmath8 at the third loop , etc . being essential these logs have to be summed giving considerable change of the results . indeed this happens in the simplest case of the @xmath9 model considered in the pioneering paper by coleman and weinberg @xcite . the summation of the leading logs to the effective potential changed the situation qualitatively leading to the disappearance of a non - trivial minimum arising at the one - loop level . the summation procedure can be naturally done with the help of the renormalization group technique , which we are going to apply to our particular case . let us remind the expression for the one - loop effective higgs potential in the mssm which takes into account the radiative corrections due to the top quark and squark loops . for the neutral higgses it has been calculated in ref.@xcite and has the form @xmath10 @xmath11-v_0 \label{5},\ ] ] where @xmath12 are the field dependent masses of the stop particles and @xmath13 is the field dependent top mass . the scale @xmath4 remains arbitrary and is usually chosen to be equal to the value of the top mass . in fact the potential is scale independent since explicit dependence on the scale is compensated by the implicit dependence of the parameters renormalized at this scale . @xmath14 is the value of the potential at @xmath15 , which has to be subtracted in order to keep the scale invariance @xcite . the field dependent squarks masses are given by the eigenvalues of the mass - squared matrix @xmath16 where @xmath17 is the conventional trilinear soft supersymmetry breaking parameter , @xmath18 is the higgs mixing parameter , and the top mass squared is given by @xmath19 . the so - called d - terms give contribution proportional to the gauge couplings and will be ignored hereafter in order to gain approximate scale independence of the potential , since we are including only top - stop contributions to @xmath20 . then the eigenvalues of the matrix ( [ 6 ] ) are @xmath21 , \ ] ] where @xmath22 . the scale independence of the effective potential ( [ 5 ] ) is given by the renormalization group equation @xmath23 where @xmath24 are the parameters of the tree - level potential and @xmath25 s denote their @xmath26 functions . according to the approximation mentioned above , we have ignored the scale dependence of the gauge couplings . the general solution of eq.(7 ) has the form : @xmath27 where the function @xmath28 is the perturbative expression with the scale chosen arbitrary , @xmath24 are the effective parameters of the potential , @xmath29 and @xmath30 s are anomalous dimensions of the fields @xmath31 . in particular one can choose @xmath4 to be equal to the top mass , as is usually done . however , the main distinction is whether we choose the top mass to be field dependent and put it equal to its numerical value after the minimization of the potential , or we take its numerical value at the very beginning and then minimize the potential . in the latter case we incorporate the perturbative corrections to the potential , while in the first case we sum all the leading logs via the renormalization group equation . proceeding the first way we get @xmath32 \nonumber,\end{aligned}\ ] ] where @xmath13 is the field dependent mass and @xmath33 and @xmath34 are the stop masses boundary values when @xmath15 . eq.[9 ] is the rg improved expression for the one - loop effective potential which corresponds to taking into account all the leading log contributions proportional to the top yukawa coupling from all the loops . to find the vacuum expectation values of the higgs fields we have to minimize the potential . using the notation @xmath35 and keeping only the terms of the first order in coupling constants the minimum of the potential ( [ 9 ] ) is given by @xmath36\tan^2\beta+(a_t^2\tan^2\beta -\mu^2 ) \frac{f(\tilde{m}^2_{t1})-f(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}- \tilde{m}^2_{t2}}\right]\bigg\ } , \label{10 } \\ 2m_3 ^ 2&=&\frac{\sin 2\beta}{1+\varepsilon_1}\bigg\{m_1 ^ 2(1-\gamma_2 ) + m_2 ^ 2(1-\gamma_1)-\varepsilon_2 \label{11 } \\ & + & \frac{3h_t^2}{16\pi^2}\left[f(\tilde{m}^2_{t1 } ) + f(\tilde{m}^2_{t2})-2f(m^2_t)+(a_t + \mu\tan\beta)(a_t + \mu\cot\beta ) \frac{f(\tilde{m}^2_{t1})-f(\tilde{m}^2_{t2})}{\tilde{m}^2_{t1}- \tilde{m}^2_{t2}}\right]\bigg\ } , \nonumber\end{aligned}\ ] ] where @xmath37 , \\ \gamma_1&=&-\frac{3}{2}(\tilde{\alpha}_2 + \frac{1}{5}\tilde{\alpha}_1 ) , \\ \gamma_2&=&\frac{3}{2}(2y_t-\tilde{\alpha}_2 -\frac{1}{5}\tilde{\alpha}_1 ) , \\ y_t&=&\frac{h_t^2}{16\pi^2 } , \tilde{\alpha}_i=\frac{\alpha_i}{4\pi}=\frac{g_i^2}{16\pi^2}.\end{aligned}\ ] ] here the values of all the mass parameters and couplings are taken at the scale equal to the top mass . having in mind eqs.([11 ] ) we are now in a position to calculate the rg improved radiative corrections to the masses . one has : @xmath38 where @xmath39 and @xmath40 is the one - loop radiative correction @xcite , @xcite @xmath41 .\ ] ] using eqs.([12 ] ) we can also calculate the corrections to the squared masses in the cp - odd neutral sector . just like in the usual case @xcite taking the second derivative of the full potential with respect to @xmath42 one has : @xmath43 where @xmath44 this gives for the cp - odd higgs mass @xmath45 where @xmath46 and @xmath47\ ] ] for the cp - even sector we have to differentiate the potential with respect to @xmath48 we find @xmath49 @xmath50 where @xmath51 , \\ \delta_6&=&6 \left[(\tilde{\alpha}_2m_2 ^ 2+\frac{1}{5}\tilde{\alpha}_1m_1 ^ 2 + \tilde{\alpha}_2\mu^2+\frac{1}{5}\tilde{\alpha}_1\mu^2)(\cot^2\beta-1 ) + 2(\tilde{\alpha}_2m_2+\frac{1}{5}\tilde{\alpha}_1m_1 ) \mu\cot\beta \right . \\ & & \left . -am_0\mu y \cot\beta \right]\end{aligned}\ ] ] and @xmath52 s are @xcite : @xmath53 ^ 2d(\tilde{m}^2_{t1},\tilde{m}^2_{t2 } ) , \\ \delta_{22}&=&\frac{3g^2}{16\pi^2}\frac{m^4_t}{\sin^2\beta m^2_w}\left [ \ln ( \frac{\tilde{m}^2_{t1}\tilde{m}^2_{t2}}{m^4_t})+ \frac{2a_t m_0 ( a_t m_0 + \mu\cot\beta ) } { \tilde{m}^2_{t1}-\tilde{m}^2_{t2 } } \ln ( \frac{\tilde{m}^2_{t1}}{\tilde{m}^2_{t2 } } ) \right.\\ & & \left.+\left [ \frac{a_t m_0(a_t m_0 + \mu\cot\beta ) } { \tilde{m}^2_{t1}-\tilde{m}^2_{t2}}\right]^2 d(\tilde{m}^2_{t1},\tilde{m}^2_{t2})\right ] , \\ \delta_{12}&=&\frac{3g^2}{16\pi^2}\frac{m^4_t}{\sin^2\beta m^2_w } \frac{\mu(a_t m_0 + \mu\cot\beta ) } { \tilde{m}^2_{t1}-\tilde{m}^2_{t2 } } \left [ \ln ( \frac{\tilde{m}^2_{t1}}{\tilde{m}^2_{t2 } } ) + \frac{a_t m_0 ( a_t m_0+\mu\cot\beta ) } { \tilde{m}^2_{t1}-\tilde{m}^2_{t2 } } d(\tilde{m}^2_{t1},\tilde{m}^2_{t2})\right ] \\\end{aligned}\ ] ] @xmath54 @xmath55 is the mass of a light squark . the diagonalization of the matrix ( [ a ] ) gives us the masses of the cp - even neutral higgses . for the charged ones one has the usual expression @xcite @xmath56 where @xmath57 and the only difference is that @xmath58 is given by eq.([15 ] ) . resulting expressions for the higgs masses differ from those obtained without rg summation . to calculate the corrections one has to perform the usual procedure @xcite , @xcite , @xcite of fitting the set of soft breaking parameters , @xmath59 as well as the top mass @xmath13 . this is not a straightforward operation , since one has to fulfill many requirements simultaneously , thus defining the optimized best fit @xcite . having performed the complete procedure with the help of computer programm we have got the best fit values of the parameters mentioned above and used them in our formufae . one of the important observation is the character of @xmath13 dependence of the results . comparing with that obtained without account of the rg summation @xcite , we find it to be smoother . fig.1 shows our results . we conclude that the account of rg summation procedure can introduce the changes in the predictions of the higgs masses . one can observe from fig.1 that there exist the lower bound on the higgs mass of about 95 gev . in the interval of top mass preferable according to the recent cdf @xcite data ( @xmath61 ) , that corresponds to the running top mass of 164 @xmath62 , the @xmath13 dependence of the lightest higgs mass is very weak and @xmath63 appears to be lighter than 100 gev . the numerical analysis was performed with the help of the computer program , developed in @xcite . we are grateful to w.de boer and r.ehret from karlshrue university for valuable discussions and for necessary modification of the program .

the one - loop radiative corrections to the higgs boson potential in the mssm , originating from the top quark and squark loops , are summed in the leading log approximation using the renormalization group . the rg improved effective potential is minimized and the corrections to the cp - odd and cp - even higgs boson masses are calculated . the resulting masses exhibit smoother top mass dependence than those calculated without rg summation . we have also found that for preferable values of the top mass the light higgs mass does not exceed 100 gev . 6.3 in .2 in .2 in jinr - e2 - 94 - 400 + october 1994 * renormalization group improved radiative corrections + to the supersymmetric higgs boson masses *

You are an expert at summarizing long articles. Proceed to summarize the following text: dependence logic @xcite is an extension of first - order logic which adds _ dependence atoms _ of the form @xmath0 to it , with the intended interpretation of `` the value of the term @xmath1 is a function of the values of the terms @xmath2 . '' the introduction of such atoms is roughly equivalent to the introduction of non - linear patterns of dependence and independence between variables of branching quantifier logic @xcite or independence friendly logic @xcite : for example , both the branching quantifier logic sentence @xmath3 and the independence friendly logic sentence @xmath4 correspond in dependence logic to @xmath5 in the sense that all of these expressions are equivalent to the skolem formula @xmath6 as this example illustrates , the main peculiarity of dependence logic compared to the others above - mentioned logics lies in the fact that , in dependence logic , the notion of _ dependence and independence between variables _ is explicitly separated from the notion of quantification . this makes it an eminently suitable formalism for the formal analysis of the properties of _ dependence itself _ in a first - order setting , and some recent papers ( @xcite ) explore the effects of replace dependence atoms with other similar primitives such as _ independence atoms _ @xcite , _ multivalued dependence atoms _ @xcite , or _ inclusion _ or _ atoms @xcite . branching quantifier logic , independence friendly logic and dependence logic , as well as their variants , are called _ logics of imperfect information _ : indeed , the truth conditions of their sentences can be obtained by defining , for every model @xmath7 and sentence @xmath8 , an imperfect - information _ semantic game _ @xmath9 between a _ verifier _ ( also called eloise ) and a _ falsifier _ ( also called abelard ) , and then asserting that @xmath8 is true in @xmath7 if and only if the verifier has a winning strategy in @xmath9 . as an alternative of this ( non - compositional ) _ game - theoretic semantics _ , which is an imperfect - information variant of hintikka s game - theoretic semantics for first order logic @xcite , hodges introduced in @xcite _ team semantics _ ( also called _ trump semantics _ ) , a compositional semantics for logics of imperfect information which is equivalent to game - theoretic semantics over sentences and in which formulas are satisfied or not satisfied not by single assignments , but by _ sets _ of assignments ( called _ teams _ ) . in this work , we will be mostly concerned with team semantics and some of its variants . we refer the reader to the relevant literature ( for example to @xcite and @xcite ) for further information regarding these logics : in the rest of this section , we will content ourselves with recalling the definitions and results which will be useful for the rest of this work . let @xmath7 be a first order model and let @xmath10 be a finite set of variables . then an _ assignment _ over @xmath7 with _ domain _ @xmath10 is a function @xmath11 from @xmath10 to the set @xmath12 of all elements of @xmath7 . furthermore , for any assignment @xmath11 over @xmath7 with domain @xmath10 , any element @xmath13 and any variable @xmath14 ( not necessarily in @xmath10 ) , we write @xmath15 $ ] for the assignment with domain @xmath16 such that @xmath17(w ) = \left\{\begin{array}{l l } m & \mbox{if } w = v;\\ s(w ) & \mbox{if } w \in v \backslash \{v\ } \end{array } \right.\ ] ] for all @xmath18 . let @xmath7 be a first - order model and let @xmath10 be a finite set of variables . @xmath19 over @xmath7 with _ domain _ @xmath20 is a set of assignments from @xmath10 to @xmath7 . let @xmath19 be a team over @xmath7 , and let @xmath10 be a finite set of variables . and let @xmath21 be a finite tuple of variables in its domain . then @xmath22 is the relation @xmath23 . furthermore , we write @xmath24 for @xmath25 . as is often the case for dependence logic , we will assume that all our formulas are in negation normal form : let @xmath26 be a first - order signature . then the set of all dependence logic formula with signature @xmath26 is given by @xmath27 where @xmath28 ranges over all relation symbols , @xmath29 ranges over all tuples of terms of the appropriate arities , @xmath30 range over all terms and @xmath14 ranges over the set @xmath31 of all variables . the set @xmath32 of all _ free variables _ of a formula @xmath8 is defined precisely as in first order logic , with the additional condition that all variables occurring in a dependence atom are free with respect to it . [ dl - ts ] let @xmath7 be a first - order model , let @xmath19 be a team over it , and let @xmath8 be a dependence logic formula with the same signature of @xmath7 and with free variables in @xmath33 . then we say that @xmath19 _ satisfies _ @xmath8 in @xmath7 , and we write @xmath34 , if and only if ts - lit : : : @xmath8 is a first - order literal and @xmath35 for all @xmath36 ; ts - dep : : : @xmath8 is a dependence atom @xmath37 and any two assignments @xmath38 which assign the same values to @xmath2 also assign the same value to @xmath1 ; ts-@xmath39 : : : @xmath8 is of the form @xmath40 and there exist two teams @xmath41 and @xmath42 such that @xmath43 , @xmath44 and @xmath45 ; ts-@xmath46 : : : @xmath8 is of the form @xmath47 , @xmath48 and @xmath49 ; ts-@xmath50 : : : @xmath8 is of the form @xmath51 and there exists a function @xmath52 such that @xmath53 } \psi$ ] , where @xmath54 = \{s[f(s)/v ] : s \in x\}\ ] ] ts-@xmath55 : : : @xmath8 is of the form @xmath56 and @xmath57 } \psi$ ] , where @xmath58 = \{s[m / v ] : s \in x , m \in { \texttt{dom}}(m)\}.\ ] ] the disjunction of dependence logic does not behave like the classical disjunction : for example , it is easy to see that @xmath59 is not equivalent to @xmath60 , as the former holds for the team @xmath61 and the latter does not . however , it is possible to define the classical disjunction in terms of the other connectives : [ defin : classic_or ] let @xmath62 and @xmath63 be two dependence logic formulas , and let @xmath64 and @xmath65 be two variables not occurring in them . then we write @xmath66 as a shorthand for @xmath67 [ propo : classic_or ] for all formulas @xmath62 and @xmath63 , all models @xmath7 with at least two elements whose signature contains that of @xmath62 and @xmath63 and all teams @xmath19 whose domain contains the free variables of @xmath62 and @xmath63 @xmath68 the following four proportions are from @xcite : [ propo : emptyteam ] for all models @xmath7 and dependence logic formulas @xmath8 , @xmath69 . if @xmath34 and @xmath70 then @xmath71 . if @xmath34 and @xmath72 then @xmath73 . [ dltosigma ] let @xmath74 be a dependence logic formula with free variables in @xmath21 . then there exists a @xmath75 sentence @xmath76 such that @xmath77 for all suitable models @xmath7 and for all nonempty teams @xmath19 . furthermore , in @xmath76 the symbol @xmath28 occurs only negatively . as proved in @xcite , there is also a converse for the last proposition : [ sigmatodl ] let @xmath76 be a @xmath75 sentence in which @xmath28 occurs only negatively . then there exists a dependence logic formula @xmath74 , where @xmath78 is the arity of @xmath28 , such that @xmath77 for all suitable models @xmath7 and for all nonempty teams @xmath19 whose domain contains @xmath21 . because of this correspondence between dependence logic and existential second order logic , it is easy to see that dependence logic is closed under existential quantification : for all dependence logic formulas @xmath79 over the signature @xmath80 there exists a dependence logic formula @xmath81 over the signature @xmath26 such that @xmath82 for all models @xmath7 with domain @xmath26 and for all teams @xmath19 over the free variables of @xmath8 . therefore , in the rest of this work we will add second - order existential quantifiers to the language of dependence logic , and we will write @xmath81 as a shorthand for the corresponding dependence logic expression . _ game logics _ are logical formalisms for reasoning about games and their properties in a very general setting . whereas the game theoretic semantics approach attempts to use game - theoretic techniques to _ interpret _ logical systems , game logics attempt to put logic to the service of game theory , by providing a high - level language for the study of games . they generally contain two different kinds of expressions : 1 . _ game terms _ , which are descriptions of games in terms of compositions of certain primitive _ atomic games _ , whose interpretation is presumed fixed for any given game model ; 2 . _ formulas _ , which , in general , correspond to assertions about the abilities of players in games . in this subsection , we are going to summarize the definition of a variant of dynamic game logic @xcite . from our formalism . in this , we follow @xcite . ] then , in the next subsection , we will discuss a remarkable connection between first - order logic and dynamic game logic discovered by johan van benthem in @xcite . + one of the fundamental semantic concepts of dynamic game logic is the notion of _ forcing relation : _ let @xmath83 be a nonempty set of _ states_. a _ forcing relation _ over @xmath83 is a set @xmath84 , where @xmath85 is the powerset of @xmath83 . in brief , a forcing relation specifies the abilities of a player in a perfect - information game : @xmath86 if and only if the player has a strategy that guarantees that , whenever the initial position of the game is @xmath11 , the terminal position of the game will be in @xmath19 . a ( two - player ) _ game _ is then defined as a pair of forcing relations satisfying some axioms : let @xmath83 be a nonempty set of states . a _ game _ over @xmath83 is a pair @xmath87 of forcing relations over @xmath83 satisfying the following conditions for all @xmath88 , all @xmath89 and all @xmath90 : monotonicity : : : if @xmath91 and @xmath92 then @xmath93 ; consistency : : : if @xmath94 and @xmath95 then @xmath96 ; non - triviality : : : @xmath97 . determinacy : : : if @xmath98 then @xmath99 , where @xmath100 . , this implies that the other player can force it to belong to the complement of @xmath19 . ] let @xmath83 be a nonempty set of states , let @xmath101 be a nonempty set of _ atomic propositions _ and let @xmath102 be a nonempty set of _ atomic game symbols_. then a _ game model _ over @xmath83 , @xmath101 and @xmath102 is a triple @xmath103 , where @xmath104 is a game over @xmath83 for all @xmath105 and where @xmath10 is a valutation function associating each @xmath106 to a subset @xmath107 . the language of dynamic game logic , as we already mentioned , consists of _ game terms _ , built up from atomic games , and of _ formulas _ , built up from atomic proposition . the connection between these two parts of the language is given by the _ test _ operation @xmath108 , which turns any formula @xmath8 into a test game , and the _ diamond _ operation , which combines a game term @xmath109 and a formula @xmath8 into a new formula @xmath110 which asserts that agent @xmath111 can guarantee that the game @xmath109 will end in a state satisfying @xmath8 . let @xmath101 be a nonempty set of _ atomic propositions _ and let @xmath102 be a nonempty set of _ atomic game formulas_. then the sets of all game terms @xmath109 and formulas @xmath8 are defined as @xmath112 for @xmath113 ranging over @xmath101 , @xmath114 ranging over @xmath102 , and @xmath111 ranging over @xmath115 . we already mentioned the intended interpretations of the test connective @xmath108 and of the diamond connective @xmath110 . the interpretations of the other game connectives should be clear : @xmath116 is obtained by swapping the roles of the players in @xmath109 , @xmath117 is a game in which the existential player @xmath118 chooses whether to play @xmath119 or @xmath120 , and @xmath121 is the _ concatenation _ of the two games corresponding to @xmath119 and @xmath120 respectively . let @xmath122 be a game model over @xmath83 , @xmath102 and @xmath101 . then for all game terms @xmath109 and all formulas @xmath8 of dynamic game logic over @xmath102 and @xmath101 we define a game @xmath123 and a set @xmath124 as follows : dgl - atomic - game : : : for all @xmath105 , @xmath125 ; dgl - test : : : for all formulas @xmath8 , @xmath126 , where + * @xmath127 iff @xmath128 and @xmath36 ; + * @xmath129 iff @xmath130 or @xmath36 + for all @xmath89 and all @xmath19 with @xmath131 ; dgl - concat : : : for all game terms @xmath119 and @xmath120 , @xmath132 , where , for all @xmath88 and for @xmath133 , @xmath134 , + * @xmath135 if and only if there exists a @xmath136 such that @xmath137 and for each @xmath138 there exists a set @xmath139 satisfying @xmath140 such that @xmath141 dgl-@xmath142 : : : for all game terms @xmath119 and @xmath120 , @xmath143 , where + * @xmath127 if and only if @xmath144 or @xmath145 , and * @xmath129 if and only if @xmath146 and @xmath147 + where , as before , @xmath133 and @xmath134;[multiblock footnote omitted ] dgl - dual : : : if @xmath148 then @xmath149 ; dgl-@xmath150 : : : @xmath151 ; dgl - atomic - pr : : : @xmath152 ; dgl-@xmath153 : : : @xmath154 ; dgl-@xmath39 : : : @xmath155 ; dgl-@xmath156 : : : if @xmath148 then for all @xmath8 , @xmath157 if @xmath128 , we say that @xmath8 is _ satisfied _ by @xmath11 in @xmath158 and we write @xmath35 . we will not discuss here the properties of this logic , or the vast amount of variants and extensions of it which have been developed and studied . it is worth pointing out , however , that @xcite introduced a _ concurrent dynamic game logic _ that can be considered one of the main sources of inspiration for the transition logic that we will develop in subsection [ subsect : tdl ] . in this subsection , we will briefly recall a remarkable result from @xcite which establishes a connection between dynamic game logic and first - order logic . in brief , as the following two theorems demonstrate , either of these logics can be seen as a special case of the other , in the sense that models and formulas of the one can be uniformly translated into models of the other in a way which preserves satisfiability and truth : [ theo : repfo1 ] let @xmath122 be any game model , let @xmath8 be any game formula for the same language , and let @xmath89 . then it is possible to uniformly construct a first - order model @xmath159 , a first - order formula @xmath160 and an assignment @xmath161 of @xmath159 such that @xmath162 [ theo : repfo2 ] let @xmath7 be any first order model , let @xmath8 be any first - order formula for the signature of @xmath7 , and let @xmath11 be an assignment of @xmath7 . then it is possible to uniformly construct a game model @xmath163 , a game formula @xmath164 and a state @xmath165 such that @xmath166 we will not discuss here the proofs of these two results . their _ significance _ , however , is something about which is necessary to spend a few words . in brief , what this back - and - forth representation between first order logic and dynamic game logic tells us is that it is possible to understand first order logic as a _ logic for reasoning about determined games _ ! in the next sections , we will attempt to develop a similar result for the case of dependence logic . we will now define a variant of dynamic game logic , which we will call _ transition logic_. it deviates from the basic framework of dynamic game logic in two fundamental ways : 1 . it considers _ one - player _ games against nature , instead of _ two - player games _ as is usual in dynamic game logic ; 2 . it allows for _ uncertainty _ about the initial position of the game . hence , transition logic can be seen as a _ decision - theoretic logic _ , rather than a _ game - theoretic _ one : transition logic formulas , as we will see , correspond to assertions about the abilities of a single agent acting under uncertainty , instead of assertions about the abilities of agents interacting with each other . in principle , it is certainly possible to generalize the approach discussed here to multiple agents acting in situations of imperfect information , and doing so might cause interesting phenomena to surface ; but for the time being , we will content ourselves with developing this formalism and discussing its connection with dependence logic . our first definition is a fairly straightforward generalization of the concept of forcing relation : let @xmath83 be a nonempty set of _ states_. a _ transition system _ over @xmath83 is a nonempty relation @xmath167 satisfying the following requirements : downwards closure : : : if @xmath168 and @xmath169 then @xmath170 ; monotonicity : : : if @xmath168 and @xmath171 then @xmath172 ; non - creation : : : @xmath173 for all @xmath174 ; non - triviality : : : if @xmath175 then @xmath176 . informally speaking , a transition system specifies the abilities of an agent : for all @xmath90 such that @xmath168 , the agent has a strategy which guarantees that the output of the transition will be in @xmath177 whenever the input of the transition is in @xmath19 . the four axioms which we gave capture precisely this intended meaning , as we will see : a _ decision game _ is a triple @xmath178 , where @xmath83 is a nonempty set of _ states _ , @xmath118 is a nonempty set of possible _ decisions _ for our agent and @xmath179 is an _ outcome function _ from @xmath180 to @xmath85 . if @xmath181 , we say that @xmath182 is a _ possible outcome _ of @xmath11 under @xmath183 ; if @xmath184 , we say that @xmath183 _ fails _ on input @xmath11 . let @xmath178 be a decision game , and let @xmath90 . then we say that @xmath102 _ allows _ the transition @xmath185 , and we write @xmath186 , if and only if there exists a @xmath187 such that @xmath188 for all @xmath36 ( that is , if and only if our agent can make a decision which guarantees that the outcome will be in @xmath177 whenever the input is in @xmath19 ) . a set @xmath167 is a transition system if and only if there exists a decision game @xmath178 such that @xmath189 let @xmath167 be any transition system , let us enumerate its elements @xmath190 , and let us consider the game @xmath191 , where @xmath192 suppose that @xmath168 . if @xmath193 , then @xmath194 follows at once by definition . if instead @xmath175 , by * non - triviality * we have that @xmath177 is nonempty too , and furthermore @xmath195 for some @xmath196 . then @xmath197 for all @xmath198 , as required . now suppose that @xmath186 . then there exists a @xmath196 such that @xmath199 for all @xmath36 . if @xmath175 , this implies that @xmath200 and @xmath201 . hence , by * monotonicity * and * downwards closure * , @xmath168 , as required . if instead @xmath193 , then by * non - creation * we have again that @xmath168 . conversely , consider a decision game @xmath178 . then the set of its abilities satisfies our four axioms : downwards closure : : : suppose that @xmath202 and that @xmath169 . by definition , there exists a @xmath187 such that @xmath188 for all @xmath36 . but then the same holds for all @xmath203 , and hence @xmath204 . monotonicity : : : suppose that @xmath202 and that @xmath171 . by definition , there exists a @xmath187 such that @xmath188 for all @xmath36 . but then , for all such @xmath11 , @xmath205 too , and hence @xmath206 . non - creation : : : let @xmath174 and let @xmath187 be any possible decision . then trivially @xmath188 for all @xmath207 , and hence @xmath208 . non - triviality : : : let @xmath209 , and suppose that @xmath186 . then there exists a @xmath183 such that @xmath188 for all @xmath36 , and hence in particular @xmath210 . therefore , @xmath177 is nonempty . what this theorem tells us is that our notion of transition system is the correct one : it captures precisely the abilities of an agent making choices under imperfect information and attempting to guarantee that , if the initial state is in a set @xmath19 , the outcome will be in a set @xmath177 . let @xmath83 be a nonempty set of states . a _ trump _ over @xmath83 is a nonempty , downwards closed family of subsets of @xmath83 . whereas a transition system describes the abilities of an agent to transition from a set of possible initial states to a set of possible terminal states , a trump describes the agent s abilities to reach _ some _ terminal state from a set of possible initial states : let @xmath211 be a transition system and let @xmath212 . then @xmath213 forms a trump . conversely , for any trump @xmath214 over @xmath83 there exists a transition system @xmath211 such that @xmath215 for any nonempty @xmath174 . let @xmath211 be a transition system . then if @xmath168 and @xmath169 , by downwards closure we have at once that @xmath170 . furthermore , @xmath173 for any @xmath177 . hence , @xmath216 is a trump , as required . conversely , let @xmath217 be a trump , and let us enumerate its elements as @xmath218 . then define @xmath211 as @xmath219 it is easy to see that @xmath211 is a transition system ; and by construction , for @xmath220 we have that @xmath221 , where we used the fact that @xmath214 is downwards closed . we can now define the syntax and semantics of transition logic : let @xmath101 be a set of _ atomic propositional symbols _ and let @xmath222 be a set of _ atomic transition symbols_. then a _ transition model _ is a tuple @xmath223 , where @xmath83 is a nonempty set of states , @xmath224 is a transition system over @xmath83 for any @xmath225 , and @xmath10 is a function sending each @xmath106 into a trump of @xmath83 . let @xmath101 be a set of atomic propositions and let @xmath222 be a set of atomic transitions . then the _ transition terms _ and _ formulas _ of our language are defined respectively as @xmath226 where @xmath227 ranges over @xmath222 and @xmath113 ranges over @xmath101 . let @xmath228 be a transition model , let @xmath229 be a transition term , and let @xmath90 . then we say that @xmath229 _ allows _ the transition from @xmath19 to @xmath177 , and we write @xmath230 , if and only if tl - atomic - tr : : : @xmath231 for some @xmath225 and @xmath232 ; tl - test : : : @xmath233 for some transition formula @xmath8 such that @xmath234 in the sense described later in this definition , and @xmath92 ; tl-@xmath235 : : : @xmath236 , and @xmath237 for two @xmath238 and @xmath239 such that @xmath240 and @xmath241 ; tl-@xmath242 : : : @xmath243 , @xmath244 and @xmath245 ; tl - concat : : : @xmath246 and there exists a @xmath247 such that @xmath248 and @xmath249 . analogously , let @xmath8 be a transition formula , and let @xmath250 . then we say that @xmath19 _ satisfies _ @xmath8 , and we write @xmath234 , if and only if tl-@xmath251 : : : @xmath252 ; tl - atomic - pr : : : @xmath253 for some @xmath106 and @xmath254 ; tl-@xmath39 : : : @xmath255 and @xmath256 or @xmath257 ; tl-@xmath46 : : : @xmath258 , @xmath256 and @xmath257 ; tl-@xmath156 : : : @xmath259 and there exists a @xmath177 such that @xmath230 and @xmath260 . for any transition model @xmath261 , transition term @xmath229 and transition formula @xmath8 , the set @xmath262 is a transition system and the set @xmath263 is a trump . by induction . we end this subsection with a few simple observations about this logic . first of all , we did not take the negation as one of the primitive connectives . indeed , transition logic , much like dependence logic , has an intrinsically _ existential _ character : it can be used to reason about which sets of possible states an agent _ may _ reach , but not to reason about which ones such an agent _ must _ reach . there is of course no reason , in principle , why a negation could not be added to the language , just as there is no reason why a negation can not be added to dependence logic , thus obtaining the far more powerful _ team logic _ @xcite : however , this possible extension will not be studied in this work . the connectives of transition logic are , for the most part , very similar to those of dynamic game logic , and their interpretation should pose no difficulties . the exception is the _ tensor operator _ @xmath264 , which substitutes the game union operator @xmath117 and which , while sharing roughly the same informal meaning , behaves in a very different way from the semantic point of view ( for example , it is not in general idempotent ! ) the decision game corresponding to @xmath264 can be described as follows : first the agent chooses an index @xmath265 , then he or she picks a strategy for @xmath266 and plays accordingly . however , the choice of @xmath111 may be a function of the initial state : hence , the agent can guarantee that the output state will be in @xmath177 whenever the input state is in @xmath19 only if he or she can split @xmath19 into two subsets @xmath238 and @xmath239 and guarantee that the state in @xmath177 will be reached from any state in @xmath238 when @xmath267 is played , and from any state in @xmath239 when @xmath268 is played . it is also of course possible to introduce a `` true '' choice operator @xmath269 , with semantical condition tl-@xmath142 : : : @xmath270 iff @xmath244 or @xmath245 ; but we will not explore this possibility any further in this work , nor we will consider any other possible connectives such as , for example , the iteration operator tl-@xmath271 : : : @xmath272 iff there exist @xmath273 and @xmath274 such that @xmath275 , @xmath276 and @xmath277 for all @xmath278 . this subsection contains the central result of this work , that is , the analogues of theorems [ theo : repfo1 ] and [ theo : repfo2 ] for dependence logic and transition logic . + representing dependence logic models and formulas in transition logic is fairly simple : [ defin : dl2tl - mod ] let @xmath7 be a first - order model . then @xmath279 is the transition model @xmath280 such that * @xmath83 is the set of all teams over @xmath7 ; * the set of all atomic transition symbols is @xmath281 , and hence @xmath222 is @xmath282 ; * for any variable @xmath14 , @xmath283 \subseteq y \}$ ] and @xmath284 \subseteq y\}$ ] ; * for any first - order literal or dependence atom @xmath285 , @xmath286 . [ defin : dl2tl - form ] let @xmath8 be a dependence logic formula . then @xmath287 is the transition term defined as follows : 1 . if @xmath8 is a literal or a dependence atom , @xmath288 ; 2 . if @xmath255 , @xmath289 ; 3 . if @xmath258 , @xmath290 ; 4 . if @xmath291 , @xmath292 ; 5 . if @xmath293 , @xmath294 . [ theo : tl - rep1 ] for all first - order models @xmath7 , teams @xmath19 and formulas @xmath8 , the following are equivalent : * @xmath34 ; * @xmath295 ; * @xmath296 ; * @xmath297 . we show , by structural induction on @xmath8 , that the first condition is equivalent to the last one . the equivalences between the last one and the second and third ones are then trivial . 1 . if @xmath8 is a literal or a dependence atom , @xmath298 if and only if @xmath299 , that is , if and only if @xmath34 ; 2 . @xmath300 if and only if @xmath237 for two @xmath301 such that @xmath302 and @xmath303 . by induction hypothesis , this can be the case if and only if @xmath304 and @xmath305 , that is , if and only if @xmath306 . 3 . @xmath307 if and only if @xmath308 and @xmath309 , that is , by induction hypothesis , if and only if @xmath310 . 4 . @xmath311 if and only if there exists a @xmath177 such that @xmath312 $ ] for some @xmath313 and @xmath314 . by induction hypothesis and downwards closure , this can be the case if and only if @xmath53 } \psi$ ] for some @xmath313 , that is , if and only if @xmath315 ; 5 . @xmath316 if and only if @xmath317 for some @xmath318 $ ] , that is , if and only if @xmath57 } \psi$ ] , that is , if and only if @xmath319 . one interesting aspect of this representation result is that dependence logic _ formulas _ correspond to transition logic _ transitions _ , not to transition logic _ formulas_. this can be thought of as one first hint of the fact that dependence logic can be thought of as a logic of transitions : and in the later sections , we will explore this idea more in depth . representing transition models , game terms and formulas in dependence logic is somewhat more complex : let @xmath320 be a transition model . furthermore , for any @xmath225 , let @xmath321 , and , for any @xmath106 , let @xmath322 . then @xmath323 is the first - order model with domain for the _ disjoint union _ of the sets @xmath324 and @xmath325 . ] @xmath326 whose signature contains * for every @xmath225 , a ternary relation @xmath327 whose interpretation is @xmath328 ; * for every @xmath106 , a binary relation @xmath329 whose interpretation is @xmath330 . for any transition formula @xmath8 and variable @xmath331 , the dependence logic formula @xmath332 is defined as 1 . @xmath333 is @xmath251 ; 2 . for all @xmath106 , @xmath334 is @xmath335 ; 3 . @xmath336 is @xmath337 , where @xmath338 is the classical disjunction introduced in definition [ defin : classic_or ] ; 4 . @xmath339 is @xmath340 ; 5 . @xmath341 is @xmath342 , where for any transition term @xmath229 , variable @xmath331 and unary relation symbol @xmath343 , @xmath344 is defined as 1 . for all @xmath225 , @xmath345 is @xmath346 ; 2 . for all formulas @xmath8 , @xmath347 is @xmath348 ; 3 . @xmath349 ; 4 . @xmath350 ; 5 . @xmath351 for a new and unused variable @xmath352 . [ theo : tl - rep2 ] for all transition models @xmath320 , transition terms @xmath229 , transition formulas @xmath8 , variables @xmath331 , sets @xmath353 and teams @xmath19 over @xmath323 with @xmath354 , is a set of states of the transition model . ] @xmath355 and @xmath356 the proof is by structural induction on terms and formulas . let us first consider the cases corresponding to formulas : 1 . for all teams @xmath19 , @xmath357 and @xmath358 , as required suppose that @xmath359 . then there exists a @xmath360 such that @xmath361 } v_p(j , x)$ ] . hence , we have that @xmath362 ; and , by downwards closure , this implies that @xmath363 , and hence that @xmath364 as required . + conversely , suppose that @xmath364 . then @xmath363 , and hence @xmath365 for some @xmath366 . then we have by definition that @xmath361 } v_p(j , x)$ ] , and finally that @xmath367 . 3 . by proposition [ propo : classic_or ] , @xmath368 if and only if @xmath369 or @xmath370 . by induction hypothesis , this is the case if and only if @xmath371 or @xmath372 , that is , if and only if @xmath373 . 4 . @xmath374 if and only if @xmath369 and @xmath375 , that is , by induction hypothesis , if and only if @xmath376 . @xmath377 if and only if there exists a @xmath343 such that @xmath378 and @xmath379 } \lnot py \vee ( \psi)^{dl}_y$ ] . by induction hypothesis , the first condition holds if and only if @xmath380 . as for the second one , it holds if and only if @xmath381 = y_1 \cup y_2 $ ] for two @xmath41 , @xmath42 such that @xmath382 and @xmath383 . but then we must have that @xmath384 and that @xmath385 ; therefore , by downwards closure , @xmath386 and finally @xmath387 . + conversely , suppose that there exists a @xmath343 such that @xmath380 and @xmath388 ; then by induction hypothesis we have that @xmath389 and that @xmath379 } \lnot py \vee ( \psi)^{dl}_x$ ] , and hence @xmath390 . now let us consider the cases corresponding to transition terms : 1 . suppose that @xmath391 . if @xmath193 then @xmath392 , and hence by * non - creation * we have that @xmath393 , as required . + let us assume instead that @xmath175 . then , by hypothesis , there exists a @xmath360 such that * there exists a @xmath313 such that @xmath394[f / y ] } r_t(i , x , y)$ ] ; * @xmath394[t^{dl}/y ] } \lnot r_t(i , x , y ) \vee py$ ] . + from the first condition it follows that for every @xmath395 there exists a @xmath396 such that @xmath397 : therefore , by the definition of @xmath327 , every such @xmath113 must be in @xmath398 . + from the second condition it follows that whenever @xmath397 and @xmath399 , @xmath400 ; and , since @xmath401 , this implies that @xmath402 by the definition of @xmath327 . + hence , by * monotonicity * and * downwards closure * , we have that @xmath403 and that @xmath404 , as required . + conversely , suppose that @xmath405 for some @xmath406 . if @xmath392 then @xmath193 , and hence by proposition [ propo : emptyteam ] we have that @xmath407 , as required . otherwise , by * non - triviality * , @xmath408 let now @xmath409 be any of its elements and let @xmath410 for all @xmath411 $ ] : then @xmath412[f / y ] } r_t(i , x , y)$ ] , as any assignment of this team sends @xmath331 to some element of @xmath398 and @xmath352 to @xmath413 . furthermore , let @xmath414 , and let @xmath396 be such that @xmath415 : then @xmath416 , and hence @xmath412[t^{dl}/y ] } \lnot r_t(i , x , y ) \vee py$ ] . so , in conclusion , @xmath417 , as required . 2 . @xmath418 if and only if @xmath419 and @xmath420 , that is , if and only if @xmath421 . 3 . @xmath422 if and only if @xmath237 for two @xmath423 such that * @xmath237 , and therefore @xmath424 ; * @xmath425 , that is , by induction hypothesis , @xmath426 ; * @xmath427 , that is , by induction hypothesis , @xmath428 ; + hence , if @xmath429 then @xmath430 . + conversely , if @xmath431 for two @xmath324 , @xmath325 such that @xmath432 and @xmath433 , let @xmath434 clearly @xmath237 , and furthermore by induction hypothesis @xmath425 and @xmath427 . hence , @xmath429 , as required . @xmath435 if and only if @xmath436 and @xmath437 , that is , by induction hypothesis , if and only if @xmath438 . 5 . @xmath439 if and only if there exists a @xmath440 such that @xmath441 and there exists a @xmath442 such that @xmath443 . by downwards closure , if this is the case then @xmath444 too , and hence @xmath445 , as required . + conversely , suppose that there exists a @xmath440 such that @xmath441 and @xmath444 . then , by induction hypothesis @xmath446 ; and furthermore , @xmath381 $ ] can be split into @xmath447 : s(y ) \not \in q\}\ ] ] and @xmath448 : s(y ) \in q\}\ ] ] it is trivial to see that @xmath449 ; and furthermore , since @xmath450 and @xmath444 , by induction hypothesis we have that @xmath451 . thus @xmath379 } \forall y ( \lnot qy \vee ( \tau_2)^{dl}_y(p))$ ] and finally @xmath452 , and this concludes the proof . hence , the relationship between transition logic and dependence logic is analogous to the one between dynamic game logic and first - order logic . in the next sections , we will develop variants of dependence logic which are syntactically closer to transition logic , while still being first - order : as we will see , the resulting frameworks are expressively equivalent to dependence logic on the level of satisfiability , but can be used to represent finer - grained phenomena of _ transitions _ between sets of assignments . now that we have established a connection between dependence logic and a variant of dynamic game logic , it is time to explore what this might imply for the further development of logics of imperfect information . if , as theorems [ theo : tl - rep1 ] and [ theo : tl - rep2 ] suggest , dependence logic can be thought of as a logic of imperfect - information decision problems , perhaps it could be possible to develop variants of dependence logic in which expressions can be interpreted directly as transition systems ? in what follows , we will do exactly that , first with _ transition dependence logic _ a variant of dependence logic , expressively equivalent to it , which is also a quantified version of transition logic and then with _ dynamic dependence logic _ , in which _ all _ expressions are interpreted as transitions ! but why would we interested in such variants of dependence logic ? one possible answer , which we will discuss in this subsection , is that transitions between teams are _ already _ a central object of study in the field of dependence logic , albeit in a non - explicit manner : after all , the semantics of dependence logic interprets quantifiers in terms of transformations of teams , and disjunctions in terms of decompositions of teams into subteams . this intuition is central to the study of issues of interdefinability in dependence logic and its variants , like for example the ones discussed in @xcite . as a simple example , let us recall definition [ defin : classic_or ] : @xmath453 where @xmath64 and @xmath65 are new variables . as we said in proposition [ propo : classic_or ] , @xmath454 if and only if @xmath48 or @xmath49 . we will now sketch the proof of this result , and as we will see this proof will hinge on the fact that the above expression can be read as a specification of the following algorithm : 1 . choose an element @xmath455 and extend the team @xmath19 by assigning @xmath456 as the value of @xmath64 for all assignments ; 2 . choose an element @xmath457 and further extend the team by assigning @xmath458 as the value of @xmath65 for all assignments ; 3 . split the resulting team into two subteams @xmath41 and @xmath42 such that 1 . @xmath62 holds in @xmath41 , and the values of @xmath64 and @xmath65 coincide for all assignments in it ; 2 . @xmath63 holds in @xmath42 , and the values of @xmath64 and @xmath65 differ for all assignments in it . since the values of @xmath64 and @xmath65 are chosen to always be respectively @xmath456 and @xmath458 , one of @xmath41 and @xmath42 is empty and the other is of the form @xmath459 $ ] , and since @xmath64 and @xmath65 do not occur in @xmath62 or @xmath63 the above algorithm can succeed ( for some choice of @xmath456 and @xmath458 ) only if @xmath48 or @xmath49 . as another , slightly more complicated example , let us consider the following problem . given four variables @xmath460 , @xmath461 , @xmath462 and @xmath463 , let @xmath464 be an _ exclusion atom _ holding in a team @xmath19 if and only if for all @xmath38 , @xmath465 that is , if and only if the sets of the values taken by @xmath466 and by @xmath467 in @xmath19 are disjoint . by theorem [ sigmatodl ] , we can tell at once that there exists some dependence logic formula @xmath468 such that for all suitable @xmath7 and @xmath19 , @xmath469 if and only if @xmath470 ; but what about the converse ? for example , can we find an expression @xmath471 , in the language of first order logic augmented with these exclusion atoms ( but with no dependence atoms ) , such that for all suitable @xmath7 and @xmath19 @xmath472 if and only if @xmath473 ? as discussed in @xcite in a more general setting , the answer is positive , and one such @xmath471 is @xmath474 , where @xmath475 is some variable other than @xmath331 and @xmath352 . in the second disjunct can be removed , but for simplicity we will keep it . ] why is this the case ? well , let us consider any team @xmath19 with domain containing @xmath331 and @xmath352 , and let us evaluate @xmath476 over it . as shown graphically in figure [ fig : f1 ] , the transitions between teams occurring during the evaluation of the formula correspond to the following algorithm : 1 . first , assign all possible values to the variable @xmath475 for all assignments in @xmath331 , thus obtaining @xmath477 = \{s[m / z ] : s \in x , m \in { { \texttt{dom}}}(m)\}$ ] ; 2 . then , remove from @xmath477 $ ] all assignments @xmath11 for which @xmath478 , keeping only the ones for which @xmath479 ; 3 . then , verify that for any possible fixed value of @xmath331 , the possible values of @xmath352 and @xmath475 are disjoint . this algorithm succeeds only if @xmath352 is a function of @xmath331 . indeed , suppose that instead there are two assignments @xmath38 such that @xmath480 , @xmath481 and @xmath482 for three @xmath483 with @xmath484 . now we have that @xmath485 , s[c / z ] , s'[b / z ] , s'[c / z]\ } \subseteq x[m / z]$ ] : and since @xmath484 , we have that the assignments @xmath486 $ ] and @xmath487 $ ] are not removed from the team in the second step of the proof . but then @xmath486(xz ) = a c = s'[b / z](xy)$ ] , and therefore it is not true that @xmath488 . and , conversely , if in the team @xmath19 the value of @xmath352 is a function of the value of @xmath331 then by splitting @xmath477 $ ] into the two subteams @xmath489 : s \in x , s(y ) = s(z)\}$ ] and @xmath490 : s(y ) \not = s(z)\}$ ] we have that @xmath491 , @xmath492 and @xmath493 ( since for all @xmath494 , @xmath495 ) . on the other hand , one dependence logic expression corresponding to @xmath464 is @xmath496 where @xmath497 , @xmath498 , @xmath64 and @xmath65 are new variable . we encourage the interested reader to verify that this is the case by examining the transitions between teams corresponding to the formula : in brief , the intuition is that first we extend our team by picking all possible pairs of values for @xmath497 and @xmath498 , then for any such pair we flag through our choice of @xmath64 and @xmath65 whether @xmath499 is different from @xmath466 or from @xmath467 . this implies that no such pair is equal to both @xmath466 and @xmath467 , or , in other words , that @xmath466 and @xmath467 have no value in common . more and more complex examples of definability results of this kind can be found in @xcite ; but what we want to emphasize here is that all these examples , like the one we discussed in depth here , have a natural interpretation in terms of algorithms which transform teams and apply simple tests to them , as the above one . hence , we hope that the development of variants of dependence logic in which these transitions are made explicit might prove itself useful for the further study of this interesting class of problems . as stated , we will now define a variant of dependence logic which can also be seen as a quantified variant of transition logic . we will then prove that the resulting transition dependence logic is expressively equivalent to dependence logic , in the sense that any dependence logic formula is equivalent to some transition dependence logic formula and vice versa . let @xmath26 be a first - order signature . then the sets of all _ transition terms _ and of all _ formulas _ of dependence transition logic are given by the rules @xmath500 where @xmath14 ranges over all variables in @xmath31 , @xmath28 ranges over all relation symbols of the signature , @xmath29 ranges over all tuples of terms of the required arities , @xmath501 ranges over @xmath502 and @xmath30 range over the terms of our signature . let @xmath7 be a first - order model , let @xmath229 be a first - order transition term of the same signature , and let @xmath19 and @xmath177 be teams over @xmath7 . then we say that the transition @xmath503 is _ allowed _ by @xmath229 in @xmath7 , and we write @xmath504 , if and only if tdl-@xmath50 : : : @xmath229 is of the form @xmath505 for some @xmath506 and there exists a @xmath313 such that @xmath507\subseteq y$ ] ; tdl-@xmath55 : : : @xmath229 is of the form @xmath508 for some @xmath506 and @xmath509 \subseteq y$ ] ; tdl - test : : : @xmath229 is of the form @xmath108 , @xmath34 in the sense given later in this definition , and @xmath92 ; tdl-@xmath235 : : : @xmath229 is of the form @xmath264 and @xmath237 for some @xmath238 and @xmath239 such that @xmath510 and @xmath511 ; tdl-@xmath242 : : : @xmath229 is of the form @xmath512 , @xmath513 and @xmath514 ; tdl - concat : : : @xmath229 is of the form @xmath515 and there exists a team @xmath136 such that @xmath516 and @xmath517 . similarly , if @xmath8 is a formula and @xmath19 is a team with domain @xmath31 . then we say that @xmath19 _ satisfies _ @xmath8 in @xmath7 , and we write @xmath34 , if and only if tdl - lit : : : @xmath8 is a first - order literal and @xmath35 in the usual first - order sense for all @xmath36 ; tdl - dep : : : @xmath8 is a dependence atom @xmath37 and any two @xmath38 which assign the same values to @xmath2 also assign the same value to @xmath1 ; tdl-@xmath39 : : : @xmath8 is of the form @xmath518 and @xmath519 or @xmath520 ; tdl-@xmath46 : : : @xmath8 is of the form @xmath521 , @xmath519 and @xmath520 ; tdl-@xmath156 : : : @xmath8 is of the form @xmath522 and there exists a @xmath177 such that @xmath504 and @xmath71 . as the next theorem shows , in this semantics formulas and transitions are interpreted in terms of trumps and transition systems : for all transition dependence logic formulas @xmath8 , all models @xmath7 and all teams @xmath19 and @xmath177 , we have that downwards closure : : : if @xmath34 and @xmath70 then @xmath73 ; empty team property : : : @xmath69 . furthermore , for all transition dependence logic transition terms @xmath229 , all models @xmath7 and all teams @xmath19 , @xmath177 and @xmath136 , downwards closure : : : if @xmath504 and @xmath523 then @xmath524 ; monotonicity : : : if @xmath504 and @xmath525 then @xmath526 ; non - creation : : : for all @xmath177 , @xmath527 ; non - triviality : : : if @xmath175 then @xmath528 . the proof is by structural induction over @xmath8 and @xmath229 , and presents no difficulties whatsoever . also , it is not difficult to see , on the basis of the results of the previous section , that this new variant of dependence logic is equivalent to the usual one : for every dependence logic formula @xmath8 there exists a transition dependence logic transition term @xmath529 such that @xmath530 for all first - order models @xmath7 and teams @xmath19 . @xmath529 is defined by structural induction on @xmath8 , as follows : 1 . if @xmath8 is a first - order literal or a dependence atom then @xmath531 ; 2 . if @xmath8 is @xmath518 then @xmath532 ; 3 . if @xmath8 is @xmath521 then @xmath533 ; 4 . if @xmath8 is @xmath51 then @xmath534 ; 5 . if @xmath8 is @xmath56 then @xmath535 it is then trivial to verify , again by induction on @xmath8 , that @xmath34 if and only if @xmath536 , as required . this representation result associates dependence logic _ formulas _ to transition dependence logic _ transition terms_. this fact highlights the dynamical nature of dependence logic operators , which we discussed in the previous subsection : in this framework , quantifiers describe _ transformations _ of teams , the dependence logic connectives are operations over games , and the literals are interpreted as tests . in fact , one might wonder what is the purpose of transition dependence logic formulas : could we do away with them altogether , and develop a variant of transition dependence logic in which _ all _ formulas are transitions ? later , we will explore this idea further ; but first , let us verify that transition dependence logic is no more expressive than dependence logic . for every transition dependence logic formula @xmath8 there exists a dependence logic formula @xmath537 such that @xmath538 for all first - order models @xmath7 and teams @xmath19 . furthermore , for every transition dependence logic transition term @xmath229 and dependence logic formula @xmath211 there is a dependence logic formula @xmath539 such that @xmath540 again for all first - order models @xmath7 and teams @xmath19 . we prove the two claims together , by structural induction over @xmath8 and @xmath229 . first , let us consider the cases corresponding to formulas : 1 . if @xmath8 is a first order literal or a dependence atom , let @xmath537 be @xmath8 itself . as the interpretation of these expressions is the same in dependence logic and in transition dependence logic , there is nothing to prove . 2 . if @xmath8 is of the form @xmath40 , let @xmath537 be @xmath541 . this expression holds in a team if and only if @xmath542 or @xmath543 hold , that is , by induction hypothesis , if and only if @xmath62 or @xmath63 do . if @xmath8 is of the form @xmath47 , let @xmath537 be @xmath544 . then @xmath537 holds if and only if @xmath62 and @xmath63 do , that is , if and only if @xmath8 does . 4 . if @xmath8 is of the form @xmath522 , let @xmath21 be the tuple of all variables occurring in @xmath545 , let @xmath28 be a new @xmath78-ary relation , and let @xmath537 be @xmath546 . indeed , suppose that @xmath547 : then for some relation @xmath28 , there exists a @xmath177 such that @xmath504 and @xmath548 . furthermore , @xmath549 , and therefore for the set @xmath550 we have that @xmath551 . but then , by downwards closure and locality , @xmath552 , and therefore @xmath553 . + conversely , suppose that @xmath554 : then there exists a @xmath177 such that @xmath504 and @xmath71 . now let @xmath28 be @xmath555 : clearly @xmath556 , since @xmath557 , and furthermore @xmath558 , by locality and by the fact that ( by induction hypothesis ) @xmath552 . now let us consider the cases corresponding to transitions : 1 . if @xmath229 is of the form @xmath505 for some variable @xmath14 , let @xmath559 be @xmath560 . indeed , suppose that @xmath561 : then @xmath53 } \theta$ ] for some @xmath313 , and by choosing @xmath562 $ ] we have that @xmath563 and @xmath564 , as required . conversely , suppose that for some @xmath177 , @xmath563 and @xmath564 : then for some @xmath313 , @xmath507 \subseteq y$ ] , and by downwards closure we have that @xmath53 } \theta$ ] . 2 . if @xmath229 is of the form @xmath508 for some variable @xmath14 , let @xmath559 be @xmath565 . indeed , suppose that @xmath566 : then @xmath57 } \theta$ ] , and if we choose @xmath567 $ ] we have at once that @xmath568 and @xmath564 . conversely , if for some @xmath177 @xmath568 and @xmath564 then @xmath509 \subseteq y$ ] and , by downwards closure , @xmath57 } \theta$ ] . 3 . if @xmath229 is of the form @xmath108 , let @xmath559 be @xmath569 . indeed , suppose that @xmath570 : then by induction hypothesis @xmath34 , and , for @xmath571 , we have that @xmath572 . furthermore , @xmath564 , as required . conversely , suppose that for some @xmath177 , @xmath572 and @xmath564 . then @xmath34 , and therefore @xmath547 ; and furthermore @xmath92 , and hence by downwards closure @xmath573 . hence , @xmath574 . 4 . if @xmath229 is of the form @xmath264 and @xmath21 is the tuple of all free variables of @xmath211 then let @xmath559 be @xmath575 , where @xmath28 is a new @xmath576-ary relation symbol . indeed , suppose that @xmath577 : then there exists a relation @xmath28 and two subteams @xmath238 and @xmath239 of @xmath19 such that @xmath237 , @xmath578 and @xmath579 . hence , there are two teams @xmath41 and @xmath42 such that @xmath580 , @xmath581 , @xmath582 and @xmath583 . now , let @xmath177 be @xmath584 : by monotonicity , we have that @xmath510 and @xmath511 , and furthermore @xmath557 too ( that is , for all @xmath585 , @xmath586 is in @xmath28 ) . since @xmath587 , this implies that @xmath564 , by locality and downwards closure . + conversely , suppose that there is a @xmath177 such that @xmath588 and @xmath564 . then let @xmath28 be @xmath589 . now @xmath237 for two @xmath238 and @xmath239 such that @xmath510 and @xmath511 , and by induction hypothesis we have that @xmath590 and @xmath591 . but then @xmath592 ; and furthermore , by locality we have that @xmath587 . hence , @xmath593 , as required . if @xmath229 is of the form @xmath512 and @xmath21 is the tuple of all variables of @xmath211 then let @xmath559 be @xmath594 . indeed , suppose that @xmath577 : then for some relation @xmath28 , by induction hypothesis , there exist teams @xmath41 and @xmath42 such that @xmath595 , @xmath596 , @xmath582 and @xmath583 . now let @xmath177 be @xmath584 : as before , by monotonicity we have that @xmath513 and @xmath597 , and hence @xmath598 . finally , since @xmath587 we have that @xmath599 , as required . + conversely , suppose that there is a @xmath177 such that @xmath598 and @xmath564 . since @xmath598 , @xmath513 and @xmath597 . now let @xmath28 be @xmath555 . by induction hypothesis , @xmath600 and @xmath601 ; and furthermore , since @xmath564 we have that @xmath587 . if @xmath229 is of the form @xmath515 let @xmath559 be @xmath602 . indeed , @xmath603 if and only if there is a @xmath177 such that @xmath604 and @xmath605 , that is , if and only if there are a @xmath177 and a @xmath136 such that @xmath513 , @xmath606 and @xmath607 . however , in a sense , transition dependence logic allows one to consider subtler distinctions than dependence logic does . the formula @xmath608 , for example , could be translated as any of * @xmath609 ; * @xmath610 ; * @xmath611 ; * @xmath612 . the intended interpretations of these formulas are rather different , even though they happen to be satisfied by the same teams : and for this reason , transition dependence logic may be thought of as a proper refinement of dependence logic even though it has exactly the same expressive power . _ dynamic semantics _ is the name given to a family of semantical frameworks which subscribe to the following principle ( @xcite ) : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the meaning of a sentence does not lie in its truth conditions , but rather in the way it changes ( the representation of ) the information of the interpreter . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ in various forms , this intuition can be found prefigured in some of the later work of ludwig wittgenstein , as well as in the research of philosophers of language such as austin , grice , searle , strawson and others ( @xcite ) ; but its formal development can be traced back to the work of groenendijk and stokhof about the proper treatment of pronouns in formal linguistics ( @xcite ) . + we refer to @xcite for a comprehensive analysis of the linguistic issues which caused such a development , as well as for a description of the ways in which this framework was adapted in order to model presuppositions , questions / answers and other phenomena ; here we will only present a formulation of _ dynamic predicate semantics _ , the alternative semantics for first - order logic which was developed in the above mentioned paper by groenendijk and stokhof . let @xmath8 be a first - order formula , let @xmath7 be a suitable first - order model and let @xmath11 and @xmath182 be two assignments . then we say that the transition from @xmath11 to @xmath182 is _ allowed _ by @xmath8 in @xmath7 , and we write @xmath613 , if and only if dpl - atom : : : @xmath8 is an atomic formula , @xmath614 and @xmath35 in the usual sense ; dpl-@xmath153 : : : @xmath8 is of the form @xmath615 , @xmath616 and for all assignments @xmath617 , @xmath618 ; dpl-@xmath46 : : : @xmath8 is of the form @xmath47 and there exists an @xmath617 such that @xmath619 and @xmath620 ; dpl-@xmath39 : : : @xmath8 is of the form @xmath40 , @xmath614 and there exists an @xmath617 such that @xmath619 or @xmath621 ; dpl-@xmath622 : : : @xmath8 is of the form @xmath623 , @xmath614 and for all @xmath617 it holds that @xmath624 dpl-@xmath50 : : : @xmath8 is of the form @xmath625 and there exists an element @xmath626 such that @xmath627 \rightarrow s ' } \psi$ ] ; dpl-@xmath55 : : : @xmath8 is of the form @xmath628 , @xmath614 and for all elements @xmath626 there exists an @xmath617 such that @xmath627 \rightarrow h } \psi$ ] . a formula @xmath8 is _ satisfied _ by an assignment @xmath11 if and only if there exists an assignment @xmath182 such that @xmath613 ; in this case , we will write @xmath35 . we will discuss neither the formal properties of this formalism nor its linguistic applications here . all that is relevant for our purposes is that , according to it , formulas are interpreted as _ transitions _ from assignments to assignments , and furthermore that the rule for conjunction allows us to bind occurrences of a variable of the second conjunct to quantifiers occurring in the first one . : by the rules given , it is easy to see that @xmath629 if and only if @xmath630 , that is , if and only if @xmath631 , differently from the case of tarski s semantics . ] the similarity between this semantics and our semantics for transition terms should be evident . hence , it seems natural to ask whether we can adopt , for a suitable variant of dependence logic , the following variant of groenendijk and stokhof s motto : + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the meaning of a formula does not lie in its satisfaction conditions , but rather in the team transitions it allows . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ from this point of view , _ transition terms _ are the fundamental objects of our syntax , and formulas can be removed altogether from the language although , of course , the tests corresponding to literals and dependence formulas should still be available . as in groenendijk and stokhof s logic , satisfaction becomes then a derived concept : in brief , a team @xmath19 can be said to satisfy a term @xmath229 if and only if there exists a @xmath177 such that @xmath229 allows the transition from @xmath19 to @xmath177 , or , in other words , if and only if _ some _ set of non - losing outcomes can be reached from the set @xmath19 of initial positions in the game corresponding to @xmath229 . in the next section , we will make use of these intuitions to develop another , terser version of dependence logic ; and finally , we will discuss some implications of this new version for the further developments and for the possible applications of this interesting logical formalism . we will now develop a formula - free variant of transition dependence logic , along the lines of groenendijk and stockhof s dynamic predicate logic . let @xmath26 be a first - order signature . the set of all formulas of dynamic dependence logic over @xmath26 is given by the rules @xmath632 where , as usual , @xmath28 ranges over all relation symbols of our signature , @xmath29 ranges over all tuples of terms of the required lengths , @xmath501 ranges over @xmath502 , @xmath30 range over all terms , and @xmath14 ranges over @xmath31 . the semantical rules associated to this language are precisely as one would expect : [ ddl - tts ] let @xmath7 be a first - order model , let @xmath229 be a dynamic dependence logic formula over the signature of @xmath7 , and let @xmath19 and @xmath177 be two teams over @xmath7 with domain @xmath31 . then we say that @xmath229 _ allows _ the transition @xmath185 in @xmath7 , and we write @xmath504 , if and only if ddl - lit : : : @xmath229 is a first - order literal , @xmath633 in the usual first - order sense for all @xmath36 , and @xmath92 ; ddl - dep : : : @xmath229 is a dependence atom @xmath37 , @xmath92 , and any two assignments @xmath38 which coincide over @xmath2 also coincide over @xmath1 ; ddl-@xmath50 : : : @xmath229 is of the form @xmath505 for some @xmath506 , and @xmath507 \subseteq y$ ] for some @xmath52 ; ddl-@xmath55 : : : @xmath229 is of the form @xmath508 for some @xmath506 , and @xmath509 \subseteq y$ ] ; ddl-@xmath235 : : : @xmath229 is of the form @xmath264 and @xmath237 for two teams @xmath238 and @xmath239 such that @xmath510 and @xmath511 ; ddl-@xmath242 : : : @xmath229 is of the form @xmath512 , @xmath513 and @xmath597 ; ddl - concat : : : @xmath229 is of the form @xmath515 , and there exists a @xmath136 such that @xmath516 and @xmath517 . a formula @xmath229 is said to be _ satisfied _ by a team @xmath19 in a model @xmath7 if and only if there exists a @xmath177 such that @xmath504 ; and if this is the case , we will write @xmath634 . it is not difficult to see that dynamic dependence logic is equivalent to transition dependence logic ( and , therefore , to dependence logic ) . let @xmath8 be a dependence logic formula . then there exists a dynamic dependence logic formula @xmath635 which is equivalent to it , in the sense that @xmath636 for all suitable teams @xmath19 and models @xmath7 we build @xmath635 by structural induction : 1 . if @xmath8 is a literal or a dependence atom then @xmath637 ; 2 . if @xmath8 is @xmath40 then @xmath638 ; 3 . if @xmath8 is @xmath47 then @xmath639 ; 4 . if @xmath8 is @xmath625 then @xmath640 ; 5 . if @xmath8 is @xmath628 then @xmath641 . let @xmath229 be a dynamic dependence logic formula . then there exists a transition dependence logic transition term @xmath642 such that @xmath643 for all suitable @xmath19 , @xmath177 and @xmath7 , and such that hence @xmath644 build @xmath642 by structural induction : 1 . if @xmath229 is a literal or dependence atom then @xmath645 ; 2 . if @xmath229 is of the form @xmath505 or @xmath508 then @xmath646 ; 3 . if @xmath229 is of the form @xmath264 then @xmath647 ; 4 . if @xmath229 is of the form @xmath512 then @xmath648 ; 5 . if @xmath229 is of the form @xmath515 then @xmath649 . dynamic dependence logic is equivalent to transition dependence logic and to dependence logic follows from the two previous results and from the equivalence between dependence logic and transition dependence logic . in this work , we established a connection between a variant of dynamic game logic and dependence logic , and we used it as the basis for the development of variants of dependence logic in which it is possible to talk directly about transitions from teams to teams . this suggests a new perspective on dependence logic and team semantics , one which allow us to study them as a special kind of _ algebras of nondeterministic transitions between relations_. one of the main problems that is now open is whether it is possible to axiomatize these algebras , in the same sense in which , in @xcite , allen mann offers an axiomatization of the algebra of trumps corresponding to if logic ( or , equivalently , to dependence logic ) . furthermore , we might want to consider different choices of connectives , like for example ones related to the theory of database transactions . the investigation of the relationships between the resulting formalisms is a natural continuation of the currently ongoing work on the study of the relationship between various extensions of dependence logic , and promises of being of great utility for the further development of this fascinating line of research . the author wishes to thank johan van benthem and jouko vnnen for a number of useful suggestions and insights . furthermore , he wishes to thank the reviewers for a number of highly useful suggestions and comments . hintikka , j. and g. sandu : 1989 , ` informational independence as a semantic phenomenon ' . in : j. fenstad , i. frolov , and r. hilpinen ( eds . ) : _ logic , methodology and philosophy of science_. elsevier , pp . 571589 . kontinen , j. and v. nurmi : 2009 , ` team logic and second - order logic ' . in : h. ono , m. kanazawa , and r. de queiroz ( eds . ) : _ logic , language , information and computation _ , vol . 5514 of _ lecture notes in computer science_. springer berlin / heidelberg , pp . 230241 . parikh , r. : 1985 , ` the logic of games and its applications ' . in : _ selected papers of the international conference on `` foundations of computation theory '' on topics in the theory of computation_. new york , ny , usa , pp . 111139 . vnnen , j. : 2007b , ` team logic ' . in : j. van benthem , d. gabbay , and b. lwe ( eds . ) : _ interactive logic . selected papers from the 7th augustus de morgan workshop_. msterdam university press , pp .

we examine the relationship between dependence logic and game logics . a variant of dynamic game logic , called _ transition logic _ , is developed , and we show that its relationship with dependence logic is comparable to the one between first - order logic and dynamic game logic discussed by van benthem . this suggests a new perspective on the interpretation of dependence logic formulas , in terms of assertions about _ reachability _ in games of imperfect information against nature . we then capitalize on this intuition by developing expressively equivalent variants of dependence logic in which this interpretation is taken to the foreground .

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Proceed to summarize the following text: random molecular interactions can have profound effects on gene expression . because the expression of a gene can be regulated by a single promotor , and because the number of mrna copies and protein molecules is often small , deterministic models of gene expression can miss important behaviors . a deterministic model might show multiple possible stable behaviors , any of which can be realized depending on the initial conditions of the system . different stable behavior that depend on initial conditions allows for variability in response and adaptation to environmental conditions @xcite . although in some cases , noise from multiple sources can push the behavior far from the deterministic model , here we focus on situation where the system fluctuates close to the deterministic trajectory ( i.e. , weak noise ) . of particular interest is behavior predicted by a stochastic model that is qualitatively different from its deterministic counterpart @xcite , even if the fluctuations are small . several interesting questions emerge when including stochastic effects in a model of gene expression . for example , what are the different sources of fluctuations affecting a gene circuit ? can noise be harnessed for useful purpose , and if so , what new functions can noise bring to the gene - regulation toolbox ? one way in which noise can induce qualitatively different behavior occurs when a rare sequence of random events pushes the system far enough away from one of the stable deterministic behaviors that the system transitions toward a different stable dynamic behavior , one that would never be realized in the deterministic model without changing the initial conditions . for example , if the deterministic model is bistable , fluctuations can cause the protein concentration to shift between the different metastable protein concentrations . this happens when fluctuations push the system past the unstable fixed point that separates two stable fixed points . while often times a spontaneous change in gene expression might be harmfull , it might also be beneficial . for example , in certain types of bacteria , a few individuals within a population enter a slow - growth state in order to resist exposure to antibiotics . in a developing organism , a population of differentiating cells might first randomly choose between two or more expression profiles during their development and then later segregate into distinct groups by chemotaxis . in both examples , switching between metastable states leads to mixed populations of phenotypic expression @xcite . this leads to the question of how cells coordinate and regulate different sources of biochemical fluctuations , or noise , to function within a genetic circuit . in many cases , the genes within a given circuit are turned on and off by regulator proteins , which are often the gene products of the circuit . if a gene is switched on , its dna is transcribed into one or more mrna copies , which are in turn translated into large numbers of proteins . typically , the protein products form complexes with each other or with other proteins that bind to regulatory dna sequences , or operators , to alter the expression state of a gene . for example , a repressor binds to an operator which blocks the promotor the region of dna that a polymerase protein binds to before transcribing the gene so that the gene is turned off and no mrna are transcribed . this feedback enables a cell to regulate gene expression , and often multiple genes interact within groups to form gene circuits . understanding how different noise sources affect the behavior of a gene circuit and comparing this with how the circuit behaves with multiple noise sources is essential for understanding how a cell can use different sources of noise productively . fluctuations arising from the biochemical reactions involving the dna , mrna , and proteins are commonly classified as `` intrinsic '' noise @xcite . one important source of intrinsic noise is fluctuations from mrna transcription , protein translation , and degradation of both mrna and protein product . this type of noise is common among many of the biochemical reactions within a cell , and its effect is reduced as the number of reacting species within a given volume grows large . another source of intrinsic noise is in the expression state of the genes within the circuit . typically there is only one or two copies of a gene within a cell , which means that thermal fluctuations within reactions with regulatory proteins have a significant effect on mrna production . here , we consider the situation where transitions in the behavior of a gene circuit are primarily driven by fluctuations in the on / off state of its promotor and examine the effect of removing all other sources of noise . stochastic gene circuits are typically modelled using a discrete markov process , which tracks the random number of mrna and/or proteins along with the state of one or more promotors @xcite ( but see also @xcite ) . monte - carlo simulations using the gillespie algorithm can be used to generate exact realizations of the random process . the process can also be described by its probability density function , which satisfies a system of linear ordinary differential equations known as the master equation . the dimension of the master equation is the number of possible states the system can occupy , which can be quite large , leading to the problem of dimensionality when analyzing the master equation directly . however , for the problem considered here , the full solution to the master equation is not necessary in order to understand metastable transitions . the motivating biological question we consider here is what percentage of a population of cells can be expected to exhibit a metastable transition within a given timeframe . if a spontaneous transition is harmfull to the cell , one expects that reaction rates and protein / dna interactions should evolve so that transition times are likely to be much larger than the lifetime of the cell . on the other hand , if spontaneous transition are functional , transition times should be tuned to achieve the desired population in which the transition occurs . in either case , the key quantity of interest is the distribution of transition times between metastable states , regardless of the noise source driving the transition . except for a few special cases @xcite exact results , even for the mean transition time , are not possible and approximation techniques or monte - carlo simulations must be used . however , because rare events typically involve long simulation times where large numbers of jumps occur , monte - carlo simulations are computationally expensive to perform , leaving perturbation analysis ideally suited for the task . past studies of metastable transitions , where perturbation methods are applied to the master equation , have used a simplifying assumption so that the state of the promotor is not accounted for explicitly @xcite . the assumption is that proteins are produced in `` bursts '' during which one or more mrna copies are translated to rapidly produce many proteins . in these models , production bursts occur as instantaneous jumps , with a predefined distribution determining the number of proteins produced during a given burst . more recently , assaf and coworkers analyzed a model where the on / off state of a single stochastic promotor is accounted for explicitly , and mrna copies are produced stochastically at a certain rate when the promotor is turned on @xcite . however , the case where the model contains an arbitrary number of promotors or promotor states has not been addressed , and as we show in this paper , accounting for even just three promotor states is nontrivial . similar asymptotic methods have also been developed to study metastable transitions in continuous markov processes @xcite , but can not be applied to a discrete chemical reaction system because continuous approximations , such as the system - size expansion , of discrete markov processes do not , in general , accurately capture transition times @xcite . another source of difficulty that arises from isolating promotor state fluctuations as the only source of noise is that the resulting state space of the markov process is both continuous and discrete . after removing all sources of intrinsic noise except for the fluctuating promotor by taking the thermodynamic limit the protein levels change deterministically and continuously , and the promotor s state jumps at exponentially - distributed random times . the random jumps in the promotor s state makes the protein levels appear random , even though they are only responding deterministically to changes in the promotor state . such random processes are sometimes called hybrid systems , or piecewise deterministic @xcite . here we refer to it as the quasi - deterministic ( qd ) process because we are taking part of the randomly fluctuating discrete state of the system ( the number of protein molecules ) and replacing it with a deterministically - changing continuous state . recently , we have developed asymptotic methods for metastable transitions similar to those applied to the discrete master equation for markov processes with both discrete and continuous state spaces @xcite . however , these methods do not account for two or more continuous state variables , which restricts the genetic circuit the method can analyze to one with a single protein product . in this paper , we develop new perturbation methods so that we can study metastable transitions in genetic circuits driven by promotor fluctuations . these methods are based on previous theory developed for one - dimensional velocity jump processes , and are generalized to account for the multiple continuous states representing the quantity of proteins produced by the genetic circuit . they also fit within a larger framework of methods to study metastable transitions in continuous markov processes @xcite and in discrete markov processes @xcite . for illustration , we use a simple model known as the `` mutual repressor '' model @xcite , which contains two genes , two promotors , and three promotor states . although our example considers only three promotor states , the methods presented are general and can account for an arbitrary number of promotor states . for a range of parameter values , the deterministic limit of the mutual repressor model is bistable , having two stable fixed points separated by an unstable saddle point . for the stochastic model , the deterministic forces create two confining wells surrounding each stable fixed point , separated by a stability barrier along the separatrix that contains the unstable saddle . this geometric interpretation is given by taking the logarithm of the stationary probability density function , which we refer as the `` stability landscape . '' an approximation of the first exit time density is found for the random process to escape over the stability barrier from one of the stability wells to the other . using this model , we seek to answer the following question . how does the random process change when protein noise is removed , leaving the state of the promotor as the only source of randomness ? that is , are there any qualitative differences in the behavior of the system without other sources of intrinsic noise ? the paper is organized as follows . in section [ sec : model ] the mutual repressor model is presented along with its reduction to the qd process , and then in section [ sec : trt ] , perturbation methods for estimating the exit time density are applied to the qd process . for comparison , the stability landscape is also computed for the full process in section [ sec : full ] , which includes fluctuations in protein production / degradation . finally , results are presented in section [ sec : results ] , and the qd process is compared to the full process , using analytical / numerical approximations and monte - carlo simulations . the mutual repressor model @xcite is a hypothetical gene circuit consisting of a single promotor driving the expression of two genes : @xmath0 and @xmath1 . each protein product can dimerize and bind to the promotor to repress the expression of the other . when no dimer is bound to the promotor , both genes are expressed equally . thus , the promotor can be in one of three states : bound to a dimer of protein one @xmath2 , unbound @xmath3 , or bound to a dimer of protein two @xmath4 . let the number of protein product of gene @xmath0 and @xmath1 be @xmath5 and @xmath6 , respectively . it is assumed that the mrna and protein production steps can be combined into a single protein production rate and that the dimerization reaction is fast so that it can be taken to be in quasi - steady - state . we then have the following transition between the three promotor states @xmath7 where @xmath8 is a rate and @xmath9 is a nondimensional dissociation constant . protein @xmath0 ( @xmath1 ) is produced at a rate @xmath10 while the promotor is in states @xmath11 ( @xmath12 ) , and both proteins are degraded at a rate @xmath13 in all three promotor states . the probability density function @xmath14 , for promotor state @xmath15 and protein numbers @xmath16 , satisfies the master equation @xmath17\mathbf{p},\ ] ] where @xmath18\ ] ] is the matrix responsible for promotor state transitions . the diagonal matrix @xmath19 , responsible for changes in protein numbers , has elements @xmath20 with @xmath21.\ ] ] the shift operators @xmath22 are defined according to @xmath23 we now introduce the nondimensional variables @xmath24 , @xmath25 , and @xmath26 , where @xmath27 . then , the master equation ( [ eq:2 ] ) for the rescaled probability density , @xmath28 , becomes @xmath29\bm{\rho},\ ] ] with dimensionless parameters @xmath30 and @xmath31 . the matrices are given by @xmath32,\ ] ] and @xmath33 the operators @xmath34 and @xmath34 are defined in terms of taylor series expansions ( in small @xmath35 ) which replace the shift operators @xmath36 with @xmath37 assume that @xmath38 is a small parameter , so that there is a large average number of proteins , and assume also that the parameter @xmath39 is small , which reflects rapid switching between promotor states compared to the rate of protein production / degradation . because we have two small parameters in our system , @xmath35 and @xmath40 , when perusing an asymptotic solution , we must carefully consider how the limit @xmath41 and @xmath42 is taken , or more practically , how large @xmath40 is compared to @xmath35 . the fluctuations in the promotor state are controlled by @xmath40 , and in the limit @xmath41 , the transitions are infinitely fast so that the promotor behaves deterministically . the fluctuations in protein levels is controlled by @xmath35 , and in the limit @xmath42 the protein production / degradation behaves deterministically . since we are concerned primarily with rare transitions driven by promotor fluctuations and not by fluctuations in the protein production / degradation reaction , we assume that @xmath43 ( i.e. , @xmath44 ) . taking both limits , @xmath45 and @xmath46 , yields the fully - deterministic dynamics , @xmath47 where @xmath48 note the symmetry in the problem ; the deterministic system is unchanged if we exchange @xmath49 . dynamically , the system is bistable for @xmath50 . at @xmath51 there is a saddle - node bifurcation , and for @xmath52 there is a single stable fixed point . we consider the case of bistability and chose @xmath53 . in fig . [ fig : det ] the nullclines and fixed points are shown . . the black curve shows the @xmath54-nullcline and the grey curve shows the @xmath55-nullcline . the green circles show the stable fixed points , the red circle shows the unstable saddle . the blue curve shows a stochastic trajectory leaving the lower basin of attraction to reach the separatrix.,width=377 ] the two stable fixed points are located near the corners , and the unstable saddle point is located along the separatrix . arrows show the eigenvectors of the jacobian with their direction determined by the sign of the eigenvalues , all of which are real . a stochastic trajectory that starts at the lower stable fixed point remains nearby for a long period of time until a rare sequence of jumps carries it to the separatrix . because the separatrix is the stable manifold , trajectories are most likely to exit near the unstable saddle point . to remove protein noise from the system , consider the limit @xmath42 , with @xmath39 fixed , so that the protein production / degradation process is deterministic within each promotor state , while the promotor state remains random . the master equation ( [ eq:7 ] ) becomes @xmath56 where @xmath57 and @xmath58.\ ] ] the focus of the remaining analysis is to obtain an accurate approximation of the first exit time density function ( fetd ) for the qd process to evolve from one metastable state to the another . to obtain the fetd , we supplement an absorbing boundary condition to the governing equation along the separatrix , @xmath59 of the deterministic dynamics , which is the barrier the process must surmount in order to transition to the other metastable state . for the master equation ( [ eq:2 ] ) , the absorbing boundary condition is simply @xmath60 for the qd ck equation ( [ eq:13 ] ) , the absorbing boundary condition is @xmath61 where @xmath62 is the unit vector normal to the boundary @xmath63 . then , the domain for the qd process with the absorbing boundary is given by @xmath64 note the choice of the lower triangular region ( instead of the upper triangular region with @xmath65 ) is arbitrary due to the symmetry in the problem . to see how the absorbing boundary on @xmath63 sets up the exit time problem , define @xmath66 to be the random time at which the separatrix is reached for the first time , given that the process starts at the stable fixed point @xmath67 . consider the survival probability @xmath68 which is the probability that @xmath69 . the fetd ( or probability density function for @xmath66 ) is then @xmath70 the fetd for the qd process can be approximated using perturbation methods as follows . suppose we have a ck equation of the form @xmath71 where @xmath72 is a linear operator acting on the continuous and discrete state variables of the density function . in the case of the qd ck equation ( [ eq:13 ] ) , we have @xmath73 assume that @xmath72 has a complete set of eigenfunctions , @xmath74 , so that the solution can be written @xmath75 for some constants @xmath76 , and that all of the eigenvalues , @xmath77 , are nonnegative . assume further that if we impose a reflecting boundary condition on @xmath63 then the principal eigenvalue , @xmath78 , is the only zero eigenvalue and the eigenfunction , @xmath79 , is the stationary density of the process ( after appropriate normalization ) . furthermore , we assume that the stationary density is exponentially small on the boundary . then , if instead we place an absorbing boundary condition on @xmath63 , the stationary density no longer exists and the principal eigenvalue is exponentially small in @xmath40 , while the remaining eigenvalues are much larger so that the solution resembles the stationary density after some initial transients . it is this difference in time scales that we exploit to approximate the fetd . a universal feature of the fetd for rare events is its exponential form ( which follows from the separation of time scales ) because the time dependence is @xmath80 , for @xmath81 . indeed the fetd ( [ eq:24 ] ) is @xmath82 thus , for large times the fetd is completely characterized by the principal eigenvalue , @xmath83 . the mean exit time is simply @xmath84 , which means that the eigenvalue also has the physical interpretation of the rate at which metastable transitions occur . to obtain an approximation of this eigenvalue , we use a spectral projection method , which makes use of the adjoint operator @xmath85 . consider the adjoint eigenfunctions , @xmath86 , @xmath87 , satisfying @xmath88 and @xmath89 so that the two sets of eigenfunctions are biorthogonal . soppose that the boundary condition is ignored and @xmath79 is approximated by the stationary density , @xmath90 . by application of the divergence theorem , the adjoint operator is such that @xmath91 where the boundary contribution is nonzero because @xmath90 does not satisfy the absorbing boundary condition . then , since @xmath92 , the principal eigenvalue is @xmath93 in the remainder of this section , we use ( [ eq:30 ] ) to approximate the principle eigenvalue by approximating the stationary density , @xmath90 , and the adjoint eigenfunction , @xmath94 . in this section , we obtain an approximation to the quasi - stationary density , @xmath95 , using a wkb approximation method . we begin by illustrating the procedure for the qd process . that is , we seek an approximation of the solution to the equation @xmath96{\bar{\bm{\rho}}}(x , y ) = 0.\ ] ] consider the anzatz @xmath97},\ ] ] where @xmath98 are @xmath99-vectors and both @xmath100 and @xmath101 are scalar functions . note that in other studies of gene regulation models where similar methods are used , the small parameter in the exponential is @xmath35 . this difference in scaling arrises from the assumption that the metastable transitions are driven by fluctuation in the promotor and not the production of protein . substituting ( [ eq:32 ] ) into ( [ eq:31 ] ) and collecting leading order terms in @xmath40 yields @xmath102\mathbf{r}_{0 } = 0,\ ] ] where @xmath103 the prefactor term @xmath104 ( up to a normalization factor ) is simply the nullspace of the matrix @xmath105 $ ] , and we assume that it is normalized so that @xmath106 . using theorem 3.1 in ref . @xcite , we can provide necessary and sufficient conditions for @xmath104 to be unique and positive . for any fixed @xmath107 , there exists a unique vector @xmath108 , satisfying ( [ eq:33 ] ) if and only if the diagonal matrix @xmath109 is such that at least two of its elements have oposite sign . that is , there exist @xmath110 , with @xmath111 , such that @xmath112 . it is interesting to speculate that once the solution @xmath113 to ( [ eq:61 ] ) is substituted into the matrix @xmath1 that this requirement is satisfies for all @xmath114 . however , this is not necessarily the case , which means that the quasi - stationary density is restricted to a subdomain where @xmath115 . it is obvious that the protein levels must be bounded within the domain @xmath116 when protein production / degradation is deterministic . that is , if the gene remains in the unrepressed state , the protein level tends toward the mean value ( @xmath117 or @xmath118 ) but never exceeds it since protein levels do not fluctuate ( unless the promotor state fluctuates ) . however , it is not as obvious that the total amount of protein is further bounded so that @xmath119 , which means the domain is further restricted to the upper triangular portion of the unit square . this means that once a trajectory enters , it remains in this domain for all time and can not escape . to show this , we need simply look at the rate of protein production / degradation for each state normal to the line @xmath120 . this gives the rate for each of the promotor states ( @xmath55 on , both on , @xmath54 on ) when both protein levels satisfy @xmath120 . these rates are given by the diagonal components of the matrix @xmath121 , where @xmath122 and @xmath123 are defined in ( [ eq:14 ] ) . it is evident that when no repressor is bound and both proteins are produced , the flux across this line is in the positive direction , and when one repressor is bound , there is no flux across this line . the leading order result ( [ eq:33 ] ) defines a nonlinear partial differential equation for @xmath124 , @xmath125 = 0.\ ] ] the function @xmath126 is referred to as the hamiltonian for the system , due to the similarity to classical hamiltonian dynamics . an implicit assumption , @xmath127 is present to ensure that @xmath128 is the gradient of the scalar field @xmath124 . the above system can be solved by the method of characteristics to obtain the system of ordinary differential equations ( see @xcite pg . 360 ) , @xmath129 where each variable is parameterized by @xmath130 , which should not be confused with physical time . the above system of ordinary differential equations is supplemented with an equation for the stability lanscape @xmath131 and solutions specify @xmath124 along a curve in the plane @xmath132 . a family of curves is defined by specifying cauchy data @xmath133 along a curve parameterized by @xmath134 . one of the difficulties found in this method is determining cauchy data . at the stable fixed points , the value of each of the variables is known ( i.e. , @xmath135 and @xmath136 ) but data at a single point can not hope to generate a family of rays . therefore , data must be specified on an ellipse surrounding the fixed point , using the hessian matrix , @xmath137.\ ] ] expanding the function @xmath124 in a taylor series around the fixed point yields the quadratic form , @xmath138 as its leading order term . cauchy data is specified on the ellipse @xmath139 for some suitably small @xmath140 . in practice , @xmath141 must be small enough to generate accurate numerical results , but large enough so that trajectories can be generated to cover the domain . on the elliptical contour , the initial values for @xmath142 are @xmath143 it can be shown @xcite that the hessian matrix is the solution to the algebraic riccati equation , @xmath144 where @xmath145,\quad c = \left [ \begin{array}{c c } \frac{\partial^{2 } \mathcal{h}}{\partial p \partial x } & \frac{\partial^{2 } \mathcal{h}}{\partial p \partial y } \\ \frac{\partial^{2 } \mathcal{h}}{\partial q \partial x } & \frac{\partial^{2 } \mathcal{h}}{\partial q \partial y } \end{array}\right],\ ] ] evaluated at @xmath135 , @xmath146 , and @xmath147 . this equation can be transformed into a linear problem ( in order to actually solve it ) by making the substitution @xmath148 to get @xmath149 an equation for the scalar function @xmath101 is found by substituting ( [ eq:32 ] ) into ( [ eq:31 ] ) and keeping second order terms in @xmath40 , to get @xmath150 \mathbf{r}_{1 } = { \frac{\partial } { \partial x}}(f\mathbf{r}_{0 } ) + { \frac{\partial } { \partial y}}(g\mathbf{r}_{0 } ) - \left({\frac{\partial { k}}{\partial x}}f - { \frac{\partial { k}}{\partial y}}g\right)\mathbf{r}_{0}.\ ] ] for solutions @xmath151 to exist , the fredholm alternative theorem requires that for @xmath152 = 0\ ] ] it must be that @xmath153 = 0.\ ] ] note that if @xmath104 spans the right nullspace of @xmath154 , then the left nullspace is also one dimensional . after rewriting ( [ eq:49 ] ) , we have the pde for @xmath155 given by @xmath156 although the solution to this equation can be formulated by the method of characteristics , it requires values of the vectors @xmath104 and @xmath157 , which in turn require the solution to the ray equations ( [ eq:38 ] ) . since rays must be integrated numerically in most cases , solving ( [ eq:50 ] ) along its own characteristics is impractical . however , ( [ eq:50 ] ) can be computed along the characteristic curves of ( [ eq:38 ] ) as follows . first , differentiating ( [ eq:50 ] ) along characteristics yields @xmath158 using the fact that @xmath159 along characteristics , we can define @xmath160 then , after combining ( [ eq:50 ] ) and ( [ eq:51 ] ) we have that @xmath161.\ ] ] the above requires values of @xmath162 and @xmath162 , which are not provided by the system ( [ eq:38 ] ) . to obtain these , a formula is needed to relate the hessian matrix , @xmath163 , of @xmath100 to @xmath164 . then , the hessian matrix can be computed by expanding the system of ray equations ( [ eq:38 ] ) . first , differentiate both sides of equation ( [ eq:33 ] ) to get @xmath165\nabla \mathbf{r}_{0 } = -\left ( \nabla a + \nabla ( p f ) + \nabla ( q g ) \right ) \mathbf{r}_{0},\ ] ] the fredholm alternative theorem requires that @xmath166 which is always true since @xmath167 and @xmath168 the general solution to ( [ eq:54 ] ) is @xmath169 where @xmath170 is the pseudoinverse of the matrix @xmath1 and @xmath10 is an unknown constant . since the vector @xmath104 is normalized so that its entries sum to one , it follows that @xmath171 . summing over both sides of equation ( [ eq:114 ] ) then yields @xmath172 thus , we have that @xmath173 equation ( [ eq:125 ] ) gives a relationship between @xmath174 , @xmath104 , and @xmath164 . to obtain the hessian matrix , @xmath163 , away from the fixed point , the ray equations are extended to include the variables @xmath175 for @xmath176 . a good choice for the current problem is to take @xmath177 and @xmath178 , where @xmath179 is a point on the initial curve defined by ( [ eq:42 ] ) . the hessian matrix is then obtained using @xmath180 = z \left[\begin{array}{c c } x_{1 } & x_{2}\\ y_{1 } & y_{2 } \end{array}\right].\ ] ] as long as the matrix on the rhs is invertible , the matrix @xmath174 can be obtained along characteristics and @xmath155 can be integrated numerically using ( [ eq:53 ] ) and ( [ eq:54 ] ) . the dynamics for the extended variables ( [ eq:55 ] ) is given by @xmath181 where @xmath182 and @xmath183 is the jaciobian matrix for the system ( [ eq:38 ] ) . one can choose different variables @xmath184 with which to extend the system based on what works in practice , and the only thing one must change are the initial conditions . for our choice , the initial conditions are @xmath185 where the matrix @xmath186 is the solution to ( [ eq:44 ] ) . the above analysis can be repeated to obtain a hamiltonian system for the full process ( with protein noise ) , but a choice must be made for how the limit @xmath45 , @xmath46 is taken . first consider the equation for the quasi - stationary density , @xmath90 , for the full process ( [ eq:7 ] ) @xmath187{\bar{\bm{\rho}}}(x , y ) = 0.\ ] ] here , the domain is the cone @xmath188 . the quasi - stationary density is assumed to have the form @xmath189},\ ] ] where again @xmath190 is a @xmath99-vector and @xmath124 is a scalar function representing the stability landscape . note that we have ignored higher order terms here because we only want the hamiltonian function for comparison to the qd process . substituting ( [ eq:58 ] ) into ( [ eq:57 ] ) does not lead to any meaningful equation for @xmath124 at leading order unless we make an assumption about how the limit @xmath46 is taken . there are two relevant cases : @xmath191 and @xmath192 . in the former case , one recovers the qd process ( [ eq:33 ] ) , and in the latter case , collecting terms of leading order in @xmath40 , with @xmath193 , yields @xmath194\mathbf{r } = 0,\ ] ] where @xmath195 and @xmath196 thus , the hamiltonian for the full process is @xmath197,\ ] ] which we refer to as the full hamiltonian the differences between the full process and the qd process is nicely illustrated by comparing their associated hamiltonians . notice that the full hamiltonian ( [ eq:61 ] ) is a transcendental function of @xmath198 and @xmath199 , whereas the hamiltonian for the qd process ( [ eq:36 ] ) is a cubic polynomial in @xmath198 and @xmath199 . one can view this as a taylor series expansion of the full hamiltonian about @xmath200 . for this reason , the qd process , as an approximation for the full process with a small amount of protein noise , is only valid within a neighborhood of a deterministic fixed point . this is , of course , just a reflection of the fluctuation dissipation theorem @xcite . an example of numerical integration ( for details regarding numerics see the appendix ) of the ray equations ( [ eq:38 ] ) for the qd ( [ eq:36 ] ) and full ( [ eq:61 ] ) hamiltonian is shown in fig . [ fig : rays ] . . the triangular restricted domain boundary for the qd process is shown with black lines . the two symmetric stable fixed points are represented by green circles , and the unstable fixed point by a red circle . the separatrix along which an absorbing boundary condition is imposed is shown by a dashed black line . parameter values used are @xmath201 , @xmath202 for the qd hamiltonian , and @xmath203 for the full hamiltonian.,width=453 ] the qd rays are shown above the separatrix for comparison . notice that the qd rays are contained within a triangular domain , while the rays from the full hamiltonian cover the entire domain . this is due to the domain restriction that occurs when removing the protein fluctuations from the process . up to terms that are exponentially small in @xmath40 , the adjoint eigenfunction , @xmath94 , satisfies @xmath204{\bm{\xi}_{0 } } = 0.\ ] ] to make things easier , we change coordinates with @xmath205 so that @xmath206 this transforms the absorbing boundary , @xmath207 , to the vertical line , @xmath208 . then , ( [ eq:66 ] ) becomes @xmath209{\bm{\xi}_{0}}(\tau,\sigma ) = 0.\ ] ] where @xmath210 where @xmath211 , @xmath122 , and @xmath123 are defined in ( [ eq:14 ] ) and ( [ eq:15 ] ) . the absorbing boundary condition is then @xmath212 before proceeding , it is convenient to make the following definitions . in the rest of this section , we make frequent use of the eigenvectors ( right eigenvectors @xmath213 and left eigenvectors @xmath214 ) and eigenvalues , @xmath215 , satisfying @xmath216 we normalize the two sets of eigenvectors ( which are biorthogonal ) so that @xmath217 . note that because the matrices @xmath218 and @xmath219 are functions of @xmath220 , so are the eigenpairs . it is easily shown that one of the eigenvalues is zero for all values of @xmath220 , which we set to @xmath221 . the right eigenvector , @xmath222 , is then given by the nullspace of the matrix @xmath218 @xmath223 furthermore , the corresponding left eigenvector is simply @xmath224 it is convenient to define distinct notation for the eigenpairs evaluated on @xmath63 , with @xmath225 at the boundary one of the eigenvalues , @xmath226 say , vanishes and the eigenspace for the zero eigenvalue is degenerate ( i.e. there are two zero eigenvalues but the nullspace is one dimensional ) which means that @xmath227 , @xmath228 and @xmath229 . the approximation of the adjoint eigenfunction proceeds using singular perturbation methods , along the lines of @xcite . three solutions are found which are valid in different regions of the domain : an outer solution , a boundary layer solution for the @xmath230 strip near the absorbing boundary , and a transition layer solution in the @xmath231 overlap region between the other two . away from the boundary , the exact solution ( that does not satisfy the boundary condition ) is @xmath232 to obtain a uniform asymptotic approximation that also satisfies ( [ eq:71 ] ) , a boundary - layer solution is needed . consider the stretched variable @xmath233 . to leading order in @xmath40 , the boundary - layer solutions , @xmath234 , satisfies @xmath235 where @xmath236 . the solution has the form @xmath237 where @xmath238 , @xmath239 is the only eigenpair ( on the boundary ) with a nonzero eigenvalue . however , the eigenvalue , @xmath240 , is negative for all values of @xmath241 , and in order to obtain a bounded solution in the limit @xmath242 we set @xmath243 . the vector @xmath244 is the generalized left eigenvector satisfying @xmath245 and is given by @xmath246 at the boundary , the solution is @xmath247 and the boundary condition ( [ eq:71 ] ) requires @xmath248 so that @xmath249 . thus , up to a single unknown constant , @xmath250 , which must be determined by matching , the boundary - layer solution is @xmath251 because @xmath234 is unbounded in the limit @xmath252 , it is not possible to match it to the outer solution , @xmath253 . we can think of the term , @xmath254 , in the boundary - layer solution is a truncated taylor series expansion of the true solution around @xmath255 . to match the boundary - layer and outer solutions , a transition - layer solution is required for the strip of width @xmath256 along the boundary . consider the stretched coordinate @xmath257 . keeping terms to @xmath231 , the transition - layer solution , @xmath258 , satisfies @xmath259 it is less clear how to truncate the above equation to obtain a leading order transition - layer solution . because we must match the outer solution , @xmath260 , to the boundary layer solution that has the generalized eigenvector @xmath244 , we try a solution of the form @xmath261 where @xmath262 and @xmath263 are unknown scalar functions . in the limit @xmath264 , the deterministic flux across the boundary , @xmath265 ( with @xmath266 given by ( [ eq:12 ] ) ) vanishes and the eigenvector @xmath267 . that is , the eigenvalue @xmath226 , corresponding to the eigenvector @xmath268 , vanishes on the boundary . furthermore , it can be shown that @xmath269 where @xmath270 substituting ( [ eq:84 ] ) into ( [ eq:83 ] ) yields @xmath271 to obtain the unknown functions @xmath272 we project ( [ eq:87 ] ) with the right eigenvectors , @xmath273 , @xmath274 . after applying these projections ( using the fact that @xmath275 , @xmath276 , @xmath277 , and @xmath278 ) and collecting leading order terms in @xmath40 , we get @xmath279 where @xmath280 it turns out that @xmath281 is related to the curvature of the stability landscape normal to the separatrix ; that is , if we define @xmath282 then @xmath283 at @xmath284 , the curvature vanishes and changes sign but is always negative ( with @xmath285 ) at the unstable fixed point , @xmath286 . divide the separatrix ( @xmath208 and @xmath287 ) into three regions : @xmath288 , @xmath289 , and @xmath290 . the first region is ignored because it is in part of the domain @xmath291 excluded from the stationary density function ( see sec . [ sec : qsd ] ) . the second region contains the unstable fixed point , and the third we can ignore as only extremely rare trajectories cross the separatrix in this region . up to an unknown constant , the solutions to ( [ eq:88 ] ) and ( [ eq:89 ] ) are @xmath292}du,\\ a_{1}(\tau , s ) & \sim & \hat{a}{\exp\left [ -\frac{1}{2}\tilde{\mu}_{1}^{(\sigma)}(\tau ) s^{2 } \right]}.\end{aligned}\ ] ] the transition layer solution is then @xmath293}du{\mathbf{1}}\right . \\ \nonumber & & \qquad\left . + { \exp\left [ -\frac{1}{2}\tilde{\mu}_{1}^{(\sigma)}(\tau ) s^{2 } \right]}\bm{\chi}(\tau , s)\right).\end{aligned}\ ] ] the three solutions can now be matched . first , matching the transition layer solution to the boundary layer solution is done using the van - dyke rule . in terms of the boundary layer variable , @xmath294 , the transition layer solution is @xmath295 matching terms with the boundary layer solution yields @xmath296 the composite boundary / transition layer solution is then @xmath297}du{\mathbf{1}}\right.\right.\\ \nonumber & & \qquad\qquad \left.\left.+{\exp\left [ -\frac{1}{2}\tilde{\mu}_{1}^{(\sigma)}(\tau ) \frac{\sigma^{2}}{\epsilon } \right]}\frac{\tilde{{\nu}}^{(\sigma)}(\tau)}{\tilde{\mu}_{1}^{(\sigma)}(\tau ) } \bm{\chi}(\tau,\frac{\sigma}{\sqrt{\epsilon}})\right)\right]\end{aligned}\ ] ] the final unknown constant , @xmath250 , is determined by matching to the outer solution so that @xmath298 which implies that @xmath299 in order to evaluate the term in the numerator of the eigenvalue formula ( [ eq:30 ] ) , we require the adjoint eigenfunction evaluated on the boundary ( in a neighborhood of the unstable fixed point ) which is @xmath300 we now have all of the components necessary to approximate the principal eigenvalue , using the formula ( [ eq:30 ] ) . first , for the term in the denominator , we can approximate the adjoint eigenfunction with the outer solution @xmath301 and the ( unnormalized ) stationary density with ( [ eq:32 ] ) ( the higher order term @xmath151 can be ignored ) . then the term in the denominator is simply the normalization factor , which can be approximated using laplace s method to get @xmath302 } da \sim \frac{2\pi\epsilon}{\sqrt{\mbox{det}(z)}},\ ] ] where @xmath174 is the hessian matrix ( [ eq:40 ] ) of @xmath124 at the stable fixed point @xmath303 . note that we have used the fact that @xmath304 and that the vector @xmath104 is normalized so that its entries sum to one . the term in the numerator of ( [ eq:32 ] ) requires the approximation ( [ eq:99 ] ) of the adjoint eigenfunction on the absorbing boundary . the integral can also be approximated using laplace s method , with @xmath305}d\tau \\ \nonumber & \sim & \frac{\epsilon b\sqrt{\pi}e^{-{k}(x_{u},y_{u})}}{b\sqrt{\pi } - \epsilon \sqrt{2\tilde{\mu}_{1}^{(\sigma)}(\tau_{u } ) } } \sqrt{\frac{2\tilde{\mu}_{1}^{(\sigma)}(\tau_{u})}{\tilde{\phi}''(\tau_{u})}}\bm{\zeta}^{t}\hat{g}(0)\tilde{\mathbf{r}}_{0}(\tau){\exp\left [ -\frac{1}{\epsilon}\phi(x_{u},y_{u } ) \right]}.\end{aligned}\ ] ] for convenience , we have defined functions on the boundary in the variable @xmath306 with @xmath307 where @xmath308 and @xmath309 are defined in ( [ eq:67 ] ) . although the quantities @xmath310 and @xmath311 must be computed numerically , the remaining unknown terms can be computed analytically by exploiting the reflection symmetry of the problem . along @xmath63 , we have that @xmath312 and @xmath313 so that the equation for @xmath124 and @xmath104 ( [ eq:33 ] ) can be written as @xmath314\tilde{\mathbf{r}}_{0}(\tau ) = 0,\ ] ] where we have defined @xmath315 the stability landscape function on the boundary is then @xmath316 the above is just an eigenvalue problem with three possible solutions , one of which can be excluded because there is a zero eigenvalue corresponding to the nullspace of @xmath218 . it can be shown @xcite that if the diagonal elements of @xmath317 are such that at least two have oposite sign then only one of the remaining two eigenvalues has a corresponding positive eigenvector ( @xmath318 must have positive elements ) making the solution to ( [ eq:103 ] ) unique . it turns out that this is only true for @xmath319 , which is due to the domain restriction caused by removing protein fluctuations . the result is @xmath320 we also have that @xmath321 and finally , using ( [ eq:79 ] ) , ( [ eq:70 ] ) , and ( [ eq:106 ] ) we get @xmath322 combining these components together , we have the final result that @xmath323 \sqrt{\frac{\tilde{\mu}_{1}^{(\sigma)}(\tau_{u})\mbox{det}(z)}{\tilde{\phi}''(\tau_{u } ) } } { \exp\left [ -\frac{1}{\epsilon}\phi(x_{u},y_{u } ) \right]},\ ] ] where @xmath324 , and @xmath325 is given by ( [ eq:90 ] ) . as expected , the eigenvalue is exponentially small in @xmath40 , which means that the height of the stability lanscape at the unstable fixed point , @xmath310 , must be approximated as accurately as possible . the remaining terms are often referred to as the ` prefactor ' , and except for the quantity @xmath311 , all of the terms in the prefactor ( the derivatives of the stability landscape function and ( [ eq:108 ] ) ) represent properties local to the fixed points . the remaining term in the prefactor depends on the function @xmath101 , which depends on properties of the process not local to the fixed points and must be computed numerically . in this section , the results gathered throughout this paper are used to explore how removing the intrinsic noise that arises from protein production / degradation effects the random process . in particular , we examine the stability landscape and the metastable transition times . first , in fig . [ fig : level_curves ] the numerical solutions to the ray equations ( [ eq:38 ] ) are used to generate level curves of the stability landscape function , @xmath124 , for both the qd ( [ eq:36 ] ) and the full ( [ eq:61 ] ) hamiltonians . . black lines show the domain boundary for the qd process , and the dashed line shows the boundary of positive protein levels . ( a ) the qd result is compared to the full process with @xmath326 so that the protein noise is weak compared to promotor noise . ( b ) same as ( a ) but with @xmath327 so that the protein noise strength is comparable to promotor noise . parameter values are the same as fig . [ fig : rays].,width=453 ] for presentation , the level curves are shown in the @xmath328 plane , with the separatrix along the left edge ( @xmath207 ) of each frame . in fig . [ fig : level_curves]a , level curves for the full process are shown for @xmath326 so that the protein noise is small compared to promotor noise . recall that the parameter @xmath329 controls the strength of protein noise relative to the strength of promotor noise so that there is no protein noise in the limit @xmath330 . the resulting curves match closely in a neighborhood of the stable fixed point and extend out toward the unstable saddle point . as expected , the level curves begin to diverge the farther away from either fixed point they are . in fig . [ fig : level_curves]b , the strength of the protein noise is increased , with @xmath331 , and in this case , the stability landscape of the two processes are quite different , matching only near the fixed points of the deterministic dynamics . as a result of the domain restriction effect , level curves of the full process extend into regions the qd process is excluded from . this is because protein levels can only cross above the line @xmath120 , not below it , when there is no protein noise . this is also true of the lines @xmath332 and @xmath333 . the domain restriction effect is eliminated when protein noise is added back into the process , even if it is very small compared to promotor noise . for the model considered here , the effect is of no serious consequence for metastable transitions as the restricted domain still contains all three fixed points . however , a model of a more complex gene circuit might be significantly affected by removing protein noise especially if this restricts the domain for the protein levels in such a way as to generate qualitatively different behavior , which would imply a nontrivial contribution of protein noise , no matter how negligible it may be . because of the symmetry in the problem , we obtained analytical results for various quantities on the separatrix , including the shape stability landscape . we can use these results to obtain an analytical approximation of the probability density for the position along the separatrix a trajectory passes through as it transitions from one basin of attraction to another . using the results of sec . [ sec : eval ] , the stationary density along the separatrix is given by @xmath334}}{\int_{0}^{1}e^{-\tilde{{k}}(\tau)}{\exp\left [ -\frac{1}{\epsilon}\tilde{\phi}(\tau ) \right]}d\tau},\ ] ] where we remind the reader that @xmath335 and @xmath312 along the separatrix . the only term that can not be obtained analytically is the function @xmath336 , which can be ignored as a first approximation . for simplicity , we also average over the promotor state to get the scalar marginal probability density for the exit point . then , using laplace s method , the exit density is @xmath337},\quad \tau\in(0,1),\ ] ] where @xmath338 is given by @xmath339 the qd exit density approximation is shown in fig . [ fig : exitdist ] along with two histograms obtained from monte - carlo simulations . , with @xmath312 . the black curve shows the analytical approximation for the qd ( @xmath340 ) process , and the symbols show histograms from @xmath341 monte - carlo simulations for different values of @xmath329.,width=377 ] while the histogram for @xmath326 is close to the qd approximation , it is evident that some trajectories pass through the separatrix in the interval @xmath342 , which is impossible without protein noise . this effect becomes negligible when the protein noise is reduced to @xmath343 . the fetd for a trajectory , starting from a stable fixed point , to reach the separatrix is asymptotically exponential in the large time limit , and the timescale is determined by the principal eigenvalue , @xmath83 , from equation ( [ eq:109 ] ) . we can then approximate the mean exit time with @xmath344 . in fig . [ fig : mfpt ] the mean exit time is shown on a log scale as a function of @xmath345 along with results from monte - carlo simulations . . symbols represent @xmath346 averaged monte carlo simulations for various values of @xmath329 . the dashed black line is the approximation , @xmath347 , for the qd process ( @xmath340).,width=453 ] notice that the approximation and the monte - carlo simulations are asymptotically linear as @xmath348 . for the approximation , the slope of this line is determined by the height of the stability barrier , @xmath310 , while the prefactor affects the vertical shift . from the monte carlo results ( symbols with grey lines ) we see that the mean exit time converges to the approximation as @xmath349 . however , it is clear that the slope of the analytical curve is slightly different then that of the monte carlo results even when @xmath329 is small . thus , we may think of the mean exit time approximation for the qd process as an asymptotic approximation of the full process in terms of the small parameter @xmath350 so long as @xmath40 is also small but not too small . understanding how different noise sources affect the dynamics of a gene circuit is essential to understand how different regulatory components interact to produce the complex variety of environmental responses and behaviors . even if one excludes extrinsic noise sources such as environmental and organism - to - organism variations there are several sources of intrinsic noise , such as fluctuations in translation , transcription , and the conformational state of dna regulatory units . the behavior we are interested in understanding is a transition from one metastable state to another . each metastable state corresponds to the stable steady - state solutions of the underlying deterministic system . the bistable mutual repressor model has two identical stable steady states separated by an unstable saddle node . on small time scales , the protein levels fluctuate near one of the two stable steady states . on large time scales , fluctuations cause a metastable transition to occur , where the protein levels shift to the other steady state by crossing the separatrix containing the unstable saddle point . to understand how metastable transitions can be induced by promotor noise , we consider a discrete stochastic model of a mutual repressor circuit where the protein levels change deterministically , which we call the qd process . this is done by fixing the promotor state and then taking the thermodynamic , or large system size , limit of the protein production / degradation reactions . we then compare the qd process to the full process that includes protein noise . we find important qualitative differences that persist even when the magnitude of the protein noise is small compared to promotor noise . in particular , without protein noise and after initial transients , the protein levels are restricted such that the total amount of protein is never less than half its maximum possible value ; that is , the total number of proteins is such that @xmath351 , where @xmath10 is the protein production rate and @xmath13 is the degradation rate . said another way , assuming that the random process starts at one of the deterministic stable fixed points , promotor fluctuations could never push the protein copy numbers so that , for example , only a single copy of each protein is present . in contrast , the stability landscape for the full process and monte - carlo simulations show that protein levels are able to reach all positive values . while this restriction does not stop the qd process from exhibiting metastable transitions , more complex gene circuits may require protein fluctuations , even if they are very small , in order to function correctly . in this appendix , we summarize the numerical methods and tools used throughout the paper . most numerical work is performed in python , using the numpy / scipy package . for more computationally - expensive tasks , we use scipy s weave package to include functions written in c , which allows us to use the gnu scientific library for numerical integration of the ray equations ( [ eq:38 ] ) , and for random number generators used in monte carlo simulations . there are a few notable observations regarding integration of the ray equations . first , characteristic projections , @xmath132 , have a tendency to stick together along certain trajectories , peeling off one at a time ( see fig . [ fig : rays ] ) . to adequately cover the domain with rays , a shooting method must be used to select points on the cauchy data . for more details on this see ref . we found that the simplest method was to use the secant method ( we use the `` brentq '' function in the scipy.integrate package ) to minimize the euclidian distance between the final value of @xmath132 along the separatrix and the saddle node . this method is convenient since it does not require knowledge of the hessian matrix , @xmath163 . second , the value of @xmath141 used to generate cauchy data must be chosen small enough to get accurate results . however , we found that if it is chosen too small , rays are no longer able to cover the domain , and more importantly , we could no longer generate a ray that reaches the unstable fixed point on the separatrix . for the mutual repressor model , the trajectory connecting the stable fixed point to the saddle is one of the curves along which characteristics tend to stick to each other . suppose that @xmath352 is the point on the cauchy data ( [ eq:126 ] ) that generates the ray that connects the fixed points . then small perturbations @xmath353 , @xmath354 , cause the characteristic , @xmath132 , to diverge sharply away from the saddle . this not only makes it difficult to compute @xmath352 , but also creates difficulties for computing the function @xmath355 ( see sec . [ sec : prefac ] ) . since the expanded set of ray equations ( [ eq:110 ] ) track the derivatives of @xmath55 , @xmath54 , @xmath198 , and @xmath199 with respect to the point on the cauchy data , which , for values of @xmath134 near @xmath352 , becomes very large as the ray approaches the saddle point . as the expanded variables become very large , computing @xmath174 using equation ( [ eq:56 ] ) is unstable . furthermore , this effect becomes worse as the initial value , @xmath356 , goes to zero . jmn would like to thank james p. keener for valuable discussions throughout this project . this work was supported by award no kuk - c1 - 013 - 4 made by king abdullah university of science and technology ( kaust ) . 10 charles doering , khachik sargsyan , and leonard sander . extinction times for birth - death processes : exact results , continuum asymptotics , and the failure of the fokker planck approximation . , 3(2):283299 , 2005 .

the stochastic mutual repressor model is analysed using perturbation methods . this simple model of a gene circuit consists of two genes and three promotor states . either of the two protein products can dimerize , forming a repressor molecule that binds to the promotor of the other gene . when the repressor is bound to a promotor , the corresponding gene is not transcribed and no protein is produced . either one of the promotors can be repressed at any given time or both can be unrepressed , leaving three possible promotor states . this model is analysed in its bistable regime in which the deterministic limit exhibits two stable fixed points and an unstable saddle , and the case of small noise is considered . on small time scales , the stochastic process fluctuates near one of the stable fixed points , and on large time scales , a metastable transition can occur , where fluctuations drive the system past the unstable saddle to the other stable fixed point . to explore how different intrinsic noise sources affect these transitions , fluctuations in protein production and degradation are eliminated , leaving fluctuations in the promotor state as the only source of noise in the system . perturbation methods are then used to compute the stability landscape and the distribution of transition times , or first exit time density . to understand how protein noise affects the system , small magnitude fluctuations are added back into the process , and the stability landscape is compared to that of the process without protein noise . it is found that significant differences in the random process emerge in the presence of protein noise .

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Proceed to summarize the following text: solar flares and coronal mass ejections ( cmes ) involve a sudden release of magnetic energy stored in complex active regions through magnetic reconnection @xcite . the several mechanisms may compel the energy build - up in the flaring and eruptive regions that later released in form of bulk mass motion , heating , as well as acceleration of the energetic particles . these processes may involve magnetic instabilities , flux and helicity emergence , building of the magnetic field complexity , etc . * ; * ? ? ? * ; * ? ? ? * and references cited therein ) . the two main space weather consequences of cmes are solar energetic particle ( sep ) events and geomagnetic storms ( gmss ) . both flares and cmes may contribute to sep events . according to @xcite , there are two types of sep events , impulsive and gradual . impulsive sep events , as characterized by enhanced @xmath4he , heavy ions and electrons and high charge states , are attributed to impulsive solar flares . the solar sources of impulsive sep events are identified with coronal jets and narrow cmes @xcite . gradual sep events , which are far more important in the context of space weather , are due to acceleration of particles at cme - driven shocks @xcite . fast moving shocks associated with cmes accelerate seps in the interplanetary medium near the sun @xcite and at 1 au in energetic storm particle ( esp ) events @xcite . cme interaction has been reported to increase the sep acceleration efficiency @xcite . the interaction between cmes results in a new cme with the magnetic field and plasma from both the cmes . the process is described as `` cme cannibalism '' @xcite . recently , @xcite have reported on the implications of cme - cme interaction for shock propagation , particle acceleration and space weather forecasting . such interactions have been modeled by three - dimensional mhd simulation @xcite . some authors have made statistical argument questioning the importance of cme interaction for sep production @xcite . however , if we consider cmes from the same active region , the physical interaction is inevitable , so there must be some influence of the cme interaction on the sep intensity profile @xcite . intense geomagnetic storms occur when the southward field in the interplanetary cmes ( icmes ) reconnect with the earth s magnetic field @xcite . intense storms can also occur when the sheath region between the icme and the shock has southward component of the magnetic field . it has been found that nearly 90% of intense storms ( dst @xmath5 -100 nt ) result from earth - directed halo cmes @xcite . in this paper we study multiple flares and cmes from the same active region on january 23 , 2012 that resulted in one of the largest sep events of solar cycle 24 . the sequence of eruptions also resulted in a moderate gms . we study the interacting cmes for the first time using multiple view points and extended field of view ( fov ) available from the solar terrestrial relations observatory ( stereo ) and solar and heliospheric observatory ( soho ) . in sec . 2 , we describe the data set used . we report the observational results in sec . 3 . in the last section ( sec . 4 ) , we summarize the results . we use imaging observations from the stereo secchi suites @xcite , including coronagraphs cor1 and cor2 , and the extreme ultraviolet imager ( euvi , @xcite ) , the large angle and spectrometric coronagraph ( lasco , @xcite ) on board soho , the atmospheric imaging assembly ( aia , @xcite ) and the helioseismic magnetic imager ( hmi , @xcite ) on board the solar dynamics observatory ( sdo , @xcite ) . radio spectral observations from wind / waves @xcite are used to study interplanetary radio bursts . we use h@xmath6 images from aryabhatta research institute of observational sciences ( aries ) @xcite and global oscillation network group ( gong ) for chromospheric flare information . for the radio emission analysis , the nobeyama radioheliograph ( norh ) images at 17 and 34 ghz have been used @xcite . radio spectral data from the hiraiso radio spectrograph ( hiras ) are used for metric type ii radio burst information . interplanetary scintillation ( ips ) observations from the ooty radio telescope ( ort ) , rac , ncra - tifr has been used to study the interplanetary ( ip ) propagation of cmes @xcite . we use ace and wind spacecraft data available at omni web ( http://omniweb.gsfc.nasa.gov ) for solar wind ( in - situ ) plasma ( temperature , density , speeds , plasma beta ) and magnetic field information . information on cmes observed by lasco is basically taken from the online cme catalog ( http://cdaw.gsfc.nasa.gov , @xcite ) . soft x - ray flare and sep information is obtained from geostationary operational environmental satellite ( goes ) . the dst ( in nt ) index has been obtained from the world data center for geomagnetism , kyoto university , japan ( e.g. , http://wdc.kugi.kyoto-u.ac.jp/dstdir/ ) . on january 23 , 2012 three solar flares ( c2.5 , m1.1 and m8.7 ) occurred , and the last two were associated with two major cmes . the time line of the whole event is presented in table 1 . the upper panel of fig . 1 shows the white light image and magnetogram of active region ar 11402 , which is the source region of the eruptions . the active region was located at @xmath0 n28 w36 , and had a simple @xmath7 magnetic field configuration . the bottom panel of fig . 1 shows the goes soft x - ray plot in the @xmath8 and @xmath9 wavelength bands during 00:00 ut to 09:30 ut . the soft x - ray flux shows three flares ( indicated by the arrows ) : first flare ( c2.5 ) peaked around 01:44 ut , the second flare ( m1.1 ) around 03:13 ut and the third one ( m8.7 ) around 03:59 ut . since all three flares occurred in quick succession , it was not possible to determine the exact onset time of the m1.1 and m8.7 flares . for the onset of the m1.1 flare we have used the time of the dip between the c2.5 and m1.1 flares . for the onset time of the m8.7 flare we have used the time when the d@xmath10/d@xmath11 turns positive ( i is the soft x - ray intensity of the flare ) . the m1.1 and m8.7 are each associated with a cme . the second cme is associated with large filament eruption . in the upcoming subsections we discuss the details on the flares , associated cmes and their interplanetary consequences . 2 shows images during the m1.1 flare in sdo / aia 304 , 171 and 94 wavelengths . the 304 channel gives information about the upper transition region , while 171 and 94 provides information about the coronal region and the flaring region , respectively . the charterstic log temperature ( in mk ) for 304 , 171 and 94 lines are around 4.7 , 5.8 and 6.8 . the three columns show the images during the rise , maximum and the decay phase respectively . it is clear from the observations that the flare starts with two bright sources in 304 wavelength ; one lies near the sunspot and other away from the sunspot . overplotted magnetogram ( fig . 2e ) shows that the kernels lie on the opposite magnetic polarity regions . we have overplotted the microwave sources ( i.e. , 17 and 34 ghz ) over the sdo / aia 304 images ( see figs . 2a , 2b and 2c ) . these sources lie over the bright kernels of the flare , which are part of the flare ribbons . post flare loops connecting the flare ribbons have been observed in aia 171 and 94 images around 03:29 ut ( figs . 2f and 2i ) . the m1.1 flare was associated with the first cme . the cme has an extended hot core ( @xmath126.3 mk ) which will be discussed later . the long duration m8.7 flare starts with the eruption of the northern part of the active region filament . 3(a - c ) show the eruption of the filament in aia 304 wavelength . 3(d - f ) show the same filament in stereo - a euvi 304 wavelength . we show in fig . 3 g a height - time measurement of the filament on aia 304 images . the filament shows slow rise with an average speed of @xmath06 km s@xmath1 between 03:00 - 03:31 ut followed by a fast rise with an asymptotic speed of @xmath0590 km s@xmath1 during 03:40 - 03:46 ut . the average speed calculated from images in fig . 3(d - f ) is @xmath13280 km s@xmath1 , consistent with speed derived form sdo / aia observations of the filament corrected for projection effects during 03:26 ut to 03:46 ut . g clearly shows that the rise of the filament triggers the soft x - ray flare . 4 shows the evolution of the m8.7 flare in the sdo / aia 304 , 171 and 94 wavelengths . the first , second and third columns of this figure show the images from the rise , the maximum and the decay phases of flare respectively . the flare starts with two bright kernels . we have overplotted the 17 ghz and 34 ghz microwave flare sources over the aia 304 flare arcade during the rise , maximum and decay phases ( figs . 4a , 4b and 4c ) . the microwave emission is from a subset of the flare loops observed in aia 304 . the hmi magnetogram contours plotted over the sdo / aia 171 image ( fig . 4d ) , show that the flare ribbons are located at the opposite magnetic polarity regions of the active region . the flare ribbons and loops can also be seen in gong and aries h@xmath6 images ( fig . 5 ) . in summary , both the m1.1 and m8.7 flares show typical flare arcades . the filament eruption is clear in the m8.7 flare , whereas in the m1.1 flare there was a hot ejecta which became the cme core in coronagraphic images ( see details below ) . the m1.1 and m8.7 flares were accompanied by cmes observed by soho and stereo coronagraphs . stereo b and a were located at -114 degrees and + 108 degrees , respectively relative to earth . stereo - a cor1 and cor2 observed the two cmes above the north - east limb . we used stereo a data for height - time measurements , since the cmes were limb events in stereo - a view ( less projection effects ) . 6(a - c ) show the two cmes ( cme1 and cme2 ) from stereo - a cor1 and euvi 195 at 03:05 ut , 03:45 ut and 03:50 ut . we were able to image the cmes in their early phase because of the extended fov of stereo cor1 ( 1.4 - 4 r@xmath14 ) . the height time plots of the cmes obtained using stereo - a cor1 and cor2 data give speeds of cme1 and cme2 as @xmath01400 km s@xmath1 and @xmath02000 km s@xmath1 , respectively ( fig . the cmes occurred in quick succession : cme2 entered into the aftermath of cme1 at the first appearance of cme2 ( fig . the cme interaction took place due to the difference in speed , their origin from the same active region and their propagation in roughly the same direction . the interaction completed at a height of 11 - 12 @xmath2 at @xmath13 04:48 ut ( fig . 6 g ) . a sequence of lasco c2 and c3 difference images in fig . 7 , shows the further evolution of the two cmes . cme1 appeared in the lasco c2 fov at @xmath1303:12 ut at a position angle of 329@xmath15 with a width of 221@xmath15 . the cme1 had a linear speed of 685 km s@xmath1 , which is much smaller than the speed estimated from the stereo measurements at lower heights . this is likely to be due to projection effects because the solar source at n28 w36 in earth view , whereas the source was at the limb in stereo - a view . cme1 had an extended core ( see fig . 7b ) . in cor1 fov the core was also seen as a bright feature , which can be traced in euv images as a hot ejecta . this is somewhat unusual because normally the cme core is a cooler filament . 8a shows the hot ejecta ( dotted line ) in the aia 94 image at 02:22 ut . the hot ejecta was surrounded by cme1 ( solid line ) as can be seen in the aia 193 difference image . this can also be seen in the stereo - a euvi 195 image at 02:35:30 ut ( fig . cme2 appeared at a height of 2.01 @xmath2 in the cor1 fov at around 03:40 ut . at this time , the leading edge of the cme1 was at 5 @xmath2 . when cme2 appeared in the lasco / c2 fov at 04:00 ut its leading edge had already reached the core of cme1 ( core1 ) ( see fig . the linear speed of cme2 was @xmath02175 km s@xmath1 in the lasco fov , which is comparable to the stereo - a speed ( the speed difference is within measurement error ) . the lasco cme also seems to be closer to the sky plane . 7(c - f ) shows that the two cmes merged at 04:12 ut and thereafter moved out as a single compound cme . this can also be seen in the three frames of stereo - a cor2 shown in fig . 6(d - f ) . even though a metric type ii bursts was not reported in the solar geophysical data our examination of the dynamic spectrum from the hiras radio spectrograph , shows a type ii burst feature between 03:40 and 03:50 ut . given the high speed of cme2 it is unusual that the type ii feature is very weak . this may be due to the fact that cme2 is running into the moving material of cme1 making it difficult to form a strong shock . furthermore , the starting frequency of the type ii burst was at 200 mhz . assuming that the radio emission is at harmonic of the local plasma frequency , we see that the 200 mhz plasma level is at an unusually large height of 2.0 @xmath2 . this may be due to the fact that the shock is formed in the body of cme1 . the interplanetary counterpart of the type ii burst can be seen in the wind / waves dynamic spectrum shown in fig . . an intense type iii burst , which is due to electrons accelerated at the flare site , can also be seen . the dh type ii burst stats around @xmath1304:00 ut , which is roughly the time the leading edge of cme2 reaches the core of cme1 ( fig . 6 ( d - f ) and fig . 7(b ) ) . at this time the shock seems to have become very strong due to the decline of the alfven speed upstream of the shock . the type ii is observed as a broad band feature at lower frequencies ( fig . 9(a - b ) ) . the same shock accelerated protons as indicated by the sep onset at @xmath1304:20 ut , which is about 40 minutes after the metric type ii start ( see fig . however , if we consider the fact that @xmath12100 mev protons take about 15 minutes to reach earth , the sep release at the sun is close to the onset of the dh type ii bursts . as noted before cme2 has already interacting with the hot core of cme1 . there is also a second increase in the sep intensity around 05:30 ut , which is right after the formation of compound cme . it is possible that some particles were trapped in the interaction region and released after the interaction ends . in the @xmath1610 mev channel , the sep intensity reaches a plateau of about @xmath03000 pfu by 12:00 ut on january 23 , 2012 until the shock arrival at 1 au ( on january 24 , 2012 at 14:33 ut ) and causing an esp event ( see fig . the esp event attained a peak proton flux of @xmath06263 pfu at 15:30 ut on january 24 , 2012 . the sep event was the largest in solar cycle 24 @xcite as of this writing . the formation of shock in the near - sun region ( as shown by the type ii radio emission ) and shock signatures observed at 1 au imply that the compound cme was able to drive a shock and accelerate particles in the entire sun - earth distance . from the onset of cme2 in cor1 fov ( i.e. , 03:40 ut on january 23 , 2012 ) to the shock arrival at 1 au ( @xmath1314:33 ut on january 24 , 2012 ) , we see that the shock transit time is @xmath035 hour . 10 shows interplanetary scintillation ( ips ) data of the cme event on january 23 , 2012 observed by the ort . the ips technique provides information on the turbulent plasma at the front of the moving cme . the ips observations describe the evolution of the compound cme beyond 50 r@xmath14 . in the ips fov , between 100 and 225 @xmath2 , the cme decelerates from @xmath0900 to @xmath0700 km s@xmath1 . the cme deceleration is mostly due to the interaction with the background solar wind flow . 11 shows the variation of interplanetary field parameters ( total magnetic field ( b ) , z component of magnetic field ( bz ) , electric field ( ey ) ) , plasma parameters ( solar wind speed ( v ) , proton density ( n ) , proton temperature ( t ) , plasma beta ( @xmath7 ) ) and dst index . the shock arrival is at @xmath014:33 ut on january 24 , 2012 , which is followed by an extended sheath and a narrow interval of icme material . the shock speed ( @xmath13700 km s@xmath1 ) is consistent with the speed derived from ips observations . the plasma beta and magnetic field plots do not show the classical signatures of a magnetic cloud . the narrow icme suggests that its impact on earth is at an angle rather than direct . the dst index shows that the icme resulted in a moderate gms ( dst @xmath13 -73 nt ) . a peak negative dst of -73 nt was reached at @xmath011:00 ut on january 25 , 2012 which corresponds to the main phase of the moderate gms . actually there are two dips in dst indicating two different interactions with the earth magnetic field . first is due to the interaction of the sheath region with negative bz with earth s magnetic field ; the second one is due to the negative bz within the icme . in this paper we presented the case study of the january 23 , 2012 solar eruptions from noaa ar 11402 ( n28 w36 ) that resulted in a large sep event and a moderate gms . the eruptions consisted of two fast interacting cmes associated with m1.1 and m8.7 flares . we have shown that the interaction of the fast primary cme with the preceding cme resulted in the huge sep event . the results presented in this study are consistent with the earlier suggestions that the interaction between cmes has important implications for large sep events @xcite . the january 23 , 2012 events were observed by the stereo mission taking advantage of its extended fov much closer to the sun , capturing cme interaction in more detail . it is significant that the metric type ii burst was very weak but the interplanetary type ii burst was very intense . this may be attributed to different combinations of cme speed , alfven speed and the physical conditions of the ambient medium . the temporal coincidence of the cme interaction with the onset of the intense interplanetary type ii burst and the large sep event is significant and suggests the plausible increase in the efficiency of particle acceleration . one of the interesting features of these eruptions is the fact that the core of cme1 was unusually hot because it was observed only in the hot plasma images ( 94 and 193 ) . the temperature exceeded 6 mk which is a few times the average temperature of the corona . the sep event started when the shock of cme2 propagated through the hot ejecta . the particles in the hot plasma were already energized by virtue of the high temperature , which makes it easy for the shock to accelerate these particles . the cmes presented in this paper resulted in a huge sep event but not a strong gms . one of the possible causes for the moderate gms may be that only a small section ( about 6 hours ) of the icme interacted with earth . following the cme motion using movies of stereo cor2 images confirms that most of the compound cme propagated above the ecliptic . this is also seen in the stereo and lasco images in figs . 6 and 7 . in this paper we have described multi - wavelength observations of flares and cmes that are important in understanding how solar eruptions cause space weather . deeper insights into the link between the solar origin of cmes and their interplanetary and geo - space consequences can be obtained by a detailed modeling of cme initiation and propagation through extreme ambient conditions . + the key results of this study can be summarized as follows : + 1 . this study demonstrated that interaction between fast cmes from the same active region can occur very close to the sun as revealed by stereo observations . the gms was of double - dip nature because the shock sheath and the icme contained negative bz values and caused the two dips in the dst index . + * acknowledgments * we thank the anonymoud referees for their valuable comments and suggestions . we thank iusstf / jc - solar eruptive phenomena/99 - 2010/2011 - 2012 project on `` multiwavelength study of solar eruptive phenomena and their interplanetary responses '' for its support to this study during our bilateral collaboration . we acknowledge the wind / waves and sdo s aia and hmi teams for providing their data . soho is a project of international cooperation between esa and nasa . the stereo science center made available the data used in this work . n.c.j thanks aryabhatta research institute of observational sciences ( aries ) , nainital for providing post doctoral grant . this work was partly supported by nasa lws program . rc , aks and wu acknowledge isro / respond project no . isro / res/2/379/12 - 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we use multiwavelength data from space and ground based instruments to study the solar flares and coronal mass ejections ( cmes ) on january 23 , 2012 that were responsible for one of the largest solar energetic particle ( sep ) events of solar cycle 24 . the eruptions consisting of two fast cmes ( @xmath01400 km s@xmath1 and @xmath02000 km s@xmath1 ) and m - class flares that occurred in active region 11402 located at @xmath0n28 w36 . the two cmes occurred in quick successions , so they interacted very close to the sun . the second cme caught up with the first one at a distance of @xmath011 - 12 @xmath2 . the cme interaction may be responsible for the elevated sep flux and significant changes in the intensity profile of the sep event . the compound cme resulted in a double - dip moderate geomagnetic storm ( @xmath3 ) . the two dips are due to the southward component of the interplanetary magnetic field in the shock sheath and the icme intervals . one possible reason for the lack of a stronger geomagnetic storm may be that the icme delivered a glancing blow to earth . sun , solar flares , coronal mass ejections , solar energetic particles , geomagnetic storm

You are an expert at summarizing long articles. Proceed to summarize the following text: the analysis of the epic data from a 30 ks observation of the seyfert 1 mrk 335 ( longinotti et al . 2006 ) shows that the 2 - 10 kev spectrum can be phenomenologically fitted with a power law , a broad gaussian line with [email protected] kev , @[email protected] kev , ew 490 ev , and a narrow ( @xmath2 = 1ev ) absorption line with [email protected] kev and ew=50 ev . these features are visible in the residuals of the spectrum plotted in fig.1 . the significance of the absorption line estimated through monte carlo simulations is 99.7% . the most obvious identification for the absorption feature is with redshifted iron k@xmath0 resonance absorption . the identification with iron is favoured since the observed energy of the line is too high to be readily explained by k@xmath0 absorption in any of the other astrophysically abundant elements . we present a simple model for the inflow ( centre and right panels of fig.1 ) , accounting approximately for relativistic and radiation pressure effects , and use monte carlo methods to compute synthetic spectra for qualitative comparison with the data . this modeling was developed following sim ( 2005 ) and assuming spherically symmetric radially infalling gas . it does show that the absorption feature can plausibly be reproduced by infalling gas providing that the feature is identified with fe xxvi . a smooth continuous flow is ruled out by the poor agreement with the data : although the presence of a broad inverse p cygni fe xxvi k@xmath0 line profile is predicted , the absorption line is insufficiently redshifted and too broad . an inflow over a limited range of radii ( as discrete blobs or section of infalling gas ) is more consistent with the data ( fig.1 ) . the narrowness of the absorption line tends to argue against a purely gravitational origin for the redshift of the line , but given the current data quality we stress that such an interpretation can not be ruled out . a longer ( 100 ks ) xmm - newton observation of mrk 335 has been performed this year . the analysis of the integrated spectrum reveals a double - peaked fe line . a broad accretion disc line is likely to be present but the peaks are due to narrower components at 6.4 and 7 kev ( oneill et al . in prep . ) . a preliminary time - resolved analysis shows that the narrow peaks are variable . 2 shows the spectra from three portions of the light curve . the narrow peak at 6.4 kev , clearly present in the first 30 ks , disappears in the central portion . the 7 kev peak instead shows up only in the last 60 ks . the variable narrow lines could be tentatively associated to the presence of flares in the light curve , but a much more detailed and careful analysis is necessary before speculating on their origin . as regard to the absorption line observed in the archival 30 ks observation , it is marginally detected only in the first portion of the longer exposure . further investigation on these data will hopefully clarify all the issues reported here .

the analysis of hard x - ray features in _ xmm - newton _ data of the bright sy 1 galaxy mrk 335 is reported here . the presence of a broad , ionised iron k@xmath0 emission line in the spectrum , first found by gondoin et al.(2002 ) , is confirmed . the broad line can be modeled successfully by relativistic accretion disc reflection models . regardless of the underlying continuum we report , for the first time in this source , the detection of a narrow absorption feature at the rest frame energy of 5.9 kev . if the feature is identified with a resonance absorption line of iron in a highly ionised medium , the redshift of the line corresponds to an inflow velocity of 0.11 - 0.15 c. preliminary results from a longer ( 100ks ) exposure are also presented .

You are an expert at summarizing long articles. Proceed to summarize the following text: as the scale of architectures in integrated circuit design continues to be reduced , the dimensions of present - day interconnects are on the scale of tens of nanometers , and the interconnects for next - generation nanoelectronic devices may well be just a few nanometers in lateral dimension . furthermore , the incorporation of nanometer - scale components , such as active molecules , into integrated circuits will require interconnects of a similar scale . on these scales , self - assembled nanowire systems are becoming increasingly interesting@xcite , as conventional lithographic techniques reach their limits around 10 - 15nm@xcite and spm - based nanolithography@xcite methods lack scalability . moreover , the introduction of such `` bottom - up '' technology , based on naturally nanometer - scale components to complement or even replace the current `` top - down '' technology is likely to require completely different architectures . one example is the `` crossbar '' architecture@xcite , in which active molecules are used as devices at the junctions between two perpendicular nanowires . this architecture is designed to take advantage of the typical product of self - assembly schemes an array of parallel wires rather than relying on controlled positioning of individual wires . over the last ten to fifteen years , a number of systems have been identified which yield self - assembled nanowires on semiconductor surfaces , and stem from a mixture of serendipity and deliberate design . there are only a few systems which provide long , straight nanolines on flat terraces . in / si(111)@xcite ( amongst other metals ) and pt / ge(001)@xcite are examples of systems which undergo surface reconstructions leading to 1d atomic chains with metallic properties . on the si(001 ) surface , the bi nanolines@xcite appear to be the only example of an atomically - perfect , straight , self - assembled 1d nanostructure , but are not metallic . aside from fortuitous discoveries , symmetry - breaking by use of a vicinal surface to produce and array of steps can be used to grow long , straight nanowires of many metals , including au , bi et al.@xcite , and 1-d nanoscale structures can be designed by taking advantage of anisotropy in the heteroepitaxial strain between the si(001 ) surface and an appropriate deposited material . the rare - earth silicides are the archetype of this method@xcite . within the family of rare - earths ( as well as scandium and yttrium ) , a close lattice match can be obtained in one direction , with a variety of positive and negative lattice mismatchs in the orthogonal direction . by choosing the right material , metastable 1d nanoscale structures 3 - 10 nm wide and hundreds of nanometres long can be formed . these have some phenomenological similarities with the bi nanolines , as will be described . this review will survey the growing field of nanowires on semiconductor surfaces , with the focus on the bi / si(001 ) nanoline system , whose importance is explained in the next section . the study of semiconductor surfaces has been revolutionised over the last twenty years by the advent of two techniques which give real - space atomic scale information : scanning tunneling microscopy ( stm ) and electronic structure modelling ( in particular , density functional theory , or dft ) . the two techniques are complementary , in that stm gives an approximate answer to an exact question ( i.e. the structure of the real si(001 ) surface ) , while dft gives an exact answer to an approximate problem ( i.e. the lowest - energy structure of a computation cell containing a few si dimers , and a few layers of bulk - like si ) . while these techniques between them enabled the clear identification of the si(001 ) reconstruction , even today there is some controversy about the exact nature of the buckling of dimers on si(001 ) at low temperatures@xcite . in the same way , the understanding of the structure and properties of the nanolines surveyed in this paper has been greatly enhanced by a close collaboration between experiment and theory . the benefits of this synergistic approach will be drawn out throughout the discussion . the bi : si(001 ) nanoline system , which will be the focus of this review , was discovered about ten years ago . as can be seen from the examples in fig . [ fig : binanoline ] , they grow perfectly straight along the @xmath0 directions on the si(001 ) surface for hundreds of nanometres , apparently limited only by the terrace size of the underlying substrate . when they encounter a step edge , they will either grow out over lower terraces in a long , narrow promontory , or they will burrow into higher terraces , until a deep inlet is formed . they have a constant width , 1.5 nm or four substrate dimers , and are very stable . so long as the temperature is not so high that the bi can evaporate , they are stable against prolonged annealing , maintaining the same width . this is quite unlike the rare earth silicide systems which in most cases ( yttrium being the probable exception ) will coarsen into 3d epitaxial islands with long anneals . there are no other elements which produce perfect nanolines on the si(001 ) surface , although there is some evidence to suggest that sb nanowires with the same appearance as the bi nanoline form in the presence of surface hydrogen@xcite , and dft modelling that indicates that a sb nanoline would be stable@xcite . there has also been a suggestion that er may form nanolines of the same structure@xcite ( as opposed to silicide wires ) , but this remains unconfirmed . m@xmath11@xmath2 m . the bi nanolines run along @xmath0 directions , perpendicular to the si dimer rows , and can grow to over 1@xmath2 m in length . in the inset , the atomic structure , comprising a pair of bi dimers , about 6.5 apart , is shown . hydrogen termination of the bi nanolines has been used to enhance the apparent height of the nanoline . ] structurally , the bi nanoline is unique amongst nanoline systems it is neither a periodic reconstruction of the surface , nor the result of anisotropically strained heteroepitaxial growth of one bulk structure on another . in fact , it is somewhere in between . the nanoline structure is built around a pair of bi ad - dimers , on top of a complex subsurface reconstruction which is responsible for many of the nanoline s remarkable properties , such as the extreme straightness ; this will be explained in detail below . in this article , we intend to describe the formation , structure and properties of this system , setting it in the context of other nanoline systems , particularly the rare - earth silicide family . the paper is laid out as follows : in section [ sec : methods ] we consider the methods used to examine the systems , both experimental and theoretical ; in section [ sec : other - nanowires ] self - assembled nanowire systems _ other _ than the bi : si(001 ) nanowire system are described , with particular reference to the rare - earth silicide family ; in section [ sec : bi - nanowires ] the bi : si(001 ) nanowire system is examined in detail , and in section [ sec : conclusions ] the state of the field is considered and conclusions are drawn . in considering the nanowires in sections [ sec : other - nanowires ] and [ sec : bi - nanowires ] we consider first the formation and structure of the systems ( including the effects of annealing ) , followed by the electronic properties ( including conductance data where available ) and finish with the reactivity of the systems . since its invention in 1982 by binnig and rohrer at ibm , stm has allowed researchers to image dynamic surface processes , in real - space , with near - atomic resolution . stm as a technique bridges a variety of disciplines . in surface physics , stm is unique as a real - space probe of electronic density of states of individual species ; the stm tip can be a passive observer of surface chemical reactions , and also an active catalyst of these reactions ; and stm allows for the direct observation of many phase transformation processes , albeit in 2d rather than 3d . however , this is not meant to be a review of capabilities of stm ; for that , the reader is pointed to classic books on the subject@xcite . here a few comments are made , which are pertinent to the particular challenges of elevated - temperature ( ` hot ' ) stm for the study of epitaxial growth . surface reactions can be followed at room temperature by using quench experiments , in which a high - temperature surface is cooled rapidly to freeze in a snapshot of an evolving surface , for example in the growth of iii - v semiconductor surfaces@xcite . however , the quenching process may introduce spurious surface features , and so it is preferable to image the surface _ in - situ _ , i.e. at temperature . the ability to vary the sample temperature makes it possible to image a surface in the process of changing , rather than after it has changed . to give just one example , the phase transition between the ( 1@xmath11 ) reconstruction and the ( 7@xmath17 ) reconstruction on the si(111 ) surface , at around 1100k , has been observed directly in both directions , by careful control of the temperature across the transition@xcite . for useful observations of dynamic processes , the speed of the reaction must be matched to the rather slow imaging rate of an stm . one example is the motion of step edges@xcite . at around 300@xmath3c , step motion can be imaged as the addition or subtraction of 4 atoms between stm images . however , above about 500@xmath3c , the step edge will have moved significantly in the time taken for the stm to scan one line . thus from line to line , the step will be in different places , as it oscillates about a mean position . in stm , therefore , fast - moving steps have a streaky appearance above 500@xmath4c , as may be seen in fig . [ fig : nucleationstm ] . furthermore , mobile species may be invisible at elevated temperatures , either due to the decrease in contrast , or due to their mobility . surface chemical processes , such as the diffusion of si atoms on si(001)@xcite or the decomposition of a molecule@xcite , can be studied at the single - species level , and complete chemical reaction pathways from initial adsorption can be followed through to complete decomposition@xcite . kinetic information such as activation barriers can be extracted , and any metastable intermediate structures can be imaged , giving a unique insight into the reaction process . the ability to control the substrate temperature is particularly important in studies of epitaxial growth , where the kinetic pathways available at different temperatures will determine the surface morphology . in the case of si / si(001 ) and ge / si(001 ) epitaxy , it has been possible to image the surface simultaneously with the deposition of material , and therefore observe directly the nucleation of islands , and the growth of these nuclei into 1d , 2d and 3d islands , along with associated processes , such as the ripening of epitaxial islands , surface stress relief mechanisms , and the transition from island growth to step flow growth modes@xcite , all of which processes occur at different sample temperatures . however , these added abilities bring with them a penalty in sensitivity . there is generally a loss of resolution due to thermal noise above room temperature , and a loss of stability , with thermal drift making it hard to image the same area for extended periods . in many of the high - temperature stm images shown here , the nanolines have a curved appearance . they have no curvature themselves , the appearance comes from the drift during scanning . furthermore , imaging of epitaxial growth necessarily involves imaging a surface where there are many atoms loosely bound to the surface , or in the gas phase being deposited . in this situation , it is very easy for atoms to stick to the stm tip , which can have adverse effects on the imaging , and it is usually preferable to obtain stable imaging conditions rather than achieve the highest possible resolution , which is generally a more unstable situation . for the same reason , scanning tunneling spectroscopy ( sts ) , or current imaging tunneling spectroscopy(cits ) , becomes unfeasible at elevated temperatures . it is generally better to take a series of images of the same area at different bias voltages , giving voltage contrast of an object , although as the feedback loop is active in this situation , the tip - sample distance will vary , and so the information gained is not equivalent to sts / cits . thus these limitations mean that while high - temperature stm brings the unique ability to study evolving surface morphology in real - space , it is not always the best way to study growth surfaces . a combination of high - temperature studies , and room - temperature studies of quenched surfaces , is likely to provide a fuller description of a system . there have been a number of books on modelling techniques published recently@xcite , so this section will feature only a brief description of the techniques used for modelling of semiconductor surfaces , concentrating on electronic structure calculations ( both semi - empirical and _ ab initio _ ) . these techniques retain the quantum mechanics of the electrons while treating the ions classically . the first problem to consider is how to model a semi - infinite piece of material ( that is , a surface ) . the approximation that is used is to consider a small piece of appropriate material with boundary conditions . generally , one of two solutions is used : either a cluster of atoms , with the dangling bonds on the edges terminated in some suitable way ( e.g. hydrogen ) , or a slab of atoms with periodic boundary conditions in two or three directions . a cluster allows a smaller number of atoms to be used than a slab , but has the drawback that it is hard to allow for long - range strain effects ( such as those caused by reconstructions or steps on surfaces ) . a slab approach requires care : the vacuum gap between repeating images must be large enough to prevent interactions , and the slab itself must be sufficiently thick to allow strain relaxation . the slab approach is more commonly used , particularly with the methods described below . once the system to be modelled has been defined , the calculation technique must be chosen . putting aside quantum chemical methods for the sake of brevity ( they scale rather poorly with system size for more details , the reader is directed to one of many excellent books on the subject@xcite ) , we will consider the semi - empirical tight binding ( tb ) technique , and the _ ab initio _ density functional theory ( dft ) technique . tb@xcite postulates a basis set of atom - centred orbitals for the wavefunctions ; the schrdinger equation can then be rewritten as a matrix equation , with elements of the hamiltonian matrix formed from integrals between orbitals on different atoms . these hamiltonian matrix elements are _ fitted _ to either _ ab initio _ calculated data or experimental results . for simplicity , the basis is often assumed to be nearest neighbour only ( with a cutoff defined on the range of interactions ) and orthogonal . once the hamiltonian has been defined , the band energy of the system can be found by diagonalisation of the hamiltonian or other methods . the other energetic terms ( e.g. hartree correction , exchange and ion - ion interactions ) are represented by a repulsive potential ( often a pair potential ) . despite its apparent simplicity , tb is extremely effective , and generally qualitatively accurate , if not approaching quantitative . it has been shown@xcite that this tb formalism can be derived from dft via a set of well - defined approximations , which explains to some extent its success . it is well - suited for a rapid exploration of configuration space ( e.g. possible structures for some new surface feature@xcite ) provided that a suitable parameterisation exists for the bonds between different species . the fitting of parameterisations is non - trivial , and the transferrability of a given parameterisation ( i.e. its accuracy in environments far from those in which the fitting was performed ) is never guaranteed . generally the matrix elements themselves are fitted to a band structure ( or energy levels of a molecule ) , while their scaling with distance is fitted to elastic constants or normal modes of the system . tb retains quantum mechanics ( since the energy is obtained by solving the schdinger equation ) while approximating the most complex problems . dft@xcite starts with an exact reformulation of the quantum mechanics of a system of interacting electrons in an external potential : the result is a set of equations for non - interacting electrons moving in an _ effective _ potential with all the complex electron - electron interactions in a single term , known as the exchange - correlation functional ( it is a function of the charge density , which is itself a function of position , hence `` functional '' ) . unfortunately , the form of this functional is not known , and must be approximated , for instance with the local density approximation ( lda ) or one of the generalised gradient approximations ( gga ) . dft has been extremely successful in many areas of physics , chemistry , materials and , increasingly , biochemistry , particularly when combined with the pseudopotential approximation . the hard nuclear potential and the core electrons of each atom are replaced with a single `` pseudopotential '' , and only the valence electrons are considered , leading to a softer potential . this approximation is most effective for atoms where there is considerable screening of the valence electrons by the core electrons ( e.g. in si the 3s electrons are screened by the 1s and 2s shells , while the 3p electrons are screened by the 2p shell ) ; in first row elements and first row transition metals , this is a much smaller effect , leading to the development of `` ultrasoft '' pseudopotentials . these issues are discussed in more detail in many excellent books and reviews@xcite . the essential point to note is that dft is widely used for electronic structure calculations , and provides results which are accurate to within approximately 0.1ev . an important development in electronic structure techniques over the last ten years is that of linear scaling techniques@xcite . standard techniques for solving for the ground state , whether tb or dft , scale with the cube of the number of atoms in the system ( either because matrix diagonalisation scales with the cube of the matrix size , or because the eigenfunctions spread over the whole simulation cell , leading to cubic scaling when they are orthonormalised ) . this scaling places rather strong restrictions on the sizes of system which can be modelled , even on massively parallel machines : for _ ab initio _ methods , going beyond 1,000 atoms rapidly becomes prohibitive , though somewhat larger systems can be addressed . however , since electronic structure is fundamentally local ( consider bonding as an example ) , the amount of _ information _ in the system should be proportional to the number of _ atoms _ in the system . tight binding methods which take advantage of this ( e.g. refs.@xcite ) have been widely used for some time , allowing calculations on many thousands of atoms . the implementation of linear scaling dft algorithms has proved significantly harder , though recent efforts in nearly linear@xcite and linear scaling methods@xcite suggest that these techniques are starting to produce useful and general results . one of the challenges of using stm to examine semiconductor surfaces is that the current arises from both geometric and electronic structure , though this is less true at high biases ( where height or geometric structure will dominate ) . some technique is required to understand these changes , and to test proposed structures against experiment . the field of stm simulation is a complex one ( in part because the structure and composition of the tip is unknown ) ; the interested reader is referred to excellent reviews for further information@xcite . we will briefly summarise the simplest and most common approximation used , and discuss how the experimental - theoretical interaction can best proceed . the most commonly used approach is the tersoff - hamann approximation@xcite . this asserts ( via a careful series of approximations ) that the tunneling current is proportional to the local density of states ( ldos ) due to the sample at the position of the tip . in effect , the tip is assumed to have a flat density of states . this approach is directly equivalent to considering the projected charge density ( i.e. the partial charge density due to each band for bands within @xmath5 of the fermi level ) . it is qualitatively accurate , though will not reproduce the observed corrugation values correctly@xcite . the authors are of the opinion that for true success in investigating nanowires ( and other systems ) on semiconductor surfaces , a close collaboration between experiment and modelling is required . any successful collaboration depends on a number of factors , but the key factors , we believe , are frequent , easy correspondence , an understanding of the limitations and abilities of the appropriate techniques , and trust between the different sides in the collaboration . the last point extends to sharing of unpublished data , and extensive discussion of possible courses of action . the three authors have been working together in different combinations for ten years ( leading to around 20 joint publications ) , and we find that our collaboration is still immensely fruitful . while the bi / si(001 ) nanolines have remarkable structural qualities , they are not the only self - assembled nanowire system on semiconductor surfaces . in this section we present an overview of these other nanowire systems ( more details are presented elsewhere@xcite ) , and we will consider in more detail the most important of these systems , the rare - earth silicide family , focussing on their structure and formation , electronic properties and reactivity . a variety of metals have been found to self - assemble into nanowires or nanolines on the si(001 ) surface , though none with the perfection of the bi nanolines . the simplest way in which to form 1d structures on the si(001 ) surface is by epitaxial growth . an island nucleus is essentially an ad - dimer , which sits between the dimer rows . the two pairs of si dimers which support the ad - dimer are distorted by its presence , and so are attractive adsorption sites for further ad - atoms or ad - dimers@xcite . this process has been described as a surface polymerisation reaction@xcite . the long sides of the string , which are equivalent to a - type steps on this surface , have a very low sticking coefficient which keeps the island from broadening at lower growth temperatures . deposition of group iii and group iv elements on si(001 ) will therefore result in long 1d chains of dimers@xcite . however , there is little control over the length or structure of these wires , and they are unstable against annealing , eventually reorganising into compact islands . the pt / ge(001 ) system@xcite forms nanowires through a complex surface reconstruction , resulting in a system with a high degree of perfection , which approaches that of the bi nanolines . when annealed to high temperatures , arrays of atomically perfect nanowires of pt / ge form , each of which is 0.4 nm wide with a spacing of 1.6 nm between the wires . sideways growth of two - dimensional islands can also be blocked by introduction of missing dimer trenches . these form to relieve surface stress , either because of contamination by a small amount of a transition metal such as ni , or through heteroepitaxial growth of another material such as ge . the trenches tend to line up in a semiregular array , giving an approximate ( 2@xmath1n ) reconstruction in leed , with n@xmath68 - 12 ( though the regularity of the trenches is rather poor ) . this method has been used to deposit a variety of metals , such as fe@xcite , ga@xcite and in@xcite ( using the ni - based technique ) and the molecule styrene@xcite . the result is long - range , reasonably well ordered wires of the metals , though their properties have not been characterised . the 2d symmetry of the surface can also be broken deliberately by the use of vicinal substrates to produce regular arrays of steps@xcite which can then be decorated with materials such as gold@xcite and bi@xcite . this approach is used primarily on si(111 ) , although with an extreme orientation towards ( 112 ) directions the surface might be labelled si(557 ) or si(5 5 12 ) depending on the angle . the resulting step reconstructions comprise chains which are metallic and one - dimensional ; early observations suggested possible observations of luttinger liquid behaviour@xcite , though this has been disputed@xcite . their spacing can be controlled to some extent by varying the angle of miscut of the surface . as an atomic - scale extension of the idea of lithography , the si(001 ) surface has been passivated with atomic hydrogen ( forming the monohydride phase ) and individual hydrogen atoms removed with an stm tip to form atomic - scale patterns . the wire formed by removal of hydrogen atoms along ( or across ) a dimer row is known as a `` dangling bond '' wire ( or db wire)@xcite and is predicted to show conduction effects similar to conjugated polymers with polaronic and solitonic effects@xcite . an example of a dangling - bond wire is shown in fig . [ fig : hashizume](a ) . these wires have been reacted with a variety of adsorbates including iron@xcite , gallium@xcite , aluminium@xcite , silver@xcite and organic molecules such as norbornadiene@xcite . however , although for a single row of dangling bonds , a perfect atomic wire can be formed , as with ga in fig . [ fig : hashizume](b ) , for greater widths , as in fig . [ fig : hashizume](c , d ) , the dangling bond wire is more ragged , and the resulting wires ( here ag ) are typically composed of a large density of small , roughly spherical crystals , which exhibit many imperfections and boundaries . a more elegant implementation stems from careful application of molecular chemistry to produce a directed reaction : when the monohydride surface was exposed to styrene and _ one _ hydrogen atom removed , self - directed growth of lines of styrene resulted from a chain reaction between each adsorbed molecule and an adjacent hydrogen@xcite . such self - organised methods are important for the growth of molecular nanowires on surfaces . however , for all these systems there is a lack of control over the size and length of the nanowires . single nws grow in both @xmath7110@xmath8 directions , blocking each other s growth . finally , the nws can form bunches . both the single nws and the bunches can grow a second layer of material , unlike the bi nanolines . _ _ image courtesy prof . j.nogami , u. toronto , canada . _ _ ] a more promising materials system for nanowire growth on si(001 ) is that of the rare - earth metal silicides . there has been extensive research on these systems because of their good conductivity and low schottky barrier with silicon@xcite , though this work concentrated on the si(111 ) surface . moreover , in the hexagonal phase some rare - earth silicides show a good lattice match with the si(001 ) substrate in one direction , with a large mismatch in the other . heteroepitaxial growth of these materials would therefore be expected to be constrained in the high mismatch direction , and facile in the other , resulting in long 1d islands . this is indeed what happens . the behaviour of the rare - earth nanowire systems on si(001 ) has been investigated extensively , and nanowires have been observed under certain growth conditions . as the other major family of self - assembled nanowires on the si(001 ) surface , we give a fuller description of this family for comparison to the bi nanolines . .lattice mismatch between hexagonal phase of various rare - earth silicides and si(001 ) . data is presented in terms of lattice constant and percentage mismatch ( the si(001 ) surface has a lattice constant of 3.84 ) . where the indication `` hexagonal '' is given , it indicates that other phases are possible . _ courtesy of dr c. ohbuchi , nims , japan._. [ cols="<,<,<,<,<",options="header " , ] the most common form of rare - earth nanowire ( renw ) formed on si(001 ) is thought to result from the anisotropic strain between the alb@xmath9 crystal structure of the nanowire and the si(001 ) substrate , leading to fast growth along the less - strained direction , and extremely limited growth along the more - strained direction . the mismatch for various silicides in the alb@xmath9 hexagonal structure is given in table [ tab : rare - earth - mismatch ] . renws of this type have been reported for er@xcite , dy@xcite , gd@xcite and ho@xcite . for sm@xcite nws only appear on vicinal substrates ; other reports@xcite find that on flat surfaces only rectangular , 3d islands are formed after a ( 2@xmath13 ) layer ) . bundles of nws have been reported for yb@xcite ; again , early reports@xcite suggested that nws were not formed , and 3d islands resulted from annealing . though they are not rare - earth metals , similar nws have been reported for both sc@xcite and y@xcite , resulting from the same combination of crystal structure and anisotropic strain . although it forms a similar wetting layer on the substrate to yb , eu does not form nws@xcite . the resulting renws are found both individually and in bundles ; the individual wires have sizes which vary from element to element but are in general 5 - 10 nm wide and less than 1 nm high . the wires grow extremely fast , reaching lengths of up to 1@xmath2 m ; more interestingly they always run along @xmath10 directions and will grow out over step edges , drawing terraces with them , features that they share with bi nanolines . the large - scale image in fig . [ fig : silicide1 ] is very similar to that of bi nanolines at the same scale . for almost all materials , the surface of the wires shows a c(2@xmath12 ) reconstruction@xcite , though the yb nws , which are off - axis relative to the other renws , may have a ( 1@xmath11 ) surface@xcite . the si(111 ) surface matches the lattice of the re alb@xmath9 structure reasonably well in all directions ( with a mismatch of less than 2% for most metals)@xcite and so would not be expected to give anisotropic growth , but gdsi@xmath9 nws have been formed along step edges of a vicinal surface@xcite . these form because of a mismatch perpendicular to the step edge , and grow to over 1@xmath2 m long on appropriately prepared samples , with a width of 10 nm and height of 0.6 nm . other nw structures have been reported for transition metals on si(111 ) , for instance fe@xcite ( though these are rather short wires ) , ni@xcite and ti@xcite . the growth mechanism for these wires is not entirely clear : the resulting nws are 5 nm high and 5 nm wide ( nisi@xmath9 ) , 10 nm high and 40 nm wide ( tisi@xmath9 ) , though for both of these systems the nws appear to grow _ into _ the substrate somewhat . there is a technique for growing nws on si(001 ) , si(110 ) and si(111 ) explicitly relying on growth into the substrate , resulting in `` endotaxial '' nws , which has been applied to dy@xcite in the hexagonal alb@xmath9 structure , as well as co@xcite , which takes the caf@xmath9 structure . the driving force appears to be kinetic in these systems : the long direction of the nw has an interface which grows faster than the short direction , though this is partly dependent on the structure of the interface below the surface . the growth and formation of the renws is not understood at an atomic level : they form extremely quickly ( on experimental timescales ) so that , for instance , hot stm can not be used to observe formation of the lines . unlike the bi nanolines , which have a constant width , and do not change in width with annealing , the rare - earth nanoline family can take a variety of widths depending on the growth conditions . in the case of dysi@xmath11 , single nws of a consistent width form from the underlying ( 2@xmath17 ) reconstructed wetting layer , as in fig . [ fig : silicide2 ] . however , further annealing results in the formation of bunches of nanowires , which join together where they meet , and multilayer islands form . as strained coherent heteroepitaxial islands , the renws are only metastable ; prolonged annealing will cause them to coarsen , and form large 3d islands . overall , the deposition / anneal temperature must be in the window between 550@xmath4c and 650@xmath4c , and annealing time should not exceed 2 - 10 minutes . below 550@xmath4c or 2 minutes , the reaction will be incomplete and the nws formed will be immature . above 650@xmath4c or 10 minutes the systems tend towards lower energy , thermodynamically stable states . in particular , dy - si(001 ) will form 3d islands if annealed for long times ( e.g. 30 minutes for 0.86ml)@xcite with wires formed for shorter annealing periods . annealing er - si(001 ) initially results in nws , but as the annealing time is increased dislocations form in the nws@xcite . if the temperature is raised beyond 620@xmath4c the same behaviour is seen , leading ultimately to coarsening into islands@xcite . a large - scale peem study of ersi@xmath11@xcite found that large nws ( which may have been bundles of nanowires the technique does not have the resolution to distinguish ) were not affected by annealing on a coarse scale . however , gd shows more complex behaviour . annealing studies of thin films of gd found that both hexagonal and orthorhombic phases could form@xcite , and that with longer annealing times the orthorhombic phase grew at the expense of the hexagonal phase . long anneals of the gd / si(001 ) system , up to 620@xmath4c for one hour , resulted in the formation of previously unknown silicide structures , which are aligned perpendicular to hexagonal silicide nws on the same terrace@xcite . there are various useful observations of their behaviour which can be made : * a wetting layer with a characteristic reconstruction forms before the nanowires , and persists on the substrate with the nanowires ; specific reconstructions have been observed for : * * eu - si(001):(2@xmath13)@xcite , * * sm - si(001):(2@xmath13)@xcite , * * yb - si(001):(2@xmath13)@xcite and ( 2@xmath14)@xcite , * * nd - si(001):(2@xmath13 ) and ( 2@xmath14)@xcite , * * dy - si(001):(2@xmath14)@xcite , and ( 2@xmath15 ) and ( 2@xmath17)@xcite , * * gd - si(001):(2@xmath14)@xcite , and ( 2@xmath15 ) and ( 2@xmath17)@xcite * * ho - si(001):(2@xmath14 ) and ( 2@xmath17)@xcite * the only system for which this does not happen is er - si(001)@xcite . * the system must be heated during deposition or annealed post - deposition to allow reaction of the rare - earth metal with the substrate@xcite * annealing the system for too long , or depositing at too high a temperature , can result in 3d islands rather than nanowires ; this is discussed in more detail below . as different metals have different properties for each of these categories , we will discuss them briefly in turn . gd and dy@xcite form a ( 2@xmath14 ) reconstruction first , followed by a ( 2@xmath17 ) reconstruction@xcite just prior to nanoline formation . rather similar behaviour is seem for ho@xcite . the metal fraction in the two reconstructions is 3/8 and 5/14 ml respectively@xcite . for these three metals , the presence of the wetting layer is intimately connected with formation and stability of the nanowires . despite the wealth of studies of ersi@xmath9 nanowires , there are no reports of the structure and make - up of a wetting layer for this system ; there is some evidence@xcite that once more than @xmath60.05ml of er is deposited , long chains of surface dimers form , followed by nanowires , with no intermediate reconstruction . indeed , there are indications@xcite that the surrounding substrate is clean si(001 ) . another group of metals form closely related reconstructions : eu@xcite , yb@xcite , nd@xcite and sm@xcite all form ( 2@xmath13 ) and in some cases ( 2@xmath14 ) reconstructions ; by contrast to er , dy , gd and ho , these metals are strained in both surface directions on contact with si(001 ) ( which might be expected to reduce the periodicity of any substrate wetting ) and lack the main driving force seen before for nanowire formation . in general these metals do not form nanowires@xcite though there are certain conditions where yb can be made to form nanowires@xcite of some kind , which may well be grown off - axis ( in a different growth mode to the other silicide nanowires which grow with the c - axis aligned with the substrate ) . it is also possible to induce nanowire formation in situations where they would not normally form , either because of uniform lattice matching ( e.g on si(111 ) surfaces , as mentioned above ) or lattice mismatch in _ both _ surface directions using step edges . this technique has been successfully employed for sm on si(001)@xcite and gd@xcite and dy@xcite on si(111 ) . one of the original technological interests in rare - earth overlayers on si(111 ) was the good crystal growth possible due to the interface between the silicon and the overlayer , and the conductivity properties . the films have good conductivity and a low schottky barrier ( @xmath60.4 ev on n - si and @xmath60.8 ev on p - si@xcite ) . measurement of the conductivity of renws is a significant challenge . full two - point or four - point measurements requires either a unique stm instrument ( for instance one used to measure conductivity of cosi@xmath9 nanowires on si(110)@xcite ) , or alternatively the formation of nanoscale contacts to a nw@xcite . while significant progress has been made in this area recently , any measurements of conductivity will include the resistance of the interface between the nw and the contacts . scanning tunneling spectroscopy ( sts ) has been performed on various nws : dy and ho@xcite and gd@xcite . while sts does not measure the conductivity _ along _ the nw , it does measure the local electronic structure , and in all cases the nws are found to be metallic , while the surrounding reconstructed substrate is not@xcite . this is good evidence that the nws are taking on the bulk silicide structure , which is conducting , and the wetting layer is an intermediate state which relieves strain . conductivity measurements of transition metal silicides on si(111 ) and si(110 ) have been made@xcite . these studies exemplify the two techniques for measuring nw conductivity : nisi@xmath9 nws were contacted by gold pads@xcite , while four - probe stm measurements were made on cosi@xmath9 nws@xcite . these nws are relatively large compared to the renws discussed so far : 15 nm ( ni ) and 60 nm ( co ) wide . the cosi@xmath9 nws showed a high schottky barrier and conductivity equivalent to that of high quality thin films of cosi@xmath9 , while the nisi@xmath9 nws showed some signatures of quantum transport , though the conductivity of these wires was significantly lower than thin films ; this is likely due to the overgrowth of the sample with sio@xmath9 for transfer to the lithography apparatus . there is very little data on the chemical reactivity of the rare - earth nanowires . thin films on si(111 ) are susceptible to reaction with o@xcite ; the renws oxidise rapidly in air@xcite , and transition metal nws show signs of this ( confirmed by the conductivity of nisi@xmath9 nws on si(111)@xcite ) . in the tm system , the nws were overgrown with native oxide , and showed a significant decrease in conductivity relative to complete thin films of similar thickness , which is attributed to scattering at the nw surface / oxide interface . further information on the reactivity comes from the deposition of pt on a surface containing ersi@xmath9 nws on si(001)@xcite . the pt was deposited at room temperature and the sample was subsequently annealed . when stm images were taken they showed that the c(2@xmath12 ) surface reconstruction on the nws was no longer visible though the ( 2@xmath11 ) substrate reconstruction was still present . furthermore , the pt - covered nws were resistant to reactive ion etching and appeared stable when exposed to air for periods of up to 8 weeks ; the pt overlayer does appear to strain the nws , however , leading to delamination of the nws from the substrate if left untreated . clearly there is much work to be done understanding the reactivity and stability of these nws . the discovery that nanowires would form if a bi - covered si(001 ) surface was left to anneal around the bi desorption temperature was made quite by chance . in 1995 , in the materials department of oxford university , two of the authors were investigating the surfactant - assisted growth of ge / si(001 ) using bi as a surfactant , and were studying the properties of bi / si(001 ) . when a sample which had been left to anneal at _ ca . _ 500@xmath3c overnight was imaged , very large , flat terraces , and long , straight , bright lines ( longer than the scanning range of the stm ) , running across the surface were found . these were the bi nanolines . the first published image of the `` nanobelts '' was in 1997@xcite , and the first big study of their growth and properties came in 1999@xcite . in this section , we will discuss the atomic structure of the bi nanoline , and its physical and electronic properties . we show that many of the properties stem from its unusual structure . the reaction of the bi dimers with a variety of reagents , and the effect of burial of the nanoline will also be discussed . the physical structure is the key to understanding the properties of a nanoline system , and great effort has been devoted to identifying the structure of the bi nanoline . in this case , the structure was identified by a synergy of stm observations with tightbinding and dft simulations . the three different proposed models for the bi nanoline are shown in fig . [ fig : models ] . the two early models ( shown as the top two models ) share certain structural motifs . they contain a pair of bi dimers set into the surface layer of the si(001 ) crystal , whose compressive stress is relieved by missing dimer defects ( dvs ) . the first model proposed , shown in fig . [ fig : models](a ) , was based around a pair of bi dimers in the surface layer , separated by a missing dimer defect , so as to relieve the local stress of the bi dimers@xcite . this became known as the miki model . the second proposed model , shown in fig . [ fig : models](b ) , was based around a pair of bi dimers with defects on either side@xcite , which became known as the naitoh model . however , the third model is more complex : a reconstruction of several layers of si underneath the bi dimers produces the nanoline core . this was named the haiku model@xcite . of the three models , this structure is the only one to fit all the criteria which have been determined from stm observations , as well as having the lowest energy of the three . in this section , we will briefly review the historical process by which this structure was determined , and discuss the properties of the haiku structure . early stm images of the bi nanoline were taken at elevated temperature@xcite , in which the si dimers were not resolved , and hence the registry of the nanoline with the substrate was unknown . this data suggested that the nanoline contained two features , which were probably bi dimers , with a spacing of approximately 6.3 , with a total width of about 1 nm , which was approximately equivalent to the space of three si dimers . as the nanolines appeared bright at large bias voltages ( in both positive and negative bias images ) to dark at low voltages , it was also reasonable to assume that these bi dimers were situated in the surface layer , rather than in an adlayer or in a subsurface layer . it was thought that bi dimers embedded in the top surface layer would have considerable compressive stress , and on this basis , a model based around a 1dv , with bi dimers to either side , was proposed . this was the miki model@xcite . tightbinding calculations of the miki model@xcite found that it was more stable than the ( 2@xmath11 ) or ( 2@xmath1n ) bi reconstructions , and that the formation energy of a defect in the line was high : around 1.1 ev@xcite ( this number considers putting the bi which has been removed as an ad - dimer on the surface ; recent data@xcite shows that the defect energy falls to 0.11 ev if the bi dimer is placed in another miki model ; this point is discussed more fully in section [ sec : miki - model - revisited ] below ) . dft calculations@xcite agreed with those conclusions . the calculated ldos showed that the bi dimer states were further away from the fermi level than those of the si dimers and simulated stm images of the miki structure@xcite showed that the line would appear dark at low bias voltages , in agreement with stm results@xcite . testing by other methods appeared to confirm this structure . photoemission spectroscopy experiments@xcite found that the bi 5d core - level spectra of the bi nanowire was essentially identical to the spectra of the ( 2@xmath1n ) phase composed of bi ad - dimers . this suggested that the local chemical state and registry of bi adsorbates for both phases was the same , i.e. that the bi was in the form of dimers in the top layer of the structure . x - ray photo - electron diffraction ( xpd ) experiments@xcite found a good fit between the experimental xpd data and simulated xpd intensity peaks from the miki model . in particular , they confirmed the presence of bi dimers parallel to the si dimers and found the spacing between them to be @xmath66.3 , in agreement with the stm measurement , though it is important to note that the quoted distances are the result of a _ fit _ to the miki model . 10 nm stm image of the si(001):h surface . the h termination makes it easier to resolve the individual si dimers . a series of markers has been placed across the bi nanoline . from this , it can be determined that the nanoline occupies the space of _ four _ dimers . ] however , it was difficult to account for the extreme straightness of the nanoline with the miki model , as the calculated kinking energy was very small , around 0.1 ev . a kinetic argument was put forward , which suggested that there was a stronger preference for incoming bi dimers to line up at the end of the nanoline , and hence the nanoline would grow straight@xcite . furthermore , the diffusion constant of a bi surface dimer was high ( this diffusion was required for a kink to form in a miki model nanoline ) , and hence a line would remain straight once it had grown . despite this explanation , the issue of the straightness of the bi nanolines remained as a question mark over the miki model . subsequent room - temperature stm experiments@xcite proved decisively that the miki model could not be the structure of the bi nanoline , as it had the wrong registry with the surface . markers were placed on the surface of an image with the si dimers resolved , as shown in fig . [ fig : registry ] , and by counting across the nanoline , the width was determined definitively to be four dimers , and not three as in the miki model ; moreover the bi dimers lay _ between _ the si dimers of the substrate , not in the same position . ( this registry has been confirmed by every other stm measurement made@xcite ) . on the basis of the new stm data , the naitoh model was proposed@xcite . again this model was tested by atomistic calculations ; a detailed study of the dimensions and simulated stm appearance@xcite showed that again the nanoline would appear dark in stm at low bias voltages , but showed also that the spacing of the bi dimers was too narrow , approximately 5 , which could not be reconciled with the spacing of 6.3 measured from stm . moreover , the energy was considerably worse than the miki model@xcite , and the kinking energy was still low . there was therefore no satisfactory structural model at this time . inspiration for the structural model of the bi nanoline arose from the structural model proposed for the b - type double - height step of as - terminated ge@xcite . at this step edge , bond rotation forms a pair of 5 and 7-membered rings of ge , capped with as , shown in fig . [ fig : haikucartoon](a ) . while the haiku structure appears to be a complex reconstruction , in fact the construction from a cell containing two bi ad - dimers can be described schematically in two simple steps , as shown in fig . [ fig : haikucartoon](b)-(d ) . the first step is to rotate the second and third layer atoms underneath the bi ad - dimers , so that the si dimer atoms sink down into the surface . this creates the 5- and 7-membered rings also seen in the as / ge step structure . the central four atoms are then removed , and the two halves of the structure bonded together , thus creating the haiku structure . ( n.b . this is not proposed as the formation mechanism , but is simply given as a means of understanding the substructure . ) a linescan of the nanoline has been matched to the haiku model , using the background si dimers as reference marks@xcite . the positions of the bi dimers match up extremely well to the peaks in the linescan , giving strong confirmation of this model . other features of experimental linescans also agree well with simulated linescans for the haiku model@xcite . dft calculations of this new structure@xcite found that it was considerably more stable than the miki model , which has been confirmed by subsequent modelling@xcite . although the haiku structure shares several structural features with previous models the top of the structure has two bi dimers joined by rebonded second - layer si atoms , as in the miki model , while on the outside of the bi dimers , there are more rebonded si atoms , just as in the naitoh model the difference lies in the si substructure ; this mixture of 5 and 7-membered rings extends down many layers . consideration of the core of the nanoline reveals that it is a small triangular section of hexagonal silicon , embedded in the diamond cubic silicon substrate , rather like the `` endotaxial '' re nanowires mentioned in section [ sec : formation - structure ] . the \{111 } planes that delineate the core are shown in red in fig . [ fig : haikucore ] . the hexagonal core of the nanoline does not exist as a bulk structure for silicon ; it is particular to this nanoline structure . in this respect , the bi nanoline is quite unlike the epitaxial rare - earth silicide family , which are essentially one bulk crystal grown epitaxially onto another bulk crystal , taking their shape from the highly anisotropic mismatch between the two . this core structure is responsible for many of the properties of the bi nanoline . it will be difficult or impossible to kink , resulting in the extremely straight nanoline observed . as will be shown below in section [ sec : dez - sub ] , the tensile strain field results in the repulsion of other nanolines , step edges and missing dimer defects . as a result , the nanolines will not grow sideways , will not grow together , and will not coarsen into larger islands , unlike the silicide wires . the electronic structure of the haiku model has been studied in some detail@xcite , and again the nanoline becomes dark in low - bias images this effect is a property of the bi dimers rather than that of the nanoline . low - bias stm images reveal other electronic effects , which provide further confirmation of the haiku structure , while ruling out the miki structure . these features are discussed below in section [ sec : electronic - structure ] . however , despite the wealth of agreement of the predictions from the haiku model with experimental evidence , there has been no direct confirmation of the haiku structure . the bi nanolines are destroyed by exposure to air , while burial of the nanolines in a capping layer also changes their atomic structure@xcite ( though if done carefully the 1d character and dimerisation of the bi can be preserved ) , so that cross - sectional tem studies@xcite have not provided definitive confirmation of the core structure , and a recent x - ray standing wave(xsw ) study@xcite which relied upon capping with amorphous silicon and whose results cast doubt on the haiku model , can not be regarded as a counterindication . the difficulties of burial of the bi nanoline are discussed in section [ sec : burial ] . we note also that a recent suggestion@xcite , that the height of the bi dimers in the haiku model is too far above the plane of the surface si dimers based on xpd@xcite , is not a significant piece of evidence against the haiku model : the xpd data was _ fitted _ to the miki model , thus naturally giving the wrong data for the haiku . if a fit of the data to the haiku model were to give height data which contradicted the atomistic modelling , then the model would have to be revisited . despite the apparent success of the haiku model , the miki model has continued to attract a large amount of attention@xcite as a possible candidate for the nanoline structure . while in fact its incorrect registry with the si substrate unambiguously rules it out as a candidate nanoline structure@xcite , it is an energetically favourable structure , and might be expected to be present on the bi - covered si(001 ) surface . with the benefit of hindsight , it can be seen that the miki structure was indeed observed in early stm images of the bi - rich surface . [ fig : naitohmiki ] shows that annealing a bi - rich surface at 400@xmath3c produces a large density of quite straight dimer vacancy ( dv ) trenches . in the filled - states image , [ fig : naitohmiki](a ) , these appear to be composed of clean silicon . however . in the empty - states image , [ fig : naitohmiki](b ) , there is contrast between the dimers adjacent to the missing dimer trenches and the rest of the surface . this contrast does not occur on the clean surface at this bias voltage , and so it can be inferred that the trenches are decorated by bi dimers , i.e. the miki structure has been formed . as with the ge - induced dv trenches mentioned in sec . [ sec : other - nanowires ] , there is cooperative strain relief between a bi dimer and a 1dv . bi dimer / dv structures have the lowest energy / bi dimer of any structures except the bi nanolines . there is a kinking energy for these ( 2@xmath1n ) trenches of around 0.1ev@xcite . hence the formation of a semi - regular array of bi dimers reduces the energy further . c , followed by 10 min . annealing . ( a ) : -2.2 v,0.3 na . there are many straight missing dimer trenches . ( b ) : + 2 v , 0.3 na . brightening around the trenches in empty - states images indicates that they are decorated by bi dimers , i.e. the miki structure . ( c ) : a bi - decorated trench and bi nanoline on a h - terminated surface . a linescan along the dotted line ( d ) reveals that the bi - decorated trenches are significantly lower than the nanoline . ( a ) & ( b ) are reprinted from appl . 142 , naitoh et al . bismuth - induced surface structure of si ( 100 ) studied by scanning tunneling microscopy , p.38 , ( 1999 ) , with permission from elsevier.@xcite ] further proof of the bi decoration of 1dv trenches has been provided by a more recent experiment , in which a bi - rich surface was quenched from 550 @xmath3c at an early stage of the annealing process@xcite . here atomic hydrogen was adsorbed after cooling to enhance the contrast between bi and si significantly , as the h will adsorb easily to the si dimers , but does not adsorb on bi@xcite , as can be seen in fig . [ fig : registry ] . it was found that the h termination increased the relative _ apparent _ height of bi - related features by ca.150 pm , or 1.5 . an example image is shown in fig . [ fig : naitohmiki](c ) . a linescan across both a bi - decorated trench and a section of nanoline shows that the bi - decorated trench is quite similar to the nanoline , but is lower and has a deeper depression between the two bi dimers . stm linescans of the feature decorating dv trenches and the bi nanolines @xcite compare very well to simulated scans of the miki model and haiku model respectively@xcite , providing strong support for both these identifications . furthermore , in this way , it has been shown that the miki structure and the bi nanoline co - exist , demonstrating once again that the miki structure is not the nanoline , as has been recently suggested@xcite , but it is kinetically stable , even at high temperatures , due to the large activation barrier necessary to form the haiku structure . while rare - earth silicide wires have quite specific growth recipes in order that the nanowires do not coarsen , the bi nanolines are robust to a wide variation in the growth conditions . the essence of bi nanoline formation is that it is a competition between deposition and evaporation . a thick layer of bi on si(001 ) is only stable below the bi bulk evaporation temperature ( ca . 400@xmath3c ) . above this temperature , an epitaxial layer of 1 - 2 ml bi is stable , and annealing of this surface will result in a surface with bi nanolines co - existing with these bi islands , as shown in fig [ fig : bi2xn ] . the first - layer bi forms a ( 2@xmath1n ) structure , with n=4 or 5@xcite , and the second - layer bi forms as small groups of dimers , which can be either parallel or perpendicular to the underlying bi . c , and then annealed for 60 mins at 497@xmath3c , resulting in bi nanolines surrounded by a bi overlayer , which forms a ( 2@xmath1n ) reconstruction , with n=4 or 5 , to accomodate its large strain . left - hand images are empty - states , right - hand images are filled - states . the ( 2@xmath1n):bi does not grow over the bi nanoline at this temperature . ] the threshold temperature at which bi dimers on si(001 ) will start to evaporate is around 500@xmath3c , while the maximum temperature at which any bi is stable on the si(001 ) surface is around 600@xmath3c . deposition of bi in this temperature window , or deposition of bi at a lower temperature , followed by an anneal within this temperature window , will result in a surface comprising bi nanolines on an otherwise clean surface , as in fig . [ fig : recipes ] . however , the details of the growth recipe will determine the surface morphology at the end of the anneal , as is described below in sec . [ sec : recipes - sub ] . of particular technological interest is the growth of majority - domain surfaces after long anneals . the si(001 ) surface has two equivalent domains with the dimer rows running in orthogonal directions . on a typical si(001 ) surface , these two domains will have roughly equivalent total areas . single - domain surfaces form on vicinal wafers miscut along a @xmath0 direction , where the dimer rows tend to run perpendicular to the step edges . however , by long anneals of the bi - rich surface towards the high end of the temperature window , majority - domain surfaces can be formed , even on flat surfaces . in this case , extremely long bi nanolines grow , up to 1@xmath2 m , as can be seen in fig . [ fig : mcleanimage ] . c. ( b ) : deposition of 1 ml bi at around 550@xmath3c , followed by annealing at the same temperature . ( c ) : continuous exposure of 0.1 ml / min bi flux for 40 mins at 570@xmath3c . ( d ) : as ( c ) , but on a 0.5@xmath3 miscut sample . the growth recipes are summarised schematically in fig.[fig : recipesplan ] . ] high - temperature stm observations of the annealing bi - rich surface shows a consistent series of surface morphologies . immediately after the end of bi deposition , a relatively rough surface is obtained , with a high density of small islands , composed of a mixture of bi and si , as shown in fig . [ fig : recipes](a ) . these will quickly disappear , but at the same time , nanolines begin to form , either on the terraces , or on one of the mixed islands . the nanolines grow out over the step edges in long , finger - like islands known as promontories , with a bi nanoline surrounded by a thin strip of silicon . examples of the growth of promontories are seen in fig . [ fig : promontory ] , and more mature examples of this type of surface are shown in figs . [ fig : recipes]&[fig : mcleanimage ] . the nanolines will not only form long , narrow promontories , but also the reverse , a deep inlet where a nanoline in one terrance has grown through an up - step into an upper terrace . the surface between the bi nanolines contains a large density of mobile missing - dimer defects , apparent in stm as dark , fast - moving streaks . after further annealing , the missing dimer trenches begin to disappear , and the bi nanolines remain on a very flat , featureless surface . finally , the bi nanolines themselves evaporate . in the following sections , we will give describe the surface processes which drive this sequence of events , and account for these observations . the interactions of the bi nanoline and its associated strain field with defects and step edges in the terraces , and with each other , will be discussed and the possible nucleation mechanisms of this complex structure explored . as shown above , annealing a bi - rich surface at lower temperatures results in a surface containing a high density of missing dimer trenches , decorated with bi dimers , i.e. the miki structure . one feature of the bi nanoline is the high temperature required to form it , implying a significant activation barrier to its nucleation . while the large number of growth parameters , such as flux , total amount deposited , substrate temperature , miscut angle and azimuth , provides a wide range of different possible recipes , as summarised in fig . [ fig : recipesplan ] , there are essentially two routes to the growth of nanolines@xcite . one growth method for the nanolines is to deposit a monolayer of bi below this threshold temperature , and then anneal the sample until bi nanolines form . after around 1 hour at around 500@xmath3c , this will result in a surface with bi nanolines and a background bi - covered surface@xcite . continued annealing of this surface will result in the desorption of the background bi forming a surface similar to that shown in fig.[fig : recipes](a ) , and eventually a surface similar to that shown in fig . [ fig : recipes](b ) is obtained . the second growth preparation route involves the deposition of bi onto the surface within the desorption temperature window . due to the lower surface coverage of bi , and the higher temperature , a more ordered surface is obtained , and fewer , longer , bi nanolines form . the final surface is much the same , however , corresponding to that shown in fig.[fig : recipes](b ) . a high density of bi nanolines can be built up by continuous bi exposure for a long time , at a temperature in the upper half of the desorption window . in this regime , the nanolines are only stable for a few minutes in the absence of a bi flux and small differences in stability , such as the ends of a nanoline , will be significant . shorter nanolines , perpendicular to the prevailing direction , will tend to dissolve , so that a highly ordered surface with long , parallel nanolines is obtained , as in fig.[fig : recipes](c ) . by use of a vicinal surface , the preference for one domain can be enhanced further , and a surface close to a single - domain surface can be obtained , as in fig.[fig : recipes](d ) . a second method which produces a majority - domain surface , even on a surface without any intentional miscut , ( the miscut angle is @xmath12 ) @xcite , produces a surface as shown in fig . [ fig : mcleanimage ] . in this case , a substrate temperature around 530@xmath3c was used , 4.5 ml of bi were deposited , with an anneal for around 40 mins . comparison of fig.[fig : recipes](d ) and fig . [ fig : mcleanimage ] reveals that in the former case , the nanolines are parallel to the prevailing step edges , while in the latter , they are perpendicular . despite the large difference in morphology , the only differences between this recipe and that which produced fig.[fig : recipes](c , d ) are that continuous deposition for an extended period was not used , and the heating current used was ac , a strategy which was designed to reduce electromigration . whether this is is the cause of the majority - domain surface remains unclear , but such large - scale single - domain surfaces are ideal for nanoelectronics applications . c , a ml of bi is stable on the surface , and the ( 2@xmath15):bi reconstruction results . annealing this surface above 500@xmath3c , or deposition of bi onto the clean surface above this threshold results in bi nanolines on a clean background . high coverages of nanolines are obtained by higher - temperature anneals , with longer bi exposure times . ] while stm can observe only a relatively small part of the surface , the large - scale order of the surface can be determined using electron diffraction methods such as rheed . a series of rheed patterns at different azimuthal angles of a single - domain bi nanoline surface are shown in fig . [ fig : rheed ] . perpendicular to the nanolines , the rheed pattern comprises vertical lines , while parallel to the nanolines , a ring feature is seen . the strength of these features demonstrates the long - range order generated by the nanolines , and means that their growth and development can be followed in real - time . 0 ] direction , perpendicular to the nanolines , a series of long streaks is seen , as in ( a ) . along the [ 110 ] direction , parallel to the nanoline , as in ( c ) , a distinctive ring feature is seen , while at intermediate angles ( b ) , the ring opens up into a long arc . ] in contrast to the silicide nanowires , which form very rapidly at the growth temperature and are only stable for about 10 mins . at that temperature , there is a significant incubation period before the nucleation of the bi nanolines , and they will continue to grow for upwards of an hour after the deposition of bi has finished . thus the bi required to generate the nanolines must come from a reservoir of bi which is present on the surface . where is this bi located , and how does it affect the growth of the nanolines ? in images of the surface taken soon after the end of the bi deposition , for example , fig.[fig : recipes](a ) , there is often a high density of small islands on the surface . these islands have the same appearance as the rest of the surface ; they are not islands of bi dimers , as may be seen after deposition at lower temperatures , but appear to be mixtures of si and bi atoms . in general , such small islands are not stable at these temperatures , as has been seen in hot stm studies of si homoepitaxy on si(001)@xcite . during growth , these small islands are stabilised by the high flux of si atoms , but if the flux is cut off , a process similar to ostwald ripening occurs , in which these islands will decompose , and the material in them will move to step edges ( which may be regarded as a very large island ) . the similarity of the behaviour observed here suggests that these islands are sustained by a flux of si and bi , which is produced during bi deposition . for low coverages of bi , the lowest - energy sites for bi dimers which have substituted for si dimers in the top layer of the si(001 ) surface hereafter `` bi surface dimers''are bi surface dimer / missing - si - dimer complexes@xcite , where a bi dimer decorates one or both sides of a 1dv , as seen in fig . [ fig : naitohmiki ] . for each adsorbed bi dimer , therefore , up to four si atoms are ejected from the surface layer , producing a significant transient flux of si atoms ; this flux in turn produces the observed islands . at the end of bi deposition , this flux will die away , the islands will no longer be stable and will decompose , as occurs between fig.[fig : promontory](a)and ( b ) . a cartoon of this process is shown in fig . [ fig : reservoir](d ) . once the small islands have annealed away , the surface morphology is very flat , and the background between the nucleating bi nanolines appears featureless in hot stm , apart from missing dimer trenches . however , at this stage a large surface density of bi may be inferred from the fact that bi nanolines continue to nucleate and grow on this surface . the bi dimers embedded in the surface layer have approximately the same contrast as the si dimers@xcite , and are therefore invisible at elevated temperature , as in fig . [ fig : reservoir](a ) . at a large positive bias , however , the background bi becomes visible , as is shown in fig . [ fig : reservoir](b ) . top - centre of this image is a bi nanoline . there is a high density of linear grey features in the background , one of which is marked with a pair of white arrows . detail of these features can not be discerned at this temperature , although the background bi is noticeably lower than the nanoline . after quenching the sample to room temperature , and exposure to hydrogen , we can judge the true coverage and distribution of bi during the annealing process , as displayed in fig . [ fig : reservoir](c ) . there is a high density of bi surface dimers , and also many bi dimers are adsorbed to one or both sides of a missing dimer trench , forming the bi1dv or miki structures , as shown in fig . [ fig : naitohmiki ] . thus the bi reservoir consists not of mobile ad - dimers , but mostly of bi surface dimers , usually decorating the missing - dimer trenches to form the miki structure . c , with a sample bias ca . + 3.0 v. grey linear features , which run parallel to the bi nanolines , can be seen in the background . an example is marked by white arrows . ( c ) : stm image taken at room temperature , with the silicon passivated with hydrogen to increase the contrast between the si and bi . the grey linear feature can now be seen to comprise bi dimers decorating a missing - dimer trench , i.e. the miki structure . ( d ) : a cartoon of the formation of the bi reservoir . impinging bi atoms ( 1 ) exchange with si atoms , making bi surface dimers and bi!dv complexes ( 2 ) . the ejected si atoms difffuse and form islands ( 3).[fig : reservoir ] ] during the high - temperature anneal , the surface bi reservoir is continuously depleted , with some bi going to form nanolines , and the remainder evaporating back into vacuum . the activation barrier for evaporation is different for different surface adsorption sites . a rheed study@xcite found that the activation barrier for evaporation of bi from the nanoline was 0.25 ev higher than the barrier for evaporation of bi from a ( 2@xmath1n):bi overlayer . hence bi is mostly being lost from the background reservoir . however , it might be expected that missing dimers in the nanolines would be observed occasionally . in fact , this is not the case , and those nanolines which are observed at high temperatures always appear perfect . it is likely that , due to the large defect energy of the haiku structure ( 0.66 ev from dft calculations@xcite ) , any bi which evaporates from the nanoline is quickly replaced from the bi reservoir . thus at high temperatures , where bi can evaporate from the nanolines , their stability is dependent upon a sufficient quantity of bi in the reservoir to replace missing dimers . at the later stage of an anneal , no new bi nanolines nucleate , and those which are present cease to grow . this is an indication that the reservoir of bi is approaching exhaustion . finally , when the reservoir is exhausted , evaporation from the bi nanolines will take place , and they will disappear within a few scans , or sometimes within the time of one scan . a sequence showing the evaporation of nanolines is shown in fig.[fig : linebreakup ] . in general , the nanolines evaporate from the ends , rather than breaking up into pieces , which strongly suggests that the ends of nanolines are less stable than the middle of the nanoline , and that the energetic barrier to break the nanoline in the middle and create two ends is prohibitive , even at these temperatures . the most notable feature of the mature nanoline surface as seen in fig . [ fig : recipes](c ) and ( d ) as well as fig . [ fig : mcleanimage ] is known as the promontory . long , narrow islands of si surrounding bi nanolines grow out across lower terraces . a series of images showing the early stages of bi nanoline growth are shown in fig . [ fig : promontory ] . in ( a ) , a short nanoline has grown out over a lower terrace and the joining of the promontory and the pre - existing step edge has formed a hole in the upper terrace , as is marked in ( b ) . the growth of this promontory is blocked by the growth of a second nanoline , which has nucleated in the time interval between ( a ) and ( b ) , and is growing out across the next lower terrace . comparison of ( b),(c ) and ( d ) makes it it very clear that the promontories are the result of growth out over a lower step edge . by ( d ) , a common pattern of zigzagging bi nanolines which have blocked each other s growth is developing . the mechanism behind this promontory growth has not been determined ; in an early paper , it was suggested that the promontories form by etching of si around the bi nanolines@xcite . however , in fig . [ fig : promontory ] , there is no sign of etching taking place ; this is a pure growth phenomenon . it is likely that the growth of the nanoline is related to the high mobility of step edges at 500@xmath3c . nanolines will always continue to grow until they reach an obstacle . thus nanolines will stop at step edges . however , the position of the step edge at this temperature is not constant ; it fluctuates back and forth@xcite , as material moves randomly along it . this motion may be inferred from the jagged , streaky step edges in these high - temperature images , as discussed in section [ sec : hot - stm ] . therefore , each time the step edge moves forwards , the nanoline can extend to the new step position , and then when the step edge tries to move back again , its position is pinned by the presence of the nanoline . with each forward fluctuation therefore , the nanoline will grow forward , and thus a promontory will gradually form . a similar process could explain the retreat of an up step ahead of a growing nanoline . the notable feature of the promontory is that they have a fixed minimum width . at least 3 - 4 nm of si is maintained on either side of the bi nanoline . this fixed width is an example of the phenomenon known as the defect exclusion zone , or `` dez '' , which is discussed in section [ sec : dez - sub ] . since the nanoline has a strain field associated with it , it is likely to exhibit quite strong interactions with other surface features which exert a localised strain on the surface . such features might be adsorbed bi dimers , missing dimer defects , step edges , and other nanolines . using tight - binding simulations , where very large cells can be relaxed , the interaction between a nanoline and several features , such as 1dv defects and rebonded b - type step edges have been calculated . these data are plotted in fig.[fig : deznewfig ] . from elevated - temperature stm images , it is clear that there is a repulsive interaction between the nanolines and missing dimer defects and step edges . at high temperature , as in fig . [ fig : deznewfig ] , the surface is covered in black streaks , which indicate the positions of rapidly moving missing dimer defects . however , either side of the nanoline , there are no streaks . likewise , where a nanoline forms a promontory , the width of this empty area is continued as the width of the promontory . this characteristic width of defect - free silicon , around 3 - 4 nm either side of the nanoline , is known as the `` defect exclusion zone '' or dez @xcite . for a large separation , the total energy of a nanoline and a defect is the same as for the two structures independently . however , as the separation decreases , the total energy increases , so that at a distance of about 3.5 nm , there is a repulsive interaction between the nanolines of ca . 0.1 ev for both step edges and 1dvs , and at smaller distances , this energy increases rapidly . statistics about the position of 1dvs in the space between two nanolines have been gathered , as shown in fig . [ fig : deznewfig ] , taken from ref . they show that on average , most defects occur at least 3.2 nm away from the nanoline , which is in agreement with the calculated repulsion . for a pair of nanolines , there is a similar interaction , and the 0.1 ev threshold is also reached when the centres of the two nanolines are about 3.5 nm apart , or in other words , the gap between the nanolines is the width of one nanoline . clusters of nanolines with this approximate spacing are often seen , as in fig . [ fig : deznewfig ] . for nanolines adjacent to each other , the excess energy is about 1 ev per unit cell . despite this , nanolines are sometimes observed growing next to each other , as in fig . [ fig : recipes](c ) . the reason for this is that unlike defects and step edges , which are mobile , a nanoline can not move sideways . hence when two nanolines which have nucleated in different places grow past each other , they can not move sideways to reduce their interaction energy . the presence of nanolines in close proximity does indicate that this interaction energy is not so high as to present an insuperable barrier to the growth of two nanolines past each other . for bi surface dimers , which have a compressive stress field , decoration of missing dimer trenches is a natural way in which to relieve their stress . the gain in energy by this process is about 0.45 ev / bi dimer in tightbinding calculations@xcite . the interaction with the nanoline is also attractive . in images where there is a large surface density of bi dimers , they are often situated adjacent to the nanolines . this position is found to be 0.28 ev better in energy@xcite than elsewhere . this suggests that the nanolines are generally more stable in a surface with a significant amount of compressive stress , such as one which is covered in bi . indeed the energy of a nanoline in a bi - terminated surface is 0.8 ev better than on a si - terminated surface . c. a series of 81 nm stm images . during the interval from ( b ) to ( c ) of 14 min . , the bi line on the bottom extended to separate the big bi island and to make an inlet as marked by a white arrow . during the same period , a new bi line appeared as marked by a black arrow . ( d ) to ( e ) shows more details of observation corresponding this latter event , in the area marked by a white box in ( c ) . the time between images was 18 sec . in figure ( e ) a small bi island is situated at the place marked by a black arrow , in ( f ) the small bi island changed into a short bi line , and between ( f ) and ( g ) the new line extends in both directions towards the island and the inlet . it is noted that the side edges of the inlet , and the shapes of all the islands change image by image . reprinted from surf . 421 , miki et al . bi - induced structures on si(001 ) , p.397 , ( 1999 ) , with permission from elsevier . ] despite the success of the haiku structure in explaining all of the above surface phenomena , its identification raises as many questions as it answers . how would such a complex structure form ? how is it terminated ? if termination is energetically unfavourable , what would a nucleus structure look like ? experimentally , the atomistic nucleation process remains a mystery . there is an incubation time during the annealing process before nanolines start to nucleate , which suggests that there is a significant barrier to formation of these structures . on many occasions , the appearance of bi nanolines has been captured during stm observations , such as in fig . [ fig : nucleationstm ] . there is no apparent precursor state ; like athena from zeus brow , a short nanoline springs fully - formed from an empty patch of the surface . the nanolines then grow extremely rapidly . thus far , there has only been one proposed mechanism for the nucleation of a bi nanoline@xcite . it has been suggested that the initial step is for bi ad - dimers to fall into the missing dimer trenches on the hot si(001 ) surface , forming a row of pairs of bi dimers . such a structure would result in the increase in local compressive stress , which may provide a driving force for nanoline nucleation . we demonstrated earlier ( fig.[fig : haikucartoon ] ) that the 7 - 5 - 7 structural motif may be achieved by a simple bond rotation process . using this motif , the proposed mechanism suggests that the central four atoms between the two bi dimers in the proposed mechanism twist , so that a zigzag core structure is formed . from this the haiku structure evolves , with a maximum barrier in the calculation of less than 1.5 ev . the reaction pathway and associated energy diagram are shown in fig . [ fig : nucleation ] . this mechanism has a number of interesting features , but also a major limitation . the calculation takes place in a periodically repeated unit cell one dimer row wide , and is therefore considering the nucleation of an infinite nanoline , _ without _ ends . it is suggested that a long nucleus forms by the filling of several unit cells of a missing dimer trench , but this is an unstable situation@xcite . moreover , in our stm observations , we observe the appearance of very short nanolines , which then grow . the termination of these finite nuclei _ must _ therefore be considered . stm observations of nanoline evaporation show that nanolines do not break up in the middle , instead they evaporate from the ends . from this , it may be inferred that the energy of termination is quite high . while there has been no systematic study of termination , we have performed calculations on many and varied structures , all of which have been found to have a high energy . in tightbinding modelling , the energies were at least 1 ev / end , while those terminations which we have considered for the proposed zigzag nucleus structure have much higher energies , around + 1.6 ev / end , giving a total nucleus energy of over + 4 ev . this suggests both that understanding the termination of the nanolines is an important area for future research , and that nucleation structures should be considered in 2d , rather than in 1d ( which may not be feasible with a technique more accurate than tightbinding ) . given that the prevalent structures on the bi - rich surface are bi surface dimer/1dv complexes , alternative nucleation mechanisms which use these structures as a basis may be more fruitful and are being explored . the knowledge of the electronic structure of the bi nanoline has stemmed from two sources : variable - voltage stm@xcite and dft calculations of the electronic structure@xcite ( generally considered in terms of the local density of states ( ldos ) , or the tersoff - hamann approach to stm , which uses the ldos ) . knowledge of the atomic structure of the bi nanoline allows detailed calculations of its electronic structure to be made , which can then be compared to the observations made in stm . while the majority of the electronic contrast is expected to be dominated by the bi dimers in the nanoline , other details of the ldos have allowed a stronger identification of the haiku structure with the bi nanoline to be made , with the other possible candidate structures ruled out after comparison with the stm data . 10 nm images of a bi nanoline . the sample bias voltages used are -2.0v , -1.2v , -0.6v , -0.3v , + 1.5v and -0.8v , in ( a - f ) respectively . as the sample bias is reduced , between ( a ) and ( d ) , the nanoline changes contrast from light to dark relative to the surrounding si(001 ) . over this range , some of the dimers in the nanoline ( marked a in ( c ) and ( e ) ) exhibit a different voltage contrast . at very low biases , around -0.3v , an enhancement of the dimers around the nanoline , similar to that seen around a missing dimer defect in clean si(001)@xcite , is seen . this is visible ( and marked schematically as b ) in ( d ) . in ( f ) , the resolution is sufficient to see that the corrugation of the si dimers closest to the nanoline is increased , ( marked schematically by the dotted black lines ) suggesting a greater separation , and hence tensile strain . reprinted from surf . 527 , owen et al . interaction between strain and electronic structure in the bi nanoline , p.177 , ( 2003 ) , with permission from elsevier.[fig : varbias ] ] we begin by summarizing the data available from stm , which is illustrated in fig . [ fig : varbias ] : 1 . at high biases , the bi nanolines appear bright compared to the surrounding si(001 ) surface@xcite 2 . at low biases , the bi nanolines become dark compared to the surrounding si(001 ) surface@xcite 3 . at low biases , the si dimers neighbouring the bi nanoline show enhancement , or brightening@xcite similar to that seen around missing dimer defects on si(001)@xcite . these facts suggest various things : first , that the bi dimers are physically higher than the surrounding si(001 ) surface dimers , as at high biases the contrast in stm is determined primarily by geometrical structure ; second , that the _ local _ electronic structure of the bi dimers will show states further from the fermi level than the surrounding si dimers@xcite , as this determines the stm current at low biases ; finally , that the bi nanoline structure must be under tensile stress , which causes the si dimers near the nanoline to be pulled together , away from their relaxed surface structure , which is the origin of enhancement at low biases@xcite . examination of the detailed electronic structure of the miki and naitoh models@xcite and the haiku structure@xcite all show that the bi dimers become dark at low bias voltages by contrast to the surrounding silicon dimers . the calculated dos for the miki and haiku structures are shown in figure [ fig : miwados]@xcite . the @xmath13 peak comes from the up atoms of the substrate si dimers , as does the @xmath14 , while the @xmath15 peak comes from both the bi and si dimer bonds and @xmath16 from the bi atoms in the line models . when the substrate dimers are passivated with h , the peaks @xmath13 and @xmath14 almost disappear leading to peaks in the gap which arise from the bi dimers ( @xmath15 and @xmath16 ) . this data confirms earlier modelling and bias - dependent stm measurements@xcite discussed below . consideration of the electronic structure of the bi dimer reveals that it is rather unusual . in its native form , bi forms puckered hexagonal sheets , similar to the structure taken by as . full - potential lapw modelling of bi in a bulk diamond cell ( thus forcing it to take up tetrahedral bonding ) showed that there is very little hybridisation between the s- and p - orbitals@xcite . thus we expect bi to tend towards p@xmath17 bonding with a filled s - orbital ( as a lone pair ) and to be energetically most favourable with 90@xmath4 bond angles . this explains to some extent the haiku structure , where bond angles are almost 90@xmath4 . projection of the local densities of states calculated using pseudopotential - based dft calculations@xcite suggests that the bi dimer in the haiku structure has hybridisation between s and p@xmath18 , with p@xmath19 and p@xmath20 separated off . the flapw calculations showed that the relativistic effects of the full core increased the s - p splitting in the bi atom ; while relativistic effects can be incorporated in pseudopotentials , the haiku structure might be more stable , and show bonding closer to p@xmath17 , if these effects were taken fully into account . the bi - bi bond is a rather weak bond , with the bi - si bonds providing most of the energetic stability . this can be seen by considering the electronic structure of a bi ad - dimer on the clean surface . the electron localisation function ( elf)@xcite for this ad - dimer is shown in figure [ fig : bidimerelf](a)@xcite . the elf gives a quantitative analysis of the bonding properties of a system , with a value of 0.5 equivalent to the localisation seen in the homogeneous electron gas and a value of 1.0 equivalent to complete localisation . the figure shows an isosurface with elf=0.8 ( equivalent to reasonably strong covalent bonds ) . it can be seen that the si - si bonds are rather strong , as expected , and that the bi - si bonds are still visible at this level , though they are weaker than the si - si bonds . the bi - bi bond is not visible ( in fact it has a maximum value around 0.7 ) but the bi lone pairs are visible as rather diffuse , spherical objects ( by comparison to the up atoms of the substrate dimers ) . this implies that the bi - bi bond is rather weak , and the lone pairs would not be available for bonding@xcite . the energy barrier to breaking the bond in this bi dimer has been calculated with dft to be 0.17ev going from over the dimer row to over the trench between rows and 0.15ev going in the other direction , and the barrier is shown in fig . [ fig : bidimerelf](b ) . having the bi dimer over the dimer row is more stable than over the trench between rows by about 0.02ev . this implies that there should be a finite chance of imaging the haiku structure with the bi dimers out - of - phase with the underlying si dimer rows . however , we note that with the energy difference noted above , the dimer would be over the row 98% of the time at room temperature , and the likelihood that a whole line of dimers would flip , will be even smaller . there are occasional images with the dimers out - of - phase , but these are rare . , title="fig : " ] , title="fig : " ] in the next section , details of some elements which will attack the bi - bi bond will be presented , along with others that will not . here we consider the local structure of the bond . when the bi - bi bond is over the haiku core , it is almost perfectly relaxed , with bond angles extremely close to 90@xmath4 . insertion of any species into the bi - bi bond will distort these angles ( and the angles formed with the underlying si atoms ) , with a potentially large energy penalty . thus the bi - x - bi angle should be relatively close to 180@xmath4 with short bi - x bond lengths if an element x is to insert successfully into the bond . furthermore , the lone pair on the bi dimer , which might be expected to be a strong electron donor , is in fact rather passive , as it lies some distance below the fermi level@xcite . the implications of this for the chemical reactivity of the bi nanoline are discussed below in section [ sec : reactivity ] . the phenomenon of strain - enduced enhancement at low stm bias voltages has been explored before@xcite ; we reproduce the arguments here for convenience@xcite . the si(001 ) surface shows a dimer reconstruction which provides three bonds per surface si atom ( compared to two bonds per atom for the bulk terminated surface ) , but which pulls the bond angles and distances away from the ideal tetrahedral values . the net result is that the local band gap is reduced ( in simple chemical terms , the hybridisation towards sp@xmath17 is decreased , leading to a decreased splitting between the resultant hybrid bonds ) . if the surface dimers are then further distorted , their local states will move closer to the fermi level . the single missing dimer defect ( 1dv ) has second layer atoms bonding across the defect which distorts the dimers either side of the defect ( though lowering the energy because the second layer atoms have no dangling bonds ) . this distortion in turn causes the states near the fermi level on the neighbouring dimers to move towards the fermi level , leading to a brightening or enhancement at low biases . low - bias simulated images of the naitoh and miki structures@xcite show that the si dimers adjacent to the naitoh become _ darker _ than the surrounding si dimers as the fermi level is approached , while those on the miki model are unchanged . by contrast , with the haiku structure , the si dimers immediately adjacent to the nanoline become brighter closer to the fermi level@xcite , in good agreement with the stm images . a comparison of high and low - bias simulated images of the 3 structures are shown in fig . [ fig : simstm ] . however , the prediction from the miki model does agree very well with the appearance suggested from stm images@xcite , in which the shoulder dimers around a 1dv on a bi - rich surface become dark at low bias voltages . on the basis of these calculations , we suggest that the lack of enhancement at low bias voltages indicates almost complete relaxation of the strain of the 1dv by the presence of the bi dimers in the miki model . taking the principle of strain - induced enhancement further , the local subsurface strain could be determined by cross - sectional stm of the bi nanoline , of which there is none , or from ldos calculations , by projection of the charge density on a plane perpendicular to the ( 001 ) surface . we show the projected charge density from gga calculations for the clean si(001 ) surface , the haiku structure and the haiku structure with hydrogen on the si(001 ) surface in fig . [ fig : ldos]@xcite , on the ( 1@xmath210 ) plane , parallel to a si dimer row . by comparing in particular the clean si(001 ) surface and the clean haiku structure , the change in different states induced by the haiku can be seen . beginning with the clean surface , the localisation of the states close to the fermi level in the top few surface layers shows the strain induced by dimerisation . in the haiku structure , these states are even more strongly localised on the dimers immediately adjacent to the haiku structure , which causes their relative enhancement in stm . meanwhile the triangular si substructure of the haiku , c.f fig . [ fig : haikucore ] , is somewhat relaxed compared to the clean si dimers , although it is strained relative to the hydrogenated dimers and perfect bulk si . this goes some way to explaining the stability of the subsurface 5 - 7 - 5 ring structures , which serve as a highly effective relief mechanism for the epitaxial stress exerted by adsorbed bi . the absence of states close to the fermi level suggests that the nanolines are likely to block surface conduction perpendicular to the nanoline , and are not likely to act as a nanowire . the lack of any states on the bi near the fermi level is clearly the cause of the darkening of the nanoline relative to the si(001 ) surface seen in stm at low voltages . in the full charge densities(fig . [ fig : ldos](iv ) ) , the density of states associated with the bi dimers , combined with their higher physical height , explains their relative brightness in stm at higher bias voltages . in the case of the hydrogenated surface , the bi dimers have a similar charge density as in the clean surface , but in this case , the si @xmath22-bonds have been eliminated , so the bi dimer remains bright in stm at all biases@xcite . as described above in more detail , the bonding orbitals in the bi dimers have essentially a p@xmath23 bonding character , while the lone pair is a diffuse s - type orbital , some 10 ev below the fermi level@xcite . thus the bi dimer is in a very stable electronic state , and is not expected to be very reactive . moreover , this low - energy configuration is dependent upon the maintenance of bond angles close to 90@xmath3 , which will have an effect on the energetics of any insertion into the bi - bi dimer bond . however , the low - bias images shown above reveal that although the nanoline appears clean at high biases , some contrast between different dimers is visible at lower biases , suggesting that some chemical attack has occurred . we have proposed that the bi nanolines , although semiconducting themselves , might be used as templates for deposition of other materials , such as metals , active molecules , nanoparticles etc.@xcite . in order for the nanoline to have utility in this role , it must be possible to mask the substrate around the nanoline , so that adsorbants stick preferentially to the nanoline . it is also important , for any nano - electronics applications , to have some method for rendering the substrate insulating after growth of the nanoline . the chemical reactivity of the bi dimers , and the nanoline as a whole , is therefore crucial to any implementation of this concept . atomic hydrogen has been used as a mask on the bare si(001 ) surface for stm lithography experiments@xcite , in which an stm tip is used to remove lines or areas of hydrogen from the surface , and thus provide areas onto which metal will adsorb , as in fig . [ fig : hashizume ] . however , the writing process is slow , and lacks scalability . use of the bi nanolines as templates would remove the need for this writing , while providing long , and more uniform patterns . hydrogen is therefore a natural species to use as a mask around a bi nanoline . one of the most favourable properties of silicon as an electronics material is the high quality of its oxide film . oxidation of the silicon around the bi nanoline is therefore a natural way in which to isolate the nanoline electrically from the substrate . the usual method of oxidation of silicon is by exposure to molecular oxygen or to water at elevated temperatures , but this only gives microelectronics - quality oxide when grown at ca.800@xmath3c . since the bi nanoline is not stable above 600@xmath3c , ozone was considered as an alternative for low - temperature oxidation . ozone preferentially attacks the backbonds of the surface si dimers , forming a stable oxide , even at room temperature@xcite . more recently , the need for high - k dielectrics has driven the development of silicon nitride and oxynitride as alternatives to silicon dioxide . nitridation is usually done using ammonia , which breaks up on contact with the clean silicon surface above 200 k@xcite . as an added bonus , when ammonia reacts with silicon , the nitrogen moves below the surface , but the hydrogen remains on the surface , passivating it . it can therefore play a dual role , as insulating reagent , and as masking material . the reaction of these species with the bi nanoline are discussed in the sections below . the bi nanoline has been shown to be almost completely inert to attack by atomic hydrogen@xcite . fig.[fig : bihydrogenoxygen](a ) shows a close - up image of a nanoline on a si(001):h monohydride surface . in this image , nanolines were formed at 590@xmath3c , and the surface was cooled to 330@xmath3c , the surface was saturated with atomic h. the surface has the monohydride phase , with the si - si @xmath24-bond intact , and the h atoms saturating the dangling bonds . thus each si - h dimer is resolved as a pair of dots . some white dots on the si surface may be holes in the monohydride layer , stray bismuth atoms , or adsorbed -oh resulting from water impurity in the hydrogen . while the nanoline is almost completely unreacted , one dimer has become dark . this sort of defect is extremely rare . the preferential adsorption of the h onto the si is consistent with dft - lda modelling , which found that h on a bi dimer is 3 ev worse than h on a si dimer . there is therefore a strong driving force for any h which does adsorb onto the bi nanoline to diffuse off it , and onto the si . at 300 k. there is virtually no attack on the nanolines , while the substrate has been oxidised . ( c ) : after exposure to 150l o@xmath11 at 700 k and an anneal to 840 k , some gaps appear in the nanoline , but most of it is unaffected . ( d ) : exposure of a h - terminated surface with bi nanolines to air results in complete oxidation of the nanolines , which now show up dark relative to the h - terminated si.[fig : bihydrogenoxygen ] ] the oxidation properties of a monolayer of bi on si(001 ) shows an unusual changeover in behaviour with oxygen exposure . an eels / auger study@xcite found that for small exposures , the reactivity of a bi overlayer on si(001 ) is 100 times less than the reactivity of oxygen with the bare silicon surface . however , for large exposures , above ca . @xmath25 l o@xmath11 , the oxygen auger intensity continues to increase for a bi / si(001 ) surface , at a point where the signal for the clean si(001 ) surface has saturated . thus for very large exposures , the bi promotes oxidation of the silicon . moreover , the stoichiometry of the oxide formed on the bi / si(001 ) surface is closer to sio@xmath11 , which would imply better insulating properties . semiempirical calculations found that insertion of oxygen into the bi - bi bond is energetically much less favourable than for sb or as ( 0.32 ev vs ca . . moreover , insertion into a bi - si bond or a si - si bond is much more favourable ( 4.5 ev and 5.5 ev respectively ) , so that preferential attack of the silicon is expected . the reactions of molecular oxygen and ozone with the bi nanoline show a similar changeover in behaviour@xcite . the surface was exposed to ozone at room temperature and , for comparison , to molecular oxygen at 400@xmath3c . both surfaces were annealed up to 570@xmath3c , near the stability limit of the bi nanoline . the room - temperature ozone experiments , using exposures up to 20 l of o@xmath26 , revealed that the silicon background could be saturated without visible damage to the nanoline . in large - scale images , the bi nanoline appears to be completely unaffected , as shown in fig . [ fig : bihydrogenoxygen](b ) . however , in small - scale images , and by varying the imaging bias , some subtle differences between the appearance of various dimers in the nanoline could be seen@xcite . this variation may be due to attack of bi - si backbonds , as found to be favourable in mndo simulations@xcite , which would leave the bi - bi dimer intact , but perturb its electronic structure somewhat . annealing this surface introduced some defects into the nanoline . after annealing at 450@xmath3c , gaps appear in the nanoline , which comprise 1 - 2 unit cells . this trend continued with a further anneal to 570@xmath3c , as shown in fig . [ fig : bihydrogenoxygen](c ) . such piecemeal damage may be explained by the observation that the activation barrier for bi to desorb from oxidised si has been found to be 0.7 ev lower than for unoxidised si@xcite . the small contrast changes , which were seen after ozone exposure at room temperature may therefore indicate reaction with bi - si , or si - si backbonds around the nanoline , so making that unit cell more susceptible to evaporation during high - temperature annealing . molecular oxygen , when adsorbed at 400@xmath3c , behaved in a very similar way to ozone . again there was preferential attack of the silicon , leaving the nanoline mostly intact , but also gaps appeared in the bi nanoline after annealing for about one hour@xcite . while the bi nanoline is remarkably resistant to oxidation in uhv , it quickly oxidises when exposed to air , as shown by the following simple experiment . a h - terminated si(001 ) surface with bi nanolines was taken out of the imaging chamber into the uhv system loadlock , which was then vented to atmosphere for 30s , and then immediately pumped back down to uhv . the resulting surface is shown in fig . [ fig : bihydrogenoxygen](d ) . the nanolines , which previously were bright at all biases relative to the h - terminated background , are now black lines , indicating that they have suffered significant attack . this agrees well with the previous study , which found that for very large exposures to oxygen , the bi dimers promoted oxidation of the underlying silicon . the adsorption of ammonia onto si(001 ) as a precursor to silicon nitride or oxynitride growth has been studied by a variety of experimental and theoretical techniques , which concluded that at room temperature , the ammonia dissociated on the surface , forming nh@xmath11 and h groups@xcite . we have found that ammonia attacks the silicon preferentially at room temperature . an image of the bi : si(001 ) surface with ammonia termination is shown in fig.[fig : biammonia ] . within each dimer , the light grey end has been identifed with the nh@xmath11 group , while the dark grey end is the h atom@xcite . the apparent height difference in stm matches the physical height difference between these two species . there is some ordering of the nh@xmath11 groups , such that zigzag patterns and straight lines are seen running along the dimer rows . reaction of ammonia with the bi nanoline might be expected to proceed by the initial formation of a hydrogen bond between an ammonia h atom and the bi lone pair , followed by insertion into a bi - bi or bi - si bond . however , as discussed above , the lone pair on the bi dimer is essentially inert@xcite . during experiments on a surface which was pre - exposed to atomic h to saturate the si dangling bonds , for exposures of up to 70l of ammonia at substrate temperatures of up to 500k , the bi nanoline was largely unaffected . however , adsorption of ammonia onto a bi nanoline surface where the si dimers are not passivated does result in some reaction with the nanoline , even at room temperature . this difference may indicate that ammonia reacts first with the silicon , forming active nh@xmath11 species , which are mobile and can attack the bi dimers . calculations suggest that it is energetically favourable for an nh@xmath27 group to insert into the bi - bi dimer bond , and form a symmetrical bi - nh - bi feature . however , stm images of reacted bi dimers shows an asymmetric feature , not unlike the feature seen on background si dimers . examples may be seen in the inset to fig . [ fig : biammonia ] . nevertheless , the probability of attack by ammonia is low , providing the promise that ammonia can be used to grow silicon nitride without damage to the nanoline . 64 nm stm image of the ammonia - terminated bi : si(001 ) surface . the background si is completely terminated by a mixture of h and nh@xmath11 groups . dark patches on the bi nanoline reveal some attack by ammonia . the inset shows that reaction occures within the space of a single dimer , and the resulting feature is asymmetric , much like the termination of the background si . ] metal deposition onto the bi nanolines is currently under investigation . deposition of ag onto bi nanolines on the clean si(001 ) surface@xcite reveals a strong preference for adsorption onto the si , leaving the bi nanoline clean ( in complete contrast to the behaviour of ersi@xmath9 nws on si(001 ) discussed above in sec . [ sec : reactivity-1 ] ) . however , the passivity of the bi nanolines to atomic hydrogen and ammonia allows these species to be used as nanoscale masks . ag is known not to adsorb onto h - terminated si(001)@xcite as shown in fig . [ fig : hashizume ] , while in completely dewets from the si(001 ) surface , forming large droplets when h is adsorbed onto an in - covered surface@xcite . in all cases , following metal deposition onto an ammonia - terminated si(001 ) with bi nanolines , preferential adsorption onto the bi nanoline ( or onto background bi ) was seen , indicating that the ammonia - terminated surface is very effective as a mask against metal deposition . the reaction of the metal with the nanoline template is more complex , however . two types of behaviour have been seen with different metals , which may be described as wetting and non - wetting behaviour . wetting occurs when the interaction of the metal with the nanoline is stronger than the interaction of the metal with itself . nonwetting occurs when the interaction of the metal with itself is stronger . examples of each type of behaviour for different metals , in and ag , are shown in fig . [ fig : metal ] . adsorption of in results in significant intermixing of in and bi , forming a zigzag island structure , which is thought to result from a chain of in and bi atoms , as shown in the inset to fig . [ fig : metal](a ) . second - layer islands have a distinctive bright hexagonal feature , which is 19 or 2.5 dimer rows wide . this rational relationship between the in island dimension and the underlying si unit cell distance demonstrates that these in islands grow _ epitaxially _ on the nanoline template , raising the possibility of the growth of long single - crystal nanowires on the si(001 ) surface . by contrast , adsorption of ag results in the formation of small clusters of ag , around 0.5 nm in height , at a very early stage of deposition , as shown in fig . [ fig : metal](c ) . further deposition at room temperature results in an increase of the number density of these nanoclusters , but the modal size increases only slightly from 0.5 nm to 0.6 nm , with a strongly peaked height distribution . this behaviour may be thought of as non - wetting behaviour . while the bi nanoline has a number of interesting properties , for any nano - electronics device application it must be possible to passivate them against atmospheric attack . one way in which to do this is to bury them in a layer of epitaxial silicon . moreover , the burial of the bi atoms may lead to their acting as a 2d delta - doping layer , with close to atomic - layer precision in the depth placement . however , as with other nanostructures , for example burial of inas / gaas quantum dots in gaas , the process of burial is likely to change the shape and detailed structure c. most of the bi has segregated to the surface , and the nanolines have been destroyed . ( b ) : the bi nanoline surface ( a ) has been capped with a monolayer of bi at 400@xmath3c , before overgrowth . after overgrowth at 400@xmath3c up to position ` b ' , the growth layer was buried in amorphous si by further deposition of si at room temperature . in this case , nearly all the bi from the bi nanolines remains at the original surface position , ` a ' . ( c ) : as ( b ) , except that the capping layer material was sb , in place of bi . here most of the bi has segregated to the growth surface ( b ) , leaving sb at the original surface ( c ) , indicating that the bi has exchanged with sb from the capping layer . ] simple overgrowth of a surface containing bi nanolines results in the destruction of the nanolines@xcite . as shown in the secondary ion mass spectrometry(sims ) profile in fig . [ fig : sims](a ) , after overgrowth of si the vast majority of the bi has segregated to the surface of the grown film , although there is a small peak in the bi concentration at the original surface position . furthermore , x - ray analysis@xcite shows that the 1-d character of the nanoline is lost after capping , even by an amorphous overlayer deposited at room temperature . the mechanism by which the bi floats to the surface during growth is thought to involve exchange of subsurface bi atoms with surface si atoms , as bi is an effective surfactant during growth . in order to prevent this , a further surfactant layer of bi can be deposited around and over the bi nanolines , before si overgrowth@xcite . in this case , the exchange mechanism is inoperative , as the second - layer bi in the nanolines will not exchange with first - layer bi , and so it is possible to block the segregation of the bi nanolines to the growth surface . in the sims profile in fig . [ fig : sims](b ) , a thin crystalline si overlayer has been grown at 400@xmath3c after deposition of a bi layer , and this has been capped by depositing further si at room temperature , so as to bury the growth surface in amorphous si . in this case there , are two peaks in the sims profile . ` a ' refers to the bi density at the original bi nanoline surface , while ` b ' refers to the surface of the crystalline overgrowth . the surface of the overlayer has approximately 1 ml of bi on it , from which the quantity of buried bi can be determined . this is approximately 5% ml , which is typical for a bi nanoline surface . thus , using the bi surfactant layer , most of the bi nanolines have been preserved at the original surface . in order to gain more information about the exchange process , a surfactant layer of sb was also used , instead of the surfactant layer of bi . the resulting sims profile is shown in fig . [ fig : sims](c ) . in this case , there is a large peak of sb at the original nanoline surface position ` b ' , while a considerable fraction of the bi has segregated to the surface of the overlayer ` c ' . this result indicates that there is considerable exchange between the sb surfactant layer and the bi nanolines , and by this method , a layer of sb has been successfully buried , with a sharp peak at the original nanoline surface . while this exchange process is successful at burying the bi atoms , it does not provide any information about the structure of the buried bi . for device applications , it is important to retain the 1-d character of the original nanoline . it is also of interest whether the haiku structure will be preserved . stm can not observe subsurface features , and instead the structure of the buried bi nanolines was studied using an x - ray method@xcite . the results of this analysis@xcite indicate that two important elements of the bi nanoline structure have been preserved : the 1-d character , and the dimerisation of the bi atoms . tightbinding and dft calculations of the buried bi nanoline structure found that the haiku structure was not stable , and broke down into a hollow 1-d structure with bi dimers bonding to the silicon grown _ over _ the nanoline , and the si substructure reconstructing into something similar to a 2dv surface defect . a ball - and - stick model of this proposed structure is shown in fig . [ fig : buriedhaiku](a ) . it has significantly lower energy per bi atom than other proposed structures , including substitutional and interstitial bi defects . this structure may be compared to the cross - sectional tem image of buried bi nanolines in fig . [ fig : buriedhaiku](b ) . two nanolines are present within the tem image . the one on the left has strain associated with it , and a dark core , which may match with the proposed structure . the one on the right has triggered misfit dislocations along \{111 } planes , c.f . [ fig : haikucore ] . these dislocations , which indicate the presence of a large local stress , may suggest that the haiku substructure of the nanoline has survived overgrowth . a recent x - ray standing wave ( xsw ) study of a bi nanoline sample which had been capped with amorphous si at room temperature@xcite has cast doubt on the validity of the haiku model . however , more detailed analysis of lines buried with amorphous silicon@xcite showed that the dimerisation and one dimensional character of the nanolines is destroyed by burial in amorphous silicon ( and , as discussed above , burial in crystalline silicon changes the haiku reconstruction while leaving the one dimensional , dimerised character of the nanolines unaffected ) . furthermore , the presence of a significant quantity of bi at the growth surface may have an effect on the results obtained , unless it is removed@xcite . we have given a broad overview of the different material systems and experimental methods which result in self - assembled 1d nanostructures on semiconductor surfaces , particularly the technologically important si(001 ) surface . there are many different self - organising systems which result in 1d nanoscale features on the surface of si(001 ) and si(111 ) . aside from the pt / ge(001 ) system , none of these approach the length , and degree of perfection , of the rare - earth silicide wires and the bi nanolines . the metal / si(111 ) nanowire systems will tend to form 3 equivalent domains , limiting the long - range order of the nanowires , while the methods which rely on step - edge adsorption are limited by the spacing of step kinks ( though this can be controlled with careful sample preparation ) . the rare - earth silicide nanowire family are not nanowires _ per se _ ( in that they are a metastable state , not seen in a bulk structure ) , but are better regarded as a conventional heteroepitaxial system in which the lattice mismatch is anisotropic . even so , while many different rare earths superficially appear to behave in a similar fashion , review of the literature indicates that in fact several different nucleation and growth mechanisms are responsible for the formation of nanowires within this family , both kinetic and thermodynamic . the recent observation of nanowires with an orthorhombic crystal structure@xcite , rather than the hexagonal structure previously reported , suggests that there is much more to be learnt about this family of materials . experimental studies of all these systems has so far been largely confined to stm , apart from x - ray and tem studies of buried bi nanolines , and tem of the endotaxial re nanowires . a much greater understanding of the physical , chemical and electronic properties of these systems , would come from other experimental techniques , such as optical and vibrational spectroscopic techniques . ( for example , the strained haiku core of the bi nanoline may have a signature bond vibration frequency . ) the main focus of this review has been the bi nanoline system . we have described the physical and electronic structure of the bi nanoline , its reactivity with different materials , and the surface phenomena associated with it . although there has been no direct observation of the core structure of the haiku model , it provides a natural explanation for all the observed experimental phenomena , and is well - supported by experimental and theoretical data . the fundamental source of many of the properties of the bi nanoline is the subsurface core of si which has recrystallized into a hexagonal phase , lonsdaleite ( known from meteorites ) . the formation mechanism of this core structure remains unknown . a significant activation barrier is expected , given the high temperature and long incubation time required to form the nanolines , and the co - existence of the nanolines with the miki structure , which is less stable , but kinetically easy to form . while mechanisms proposed thus far have concentrated on the twisting of pairs of atoms , an alternative may well be a stacking - fault - like shift of the entire core from the cubic phase to the hexagonal phase . given their structure , it is clear that the bi nanolines are not a conventional heteroepitaxial system like the silicide family . there is no variation in width , either with annealing temperature or time only the number density of nanolines and the length changes with further deposition and annealing . the nanolines are only ever one unit cell wide , although clusters of nanolines are sometimes seen . the nanolines are not the result of simple adsorption onto symmetry - breaking features of the surface , such as step edges , or linear defects . nor are they a periodic reconstruction of the surface , as in the pt / ge wires@xcite . they might best be described as an adsorbate - stabilized surface recrystallization . the chemical properties of the nanoline stem from the behaviour of the bi dimer with its preference for p@xmath23 bonding and an inert s - type lone pair , rather than the sp@xmath23 hybridization of si dimers . although the bi - bi bond is weak , it is also passive , and only in shows any significant reaction with it , probably because the zigzag structure produced allows the bi atoms to retain their preferred 90@xmath3 bond angles . hydrogen does not attack the bi nanoline , while ozone will oxidise around the nanoline at room temperature , leaving it intact . likewise ammonia preferentially attacks the silicon , although some damage to the nanoline occurs at small fluxes . the determination of both the physical and electronic structure of the nanoline has demonstrated the importance of a close interaction between experiment and theory . we note that all the recent publications on the bi nanoline have been a collaboration between the two . the method of tightbinding has proved to be a valuable tool for quickly searching through a large variety of possible structures , as in for instance the haiku structure , or for relaxing a large number of atoms , as in modelling of the nanoline kink energies . at the same time , the tightbinding method retains a quantum mechanical description of the energetics , so that the calculated energies are in most cases in reasonable agreement with those calculated using dft . the bi nanoline has benefitted from an unusual degree of theoretical investigation . a similar level of theoretical work for the rare - earth nanowires is likely to deepen greatly the understanding of the different nucleation and growth phenomena that have been identified in that family of systems . with a significant body of work devoted to the fabrication and structure of the nanoline systems , attention should be paid to their possible applications . the length , straightness and perfection of self - assembled nanolines lend themselves to use as a nanoscale template . the most obvious candidate systems are the rare - earth nanowires , which are themselves metallic , so that an array of parallel nanowires would readily form the contacts for a nanoelectronic device , perhaps using the `` crossbar '' architecture . the er silicides appear also to act as preferential adsorption sites for metals such as pt , so that the properties of these contacts is not limited to that of the nanowires themselves . however , the growth of these nanowires on a semiconducting substrate , with no indication as yet that the substrate could subsequently be oxidised , might limit their application in nanoelectronics ( for which a high - quality insulating substrate is crucial ) . for the bi nanolines , oxidation of the surrounding silicon with ozone , followed by passivation of the surface with hydrogen , or nitridation using ammonia , is a plausible route for the fabrication of a single - nm 1d template on an insulating surface , considerably smaller than any of the rare - earth nanowires , and well within the size range where quantum scale effects might be expected to occur . by the deposition of metals , active molecules , nanoparticles , or other species onto this template , the fabrication of nanowires , or arrays of nanoparticles with interesting optical , electronic or magnetic properties could be achieved , with atomic precision . this area is being actively pursued by the authors . we are happy to acknowledge useful discussions with chigusa ohbuchi and wataru yashiro , as well as permission from various authors to reproduce figures as noted in the text . this study was performed through special coordination funds for promoting science and technology from the mext , japan ( icys and active atom - wire interconnects ) . drb is funded by the royal society .

a number of different families of nanowires which self - assemble on semiconductor surfaces have been identified in recent years . they are particularly interesting from the standpoint of nanoelectronics , which seeks non - lithographic ways of creating interconnects at the nanometre scale ( though possibly for carrying signal rather than current ) , as well as from the standpoint of traditional materials science and surface science . we survey these families and consider their physical and electronic structure , as well as their formation and reactivity . particular attention is paid to rare earth nanowires and the bi nanoline , both of which self - assemble on si(001 ) .

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Proceed to summarize the following text: in depth filtration , suspended particles in a fluid are removed during their passage through a porous medium@xcite . the basic dynamics of depth filtration is determined primarily by the pore structure of the filter , the particle size distribution , and by various physicochemical and hydrodynamic details . if the particle size is larger than the typical pore size , particles get stuck relatively quickly . the permeability of the filter decreases steadily during this process and drops to zero when clogging is reached . this process is often referred to as sieving , or straining@xcite . conversely , if particles are much smaller than the pore size and if particles are trapped only at the interfaces of the porous medium , the flow field is only slightly affected by the trapping . the goal of this paper is to provide a general understanding of this latter process of _ infiltration _ by microscopic network modeling . 2.0 cm .4 cm infiltration underlies many practical situations , such as underground waste disposal @xcite , gas mask design , or drinking water filters @xcite . typically , sub - micron size contaminant particles are suspended in a carrier fluid and flow through a porous material , such as a sand filter whose typical grain size is much larger than the contaminant particles , or an ion exchange filter@xcite where the contaminant size is molecular in scale . in such cases , one can neglect the change of the flow field due to particle trapping @xcite , an approximation which considerably simplifies theoretical analysis . the kinetics of infiltration is controlled by the microscopic mechanisms for the trapping of the invader particles . typically each pore can hold a limited number of particles due to a finite surface area or a finite range of the surface potential . when all the available surface area is covered by particles , subsequent invaders flow passively through the filter without being trapped . our basic goal is to understand the kinetics of this infiltration and the ultimate breakthrough of the invader , as well as the evolution of the invader and defender density profiles as functions of downstream position and time . previous work on infiltration in porous media has often been based on a macroscopic convection - diffusion equation description , with reaction terms introduced to account for particle trapping @xcite . another approach has been to use a single absorbing sphere to calculate the collection efficiency at the initial stage of filtration@xcite . while numerical simulations of these models have some predictive power , it is hard to develop a connection between this macroscopic approach and basic features of the microscopic process , such as the concentration profiles of the trapped and flowing particles . for filtration by straining , models based on a discrete network description of the filter medium are relatively well developed @xcite . to our knowledge , however , there has been no microscopic network modeling work on infiltration . as in the case of straining , a spatial density gradient naturally arises in infiltration , since particles begin to deposit at the upstream end of the filter and advance downstream as the filter gets used up . the density gradient is experimentally observed as the time - dependent output concentration @xcite . in this paper , we will account for this basic experimental observation by using a discrete network approach . practical questions raised by infiltration are the breakthrough time , which is defined as the time for the output concentration to reach a specified threshold level , and the filter efficiency , which is related to the fraction of the filter material actually used before breakthrough . clearly , it is desirable to use as much of the filter material as possible before breakthrough occurs . this paper is organized as follows . in sec . [ basic ] , we introduce the basic parameters that govern particle trapping and provide a qualitative picture of infiltration . in the following sections , we construct a sequence of discrete models with increasing complexity and realism to ultimately provide a lattice network description . in sec . [ 1d ] , we discuss the case of a one - dimensional ( 1d ) chain of trapping sites and in sec . [ bubble ] , we analyze infiltration in the bubble model to provide a mean - field - like description . building on these results , we then turn to simulations of infiltration on tube lattice networks in sec . [ network ] . we summarize and compare our results with experiments in sec . [ discussion ] . the two basic characteristics of particle trapping are the efficiency of an unoccupied trapping site and the number of trapping sites in a pore . we introduce the trapping probability @xmath0 as the probability that a particle is trapped upon encountering an open collector site . the parameter @xmath0 thus represents the strength of the particle - collector interaction and accounts for the possibility that contact between particles and the filter grains may not necessarily lead to deposition @xcite . while this simplifies the complicated adsorption mechanism , later we show that the basic feature such as the invasion front propagation velocity ( fig.[fig1 ] ) is independent of the interaction details . next we introduce the capacity @xmath1 as the number of particles a pore at position @xmath2 can hold at time @xmath3 . in the case of non - coagulating particles which can not get trapped on top of an already adsorbed particle , the initial capacity is proportional to the inner surface area of the pore and then decreases as the pore surface is covered by particles . for simplicity , we ignore multiple trapping on an already occupied collector site as well as particle re - launching . the key factors which determine the dynamic behavior of the system are geometric , such as the capacity of a clean filter and the pore size distribution , and kinematic , such as the particle concentration and the flow rate . more refined models for particle trapping can be incorporated within our basic modeling . consider a generic infiltration process based on the above concepts . initially a layer in the clean filter has a total capacity @xmath4 independent of the downstream position . at @xmath5 , a fluid which contains a mixture of invader and non - reacting tracer particles enters the filter whose flow rate is determined by the steady - state solution of darcy s law . tracer particles passively follow the fluid motion and advance with the average flow velocity @xmath6 . the width of the tracer density profile spreads as @xmath7 due to hydrodynamic dispersion ( fig . [ fig1 ] ) . invader particles first encounter clean collector sites . because each such encounter leads to deposition with probability @xmath0 , the survival probability of the particles in this leading invaded region decreases exponentially with downstream position as illustrated in fig . as particles advance and get trapped , the pore capacity decreases and subsequent particles are more likely to survive , giving rise to an advancing invasion front with a velocity @xmath8 . in principle , the propagation velocity and shape of the front are functions of time . however , at long times these features approach steady - state values . feature is that the trailing edge of the capacity profile decays as a power of the distance @xmath9 ( @xmath10 ) from the invasion front whose location is defined , _ e.g. _ , as the position where @xmath11 . for large @xmath9 , any reasonable definition for the front location can be used . the existence of different propagation velocities , @xmath6 for the pure fluid and @xmath8 for the contaminant , leads to purification of the liquid . the filter can be used until the invaded region reaches the outlet end . for a filter of length @xmath12 , the breakthrough time will be of the order of @xmath13 , so the amount of throughput will be approximately proportional to @xmath14 . 0.25 cm .6 cm as a preliminary , we study infiltration in a one - dimensional chain of identical pores at @xmath15 . first we consider the case where each pore can accommodate only one particle and then we generalize to multiple capacity pores . we choose a time unit such that one particle is injected at each discrete time step . multiple particle injection leads to a different particle density and will affect only the overall scale factor and not change qualitative features of the system . the carrier fluid advances by one pore distance at each time step ; that is , its velocity is unity . when particle trapping occurs in a pore at @xmath16 , the capacity @xmath17 changes permanently from 1 to 0 . at time @xmath3 , a particle at pore @xmath16 gets trapped in that pore with probability @xmath0 if @xmath18 . a particle advances to the next pore in one time step with probability 1 if @xmath19 . based on these elemental steps , we introduce the following two probability densities : * @xmath20 : the probability that a freely - moving particle is in pore @xmath16 at time @xmath3 . * @xmath21 : the probability that the pore at site @xmath16 is unoccupied ; that is @xmath18 . the corresponding master equations for @xmath20 and @xmath21 are @xmath22 where we drop the argument @xmath3 on the right hand side for simplicity . unless there is a possibility for confusion , we will not write the argument @xmath3 in related formulae . since a particle advances to the next pore in one time step , @xmath23 depends on @xmath24 . the term @xmath25 in eq . ( [ 1dmep ] ) is the probability that the particle at @xmath26 does not get trapped by an unoccupied pore also at @xmath26 . similarly , the term @xmath27 in eq . ( [ 1dmeq ] ) is the probability that pore @xmath16 does not trap a free particle at time @xmath3 . the initial and boundary conditions for these equations are : @xmath28 if the trapping probability @xmath0 is small , a particle can advance many pores without being trapped , so that @xmath20 and @xmath21 vary slowly in space and time and a continuum approximation can be applied . letting @xmath29 and @xmath30 , eqs . ( [ 1dmep ] ) and ( [ 1dmeq ] ) become , to lowest order , @xmath31 where @xmath32 , and @xmath33 is a redefinition of the trapping probability in units of the infinitesimal time increment . in a co - moving reference frame , @xmath34 , with @xmath35 the invasion front propagation velocity shown in fig . [ fig1 ] ( which is yet to be determined ) , eqs . ( [ pq ] ) become @xmath36 let us first examine the steady state solution of eqs . ( [ 1dpwave ] ) and ( [ 1dqwave ] ) . setting the time derivatives to zero , subtracting eq . ( [ 1dqwave ] ) from eq . ( [ 1dpwave ] ) , and integrating with respect to @xmath37 gives @xmath38 the integration constant can be determined by applying eqs . ( [ 1dic ] ) in the co - moving frame . as @xmath39 , @xmath40 , @xmath41 , and as @xmath42 , @xmath43 , @xmath44 . these immediately give @xmath45 . note that @xmath35 is determined entirely from the boundary condition ( and @xmath6 ) in the co - moving frame , and not from the interaction strength @xmath0 . this feature continues to hold for all the models in this paper . using eq . ( [ 1dpqsol ] ) with @xmath46 , eqs . ( [ 1dpwave ] ) and ( [ 1dqwave ] ) can be solved to give @xmath47 ( fig . [ fig2 ] ) . thus @xmath48 is the characteristic width @xmath49 of the profile . notice also that the profiles of @xmath50 and @xmath51 are symmetric about their intersection . we verified both the dependence of the width on @xmath0 and the profile shape predicted by eq . ( [ 1dpq ] ) by numerical integration of the master equations ( [ 1dmep ] ) and ( [ 1dmeq ] ) . .1 cm one subtle point is the rate of approach to the steady state . first , we find that the asymptotic propagation velocity @xmath46 is reached before the asymptotic profile is established . this arises because @xmath35 is determined by the boundary conditions , and not by interaction details . we also verified this feature numerically . adopted this asymptotic velocity , eqs . ( [ 1dpwave ] ) and ( [ 1dqwave ] ) are then symmetric in @xmath50 and @xmath51 . in fact , the system is identical to two - species annihilation , @xmath52 , where each species is ballistically injected from opposite sides with velocity @xmath53 for the @xmath54s and @xmath55 for the @xmath56s . thus most of the the time variation in fig . [ fig2 ] occurs in the reactive region of width @xmath49 , where @xmath57 at @xmath5 . integrating eq . ( [ 1dpwave ] ) from @xmath58 to @xmath59 gives , @xmath60 the integral on the left hand side is the area under the curve @xmath61 between @xmath62 and @xmath63 , whose time dependence mainly comes from the change in @xmath49 . on the right hand side , the integral is approximately proportional to @xmath49 , since @xmath64 is significantly different from zero only in the reactive region . we can then rewrite eq . ( [ 1dpint ] ) as @xmath65 where @xmath66 and @xmath67 are constants . integrating eq . ( [ 1dpint1 ] ) and applying the condition @xmath68 gives an exponential decay to the steady state @xmath69 . it is worth emphasizing that the symmetry between the invader and defender is generally responsible for the relation @xmath46 . at the inlet , invaders are injected with velocity @xmath6 , and the invasion front advances with velocity @xmath35 , with one invader particle annihilating with one defender site . in the reference frame moving with velocity @xmath6 , the situation is reversed . the invaders are at rest and defenders are injected with velocity @xmath6 from the opposite direction . therefore the invasion front advances with velocity @xmath70 . since these two reference frames describe the system in the same way , the front velocities should be the same ; that is , @xmath71 , or @xmath46 . now we consider the case where each pore can trap @xmath72 particles , that is , the initial pore capacity is @xmath73 . we again follow the previous rules of injecting a single particle and advancing a particle by one pore ( @xmath74 ) at each time step . multiple particle injection or different injection intervals again simply changes the overall concentration and time scale . in a multiple capacity pore , the probability of encountering an open trap in a pore needs be considered , in addition to the trapping probability @xmath0 upon encounter with an open trap . generally , the encounter probability decreases as more particles get trapped , since the inner pore surface area available for trapping shrinks . when fluid mixing within a pore is weak , a particle can encounter only one trap , either open or occupied . then the encounter probability is approximately proportional to the fraction of the open surface area . on the other hand , if the mixing is perfect , a particle encounters all the available traps in a pore before exiting . for practically relevant situations , pores are sufficiently short so that a particle in a pore follows streamlines without transverse diffusive mixing @xcite . in what follows , we consider this limit of weak mixing . for a pore with @xmath75 out of @xmath72 traps available , the encounter probability is @xmath76 , and the overall trapping probability of this pore is @xmath77 . in writing this expression , we ignore the possibility that a particle far from the pore wall does not encounter any traps . in sec . [ discussion ] , we argue that this volumetric effect does not change the basic behavior of infiltration . to describe the evolution of the system , we use the same single - particle probability density @xmath20 as in the single - capacity pore system , but modify the probability density for the capacity as follows : * @xmath78 : the probability that a pore at position @xmath16 contains @xmath75 open traps . this is the same as the probability that @xmath79 , for @xmath80 . following similar reasoning as that applied to deduce eqs . ( [ 1dmep ] ) and ( [ 1dmeq ] ) , the master equations for @xmath20 and @xmath78 are @xmath81\\ & & \nonumber\\ \label{mulmeq } q_k^n(t+1 ) & = & q_k^n(1-p_k{t}_n ) + p_k q_k^{n+1}{t}_{n+1}.\end{aligned}\ ] ] in eq . ( [ mulmep ] ) , @xmath82 accounts for the case that the pore @xmath26 has zero capacity . other terms in eq . ( [ mulmep ] ) correspond to cases when the capacity is different from 0 , with @xmath83 the survival probability for each case . in eq . ( [ mulmeq ] ) @xmath84 is the probability that the pore with capacity @xmath75 does not trap a particle , and @xmath85 is the probability that the capacity decreases from @xmath86 to @xmath75 by a particle trapping event . hence the last term is absent when @xmath87 . we simplify eqs . ( [ mulmep ] ) and ( [ mulmeq ] ) by introducing the _ average capacity _ of a pore at position @xmath16 , @xmath88 this gives the average number of sites still available for trapping in the pore . now by multiplying eq . ( [ mulmeq ] ) by @xmath75 , summing from 1 to @xmath72 , and using @xmath89 , we obtain @xmath90 these are identical in form to eqs . ( [ 1dmep ] ) and ( [ 1dmeq ] ) , so the same steady state analysis applies . we transform to a co - moving frame and take the continuum approximation to reduce the rate equations to eqs . ( [ 1dpwave ] ) and ( [ 1dqwave ] ) with @xmath91 and @xmath92 . the boundary conditions are also the same as in the case of single - capacity pores , except @xmath93 . combining these results give @xmath94 notice that for @xmath95 , eq . ( [ mulvw ] ) reduces to the single capacity case , while for @xmath96 , @xmath97 . this means that there is no steady state for the case of infinite capacity pores . we can generalize the symmetry argument given in the single capacity case to find the propagation velocity in eq . ( [ mulvw ] ) . at the input , the flux of invaders moving with the carrier fluid is equal to @xmath98 . similarly , in the reference frame moving with velocity @xmath6 , the flux of defenders is @xmath99 , while the invaders are at rest . because one invader annihilates with one defender , the two particles are kinetically indistinguishable . therefore , if a particle flux of @xmath98 results in a front moving with velocity @xmath35 , the front velocity produced by a flux of @xmath99 should be @xmath100 , which , in turn , equals @xmath70 in the moving reference frame . by this equivalence , eq . ( [ mulvw ] ) immediately follows . in the limit of perfect mixing , a particle encounters all traps in the pore . the overall trapping probability with @xmath75 open traps is then @xmath101 . in the limit of small @xmath0 , @xmath102 , thus the analysis is exactly the same as in the poor mixing case except without the factor @xmath103 in @xmath104 . the propagation velocity of the front is the same as in eq . ( [ mulvw ] ) , since this velocity is independent of trapping mechanism , while the width varies as @xmath105 . notice that @xmath49 is a decreasing function of @xmath72 . this arises because a particle must survive all the traps in a pore before advancing to the next pore . finally , for a mixing mechanism which is intermediate between the two limits of perfect and poor mixing , the propagation velocity will be @xmath106 , while the width of the front will lie between the limiting values of @xmath107 and @xmath108 . we now study the bubble model as a logical next step towards understanding infiltration in porous media . the bubble model was introduced to account for the breaking of fibers @xcite , extremal voltages in resistor networks @xcite , and later to filtration kinetics @xcite . the bubble model consists of @xmath12 `` bubbles '' in series , each of which is a parallel bundle of @xmath49 tubes , with each tube representing a pore ( fig . [ fig4 ] ) . a bubble can be viewed as a single layer of parallel bonds in a lattice with all the ends `` shorted '' . this model has multiple paths , as in real porous media , and is sufficiently simple to be amenable to analytic study . a useful feature of the bubble model is that for straining dominated filtration , this model predicts similar behavior to that of lattice networks @xcite . .5 cm .4 cm we choose the tube radii in the bubble model from the hertz distribution @xmath109 where @xmath110 is the characteristic pore radius . this form is often seen in experimental pore size measurements @xcite and has been used for modeling the pore size distribution in filters @xcite . for simplicity we assume identical tube lengths and measure downstream distance in units of the tube length , which is set equal to 1 . we also assume that the flow rate in a tube of radius @xmath111 is proportional to @xmath112 , where @xmath113 is the pressure gradient along the tube and @xmath114 depends on the nature of the flow , with @xmath115 corresponding to poiseuille flow and @xmath116 to euler flow . perfect mixing is assumed at each node . a particle chooses a tube in the next downstream bubble according to _ flow induced probability _ @xmath117 , @xmath118 in which the probability of choosing an outgoing tube of radius @xmath111 is proportional to the flow rate into the tube , @xmath119 @xcite . since tubes of different flow velocities give the dominant mechanism for dispersion , the radial dependence of the local flow velocity in a tube ( taylor dispersion ) is ignored . thus we assume that a particle moves with the average flow velocity @xmath120 along the tube . we now investigate the hydrodynamic dispersion of passive brownian particles which are carried by the background fluid in the bubble model , in the absence of any trapping . this will provide the concepts and tools necessary to understand infiltration in the bubble model . in the large @xmath49 limit , each bubble is nearly identical , and we can regard the particle motion as a directed random walk in which the average residence time @xmath121 in bubble @xmath16 ( @xmath15 ) is a random variable whose distribution @xmath122 is related to the flow induced entrance probability @xmath117 and the radius distribution @xmath123 . this random walk description of the continuous particle motion introduces an additional stochasticity into the system . however , we will show below that this only modifies the hydrodynamic dispersion coefficient by an overall multiplicative factor . the master equations for @xmath20 , the probability that there is a particle in the @xmath124 bubble at time @xmath3 , are @xmath125 here @xmath126 is the initial particle number concentration , @xmath127 is the ( constant ) flow rate , and the initial condition is @xmath128 for all @xmath16 . since the flow rate does not change in infiltration , constant pressure drop and constant flow rate conditions are equivalent . the particle transport properties can be obtained in terms of the residence time distribution @xmath122 , namely , the probability that a particle spends a time @xmath129 in a bubble . this residence time distribution is related to microscopic distributions by @xmath130 where @xmath131 is the dirac delta function . since the flow rate into a tube of radius @xmath111 is @xmath132 , the average flow velocity @xmath133 . using this together with eq . ( [ hertz ] ) for @xmath123 , we obtain the first two moments of @xmath129 @xmath134 where @xmath135 is the average tube volume ( recall that the tube length is fixed to be 1 ) and @xmath136 is the gamma function . we solve eq . ( [ bubbleme ] ) in the appendix by the laplace transform technique . from this solution , the average propagation velocity and the width of the front are @xmath137\right\}^{1\over2}\equiv ( d_\parallel t)^{1\over2}.\end{aligned}\ ] ] thus we see that the dispersion coefficient is proportional to the average flow velocity @xmath138 . when @xmath116 ( euler flow ) , the flow velocities in all the tubes are identical and there should be no dispersion . however , eq . ( [ hydrow ] ) gives a nonzero dispersion coefficient . as mentioned above , this arises from the stochasticity of the random walk picture for the particle motion . for the practically relevant case of @xmath139 , the effect of this stochasticity is only to change the dispersion coefficient by a factor of order unity . to describe infiltration in the bubble model , we need to specify the particle motion , the tube capacities , and particle trapping in a tube . for the particle motion we again assume that a particle chooses a tube according to flow induced probability and then advances with the average flow velocity @xmath120 of this tube . the capacity of a tube is proportional to its inner surface area , which is proportional to the tube radius , since all tubes have the same length . last , the overall trapping probability of a tube is equal to the microscopic trapping probability @xmath0 multiplied by the fraction of open traps in a tube ( sec . [ 1dmulti ] ) . to simulate this process efficiently we propagate the probability distribution function ( pdf ) of the suspended particles rather than simulating the motion of individual particles@xcite . the pdf propagation therefore provides the exact distribution of particle positions and tube capacities for a single realization of tube radii . conceptually , the pdf algorithm is equivalent to an exact integration of the master equations . to implement the pdf propagation , we define * @xmath140 : the probability that there is a particle at the _ entrance _ of tube @xmath141 in bubble @xmath16 ( @xmath142 , @xmath143 ) . * @xmath144 : the capacity of tube @xmath141 in bubble @xmath16 . since particles generally have different velocities , their positions could be anywhere within a tube . we simplify this by forcing particles to _ always _ be at the tube entrance by adjusting the time unit and the pdf propagation so that the _ average _ particle position is at the correct location along the tube , as illustrated in fig . [ fig5 ] . to construct the particle motion , let us temporarily disregard particle trapping . we set the time increment to be @xmath145 , where @xmath146 is the maximum flow velocity among all tubes . in a time @xmath147 , a particle at the entrance of the fastest tube should traverse the entire tube length which is equal to 1 . we then let a particle in a slower tube , with velocity @xmath148 , travel a distance @xmath149 with probability @xmath150 , or remain fixed with probability @xmath151 . one can regard @xmath152 as a normalized flow velocity . by construction , such a particle travels the correct average distance in time @xmath147 , @xmath153 . let us now recast this random walk into a probability propagation algorithm . consider an element of the pdf which is at the junction before the @xmath154 bubble . before any particle motion occurs , we split this probability element among the downstream bonds in this bubble according to the flow induced probability at the tube entrance . we can view the probability element as advancing infinitesimally into each bond , as indicated on the left side of fig . once this initial tube assignment is made , the probability element remains within its assigned tube until it reaches the next junction . now consider the motion of a probability element which has just entered a particular bond . after a time @xmath147 , a fraction @xmath152 of the pdf is advanced to the next bubble , while a fraction @xmath151 remains fixed at the entrance to bond @xmath141 . .4 cm due to the filtration , a fraction of the flowing pdf becomes trapped in tube @xmath141 in the @xmath124 bubble at a rate which is proportional to the tube capacity @xmath144 . the overall trapping probability of this tube is therefore @xmath155 . after trapping has occurred , the tube capacity is decremented according to the following prescription . when one unit of pdf ( equivalent to one particle ) gets trapped , we define the bond capacity to be decreased by @xmath156 . therefore @xmath156 is just the surface area covered by one particle . correspondingly , @xmath157 equals the number of particles the tube can accommodate . our algorithm for propagating an element of probability at the entrance to bond @xmath141 in the @xmath124 bubble over a time @xmath147 therefore consists of the following steps ( fig . [ fig5 ] ) : * fraction of pdf remaining at the start : @xmath158 . * fraction trapped in tube @xmath141 : @xmath159 . * fraction advancing to the next junction : + @xmath160 * capacity change of the tube by trapping : + @xmath161 $ ] . the rate equations which account for these steps are : @xmath162 the first term on the right hand side of eq . ( [ bubblepdf ] ) is the fraction of probability that does not move , and the second term is contribution from elements of probability which has moved from the previous site . the flow induced probability @xmath163 in eq . ( [ bubblepdf ] ) accounts for the fraction of pdf which enters into tube @xmath141 . to test this approach , we set @xmath164 ( no trapping ) in the above rate equations and simulate the pdf propagation . by this method , we find a traveling front whose basic properties coincide with the hydrodynamic dispersion results given by eqs . ( [ hydrov ] ) and ( [ hydrow ] ) . it is also worth mentioning that our pdf algorithm can be generalized to allow for hopping a distance which is a fraction of the tube length . in this manner one can account for different longitudinal flow velocities at different radial positions within a tube ( taylor dispersion @xcite ) . in the limit of an infinitesimal hopping distance , continuous particle motion is reproduced by the pdf algorithm . unfortunately , the gain in having a more accurate description of the motion is offset by the complexity of the algorithm and large increase in the computation time . to obtain the average invasion front profile over the tubes in each bubble , we define the bubble - average quantities @xmath165 and @xmath166 . we first derive the invasion front velocity via the symmetry argument of sec . [ 1dmulti ] . a rigorous derivation of the front velocity from the master equations for @xmath167 and @xmath168 is given in @xcite . the carrier fluid is moving with velocity @xmath6 ( eq . ( [ hydrov ] ) ) and the input flux of invaders per tube is equal to @xmath169 , since @xmath170 is equal to the number of invader particles per tube volume in the input fluid . on the other hand , in a reference frame moving with velocity @xmath6 , the input flux of defenders is equal to @xmath171 where @xmath172 is the average initial number of invaders a tube can accommodate . following the argument in sec . [ 1dmulti ] , we find the front velocities @xmath35 and @xmath70 in the two reference frames are related by @xmath173 , yielding @xmath174 good filter performance means that the breakthrough time is long or , equivalently , that the propagation velocity is slow . ( [ bubblev ] ) implies that the propagation velocity can be made small by increasing the capacity of a pore , or by decreasing either the filter grain size or the input particle concentration . notice that neither the reaction strength @xmath0 nor the nature of the flow ( through @xmath175 ) affect this propagation velocity . we now study the asymptotic density profiles . instead of working directly with the averaged quantities @xmath167 and @xmath168 , we first focus on the behavior of a single tube of radius @xmath111 , since tubes with the same radius in a bubble have identical time dependence . the asymptotic profiles can be obtained after averaging over the distribution of tube radii . therefore , we label tubes according to their radii instead of the index @xmath141 . we denote @xmath176 and @xmath177 as the pdf and capacity of a tube of radius @xmath111 in the @xmath124 bubble . let us first focus on the pdf profile in the invaded region . here , traps are mostly unoccupied , so that the tube capacity @xmath177 is approximately equal to its initial value @xmath178 , which is proportional to the tube radius and we set it equal to @xmath111 . the arbitrariness in the unit of capacity can be controlled by the magnitude of the parameter @xmath156 . then eq . ( [ bubblepdf ] ) becomes , @xmath179 where the integration over @xmath180 replaces the summation over the tube index , the flow induced probability @xmath117 and normalized velocity @xmath181 are independent of the downstream position @xmath16 because these only depend on the radius of a tube . the equation for @xmath182 can be obtained by multiplying eq . ( [ attack ] ) by @xmath123 and integrating over @xmath111 . as in sec . [ 1d ] , we take @xmath183 and consider the continuum limit . if we redefine the length of the bubble from 1 to @xmath184 , @xmath147 becomes @xmath185 . integrating eq . ( [ attack ] ) and expanding in @xmath147 and @xmath184 yields @xmath186\ ] ] here , we use @xmath187 . dividing eq . ( [ attack0 ] ) by @xmath147 changes @xmath181 back to @xmath120 . after redefining @xmath188 as before , we obtain @xmath189.\ ] ] in the steady state co - moving frame , eq . ( [ attack1 ] ) becomes @xmath190.\ ] ] since only a small number of particles have entered the invaded region , the density of moving particles is approximately proportional to @xmath119 , and we introduce the ansatz @xmath191 to factorize the pdf . in order to calculate the dominant contribution from the integral in eq . ( [ attack2 ] ) , we substitute @xmath192 , where @xmath193 is the deviation from the average carrier fluid velocity @xmath6 . since @xmath193 has zero mean , the dominant contribution to the integral over @xmath120 in eq . ( [ attack2 ] ) comes from the constant part @xmath6 . using these approximations in eq . ( [ attack2 ] ) , and using @xmath194 , we find @xmath195 since @xmath8 , we find @xmath196 $ ] . hence the profile of free particles in the invaded region @xmath197 decays exponentially in @xmath37 , with a characteristic decay length which has the same @xmath198 dependence as in the 1d model . let us now turn to the analysis of the capacity profile in the tail region . in terms of @xmath199 and @xmath200 , eq . ( [ bubblecap ] ) becomes @xmath201 since there is negligible trapping in the tail region , the particle motion follows that of the carrier fluid . thus @xmath202 , where @xmath203 is the tube volume , and the flow velocity is @xmath204 . substituting these in eq . ( [ tail ] ) and transforming into the co - moving frame gives @xmath205 where @xmath206 denotes the strength of the particle trapping reaction . we now integrate eq . ( [ tail1 ] ) from @xmath58 to @xmath207 and use the boundary condition @xmath208 to obtain @xmath209 where we drop the subscript of @xmath210 . finally , the average bond capacity as a function of position with respect to the front , @xmath211 , is @xmath212 for large @xmath9 , the integral is dominated by the smallest tubes and the initial distribution of tube radii is irrelevant in the tail region . hence the factor @xmath213 in the exponential can be ignored . performing the resulting integration gives @xmath214 where the last relation serves to define the _ profile exponent _ @xmath215 . this is one of our primary results . correspondingly , the pdf in the tail region will approach its initial value with the same power law . the existence of the power law tail in the capacity profile stems from the fact that the flow rate is not affected by trapping . thus when large pores are `` used up '' , the fluid still predominantly flows through these pores , leading to a substantial unused capacity in the smaller tubes . it is these unused smaller tubes which contribute substantially to the capacity profile in the tail region . this mechanism is quite general and only depends weakly on the form of the radius distribution . for example , for a uniform distribution in the range @xmath216 , we obtain @xmath217 . however , if there is a finite lower cutoff in the radius distribution , the pdf will have an asymptotic exponential tail . it is interesting to note that the density profile has different dependence on @xmath0 in the invaded and tail regions . from eq . ( [ tail4 ] ) , the density profile contains an overall factor @xmath218 . thus @xmath0 typically does not appear as an overall scale factor of the entire profile , as in the invaded region . however , for the practically relevant case of @xmath115 , the exponent in eq . ( [ tail4 ] ) is equal to 1 , and @xmath0 becomes the overall scale factor of the profile . in our numerical simulations , we set the input particle flux per tube @xmath219 equal to 1 , which means that @xmath49 units of pdf are injected into the system at every time step . this can be achieved by choosing @xmath220 and @xmath221 , which also makes @xmath222 ( eqs . ( [ tau1 ] ) and ( [ hydrov ] ) ) . .5 cm .4 cm we applied the pdf propagation of eqs . ( [ bubblepdf ] ) and ( [ bubblecap ] ) to a system of size @xmath223 . due to the exact nature of the pdf algorithm , a single realization provides good quality data for @xmath224 tubes . a system length of @xmath225 is sufficiently long to give the continuum functional form of the profiles . all the data shown below are results of single realization of tubes including the network simulation in the next section . the simulation is stopped before the front exits the system . _ density profiles_. fig . [ fig6 ] shows typical particle and tube capacity profiles . there are strong bubble - to - bubble fluctuations and some type of smoothing procedure is necessary . we use the savitzky - golay smoothing technique , which approximates successive windows of data points to a @xmath226 order polynomial ( solid lines in the figure ) @xcite . this technique is superior to local averaging because savitzky - golay smoothing can faithfully follow rapid changes in the profile , as can be seen in fig . this smoothing is also useful in estimating the exponents . since logarithm of the profile in the tail region amplifies the fluctuation in nonlinear way , slopes of the raw data in fig . [ fig_tail1 ] are larger than those from the smoothed data , and differs from the predicted value of the profile exponent @xmath215 . .1 cm .4 cm _ front velocity_. fig . [ fig_vel](a ) shows the front position , defined as the point where @xmath167 is half of its saturation value , versus time . notice that a constant front propagation velocity sets in almost immediately . with @xmath220 and @xmath221 , ( [ bubblev ] ) gives @xmath227 the slopes in fig . [ fig_vel](a ) agree well with eq . ( [ vel ] ) . notice that the propagation velocity does not depend on the reaction strength @xmath0 nor the exponent @xmath114 in the radius dependence of the velocity . .5 cm _ tail profile_. fig . [ fig_tail1](a ) shows the tube capacity profile @xmath211 in the tail region as a function of the distance @xmath9 ( @xmath10 ) from the front on a double logarithmic scale . the plot becomes straight for large @xmath9 and the slope in this region corresponds to the exponent @xmath228 predicted by eq . ( [ tail4 ] ) . for the uniform distribution on @xmath229 , we predicted the profile exponent to be @xmath217 . for @xmath115 , the exponent value of 2/3 agrees well with our simulations ( fig . [ fig9 ] ) . however , for a radius distribution with a lower size cutoff , we expect an exponential density profile ( inset to fig . [ fig9 ] ) . .1cm.4 cm .5 cm .4 cm as we also discussed in sec . [ bubbleanal ] , the amplitude of the density profile in the tail region typically has a power - law dependence on @xmath0 . for a hertz distribution of particle radii , this amplitude should be proportional to @xmath218 according to eq . ( [ tail4 ] ) . thus fig . [ fig_tail2](a ) shows @xmath230 versus @xmath231 for @xmath115 and @xmath232 . values of the abscissa should be proportional to @xmath198 , which is indeed the case . _ invader profile_. fig . [ fig_invade](a ) is a log - normal plot of the invader profile @xmath197 ( raw data ) versus @xmath37 . the slopes of the two data sets are in excellent agreement with the predicted value @xmath233 from eq . ( [ attack3 ] ) . in the invaded region , since there are few particles present , the corresponding pdf monotonically decreases and its fluctuation is significantly smaller compared to the tail region . -.2cm.4 cm .2 cm .4 cm we now consider infiltration on a square lattice network of tubes . here , local mixing at tube junctions occurs as opposed to the mean - field - like mixing in the bubble model . nevertheless , many of our predictions from the bubble model continue to be valid for the lattice network . for example , we expect that the propagation velocity given by eq . ( [ bubblev ] ) will continue to hold in the lattice network because it is determined _ only _ by the boundary conditions of the particle flux and initial filter capacity . we also find numerically that the network model has both the same exponential invader profile and the power law capacity profile as in the bubble model , although the values of the amplitudes and decay exponents are different . overall , it appears that the bubble model provides an excellent account of the numerical results from the lattice network . we study a square lattice of size @xmath234 which is tilted at @xmath235 . a periodic boundary condition is imposed in the transverse direction . the tube radii are drawn from the hertz distribution of eq . ( [ hertz ] ) . notice that the bubble model would arise from this system by merging together all sites at the same longitudinal position . the overall flow rate @xmath127 is set to @xmath236 and the particle density to @xmath220 , just as in the bubble model . for a given set of tube radii , the flow field is calculated by using the conjugate gradient method@xcite to solve the set of linear algebraic equations for fluid conservation at each node . the tolerance of the computation is set so that the measured average pdf and the tube capacity in the @xmath124 layer are accurate to within 0.01% . after the flow field is solved , we use the same pdf algorithm as in the bubble model to track the motion of the suspended particles . the new features due to the lattice nature of the network are that tubes are only locally connected and that the local flow direction is not always downstream . to facilitate comparisons , all our numerical results for the bubble model and the square lattice are presented side - by - side . [ fig_vel ] shows the front position versus time for both the bubble model and the square lattice . the square lattice results are in excellent agreement with the bubble model prediction for the front velocity , eq . ( [ vel ] ) . similarly , the tube capacity profile exhibits a power law tail ( fig . [ fig_tail1](b ) ) . however , the dependence of the decay exponent @xmath215 on @xmath114 is much weaker compared to the bubble model . as @xmath114 decreases from 4 to 2 , @xmath215 increases slowly from 0.95 to 1.28 . to isolate the dependence of the capacity profile on the trapping probability @xmath0 , fig . [ fig_tail2](b ) shows @xmath230 versus @xmath231 in the tail region . here , the values of @xmath215 used are obtained from fig . [ fig_tail1](b ) . unlike the bubble model , the overall amplitude has a relatively stronger dependence on @xmath114 . however , the dependence on @xmath198 still holds even in the network case . it seems that in the square lattice network , @xmath114 has more effect on the amplitude of the tail than on the decay exponent . lastly , fig . [ fig_invade](b ) shows the density profile of invaders in the invaded region . the slopes of these particles are almost identical to those of the bubble model . currently we do not have clear explanation of this fact . it seems that the network geometry does not affect the profile . in fact , the characteristic decay length , @xmath237 , of the profiles in fig . [ fig_invade ] is the same as that of the corresponding 1d model in sec . [ 1dmulti ] where @xmath170 invaders are injected with velocity @xmath6 into the chain of defenders of capacity @xmath72 . in this paper , we studied infiltration , in which suspended particles are removed from a carrier fluid as the suspension passes through a porous medium . the trapping mechanism has a built - in saturation so that once all available trapping sites are used up , subsequent particles can pass through the medium freely . the particles are assumed to be sufficiently small that their trapping does not change the flow rate . the basic dynamical properties of this infiltration process are the density profile of the invader particles and the capacity profile of the remaining active pores . when the invader profile reaches the end of the system , the output concentration of particles quickly increases to a saturation level and the filter should be discarded . thus the features of this profile are important to understand the operating characteristics of infiltration . we have developed a series of discrete network models to describe the basic characteristics of infiltration , starting with a one - dimensional model and building up to the bubble model , which is a series array of parallel , multiple - capacity tubes . the advantage of these quasi - one - dimensional models is that they remain relatively simple , even after incorporating local spatial heterogeneity . the bubble model , in particular , appears to capture many of the quantitative features that we observed in numerical simulations of infiltration on a square lattice tube network . our modeling is also flexible , so that variations can be easily implemented for case - specific situations . our main qualitative result is that basic dynamical features of the system , including the value of the front propagation velocity , the exponential profile of flowing particles in the invaded region , and the power law capacity profile of pores in the tail region , are relatively insensitive to microscopic details of the model . we have also identified the basic parameters which do affect quantitative features of the profiles . it is useful to summarize these results and to compare with experimental data , as well as with predictions from previous studies based on the reaction - diffusion equation approach @xcite . .2 cm .4 cm _ invader concentration at the output_. in typical experiments , the invader concentration at the output is measured as a function of time . a slower propagation velocity shifts this output concentration curve to a later time , as indicated in fig . [ fig_discuss](a ) . typically , the time unit is normalized by the time for passive particles to pass through the system @xcite . hence , the amount of the time shift is determined by the ratio @xmath238 rather than by @xmath35 itself . a nice set of infiltration experiments , as well as an accompanying numerical study of the reaction - diffusion equation were performed in @xcite . in these experiments , contaminant solutions with different values of the invader particle diameter @xmath239 , but with fixed mass concentration were used . this makes the corresponding number density @xmath126 of invaders in each solution proportional to @xmath240 . also , since the cross - sectional area of each particle is proportional to @xmath241 , the average initial tube capacity @xmath72 varies as @xmath242 . from eq . ( [ bubblev ] ) , we then have @xmath243 . thus the output concentration curve shifts to a later time for a solution with a larger value of @xmath239 , which is consistent with the experiment and numerical predictions in @xcite . in another set of experiments , different electrolyte concentrations of the carrier fluid were used . this mainly affects the trapping rate @xmath0 . for a larger trapping rate , the width of the output concentration curve , namely , the time range over which the output concentration changes from zero to its saturation value becomes narrower . however , there is no shift in the breakthrough time because the propagation velocity is independent of @xmath0 . these two features are illustrated in fig . [ fig_discuss](b ) . this behavior again qualitatively agrees with the experiment in @xcite . it would also be interesting to study the effect of different filter grain sizes . in our model this would be accomplished by changing the characteristic parameter @xmath244 of the pore size distribution . in turn , this affects the invader propagation velocity , and the output concentration curve will shift in time accordingly . however , since the microscopic parameters we use in our modeling may be coupled with each other in experimental situations , different sizes of filter grains or invader particles may also affect , _ e. g. _ , the reaction strength @xmath0 . we can incorporate such a coupling effect by extending our model to deal with these effects explicitly . for example , we can adapt the microscopic models of particle trapping on a single sphere or on a plane @xcite , to the tube geometry . from such an approach , we can express the reaction strength @xmath0 as a function of the invader or defender diameter . _ tail of the output concentration curve_. a slowly decaying tail in the deviation of the output concentration from its asymptotic value is generally observed in experiments @xcite . this observation is in contrast to the empirical approaches , such as that given in @xcite , which gives an exponential profile for the whole time range . their prediction agrees with experimental observations at early times but then deviates at later times , implying that the output profile at later times is not exponential . a closely related approach , based on the study of a reaction - diffusion equation , is presented in @xcite , along with experiments which measure the output concentration . unlike @xcite , here the adsorption rate depends on the local concentration of contaminants and thus is spatially inhomogeneous . a crossover from a rapid increase to a slowly decaying tail of the output concentration was numerically predicted . however , the functional forms of these two regimes in particular , whether they are exponential or power law in time were not investigated quantitatively . however , the data presented in this work seem consistent with a slower than exponential decay of the density profile . in @xcite , an exponential output concentration profile , @xmath245 , is assumed from the outset , where @xmath2 is the downstream distance and @xmath246 is the experimentally measured filter coefficient . the corresponding experimental data show that @xmath246 is constant at early times , and then sharply decreases at later times . thus the initial stage of the experiment is consistent with an exponential profile , but later the profile decays more slowly . in @xcite , this is attributed to a `` blocking effect '' in which previously - deposited particles can block the further deposition of particles onto nearby available trapping sites . _ probability of encountering an open trap_. as a last remark , let us examine the assumption that the probability of encountering an open trap is proportional to the fraction of open traps in a pore ( sec . [ 1dmulti ] ) . suppose instead that one takes into account the volumetric effect that particles far from the surface of the pore do not have chance to encounter a trap . then the fraction of particles in contact with the inner surface of a tube is proportional to @xmath247 , namely , the ratio between the surface area and the volume of the tube . this would lead to the interaction terms involving @xmath248 in eqs . ( [ bubblepdf ] ) and ( [ bubblecap ] ) being multiplied by another factor of @xmath247 . however , this modification does not affect the propagation velocity of the front , nor the power law feature of the tail . only the decay exponent changes through the steps of eqs . ( [ tail])-([tail3 ] ) with an additional factor of @xmath247 . our results can also provide practical guidelines for improving the design of a filter in two aspects , namely , the breakthrough time and the amount of filter material used before the breakthrough . a longer breakthrough time can be achieved by having a smaller filter grain size , a lower input concentration , or a larger pore capacity . while these trends may seem intuitively clear , we can quantitatively estimate the increase in the breakthrough time through the expression for the propagation velocity , eq . ( [ bubblev ] ) . when breakthrough occurs , the amount of unused filter material is determined by the shape of the tail in the density profile of the defenders . according to eq . ( [ tail4 ] ) , the amplitude of this tail is proportional to @xmath218 . from this , we can quantitatively estimate the amount of filter material left unused at the breakthrough time as a function of the reaction strength . we thank dr . jysoo lee for helpful discussions about flow field calculations . we are also grateful to grants aro daad19 - 99 - 1 - 0173 and nsf dmr9978902 for financial support . 99 c. tien and a. c. payatakes , aiche j. * 25 * , 737 ( 1979 ) . r. f. probstein , _ physicochemical hydrodynamics _ , 2nd ed . 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( [ bubbleme ] ) , by the laplace transform method . define @xmath249 , and take laplace transform of eq . ( [ bubbleme ] ) to find @xmath250 rearranging yields @xmath251 we multiply the above equations for indices 0 , 1 , @xmath252 together and rearrange terms yet again to obtain @xmath253 before taking averages , we use @xmath254(1+\tau_ks)$ ] on the left hand side of eq . ( [ laplace1 ] ) to get @xmath255 averaging over the residence time distribution @xmath122 defined by eq . ( [ ftau ] ) , we obtain @xmath256 where @xmath257 denotes an average over @xmath122 . note that @xmath56 does not depend on @xmath258 because all the @xmath259 s are independent and identically distributed in the large @xmath49 limit . for the long - time limit , we need the small-@xmath260 behavior of @xmath56 . accordingly , we expand @xmath261 in terms of the moments @xmath262 . the profile of the carrier fluid ( dashed line in fig . [ fig1 ] ) is monotonically decreasing near the front , thus can be characterized by its first derivative , which gives a bell shaped distribution centered at the front . we divide the derivative by the total sum of derivatives @xmath263 to obtain the normalized probability distribution of the front @xmath264.\ ] ] the average position of the front is , using eq . ( [ pntilde ] ) , and the steady state solution of eq . ( [ bubbleme ] ) , @xmath265 , @xmath266\nonumber\\ & = & { 1\over\langle p_0\rangle } { \cal l}^{-1 } \bigl(\sum_k k [ \langle\tilde p_{k-1}\rangle-\langle\tilde p_k\rangle]\bigr)\nonumber\\ & = & { 1\over\langle\tau\rangle}{\cal l}^{-1}\bigl({1-b\over s^2 } \sum_k k[b^{k-1}-b^k]\bigr)\nonumber\\ & \simeq&{t\over \langle{\tau}\rangle},\end{aligned}\ ] ] where the over - bar means averaging over @xmath267 , and @xmath268 is the inverse laplace transform . we use the identity @xmath269 for the last step . we assume sufficiently long chain of bubbles in the summations above to prevent finite length effect . from above , we find the propagation velocity as @xmath270 ( eq . ( [ hydrov ] ) ) . similarly , @xmath271 $ ] , and the width of the front is @xmath272^{1\over2 } \simeq\left\{{t\over\langle{\tau}\rangle}\left [ 2\gamma(1+{\mu\over2})\gamma(3-{\mu\over2})-1\right]\right\}^{1\over2},\ ] ] which gives eq . ( [ hydrow ] )

we study the kinetics of _ infiltration _ in which contaminant particles , which are suspended in a flowing carrier fluid , penetrate a porous medium . the progress of the `` invader '' particles is impeded by their trapping on active `` defender '' sites which are on the surfaces of the medium . as the defenders are used up , the invader penetrates further and ultimately breaks through . we study this process in the regime where the particles are much smaller than the pores so that the permeability change due to trapping is negligible . we develop a family of microscopic models of increasing realism to determine the propagation velocity of the invasion front , as well as the shapes of the invader and defender profiles . the predictions of our model agree qualitatively with experimental results on breakthrough times and the time dependence of the invader concentration at the output . our results also provide practical guidelines for improving the design of deep bed filters in which infiltration is the primary separation mechanism . 2

You are an expert at summarizing long articles. Proceed to summarize the following text: the electron - phonon coupling strongly affects vibrational modes of nano- and micro - electro - mechanical systems . much interest have attracted the effects of this coupling related to the reduced dimensionality of the electron system , as they make it possible to reveal interesting consequences of the electron correlations at the nanoscale , the coulomb blockade being a simple example , cf . @xcite and references therein . much less attention has been paid to the consequences of the electron - phonon coupling , which are related to the discreteness of the vibrational spectrum of a nanosystem , but emerge in the absence of size quantization of the electron motion . one of such consequences , which we study in this paper , is the coupling - induced change of the vibration nonlinearity . strong nonlinearity is a generic feature of vibrations in small systems @xcite . its easily accessible manifestation is the dependence of the mode frequencies on the vibration amplitudes . this dependence corresponds to the self - action " of the mode , and its familiar analog in bulk crystals are acoustic solitons @xcite ; however , the nonlinearity required for observing such solitons usually is sufficiently strong only for high - frequency phonons . also , the change of the eigenfrequency with the mode amplitude is of interest for modes with a discrete frequency spectrum , such as standing waves in mesoscopic systems , but not for propagating waves with a quasi - continuous spectrum . much attention have been recently attracting si - based nano- and micromechanical systems , see @xcite and references therein . in such systems there was observed an unexpectedly large change of the amplitude dependence of the vibration frequency with the varying electron density @xcite . when the doping level was increased from @xmath0 @xmath1 to @xmath2 @xmath1 , the nonlinearity parameter increased by more than an order of magnitude . moreover , the nonlinearity change was different for the vibrational modes with different spatial structure . in this paper we develop a theory of the nonlinearity of vibrational modes in semiconductor nano- and micro - mechanical systems with high electron density . we show that the electron - phonon coupling can lead to a strong self - action of the vibrational modes , which in turn significantly modifies the amplitude dependence of the mode frequencies . we find the dependence of the effect on the electron density and temperature . for bulk semiconductors , the effect of the electron - phonon coupling on the elastic properties , including the three - phonon coupling , was first analyzed by keyes @xcite . the analysis referred to @xmath3-ge and was based on the deformation potential approximation . the idea was that deformation lifts the degeneracy of the equivalent electron valleys , which leads to a redistribution of the electrons over the valleys . in turn , such redistribution changes the speed of sound depending on the direction and polarization of the sound waves and also affects the sound speed in the presence of uniaxial stress . this theory was extended to silicon and the corresponding measurements were done by hall @xcite . however , hall also observed the change of the speed of transverse sound waves and the effect of stress on sound propagation in the geometries , where these effects are due to shear deformation and do not arise in the deformation potential model . a theory of the change of the linear shear elastic constant in silicon due to the intervalley redistribution of the electrons was developed by cerdeira and cardona @xcite . as we show , in mesoscopic systems the strain - induced redistribution of the electrons over the valleys of the conduction band leads to the previously unexplored strong fourth - order nonlinearity of the vibrational modes . this nonlinearity gives a major contribution to the amplitude dependence of the vibration frequency . the redistribution also leads to a temperature dependence of the frequencies . the magnitudes of the effects sensitively depends on the mode structure . we describe them for several types of modes , including those studied in the experiment @xcite and qualitatively compare the results with the observations . the theoretical results refer to both degenerate and nondegenerate electron systems . specific calculations are done for silicon resonators . in sec . [ sec : model ] we give , for completeness , the expressions for the mode normalization and the amplitude - dependent frequency shift of coupled nonlinear modes in a nano- or micro - system . in sec . [ sec : e - ph_coupling ] and appendix [ appendixa ] we provide expressions for the electron - phonon coupling induced change of the elasticity parameters , including the parameters of quartic nonlinearity . in sec . [ sec : explicit_form ] we discuss the asymptotic behavior of the parameters of quartic nonlinearity for low and high electron density and give their explicit form for silicon . in sec . [ sec : simple_modes ] we calculate the nonlinear frequency shift for several frequently used vibrational modes in single - crystal silicon systems and show the dependence of this shift on the electron density and temperature . the explicit analytical expressions are given in appendices [ sec : lame ] and [ sec : extension ] . [ sec : conclusions ] contains concluding remarks . of primary interest for nano- and micro - mechanical systems are comparatively low - frequency modes with wavelength on the order of the maximal size of the system . examples are provided by long - wavelength flexural modes of nanotubes , nanobeams , and nano / micro - membranes , or acoustic - type modes in microplates or beams . these modes are easy to excite and detect . we will enumerate them by index @xmath4 . their dynamics is described by the elasticity theory @xcite . the spatial structure of the displacement field of a mode @xmath5 in the harmonic approximation is determined by the boundary conditions . we will choose @xmath5 dimensionless , so that in our finite - size system @xmath6 here , @xmath7 is the volume of the system . we assumed that the mode eigenfrequencies @xmath8 are nondegenerate ; including degenerate modes is straightforward . for simplicity , we also assumed that the system is spatially uniform ; an extension to spatially nonuniform systems is straightforward as well . we emphasize the distinction of the normalization ( [ eq : normalization ] ) from the conventional normalization for bulk crystals , where @xmath4 corresponds to the wave vector and the branch number , and the normalization integral is independent of the volume . the normalization ( [ eq : normalization ] ) is convenient for the analysis of low - frequency modes with the discrete spectrum characteristic of mesoscopic systems . such modes are standing waves , and therefore vectors @xmath9 can be chosen real . the low - frequency part of the displacement can be written as @xmath10 functions @xmath11 give the mode amplitudes . in the harmonic approximation the dynamics of the standing waves is described by the hamiltonian @xmath12 where @xmath13 is the momentum of mode @xmath4 and @xmath14 is the mass of the system . the anharmonicity of the crystal leads to mode - mode coupling . within the elasticity theory this coupling is described by the terms in the hamiltonian , which are cubic and quartic in the strain tensor . we will not consider higher - order terms , which are small for the mode amplitudes of interest . from the expansion , we obtain the nonlinear part of the hamiltonian in the form @xmath15 equation is essentially an expansion in the ratio of the mode amplitudes to their characteristic wavelength , which is of the order of the appropriate linear dimension of the system . this is why mesoscopic systems are of particular interest , as here vibrations of low - frequency eigenmodes become nonlinear for already small vibration amplitudes . a familiar consequence of nonlinearity in nano- and micromechanical systems is the dependence of the vibration frequency of a mode on its own amplitude and on the amplitudes of other modes , see ref . for a review . in particular , the change @xmath16 of the mode frequency due to the vibrations of the mode itself , @xmath17 , is @xcite @xmath18a_\nu^2,\end{aligned}\ ] ] where @xmath19 and we kept the terms of the first order in @xmath20 and the second order in @xmath21 . the nonlinear mode coupling ( [ eq : anharmonic_hamiltonian ] ) leads also to the frequency shift due to thermal vibrations of the modes . the dominating contribution to this shift for low - frequency modes comes from their coupling to modes with frequencies @xmath22 , which have a much higher density of states . this shift is described by an expression that is similar to eq . ( [ eq : duffing_shift ] ) with @xmath23 replaced by @xmath24 and placed under the sum over @xmath25 , in the classical limit . we will consider the vibration nonlinearity due to the electron - phonon coupling in multi - valley semiconductors with cubic symmetry , silicon and germanium being the best known examples . in such semiconductors , the energy valleys of the conduction band are located at high - symmetry axes of the brillouin zone . strain lifts the symmetry and thus the degeneracy of the valleys . the simplest mechanism of the electron - phonon coupling is the deformation potential . here , the energy shift @xmath26 of valley @xmath27 is determined by the deformation potential parameters @xmath28 and @xmath29 of the coupling to a uniaxial strain along the symmetry axis of the valley and to dilatation , respectively . in terms of the strain tensor @xmath30 we have @xmath31 , where @xmath32 , with @xmath33 being the unit vector along the symmetry axis of the valley . we use the hat symbol to indicate tensors and symbol @xmath34 " to indicate tensor products . the analysis below is not limited to the deformation potential approximation . an important extension will be discussed using silicon as an example . we assume that the strain varies in time and space slowly compared to the reciprocal rate of intervalley electron scattering and the intervalley scattering length , respectively . then the electron system follows the strain adiabatically . the electron density @xmath35 in valley @xmath27 is decreased or increased depending on whether the bottom of the valley goes up or down . in the single - electron approximation and for the deformation potential coupling , the electron free energy density for a given strain is @xmath36 + n^{(\alpha ) } ( \rb ) \xi^{(\alpha)}_{ij } \ep_{ji}(\rb)\}$ ] where @xmath37 $ ] is the free energy density for electrons with density @xmath38 in a valley in the absence of coupling to phonons . the electro - neutrality requires that the total electron density summed over the valleys be constant . the free energy density @xmath39 has to be minimized over @xmath35 to meet this constraint . this gives the change of the electron chemical potential @xmath40 due to strain @xmath41 . the resulting increment of the electron free energy density has the form of a series expansion in the strain tensor , @xmath42 here @xmath43 , and @xmath44 are tensors of ranks 2 , 4 , 6 , and 8 , respectively . they are contracted with the tensor products of the strain tensor @xmath41 . respectively , @xmath45 are the electronic contributions to the linear ( for @xmath46 ) and nonlinear ( for @xmath47 ) elasticity parameters of the crystal . these contributions are isothermal , but since the change of the mode frequencies from the electron - phonon coupling is small and the nonlinearity is also small , the difference with the adiabatic expressions can be disregarded . to the third order in @xmath41 the expression for @xmath48 in terms of the shift of the valleys was found by keyes @xcite in the analysis of sound wave propagation . however , to find the parameters of the quartic nonlinearity of resonant modes in small systems , which is of primary interest to us , we also need to keep quartic terms in eq . ( [ eq : free_energy_expansion ] ) . as seen from the explicit form of the parameters of the expansion ( [ eq : free_energy_expansion ] ) given in appendix [ appendixa ] , @xmath49^{k-1}$ ] ( @xmath50 ) , where @xmath51 is the electron chemical potential in the absence of strain ; it is determined by the total ( summed over the valleys ) electron density @xmath3 . of central importance for the analysis is that parameter @xmath52 for electron densities @xmath53 and room temperatures , i.e. @xmath54 as a consequence , the coefficients at the nonlinear in @xmath41 terms in eq . ( [ eq : free_energy_expansion ] ) quickly increase with the increasing order of the nonlinearity [ the overall series ( [ eq : free_energy_expansion ] ) is converging fast because of the smallness of the strain tensor ] . the increase of @xmath45 with @xmath55 allows us to keep in @xmath41 only the terms linear in the lattice displacement , i.e. , to set @xmath56 , where @xmath57 and @xmath58 are the components of the displacement and the coordinates , respectively . indeed , in this case a @xmath55th term of the series ( [ eq : free_energy_expansion ] ) is of order @xmath55 in the displacement . if we included the quadratic in @xmath59 term into one of the @xmath41 tensors in the @xmath55th term , this term would become of order @xmath60 in the displacement . however , for linear @xmath41 the @xmath61th term in the series ( [ eq : free_energy_expansion ] ) is also of the @xmath61th order in the displacement , but is larger by factor @xmath62 . for linear @xmath41 , the total strain is a sum of partial contributions of strain from individual modes . for mode @xmath4 , such partial contribution is expressed in terms of the scaled displacement @xmath5 [ see eq . ( [ eq : displacement ] ) ] as @xmath63 , where @xmath64 $ ] . we note that , in contrast to the dimensionless strain tensor @xmath41 , tensor @xmath65 has dimension [ length]@xmath66 . from eq . ( [ eq : free_energy_expansion ] ) we find the electronic contributions to the nonlinearity parameters @xmath67 in hamiltonian ( [ eq : anharmonic_hamiltonian ] ) , @xmath68 where @xmath69 ; tensors @xmath70 are independent of @xmath71 . similarly , the electronic contribution to the eigenfrequency is @xmath72 generally , the term @xmath73 leads to mode mixing ; however , if the mode frequencies are nondegenerate , this mixing is weak and can be disregarded , to the leading order in the electron - phonon coupling . one can see that the effect of the static stress @xmath74 can be disregarded as well . the frequency change ( [ eq : frequency_change ] ) depends on temperature because of the temperature dependence of @xmath75 . the nonlinearity ( [ eq : nonlin_params ] ) also leads to a temperature dependence of the mode eigenfrequency . together they modify the temperature dependence of the mode eigenfrequencies compared to that of undoped crystals . this modification often weakens the temperature dependence of the eigenfrequencies , which proves very important for applications of micro - mechanical systems in devices that work in a broad temperature range @xcite . equations ( [ eq : free_energy_expansion ] ) - ( [ eq : frequency_change ] ) are generic and apply beyond the deformation potential approximation . this is of particular importance for silicon . here , the electron band valleys lie on the @xmath76-axes close to the @xmath77-points on the zone boundaries where two electron energy bands cross . lattice strain can lead to a band splitting at @xmath77-points and a shift of the valleys @xcite . importantly , this shift results from a shear strain , which does not lead to a linear in the strain shift in the deformation potential approximation . the valley shift is quadratic in @xmath41 in this case , as explained in appendix [ appendixa ] , which corresponds to an effectively two - phonon coupling . the coupling parameter @xmath78 is quadratic in the strain - induced band splitting , see eq . ( [ eq : general_shift ] ) . it is large , much larger than the constant @xmath28 . therefore the arguments given below eq . ( [ eq : strong_coupling ] ) apply in this case as well . for purely shear strain in silicon , terms of odd order in @xmath41 in @xmath48 , eq . ( [ eq : free_energy_expansion ] ) , vanish . [ cols="^,^,^,^ , < " , ] [ table : tensors ] tensors @xmath79 can be obtained by minimizing the free energy density of the electron system for a given strain and expanding the result in a series in @xmath41 . a general procedure that allows one to find the components @xmath79 for @xmath80 is described in appendix [ appendixa ] . using the symmetry arguments , the elasticity tensors are conveniently written in the contracted ( voigt ) notation where the symmetric strain tensor is associated with a six - component vector . then the nonlinear elasticity tensors @xmath81 and @xmath82 become tensors of rank three and four in the corresponding vector space . we use notation @xmath83 for tensors @xmath84 in these notations to emphasize that we are calculating corrections to the nonlinear elasticity tensors due to the electron - phonon coupling . the explicit expressions for the nonlinear elasticity tensors @xmath85 are given in table [ table : tensors ] . they refer to silicon and include the contributions that come from both the deformation potential coupling and from the splitting of the electron bands due to shear strain . in the deformation potential approximation , the components of the third - rank tensor @xmath85 , which determine the cubic in the strain terms in the free energy , were found earlier @xcite . therefore we give only the components that contain a contribution from shear strain . the fourth - rank tensor @xmath85 determines the quartic in the strain terms in the free energy and has not been discussed before , to the best of our knowledge . we give all independent components of this tensor . it is expressed in terms of the derivative of the electron density @xmath3 over the chemical potential in the absence of strain @xmath51 , which is a familiar thermodynamic characteristic . it is intuitively clear that the considered effect of the change of the electron density in different valleys in response to strain should be related to the derivative @xmath86 . interestingly , because we consider nonlinear response to strain , the expressions in table [ table : tensors ] contain also higher - order derivatives of @xmath3 over @xmath51 . as we will see , this leads to a nontrivial behavior of the nonlinear frequency shift with varying temperature and density . the considered mechanism of the strain - induced inter - valley electron redistribution does not contribute to the components @xmath87 and @xmath88 , therefore @xmath89 . the expressions for @xmath85 simplify in the case of low doping ( or high temperature ) , where the electron gas is strongly nondegenerate , and in the opposite case of a strongly degenerate electron gas . for a nondegenerate gas , where the chemical potential in the absence of strain is @xmath90 , we have in table [ table : tensors ] @xmath91 with @xmath92 . the @xmath51-dependent factors @xmath93 in @xmath94 and its derivatives cancel each other in the expressions for @xmath85 and drop out from these expressions . the dependence of @xmath85 on density is then just linear , @xmath95 . parameters @xmath96 in table [ table : tensors ] depend only on temperature , @xmath97 and @xmath98 . the decrease of the nonlinear elasticity parameters with increasing temperature in a nondegenerate electron gas is easy to understand . the effect we consider is determined by the competition between the energetically favorable unequal population of the electron energy valleys in a strained crystal and the entropically more favorable equal valley population . with increasing temperature the entropic factor becomes stronger , leading to a smaller population difference and thus smaller effect of the electron system on the vibrations . for strong doping , where @xmath99 , we have @xmath100 , and then @xmath101 with @xmath92 . therefore parameters @xmath102 in table [ table : tensors ] become temperature independent , with @xmath103 , and @xmath104 . the results on the asymptotic behavior of the corrections to nonlinear elasticity are not limited to silicon . since parameters @xmath105 are given by the coefficients in the general expansion of the free energy in strain , ( [ eq : total_f ] ) , these results can be applied to the nonlinear elasticity induced by the electron - phonon coupling in other multi - valley semiconductors . to illustrate this point , in appendix [ sec : germanium ] we give @xmath85 tensor in germanium . the difference between the asymptotic behavior of the tensors @xmath85 in the limits of nondegenerate and strongly degenerate electron gas can lead to a peculiar density and temperature dependence of the nonlinear frequency shift of the vibrational modes . it comes from the coefficients @xmath102 containing higher - order derivatives of @xmath3 with respect to @xmath51 . in the transition region @xmath106 , thinking of the competition between the entropic and energetic factors does not provide a simple insight into the behavior of @xmath85 , as both the energy and the entropy are complicated functions of density and temperature . the nonlinear elasticity tensors in table [ table : tensors ] give the doping - induced contributions to the nonlinearity parameters of the eigenmodes of micro- and nanomechanical systems . these contributions are described by eq . ( [ eq : nonlin_params ] ) . as mentioned before , an important characteristic of the mode nonlinearity is the dependence of the mode frequency on the vibration amplitude . to the leading order , it is given by eq . ( [ eq : duffing_shift ] ) . this dependence has a contribution from the nonlinearity of an undoped crystal , which is quadratic in the parameters of the cubic nonlinearity ; for example , if the latter is described by the grneisen constant , the corresponding contribution is quadratic in this constant . it is typically small . there is also a contribution from the quartic nonlinearity ; the parameters of such nonlinearity are not known in undoped crystals and are not expected to be large . respectively , the amplitude dependence of the vibration frequency for low - frequency modes in weakly doped single - crystal micro - mechanical systems is relatively weak @xcite . a feature of the doping - induced nonlinearity described by table [ table : tensors ] is that the quartic in the strain term in the free energy has a large coefficient compared to the cubic term , cf . ( [ eq : strong_coupling ] ) and the discussion below this equation . therefore , in eq . ( [ eq : duffing_shift ] ) for the amplitude dependence of vibration frequency one can keep only the duffing nonlinearity constant @xmath107 . the contribution from the cubic nonlinearity terms @xmath108 can be disregarded . for a mode @xmath4 , the doping - induced contribution to @xmath107 is equal to @xmath109 in eq . ( [ eq : nonlin_params ] ) . to find the dependence of the mode frequency on the vibration amplitude we go through the following steps . first , we find the normal modes of interest for the given geometry of the system , with account taken of the boundary conditions , and normalize the displacements @xmath5 as indicated in eq . ( [ eq : normalization ] ) . we use @xmath5 to find the strain tensor @xmath110 . the result is substituted into eq . ( [ eq : nonlin_params ] ) and is convoluted with tensor @xmath82 , giving the value of @xmath107 , which is then used in eq . ( [ eq : duffing_shift ] ) to find the frequency dependence on the vibration amplitude @xmath16 . of particular interest is the relative frequency shift @xmath111 . to find this shift to the leading order , one can disregard nonlinearity when calculating the eigenfrequency @xmath8 . then , from eq . ( [ eq : duffing_shift ] ) , @xmath112 where @xmath113 is the full tensor of linear elasticity , which includes the major term of the linear elasticity of the undoped crystal and the doping - induced correction @xmath75 . an important feature of the relative shift @xmath111 is its scaling with the size of the system . the vibration amplitude @xmath114 in eq . ( [ eq : scaling ] ) can be scaled by the lateral dimension @xmath115 , for example the length of a nanobeam or a nanowire for an extension mode , or the size of the square for a lam mode , or the diameter of a disk for a breathing mode in a disk . respectively , we write @xmath116 . then , if one takes into account the explicit form ( [ eq : nonlin_params ] ) of the parameter @xmath117 , one finds from eq . ( [ eq : scaling ] ) that the ratio @xmath118 is independent of the system size for the aforementioned modes . in this estimate we used that the tensors @xmath119 are material parameters and are independent of the geometry . we also used that the modes of interest have typical wavelength @xmath120 , and therefore @xmath65 scales as @xmath121 . most of the experiments in nano- and micromechanics are done with nanobeams , nanowires , membranes , or thin plates . in such systems the thickness is much smaller than the length or , in the case of membranes or plates , the lateral dimensions . then , from the boundary condition of the absence of tangential stress on free surfaces @xcite , it follows that the strain tensor @xmath41 weakly depends on the coordinate normal to the surface . this simplifies the denominator in eq . ( [ eq : scaling ] ) , making it proportional to the thickness . similarly , from eq . ( [ eq : nonlin_params ] ) @xmath107 is also proportional to the thickness , and the thickness drops out of eq . ( [ eq : scaling ] ) . the explicit expressions for @xmath122 and @xmath107 that determine the denominator and the numerator in eq . ( [ eq : scaling ] ) , respectively , are given in appendices [ sec : lame ] and [ sec : extension ] for lam and extension modes . these expressions are cumbersome , and it is convenient to use symbolic programming to obtain them . the scaled ratio @xmath118 that characterizes the relative nonlinear frequency shift is shown in fig . [ fig : n_dependence ] for several modes that are often used in single - crystal silicon mems . this ratio depends on the type of the mode and the crystal orientation . figure [ fig : n_dependence ] refers to high - symmetry crystal orientations , in which case the modes have a comparatively simple spatial structure and the surfaces can be made smooth . we used the values @xmath123 ev @xcite , @xmath124 ev , the effective mass for density of states @xmath125 @xcite , and the temperature - dependent linear elasticity parameters given in ref . @xcite . figure [ fig : n_dependence ] shows that the electron - redistribution induced nonlinearity of vibrational modes is very strong . for the ratio of the vibration amplitude to the system size @xmath126 and the mode eigenfrequency @xmath127 mhz , the frequency change can be as a large as @xmath128 khz . this explains , qualitatively , the observations @xcite . a quantitative comparison with the experiment @xcite is complicated , as the observations refer to different samples . our preliminary results show an excellent quantitative agreement with the data obtained for the same sample at different temperatures and for different types of modes @xcite . the nonlinear frequency shift displays several characteristic features , as seen from fig . [ fig : n_dependence ] . one of them is the strong dependence of the shift on the type of the mode and the crystal orientation . for both the lam and the extension mode , the shift is much stronger for crystals cut out in @xmath76 direction than in @xmath129 direction . this is a consequence of the electron energy valleys lying along the @xmath76 axes , making the system more responsive " to the lattice displacement along these axes . interestingly , in the both configurations the shifts for the lam modes are larger than for the extension modes . a somewhat unexpected feature is the nonmonotonic dependence of the nonlinear frequency shift on the electron density and temperature . the nonmonotoncity occurs in the range where the electron system is close to degeneracy , @xmath130 , and it strongly depends on the crystal orientation . it is much stronger for crystals cut in @xmath76 than @xmath129 directions . for a crystal cut in @xmath129 direction , both the density and temperature dependence of the shift are monotonic in the case of the lam mode , whereas for the extension mode the nonmonotonicity is weak . the nonmonotonicity of the frequency shift stems from the behavior of the parameters @xmath131 in the range @xmath106 . as seen from table [ table : tensors ] , parameter @xmath132 exponentially increases with the increasing @xmath133 for negative @xmath133 , but for large positive @xmath133 it falls off as @xmath134 . it has a pronounced maximum for @xmath135 . parameter @xmath136 also displays a maximum , which occurs for @xmath137 . in contrast , parameters @xmath138 depend on @xmath133 monotonically . the results of appendices [ sec : lame ] and [ sec : extension ] show that , for the lam and extension modes in crystals cut in @xmath76 direction , the relative shift @xmath111 is determined by coefficient @xmath132 , which explains the nonmonotonicity of the shift . for crystals cut in @xmath129 , the shift of the lam mode is fully determined by coefficient @xmath139 and is monotonic , whereas for the extension mode the expression for the shift has contributions from @xmath132 , @xmath136 , and @xmath139 that partly compensate each other , leading to a comparatively small shift all together and its weak nonmonotonicity . the results of this paper show that the electron - phonon coupling strongly affects the nonlinearity of vibrational modes in semiconductor - based nano- and micromechanical systems . the mechanism of the effect is the strain - induced redistribution of the electrons between the valleys of the conduction band . the redistribution results from lifting the degeneracy of the electron energy spectrum by the strain from a vibrational mode . the analysis refers to the range of temperatures where the rate of intervalley scattering strongly exceeds the frequencies of the considered modes . in this case the valley populations follow the strain adiabatically . the change of the valley populations is a strongly nonlinear function of the strain tensor . the respective expansion of the free energy in the strain is an expansion in the strain multiplied by the ratio of the electron - phonon coupling energy ( in particular , the deformation potential ) to the chemical potential of the electron system or the temperature . this ratio is large , @xmath140 . it is this parameter that makes the nonlinearity of the vibrational modes in doped semiconductor structures strong . of special interest in nano- and micromechanical systems is the amplitude dependence of the vibration frequency . to the leading order , it is determined by the quartic terms in the expansion of the free energy in strain . these terms are comparatively large in doped crystals . we have calculated the nonlinear elasticity tensor that describes the electron contribution to the terms in the free energy , which are quartic in the strain . the explicit expressions for the tensor components refer to semiconductors with the valleys on @xmath76 axes , in particular , to silicon . we have also found this tensor for germanium . in silicon , along with the deformation potential coupling , an important role is played by the coupling to shear strain . such strain lifts the band degeneracy at the zone boundary and is effectively described by a two - phonon coupling . we show that this coupling also leads to strong nonlinearity of vibrational modes . the parameter of the electron coupling to shear strain in silicon is not easy to access in the experiment @xcite . measurements of the nonlinear frequency shift provide a direct means for determining this parameter . in particular , the nonlinear frequency shift of the fundamental lam mode in a silicon plate cut along @xmath129 axes is determined by this parameter only , except for small corrections from the nonlinearity of the undoped crystal . we found that the nonlinear frequency shift strongly depends on the type of a vibrational mode and the crystal orientation . we also found that the ratio of the frequency shift to the squared vibration amplitude can be profoundly nonmonotonic as a function of electron density and temperature . the results provide an insight into the experimentally observed strong mode nonlinearity in doped crystals @xcite . in terms of applications , they enable choosing the appropriate range of doping and the temperature regime to optimize the operation of nano- and micromechanical resonators . we are grateful to t. kenny for attracting our attention to the problem and for stimulating discussions . we benefited from useful discussions with j. atalaya , d. heinz , p. polunin , s. w. shaw , and y. yang . this research was supported in part by the us defense advanced research projects agency ( grant no . fa8650 - 16 - 1 - 7600 ) . the major effect of a strain on the electron free energy comes from the shift of the energy valleys . we will assume that valley @xmath27 is shifted in energy by @xmath26 and the shift is small , @xmath141 , where @xmath51 is the chemical potential in the absence of strain . we further assume that the vibrations are slow compared to the time it takes the electron system to come , locally , to thermal equilibrium for given values of @xmath26 , i.e. , the temperature and the chemical potential are the same in all valleys . since for high electron densities the thermal conductivity is high , the change of the temperature compared to the ambient temperature can be disregarded ; also , as mentioned in the main text , the electron density @xmath3 summed over all valleys is constant . expanding the electron free energy density to the 4th order in the strain - induced shifts @xmath26 , we find that , in an @xmath142-valley semiconductor , the change @xmath48 of the free energy density is @xmath143 + \frac{1}{6}\frac{f''_{1/2}}{f_{1/2}}\left[\overline { \delta_\ep^3 } -3\overline { \delta_\ep^2}\;\overline\delta_\ep + 2(\overline\delta_\ep)^3 \right ] \nonumber\\ & + \frac{1}{8}\frac{{f''_{1/2}}^2}{f_{1/2}f'_{1/2}}\left[(\overline { \delta_\ep^2})^2 -2\overline { \delta_\ep^2}\,(\overline\delta_\ep)^2 + ( \overline\delta_\ep)^4 \right]+ \frac{1}{24}\frac{f'''_{1/2}}{f_{1/2}}\left [ 4\overline { \delta_\ep^3}\;\overline\delta_\ep - \overline { \delta_\ep^4 } - 6\overline { \delta_\ep^2}\,(\overline\delta_\ep)^2 + 3(\overline\delta_\ep)^4 \right].\end{aligned}\ ] ] here , @xmath144 . we use the standard notation @xmath145 $ ] ; primes indicate differentiation over @xmath146 , for example , @xmath147 . function @xmath94 and its derivatives are calculated for @xmath92 . equation ( [ eq : total_f ] ) immediately gives the tensors @xmath148 of the expansion of the free energy increment ( [ eq : free_energy_expansion ] ) if one expresses the shift @xmath26 of the valleys in terms of the strain tensor . in the deformation potential approximation the relation between @xmath26 and @xmath41 is given in the main text , see also eq . ( [ eq : general_shift ] ) below . in the case of si crystals , which are often used in micromechanical resonators , an important contribution to @xmath26 comes from the shear - strain induced splitting of the electron energy bands at the zone boundary . shear strain does not lead to the valley shift in the deformation potential approximation . the overall shift of valley @xmath27 , to the lowest order in the coupling that causes it ( i.e. , to the first order in the deformation potential where its contribution is nonzero and to the second order in the band splitting for shear strain ) is @xcite : @xmath149 here we use that silicon has six valleys located at the @xmath76 axes , and we chose the coordinate axes @xmath150 along @xmath76 . respectively , the valley index @xmath27 takes on three values that correspond to the @xmath150 axes ( the valleys lying on the same axis , but in the opposite directions , are equivalent ) . the strain @xmath151 , which enters the second term in the right - hand side of eq . ( [ eq : general_shift ] ) , is a component of the strain tensor @xmath30 with @xmath152 such that @xmath153 and @xmath154 . the parameter @xmath155 is the interband matrix element of the electron - phonon coupling calculated for the electron conduction bands @xmath156 and @xmath157 at the @xmath77 point on the boundary of the brillouin zone , where the bands cross ; @xmath158 is the energy separation between the bands @xmath156 and @xmath157 at the value of the wave vector @xmath159 that corresponds to the conduction band minimum . parameter @xmath78 is the effective deformation potential of two - phonon coupling to shear strain . the numerical value of @xmath78 is not well known . the experimental data give @xmath160 ev @xcite and the numerical data on the band splitting give @xmath161 ev @xcite so that @xmath78 is in the range of @xmath162 ev ; this is essentially an order of magnitude estimate . in calculating @xmath48 in eq . ( [ eq : total_f ] ) we kept terms that are quartic in @xmath41 . the components of the tensors @xmath45 in eq . ( [ eq : free_energy_expansion ] ) are expressed in terms of @xmath48 as @xmath163 tensors @xmath84 are symmetric with respect to the interchange of indices @xmath164 and the pairs @xmath165 . for the considered long - wavelength strain , tensors @xmath45 are independent of coordinates . the corrections @xmath166 to the linear elasticity tensors were found previously @xcite and are not discussed in this paper . in this section we provide the corrections to the nonlinear elastic constants of germanium , which are due to the redistribution of the electrons over the valleys . germanium has four equivalent valleys in the conduction band , which are located on the boundary of the brillouin zone along @xmath167 axes . we use the voigt notation and write the components of the corrections to the nonlinear elasticity tensor @xmath85 in the frame where the axes @xmath168 are along the @xmath76 directions of the crystal . using the results of appendix [ appendixa ] , we obtain @xmath169 the notations are the same as in appendix [ appendixa ] and in table [ table : tensors ] . the electron - phonon coupling does not contribute to the other third- and fourth - order elastic constants . corrections @xmath170 and @xmath171 for germanium were found by keyes @xcite ; however , his final expression for @xmath171 differs from eq . ( [ eq : ge ] ) by a factor of 4 ( our expressions for @xmath170 coincide with ref . parameters @xmath172 and @xmath173 have not been found before , to the best of our knowledge . in the limiting cases , corrections @xmath172 and @xmath173 have the same dependence on temperature and electron density as constant @xmath132 discussed in sec . [ sec : limiting ] . we consider a square plate with side @xmath115 and thickness @xmath174 made out of a single crystal with cubic symmetry . if the crystal is cut out along @xmath76 or @xmath129 axes , one of the simplest modes is the first lam mode @xcite . the normalized displacement field is @xmath175 here , @xmath146 and @xmath176 axes are in the lateral plane along the sides of the square , axis @xmath177 is perpendicular to the plate and @xmath178 . calculating the strain tensor for the displacement ( [ eq : lame_displacements ] ) and substituting the expressions into eqs . ( [ eq : nonlin_params ] ) and the relation @xmath179 for the plate cut out along @xmath180 axes we obtain , in voigt notation for the elasticity tensors , @xmath181 if we consider silicon and take into account only the contribution @xmath85 to the nonlinear elasticity tensor @xmath182 , with the account taken of table [ table : tensors ] , the expression for @xmath107 simplifies to @xmath183 for the lam mode cut along the @xmath129 axis , if the tensors are calculated in the axes @xmath76 , we have @xmath184 note that only coupling to shear strain contributes to the nonlinearity parameter @xmath107 in this case . we consider the fundamental extension mode in a thin beam of length @xmath115 with a rectangular cross - section of area @xmath185 . the beam is cut along a symmetry axis , and the sides are also along symmetry planes of a cubic crystal . from the free - surface boundary conditions , the normalized displacement field is @xcite : @xmath186 this expression takes into account transverse compression that accompanies beam extension and uses the smallness of the beam cross - section ; corrections @xmath187 are disregarded . the transverse compression in a cubic crystal cut in a symmetric direction is described by poisson s ratios @xmath188 and @xmath189 . generally , they do not coincide . in eq . ( [ eq : displace_extension ] ) the transverse coordinates @xmath176 and @xmath177 are counted off from the center of the beam for the longitudinal direction of the beam @xmath76 and the sides parallel to @xmath190 planes , the poisson parameters are equal , @xmath191 and @xmath192 . in this case ( [ eq : nonlin_params ] ) and ( [ eq : frequency ] ) give @xmath193.\end{aligned}\ ] ] the expression for @xmath107 is simplified if in the nonlinear elasticity tensors we take into account only the contribution from the electron - phonon coupling as given in table [ table : tensors ] and also allow for the interrelation between different components of the tensor @xmath85 . then for a silicon beam @xmath194 for extension along @xmath129 axis , with one side parallel to @xmath190 plane and the other side parallel to @xmath195 plane , the poisson s ratios @xmath196 and @xmath197 are given in ref . @xcite . then eqs . ( [ eq : nonlin_params ] ) and ( [ eq : frequency ] ) give @xmath198.\end{aligned}\ ] ] if in the nonlinear elasticity tensor @xmath182 we take into account only the contribution @xmath85 from the electron - phonon coupling , in the case of a silicon beam the expression for @xmath107 simplifies to @xmath199 expressions ( [ eq : extension_100 ] ) and ( [ eq : extension_110 ] ) were generated using a computer code to calculate the sums and integrals in eq . ( [ eq : nonlin_params ] ) . 35ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/physrevlett.92.166801 [ * * ( ) , 10.1103/physrevlett.92.166801 ] link:\doibase 10.1103/physrevb.70.193305 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.76.165317 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.110.066804 [ * * , ( ) ] @noop _ _ ( , ) in @noop _ _ , ( , ) pp . @noop * * , ( ) http://stacks.iop.org/1742-6596/92/i=1/a=012002 [ * * , ( ) ] link:\doibase 10.1038/srep03244 [ * * , ( ) ] @noop * * , ( ) in @noop _ _ ( ) pp . @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrev.161.756 [ * * , ( ) ] @noop * * , ( ) @noop _ _ , ed . ( , ) @noop _ _ , ed . 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we study the effect of the electron - phonon coupling on vibrational eigenmodes of nano- and micro - mechanical systems made of semiconductors with equivalent energy valleys . we show that the coupling can lead to a strong mode nonlinearity . the mechanism is the lifting of the valley degeneracy by the strain . the redistribution of the electrons between the valleys is controlled by a large ratio of the electron - phonon coupling constant to the electron chemical potential or temperature . we find the quartic in the strain terms in the electron free energy , which determine the amplitude dependence of the mode frequencies . this dependence is calculated for silicon micro - systems . it is significantly different for different modes and the crystal orientation , and can vary nonmonotonously with the electron density and temperature .

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Proceed to summarize the following text: one of the most important predictions of lattice qcd is the existence of a transition from ordinary hadronic matter , where quarks and gluons are confined , to a quark gluon plasma at high temperatures . current estimates are that this transition takes place at @xmath0 mev , and is most likely a crossover at zero chemical potential . at higher chemical potential a first - order phase transition is predicted , ending in a tricritical point at @xmath1 mev . these predictions are currently being put to the test at rhic and other heavy - ion colliders . however , interpreting the results of these experiments and comparing them to lattice qcd predictions is far from straightforward . one important reason for this is that the process to a large extent takes place out of thermal equilibrium , and equilibrium field theory methods such as lattice monte carlo are therefore not sufficient . one major , still unsolved puzzle of heavy - ion physics is whether the system ever actually reaches thermal equilibrium , and if so , what the equilibration time is . many aspects of the collision can be successfully described ( eg , using hydrodynamics @xcite ) by assuming a very short equilibration time of 1 @xmath2 , but it is far from understood how this would come about . clearly , a proper understanding of the thermalisation process is essential if we are to have a coherent and reliable description of the heavy - ion collision . at late stages of the collision process , as the system expands , the particles eventually decouple . this typically occurs in two phases : first , inelastic collisions cease , causing the ratios of different particle species to be fixed ( chemical freeze - out ) ; later , also the mean free path for elastic collisions becomes too large and the momentum distribution of the particles deviates from that of thermal equilibrium ( thermal or kinetic freeze - out ) . it is at this point that the final particle yields of the collision are fixed . thus , non - equilibrium dynamics is needed to understand also this aspect of the process . another field where non - equilibrium field theory is needed , is early - universe physics . examples of non - equilibrium processes which require a field - theoretical approach include ( p)reheating , electroweak and qcd phase transitions in the early universe , and baryogenesis . field theories out of equilibrium is a notoriously difficult problem to study nonperturbatively . a large number of approaches have been employed , including hartree and large - n approximations @xcite , dyson schwinger - related approaches based on the 2pi effective action @xcite , and kinetic theory @xcite . all of these have their strengths as well as drawbacks and limitations . within its area of applicability , the classical approximation has the advantage of being fully nonperturbative , easy and relatively inexpensive to implement numerically , and straightforwardly applicable to gauge theories . the major drawback is obviously that quantum effects are not taken into account . classical statistical physics is quantum statistical physics in the limit of large occupation numbers . it follows that the classical approximation can be used when the occupation numbers of the relevant or dominant modes of the system are large . in the context of heavy - ion collisions it is arguable that the multiplicity of soft gluons in the initial ( pre - equilibrium ) stages is very high , and the classical approximation may therefore be valid . the classical approximation has also been applied to the chiral dynamics during freeze - out . in the linear sigma model and related models , the effective potential of the chiral @xmath3 field changes from symmetric to mexican - hat type as the temperature drops , giving rise to an instability where the low - momentum modes increase exponentially . in this scenario , the resulting high occupation numbers justify the use of the classical approximation . a particular hazard with simulating classical dynamics on a lattice is connected with high - momentum lattice artefacts . these will in general interact with the soft modes which carry the interesting physics and for which the classical approximation is in principle valid . if there is sufficient strength in the hard modes , they may equilibrate classically with the soft modes on a much shorter timescale than that of the interesting physics @xcite . in that case , not only the hard modes , but the entire system will be dominated by classical lattice artefacts . to avoid this problem , it is important that the high - momentum modes should be , and remain , strongly suppressed . at early stages , this can be ensured by choosing appropriate initial conditions . the initial conditions are a crucial part of the simulation . they should reflect the salient features of the system at the outset . this is also the only place where information about the quantum nature of the real world enters into the simulation . a quantum system may be represented as an ensemble of classical configurations initially distributed according to quantum statistics @xcite . one example of this may be to choose the 2-point correlators of the fields and their canonical momenta to obey the bose einstein distribution for free fields at some temperature @xmath4 , after subtracting the quantum vacuum fluctuations . for scalar fields @xmath5 with momentum fields @xmath6 and mass @xmath7 one would then have @xmath8 where @xmath9 . such an initialisation also provides an exponential cutoff for the hard modes , which will help in avoiding the dangerous lattice artefacts . following this , each configuration evolves independently according to the classical hamiltonian equations of motion , and time - dependent correlators are computed as averages over the initial conditions . given sufficient time , the system will eventually thermalise classically , resulting in classical equipartition @xmath10 and giving rise to rayleigh jeans type divergences . the hope is that this will happen on much longer time scales than those under consideration in the simulation . there is no unique definition of local particle numbers and energies for interacting fields out of equilibrium . still , the system may exhibit effective particle - like behaviour , which may be used to characterise the approach to thermal equilibrium or to an equilibrium - like distribution . given a definition of local particle numbers , these can also be used to give an effective description of the system in terms of kinetic theory . the effective particle numbers may be extracted from the two - point field correlators , which in the free - field case ( where the particle description is appropriate and well - defined ) contain all the information there is about the system . for example , for a homogeneous , free scalar field we have @xmath11 where @xmath12 is the occupation number for the mode with momentum @xmath13 and @xmath14 is the associated energy . for interacting fields , this may in turn be used as a _ definition _ of the instantaneous particle numbers and energies @xmath15 and @xmath16 @xcite : @xmath17 in the classical approximation the @xmath18 is left out . in a non - abelian gauge theory , the correlation functions will in general be gauge dependent , so the distribution functions will contain ambiguities due to the gauge choice . this ambiguity may be removed by constructing gauge invariant correlators using parallel transporters ; however , this introduces path dependence . in particular , in a lattice regularisation there is in general no one preferred path between two points . although the distribution functions are not unique , all physical observables extracted from them , such as masses , temperatures and chemical potentials , should not depend on the definition and in particular on the gauge . as long as these quantities are not well - defined on the other hand ( such as when the system is very far from equilibrium and the quasiparticle picture does not apply ) , one may expect `` masses '' and `` temperatures '' to be definition - dependent . thus , studying the gauge dependence ( or path dependence ) of distribution functions may serve the double purpose of monitoring the approach to equilibrium and verifying the validity of the approach used . one natural choice of gauge is the coulomb gauge , which is a smooth gauge . in a system with spontaneously broken gauge symmetry ( e.g. , a higgs system ) , the unitary gauge , where the ( effective ) higgs field has only one non - zero , real component , is another natural choice . other gauges , such as maximal abelian gauge , axial gauges or random gauge , may also be considered . in the coulomb gauge , the gauge potential @xmath19 ( but not its conjugate momentum @xmath20 ) is purely transverse , and it can be shown that the transverse free correlators behave analogously to the scalar case @xcite . thus the particle numbers and energies can be defined as @xmath21 here @xmath22 are the transverse @xmath23- and @xmath24-correlators respectively , constructed from the two - point functions @xmath25 according to @xmath26 + + in figure [ fig : n - gauge ] the gauge dependence of effective particle numbers is illustrated in the su(2)higgs model @xcite . in this case , the system was prepared in such a way that all the energy initially was in the higgs field , while the gauge potential was initialised to zero . since the angular modes of the higgs fields are absorbed into the gauge fields in the unitary gauge , the inital occupation numbers are very different . however , already after @xmath27 the two distributions appear almost identical . however , while the particle numbers in the coulomb gauge change very little from here on , in the unitary gauge they continue to fluctuate and it is only from @xmath28 on that one with some confidence can claim the numbers are gauge independent . this agrees roughly with the point where the dispersion relation in the unitary gauge begins to show stable , particle - like behaviour in this particular model . in a non - homogeneous system , and in general in a kinetic - theory description , it is appropriate to think in terms of _ local _ particle numbers @xmath29 . these may be related to the wigner functions constructed from gauge invariant two - point functions @xcite , or more generally to the two - point functions fourier transformed on a region @xmath30 centred on @xmath31 . in the case of a scalar theory we may have ( suppressing the common @xmath32-coordinate for brevity ) @xmath33 here @xmath34 is the volume of the region @xmath30 . this coarse - graining creates an intrinsic unsharpness in the momentum @xmath13 and position @xmath31 of the quasiparticles , given by the size ( and shape ) of the region @xmath35 . if the system under consideration is homogeneous , we may improve statistics by performing an average over all space . this can be shown to be equivalent to local averaging in momentum space , with a weight function @xmath36 depending on the size and shape of @xmath35 : @xmath37 where @xmath38 denotes the correlation function evaluated on the total volume @xmath39 , and the sum is over the discrete momenta available on this volume . in practice , it is simpler to work backwards , choosing a simple form of momentum - space averaging which may correspond to rather complicated spatial regions . for instance , binning in the absolute value of the momentum , @xmath40 corresponds to spherical shells with thickness approximately @xmath41 in position space . at the earliest stages of heavy - ion collisions , the gluon density is expected to be so high that the classical approximation can be justified . the same approximation also justifies ignoring the back - reaction of the quarks , since their number density will be much lower ; leaving us with classical yang mills equations of motion , which may be solved numerically on a lattice . the lattice equations of motion in the temporal gauge ( @xmath42 ) read @xmath43\ ] ] where @xmath44\ ] ] is the canonical momentum to @xmath45 . here , @xmath46 denotes the backward lattice derivative , while @xmath47 is the backward covariant lattice derivative . the equations of motion for @xmath48 constitute the gauss constraint , @xmath49 which must be satisfied by the initial conditions but is conserved by the equations of motion . the initial gluon fields should be related to the gluon distributions of the two colliding nuclei : in principle they should just be the superposition of two lorentz - boosted nuclear gluon distributions . simulations have been carried out over a number of years by krasnitz , nara and venugopalan @xcite ( see also @xcite ) using the `` colour glass condensate '' model of the nuclear wave function to provide the initial conditions . in these studies , the numerical work has been simplified by considering only the mid - rapidity region where the physics is assumed to be boost - invariant . this reduces the system to effectively 2 + 1 dimensions . with these assumptions , the authors have been able to provide an estimate of the initial energy density and gluon distribution which may be used as input into hydrodynamic or kinetic calculations . an alternative approach would be to determine the nuclear gluon field from e.g. a bag model , give this a lorentz boost , and perform a 2 + 1 + 1-dimensional simulation with the longitudinal lattice spacing @xmath50 , where @xmath51 is the lattice spacing in the transverse ( @xmath52 ) direction . work is underway to implement this . the classical approximation may be applied to a range of problems in non - equilibrium field theory where occupation numbers are high , such as the earliest stages of heavy - ion collisions . it has the advantage of being non - perturbative and computationally relatively inexpensive . effective particle numbers may be defined out of equilibrium in a self - consistent manner , and their gauge dependence ( or that of derived quantities such as masses and temperatures ) can be used as a check on the validity of the quasi - particle picture . this work was supported by fom / nwo . i am thankful to jan smit and anders tranberg for numerous fruitful discussions . p. f. kolb and u. heinz , nucl - th/0305084 . f. cooper _ et al . _ , phys . rev . * d50 * , 2848 ( 1994 ) [ hep - ph/9405352 ] . d. boyanovsky , h. j. de vega and r. holman , phys . rev . * d51 * , 734 ( 1995 ) [ hep - ph/9401308 ] . j. berges and j. cox , phys b517 * , 369 ( 2001 ) [ hep - ph/0006160 ] . j. berges and j. serreau , hep - ph/0302210 . r. baier , a. h. mueller , d. schiff and d. t. son , phys . lett . * b502 * , 51 ( 2001 ) [ hep - ph/0009237 ] . g. d. moore , jhep * 11 * , 021 ( 2001 ) [ hep - ph/0109206 ] . m. sall , j. smit and j. c. vink , phys . rev . * d64 * , 025016 ( 2001 ) [ hep - ph/0012346 ] . g. aarts and j. smit , phys . rev . * d61 * , 025002 ( 2000 ) [ hep - ph/9906538 ] . skullerud , j. smit and a. tranberg , jhep * 08 * , 045 ( 2003 ) [ hep - ph/0307094 ] . a. krasnitz and r. venugopalan , phys . * 84 * , 4309 ( 2000 ) [ hep - ph/9909203 ] . a. krasnitz , y. nara and r. venugopalan , nucl . phys . * a717 * , 268 ( 2003 ) [ hep - ph/0209269 ] . t. lappi , phys . rev . * c67 * , 054903 ( 2003 ) [ hep - ph/0303076 ] .

the classical approximation may be applied to a number of problems in non - equilibrium field theory . the principles and limits of classical real - time lattice simulations are presented , with particular emphasis on the definition of particle numbers and energies and on applications to the earliest stages of heavy - ion collisions . mathmargin = 0pt

You are an expert at summarizing long articles. Proceed to summarize the following text: networks of coupled oscillators have been extensively studied for many years , owing to their wide applicability in physics , chemistry , and biology . as examples we mention laser arrays , josephson junctions , populations of fireflies , etc . @xcite . the phase - only models have proved to provide useful models for systems with weak coupling . the best known model of this type is the kuramoto model in which the oscillators are described by phase variables @xmath0 and coupled to others through a sinusoidal function @xcite . these models exhibit a transition to collective synchronization as the coupling strength increases , a process that has been described as a phase transition . a general form of these systems is as follows : @xmath1 here @xmath2 is the natural frequency of oscillator @xmath3 , @xmath4 represents the coupling between oscillators @xmath3 and @xmath5 , @xmath6 is a phase lag and @xmath7 is the overall coupling strength . a general treatment of this system is not easy , and two types of simplifications are commonly used . one of them is to assume global coupling among the oscillators , and assign the natural frequencies @xmath2 randomly and independently from some prespecified distribution @xcite . the second tractable case arises when all the oscillators are assumed to be identical , and the coupling @xmath4 is taken to be local , eg . , nearest - neighbor coupling @xcite . the intermediate case of nonlocal coupling is harder but the phenomena described by the resulting model are much richer . in this paper we suppose that the oscillators are arranged on a ring . in the continuum limit @xmath8 the system is described by the nonlocal equation @xmath9\,dy\label{phase_eq}\ ] ] for the phase distribution @xmath10 . when the oscillators are identical ( @xmath11 is a constant ) and @xmath12 this system admits a new type of state in which a fraction of the oscillators oscillate coherently ( i.e. , in phase ) while the phases of the remaining oscillators remain incoherent @xcite . in this paper we think of this state , nowadays called a _ chimera _ state @xcite , as a localized structure embedded in a `` turbulent '' background . subsequent studies of this unexpected state with different coupling functions @xmath13 have identified a variety of different one - cluster and multi - cluster chimera states @xcite , consisting of clusters or groups of adjacent oscillators oscillating in phase with a common frequency . the clusters are almost stationary in space , although their position ( and width ) fluctuates under the influence of the incoherent oscillators on either side . recently , a new type of chimera state has been discovered , a traveling chimera state @xcite . in this state the leading edge plays the role of a synchronization front , which kicks oscillators into synchrony with the oscillators behind it , while the trailing front kicks oscillators out of synchrony ; these two fronts travel with the same speed , forming a bound state . it is natural to ask whether these states persist in the presence of spatial inhomogeneity , in particular , in the presence of spatial inhomogeneity in the natural frequency distribution ( @xmath14 ) . in @xcite laing demonstrated the robustness of the chimera state with respect to inhomogeneity in a two population model , which can be considered to be the simplest model exhibiting chimera states @xcite . however , in this model there is no spatial structure to either population . consequently , we focus in this paper on a ring of adjacent oscillators with a prescribed but nonuniform frequency distribution @xmath15 , where the continuous variable @xmath16 ( @xmath17 ) represents position along the ring . to be specific , two classes of inhomogeneity are considered , a bump inhomogeneity @xmath18 , @xmath19 , and a periodic inhomogeneity @xmath20 , where @xmath21 is a positive integer . in each case we follow @xcite and study the coupling functions @xmath22,\nonumber & & g^{(2)}_n(x)\equiv \cos(nx)+\cos[(n+1)x],\nonumber\end{aligned}\ ] ] where @xmath23 is an arbitrary positive integer . these choices are motivated by biological systems in which coupling between nearby oscillators is often attractive while that between distant oscillators may be repelling @xcite . there are several advantages to the use of these two types of coupling . the first is that these couplings allow us to obtain chimera states with random initial conditions . the second is that we can identify a large variety of new states for suitable parameter values , including ( a ) splay states , ( b ) stationary multi - cluster states with evenly distributed coherent clusters , ( c ) stationary multi - cluster states with unevenly distributed clusters , ( d ) a fully coherent state traveling with a constant speed , and ( e ) a single cluster traveling chimera state @xcite . in this paper , we analyze the effect of the two types of inhomogeneity in @xmath11 on each of these states and describe the results in terms of the parameters @xmath24 and @xmath25 or @xmath21 describing the strength and length scale of the inhomogeneity , while varying the parameter @xmath26 representing the phase lag @xmath6 . in section [ eff_eq ] , we briefly review the notion of a local order parameter for studying chimera states and introduce the self - consistency equation for this quantity . in sections [ rot ] , [ trav_coh ] and [ trav ] we investigate , respectively , the effect of inhomogeneity on rotating states ( including splay states and stationary chimera states ) , traveling coherent states and the traveling chimera state . we conclude in section [ conclusion ] with a brief summary of the results and directions for future research equation ( [ phase_eq ] ) is widely used in studies of chimera states . an equivalent description can be obtained by constructing an equation for the local order parameter @xmath27 defined as the local spatial average of @xmath28 $ ] , @xmath29 the evolution equation for @xmath30 then takes the form @xcite @xmath31 where @xmath32(x , t)$ ] and @xmath33 is a compact linear operator defined via the relation @xmath34(x , t)\equiv\int_{-\pi}^{\pi}g(x - y)u(y , t)\,dy.\ ] ] a derivation of eq . ( [ effective_eq ] ) based on the ott antonsen ansatz @xcite is given in the appendix . equation ( [ effective_eq ] ) can also be obtained directly from eq . ( [ phase_eq ] ) using the change of variable @xmath35.\label{changeofvar}\ ] ] an important class of solutions of eq . ( [ phase_eq ] ) consists of stationary rotating solutions , i.e. , states of the form @xmath36 whose common frequency @xmath37 satisfies the nonlinear eigenvalue relation @xmath38\tilde{z}+\frac{1}{2}\left[e^{-i\alpha}\tilde{z}(x)-\tilde{z}^2e^{i\alpha}\tilde{z}^*(x)\right ] = 0.\label{tw1}\ ] ] here @xmath39 describes the spatial profile of the rotating solution and @xmath40 $ ] . solving eq . ( [ tw1 ] ) as a quadratic equation in @xmath41 we obtain @xmath42 the function @xmath43 is chosen to be @xmath44^{1/2}$ ] when @xmath45 and @xmath46^{1/2}$ ] when @xmath47 . this choice is dictated by stability considerations , and in particular the requirement that the essential spectrum of the linearization about the rotating solution is either stable or neutrally stable @xcite . the coherent ( incoherent ) region corresponds to the subdomain of @xmath48 $ ] where @xmath49 falls below ( above ) @xmath50 . substitution of expression ( [ zx ] ) into the definition of @xmath51 now leads to the self - consistency relation @xmath52 here the bracket @xmath53 is defined as the integral over the interval @xmath54 $ ] , i.e. , @xmath55 in the following we write @xmath56 and refer to @xmath57 and @xmath58 as the amplitude and phase of the complex order parameter @xmath51 . as shown in @xcite , eq . ( [ phase_eq ] ) with constant natural frequency @xmath11 exhibits both stationary rotating solutions ( splay states and stationary chimera states ) and traveling solutions ( traveling coherent states and traveling chimera states ) for suitable coupling functions @xmath13 . in this section , we investigate the effect of spatial inhomogeneity in @xmath11 ( i.e. , @xmath14 ) on the splay states and on stationary chimera states , focusing on the case @xmath59 studied in @xcite . we find that when the inhomogeneity is sufficiently weak , the above solutions persist . however , as the magnitude of the inhomogeneity increases , new types of solutions are born . the origin and spatial structure of these new states can be understood with the help of the self - consistency relation ( [ self - consistency ] ) . since eq . ( [ self - consistency ] ) is invariant under the transformation @xmath60 with @xmath61 an arbitrary real constant , the local order parameter @xmath51 for the coupling function @xmath62 can be written as @xmath63 where @xmath64 is positive and @xmath65 with @xmath66 and @xmath67 both real . substituting the ansatz ( [ ansatz0 ] ) into eq . ( [ self - consistency ] ) we obtain the following pair of integral - algebraic equations @xmath68 where @xmath69 and @xmath70 . these equations may also be written in the more convenient form @xmath71 in the following we consider two choices for the inhomogeneity @xmath15 , a bump @xmath72 , @xmath19 , and a periodic inhomogeneity @xmath73 where @xmath21 is a positive integer . the resulting equations possess an important and useful symmetry , @xmath74 , @xmath75 . moreover , for @xmath15 satisfying @xmath76 with @xmath21 an integer , a solution @xmath51 of the self - consistency relation implies that @xmath77 is also a solution . here @xmath78 is an arbitrary integer . in the following we use eqs . ( [ sc - full12 ] ) and ( [ sc - full22 ] ) repeatedly to study the changes in both the splay states and the stationary chimera states as the magnitude @xmath79 of the inhomogeneity increases , and compare the resulting predictions with numerical simulation of @xmath80 oscillators evenly distributed in @xmath54 $ ] . we now consider the case where @xmath15 has a bump defect at @xmath82 . for the spatial profile of the defect we pick @xmath83 , where @xmath79 and @xmath19 are parameters that can be varied . when @xmath84 and @xmath11 is a constant , eq . ( [ phase_eq ] ) exhibits so - called splay state solutions with @xmath85 , where @xmath37 is the overall rotation frequency and @xmath86 is an integer called the twist number . the phase in this type of solution drifts with speed @xmath87 to the right but the order parameter is stationary . consequently we think of the splay states as a stationary rotating states . linear stability analysis for @xmath62 shows that the splay state is stable when @xmath88 @xcite , a result that is easily confirmed in simulations starting from randomly distributed initial phases . in the following we consider the case @xmath89 and simulate @xmath80 oscillators evenly distributed in @xmath54 $ ] for different values of @xmath24 and @xmath25 . for splay states observed with @xmath90 , @xmath91 and @xmath92 . ( a ) @xmath93 . ( b ) @xmath94 . ( c ) @xmath95 . ( d ) @xmath96 . the states travel to the right ( @xmath97 ) . , title="fig:",height=113 ] for splay states observed with @xmath90 , @xmath91 and @xmath92 . ( a ) @xmath93 . ( b ) @xmath94 . ( c ) @xmath95 . ( d ) @xmath96 . the states travel to the right ( @xmath97 ) . , title="fig:",height=113 ] for splay states observed with @xmath90 , @xmath91 and @xmath92 . ( a ) @xmath93 . ( b ) @xmath94 . ( c ) @xmath95 . ( d ) @xmath96 . the states travel to the right ( @xmath97 ) . , title="fig:",height=113 ] for splay states observed with @xmath90 , @xmath91 and @xmath92 . ( a ) @xmath93 . ( b ) @xmath94 . ( c ) @xmath95 . ( d ) @xmath96 . the states travel to the right ( @xmath97 ) . , title="fig:",height=113 ] figure [ fig : cosx_bump_splay_four ] shows how the splay solutions change as @xmath24 increases . when @xmath24 becomes nonzero but remains small , the splay states persist but their phase @xmath10 no longer varies uniformly in space ( fig . [ fig : cosx_bump_splay_four](a ) ) . instead @xmath98 where @xmath99 is a continuous function of @xmath16 with @xmath100 . we refer to this type of state as a near - splay state . as @xmath24 becomes larger an incoherent region appears in the vicinity of @xmath82 , with width that increases with increasing @xmath24 ( figs . [ fig : cosx_bump_splay_four](b , c ) ) . we refer to this type of state as a chimera splay state . as @xmath24 increases further and exceeds a second threshold , a new region of incoherence is born ( fig . [ fig : cosx_bump_splay_four](d ) ) . figure [ fig : cosx_bump_splay ] provides additional information about the partially coherent near - splay state in fig . [ fig : cosx_bump_splay_four](b ) . the figure shows the real order parameters @xmath57 and @xmath58 together with @xmath101 , the local rotation frequency averaged over a long time interval ( fig . [ fig : cosx_bump_splay](d ) ) , and reveals that in the coherent region the oscillation frequencies are identical ( with @xmath102 ) , with an abrupt but continuous change in the frequency distribution within the incoherent region . the figures also reveals that as @xmath24 increases the incoherent region develops a stronger and stronger asymmetry with respect to @xmath82 , the bump maximum . this is a consequence of the asymmetry introduced by the direction of travel , i.e. , the sign of the frequency @xmath37 in eq . ( [ splay_ansatz ] ) as discussed further below . for @xmath90 and @xmath103 . ( b ) the corresponding @xmath57 . ( c ) the corresponding @xmath58 . ( d ) the oscillator frequency @xmath101 averaged over the time interval @xmath104 ( solid line ) . in the coherent region @xmath101 coincides with the global oscillation frequency @xmath105 ( open circles ) . all simulations are done with @xmath92 and @xmath80.,title="fig:",height=113 ] for @xmath90 and @xmath103 . ( b ) the corresponding @xmath57 . ( c ) the corresponding @xmath58 . ( d ) the oscillator frequency @xmath101 averaged over the time interval @xmath104 ( solid line ) . in the coherent region @xmath101 coincides with the global oscillation frequency @xmath105 ( open circles ) . all simulations are done with @xmath92 and @xmath80.,title="fig:",height=113 ] for @xmath90 and @xmath103 . ( b ) the corresponding @xmath57 . ( c ) the corresponding @xmath58 . ( d ) the oscillator frequency @xmath101 averaged over the time interval @xmath104 ( solid line ) . in the coherent region @xmath101 coincides with the global oscillation frequency @xmath105 ( open circles ) . all simulations are done with @xmath92 and @xmath80.,title="fig:",height=113 ] for @xmath90 and @xmath103 . ( b ) the corresponding @xmath57 . ( c ) the corresponding @xmath58 . ( d ) the oscillator frequency @xmath101 averaged over the time interval @xmath104 ( solid line ) . in the coherent region @xmath101 coincides with the global oscillation frequency @xmath105 ( open circles ) . all simulations are done with @xmath92 and @xmath80.,title="fig:",height=113 ] these states and the transitions between them can be explained within the framework of the self - consistent analysis . to compute the solution branches and the transition thresholds , we numerically continue solutions of eqs . ( [ sc - full12 ] ) and ( [ sc - full22 ] ) with respect to the parameter @xmath24 . when @xmath106 , the splay state ( with positive slope ) corresponds to @xmath107 , @xmath108 , @xmath109 , with @xmath110 . the order parameter is therefore @xmath111 . we use this splay state as the starting point for continuation . as @xmath24 increases , a region of incoherence develops in the phase pattern in the vicinity of @xmath82 . from the point of view of the self - consistent analysis , the incoherent region corresponds to the region where the natural frequency exceeds the amplitude @xmath57 of the complex order parameter . the boundaries between the coherent and incoherent oscillators are thus determined by the relation @xmath112 . for @xmath94 , the left and right boundaries are thus @xmath113 and @xmath114 , respectively . these predictions are in good agreement with the values measured in direct numerical simulation ( fig . [ fig : cosx_bump_splay](a ) ) . figure [ fig : cosx_bump_omega0_continue_splay](a ) shows the overall frequency @xmath37 obtained by numerical continuation of the solution of eqs . ( [ sc - full12 ] ) and ( [ sc - full22 ] ) in the parameter @xmath24 , while figs . [ fig : cosx_bump_omega0_continue_splay](b , c ) show the corresponding results for the fraction @xmath115 of the domain occupied by the coherent oscillators and the extent of the ( first ) region of incoherence , i.e. , the interval @xmath116 , also as functions of @xmath24 . from the figure we can see a clear transition at @xmath117 from a single domain - filling coherent state to a `` splay state with one incoherent cluster '' in @xmath116 , followed by a subsequent transition at @xmath118 from this state to a `` splay state with two incoherent clusters '' . these transitions occur when the profiles of @xmath119 and @xmath57 touch as @xmath24 increases and these points of tangency therefore correspond to the locations where coherence is first lost . figure [ fig : transitions ] shows that tangencies between @xmath119 and @xmath57 occur when @xmath117 and @xmath120 , implying that intervals of incoherent oscillators appear first at @xmath82 ( i.e. , the bump maximum ) and subsequently at @xmath121 , as @xmath24 increases . these predictions are in excellent agreement with the direct numerical simulations shown in fig . [ fig : cosx_bump_splay_four ] . moreover , the critical values of @xmath24 predicted by the self - consistency analysis are fully consistent with the simulation results when @xmath122 is increased quasi - statically ( not shown ) . in each case we repeated the simulations for decreasing @xmath122 but found no evidence of hysteresis in these transitions . , ( b ) the fraction @xmath115 of the domain occupied by the coherent oscillators , and ( c ) the width of the incoherent region @xmath116 , all as functions of the parameter @xmath24 . the calculation is for @xmath123 , @xmath92 and @xmath80.,title="fig:",height=151 ] , ( b ) the fraction @xmath115 of the domain occupied by the coherent oscillators , and ( c ) the width of the incoherent region @xmath116 , all as functions of the parameter @xmath24 . the calculation is for @xmath123 , @xmath92 and @xmath80.,title="fig:",height=151 ] , ( b ) the fraction @xmath115 of the domain occupied by the coherent oscillators , and ( c ) the width of the incoherent region @xmath116 , all as functions of the parameter @xmath24 . the calculation is for @xmath123 , @xmath92 and @xmath80.,title="fig:",height=151 ] and @xmath57 at the critical values @xmath24 for the appearance of new regions of incoherence around ( a ) @xmath82 for @xmath124 and ( b ) @xmath125 for @xmath126 . the calculation is for @xmath123 , @xmath92 and @xmath127.,title="fig:",height=151 ] and @xmath57 at the critical values @xmath24 for the appearance of new regions of incoherence around ( a ) @xmath82 for @xmath124 and ( b ) @xmath125 for @xmath126 . the calculation is for @xmath123 , @xmath92 and @xmath127.,title="fig:",height=151 ] chimera states with @xmath128 evenly distributed coherent clusters are readily observed when @xmath84 and @xmath11 is a constant . these states persist when a bump is introduced into the frequency distribution @xmath15 . figure [ fig : cosx_bump_2cluster ] shows the phase distribution and the corresponding local order parameters @xmath57 and @xmath58 for @xmath90 when @xmath129 . the figure shows that the two clusters persist , but are now always located near @xmath130 and @xmath131 . this is a consequence of the fact that the presence of the bump breaks the translation invariance of the system . figures [ fig : cosx_bump_2cluster](b , c ) show that the local order parameter @xmath132 has the symmetry @xmath133 . this symmetry implies @xmath134 should take the form @xmath135 , where @xmath136 is real . the corresponding self - consistency equation takes the form @xmath137 the result of numerical continuation of the solutions of eq . ( [ bump_scequation_2cluster ] ) are shown in fig . [ fig : cosx_bump_omega0_continue_2cluster ] . the 2-cluster chimera state persists to large values of @xmath24 , with the size of the coherent clusters largely insensitive to the value of @xmath24 . this prediction has been corroborated using direct simulation of eq . ( [ phase_eq_discrete ] ) with @xmath80 oscillators . in a 2-cluster chimera state for @xmath90 and @xmath129 . ( b ) the corresponding order parameter @xmath57 . ( c ) the corresponding order parameter @xmath58 . note that the oscillators in the two clusters oscillate with the same frequency but @xmath138 out of phase . the calculation is done with @xmath123 , @xmath92 and @xmath80.,title="fig:",height=151 ] in a 2-cluster chimera state for @xmath90 and @xmath129 . ( b ) the corresponding order parameter @xmath57 . ( c ) the corresponding order parameter @xmath58 . note that the oscillators in the two clusters oscillate with the same frequency but @xmath138 out of phase . the calculation is done with @xmath123 , @xmath92 and @xmath80.,title="fig:",height=151 ] in a 2-cluster chimera state for @xmath90 and @xmath129 . ( b ) the corresponding order parameter @xmath57 . ( c ) the corresponding order parameter @xmath58 . note that the oscillators in the two clusters oscillate with the same frequency but @xmath138 out of phase . the calculation is done with @xmath123 , @xmath92 and @xmath80.,title="fig:",height=151 ] and the order parameter amplitude @xmath139 on @xmath24 . ( b ) the fraction @xmath115 of coherent oscillators as a function of @xmath24 . the calculation is done with @xmath123 , @xmath92 and @xmath127.,title="fig:",height=151 ] and the order parameter amplitude @xmath139 on @xmath24 . ( b ) the fraction @xmath115 of coherent oscillators as a function of @xmath24 . the calculation is done with @xmath123 , @xmath92 and @xmath127.,title="fig:",height=151 ] when @xmath11 is constant , finite size effects cause the phase pattern to fluctuate in location . in @xcite , we demonstrate that this fluctuation is well modeled by brownian motion in which the variance is proportional to @xmath140 , even though the original system is strictly deterministic . as mentioned above , when @xmath15 is spatially dependent , the translation symmetry is broken and the coherent cluster has a preferred location . figure [ fig : cosx_bump_2cluster ] suggests that local maxima of the order parameter @xmath141 can be used to specify the location of coherent clusters . consequently we plot in fig . [ fig : bump_pos_vs_t ] the position @xmath142 of the right coherent cluster as a function of time for three different values of the parameter @xmath25 . we see that the inhomogeneity pins the coherent cluster to a particular location , and that the cluster position executes apparently random oscillations about this preferred location , whose amplitude increases with increasing @xmath25 , i.e. , with decreasing width of the bump . figure [ fig : std_of_pos ] shows the standard deviation of the position @xmath142 of the coherent cluster as a function of the parameter @xmath25 . of the coherent cluster as a function of time when @xmath96 , @xmath92 , @xmath127 and ( a ) @xmath143 , ( b ) @xmath144 and ( c ) @xmath123.,title="fig:",height=132 ] of the coherent cluster as a function of time when @xmath96 , @xmath92 , @xmath127 and ( a ) @xmath143 , ( b ) @xmath144 and ( c ) @xmath123.,title="fig:",height=132 ] of the coherent cluster as a function of time when @xmath96 , @xmath92 , @xmath127 and ( a ) @xmath143 , ( b ) @xmath144 and ( c ) @xmath123.,title="fig:",height=132 ] as a function of the parameter @xmath25 for @xmath96 , @xmath92 and @xmath127.,height=151 ] is well approximated by a normal distribution when @xmath145 . , height=151 ] the behavior shown in figs . [ fig : bump_pos_vs_t ] and [ fig : std_of_pos ] can be modeled using an ornstein - uhlenbeck process , i.e. , a linear stochastic ordinary differential equation of the form @xmath146 where @xmath147 represents the strength of the attraction to the preferred location @xmath43 , and @xmath148 indicates the strength of the noise . models of this type are expected to apply on an appropriate timescale only : the time increment @xmath149 between successive steps of the stochastic process must be large enough that the position of the cluster can be thought of as the result of a large number of pseudo - random events and hence normally distributed , but not so large that nonlinear effects become significant . figure [ fig : hist ] reveals that for an appropriate interval of @xmath149 the fluctuations @xmath150 are indeed normally distributed , thereby providing support for the applicability of eq . ( [ eq : oe ] ) to the present system . equation ( [ eq : oe ] ) has the solution @xmath151 where @xmath152 , @xmath153 and @xmath154 . to fit the parameters to the data in fig . [ fig : bump_pos_vs_t ] we notice that the relationship between consecutive observations @xmath155 is linear with an _ i.i.d . _ error term @xmath156 , where @xmath157 denotes the normal distribution with zero mean and unit variance ( fig . [ fig : hist ] ) and @xmath158 is a constant . a least - squares fit to the data @xmath159 gives the parameters @xmath147 , @xmath43 and @xmath148 . we find that for @xmath160 , @xmath161 and @xmath92 , the choice @xmath162 works well and yields the empirical model parameters @xmath163 , @xmath164 and @xmath165 , a result that is in good agreement with the simulation results for @xmath161 summarized in fig . [ fig : std_of_pos ] . in this section we consider the case @xmath73 . when @xmath24 is small , the states present for @xmath166 persist , but with increasing @xmath24 one finds a variety of intricate dynamical behavior . in the following we set @xmath167 and @xmath168 , with @xmath24 and @xmath21 as parameters to be varied . when @xmath15 is a constant , splay states are observed in which the phase @xmath10 varies linearly with @xmath16 . when @xmath24 is nonzero but small , the splay states persist as the near - splay states described by eq . ( [ splay_ansatz ] ) ; fig . [ fig : cosx_inho_nearsplay_wam1234](a ) shows an example of such a state when @xmath169 . as @xmath24 becomes larger , incoherent regions appear and these increase in width as @xmath24 increases further ( e.g. , fig . [ fig : cosx_inho_nearsplay_wam1234](b)(d ) ) . we refer to this type of state as a chimera splay state , as in the bump inhomogeneity case . figures [ fig : cosx_inho_nearsplay_wam1234 ] show the phase distribution obtained for different values of @xmath24 . for these solutions , the slope of the coherent region is no longer constant but the oscillators rotate with a constant overall frequency @xmath37 . this type of solution is also observed for other values of @xmath21 . figure [ fig : cosx_inho_l_cluster_345 ] shows examples of chimera splay states for @xmath170 , @xmath171 and @xmath172 , with @xmath21 coherent clusters in each case . . chimera splay states for ( b ) @xmath173 , ( c ) @xmath174 and ( d ) @xmath175 . in all cases @xmath92 and @xmath80.,title="fig:",width=132 ] . chimera splay states for ( b ) @xmath173 , ( c ) @xmath174 and ( d ) @xmath175 . in all cases @xmath92 and @xmath80.,title="fig:",width=132 ] . chimera splay states for ( b ) @xmath173 , ( c ) @xmath174 and ( d ) @xmath175 . in all cases @xmath92 and @xmath80.,title="fig:",width=132 ] . chimera splay states for ( b ) @xmath173 , ( c ) @xmath174 and ( d ) @xmath175 . in all cases @xmath92 and @xmath80.,title="fig:",width=132 ] and ( a ) @xmath176 ( 3-cluster state ) , ( b ) @xmath177 ( 4-cluster state ) , and ( c ) @xmath178 ( 5-cluster state ) . in all cases @xmath92 and @xmath80.,title="fig:",height=132 ] and ( a ) @xmath176 ( 3-cluster state ) , ( b ) @xmath177 ( 4-cluster state ) , and ( c ) @xmath178 ( 5-cluster state ) . in all cases @xmath92 and @xmath80.,title="fig:",height=132 ] and ( a ) @xmath176 ( 3-cluster state ) , ( b ) @xmath177 ( 4-cluster state ) , and ( c ) @xmath178 ( 5-cluster state ) . in all cases @xmath92 and @xmath80.,title="fig:",height=132 ] to compute the solution branches and identify thresholds for additional transitions , we continue the solutions of eqs . ( [ sc - full12 ] ) and ( [ sc - full22 ] ) with respect to the parameter @xmath24 . figures [ fig : splay1 ] and [ fig : splay2 ] show the dependence of @xmath37 and of the coherent fraction @xmath115 on @xmath24 when @xmath179 and @xmath180 , respectively . the figures indicate that the coherent fraction @xmath115 falls below 1 at @xmath181 ( @xmath179 ) , @xmath182 ( @xmath183 ) , and @xmath184 ( @xmath185 , 4 and 5 ) . these values coincide with the parameter values at which an incoherent region emerges . in addition , fig . [ fig : splay1](b ) reveals the presence of a second transition , at @xmath186 , corresponding to the emergence of a 2-cluster state from a 1-cluster state . figures [ fig : cosx_cosx_tangent ] and [ fig : tangent_l2 ] show the profiles of @xmath187 and @xmath57 at the critical values of @xmath24 at which incoherent regions emerge . and ( b ) the fraction @xmath115 of the domain occupied by the coherent oscillators on @xmath24 , with @xmath188 and @xmath189.,title="fig:",height=151 ] and ( b ) the fraction @xmath115 of the domain occupied by the coherent oscillators on @xmath24 , with @xmath188 and @xmath189.,title="fig:",height=151 ] and ( b ) the fraction @xmath115 of the domain occupied by the coherent oscillators on @xmath24 , with @xmath190 and @xmath189.,title="fig:",height=151 ] and ( b ) the fraction @xmath115 of the domain occupied by the coherent oscillators on @xmath24 , with @xmath190 and @xmath189.,title="fig:",height=151 ] and @xmath57 at the critical values @xmath24 for the appearance of a new region of incoherence around ( a ) @xmath82 for @xmath191 and ( b ) @xmath192 for @xmath193 . parameters : @xmath179 and @xmath92.,title="fig:",height=151 ] and @xmath57 at the critical values @xmath24 for the appearance of a new region of incoherence around ( a ) @xmath82 for @xmath191 and ( b ) @xmath192 for @xmath193 . parameters : @xmath179 and @xmath92.,title="fig:",height=151 ] and @xmath57 at the critical value @xmath194 . parameters : @xmath183 and @xmath92.,height=151 ] to understand the @xmath21-cluster chimera states reported in fig . [ fig : cosx_inho_l_cluster_345 ] , we computed the local order parameter @xmath51 and found that @xmath57 is approximately constant while the phase @xmath58 varies at a constant rate , suggesting the ansatz @xmath195 . with this ansatz , eqs . ( [ sc - full12 ] ) and ( [ sc - full22 ] ) reduce to a single equation , @xmath196 for all positive integers @xmath197 . figure [ fig:3cluster ] shows the result of numerical continuation of a solution for @xmath185 . the figure reveals no further transitions , indicating that the @xmath185 chimera state persists to large values of @xmath24 . numerical simulation shows that these states ( @xmath185 , 4 , 5 ) are stable and persist up to @xmath198 . and ( b ) the coherent fraction @xmath115 on @xmath24 , with @xmath199 and @xmath189.,title="fig:",height=132 ] and ( b ) the coherent fraction @xmath115 on @xmath24 , with @xmath199 and @xmath189.,title="fig:",height=132 ] in a 1-cluster chimera state for @xmath90 and @xmath200 . ( b ) the corresponding order parameter @xmath57 . ( c ) the corresponding order parameter @xmath58 . figure ( a ) shows that the presence of a nearly coherent region near @xmath82 with oscillators that oscillate @xmath138 out of phase with the coherent cluster . the calculation is done with @xmath92 and @xmath80.,title="fig:",height=132 ] in a 1-cluster chimera state for @xmath90 and @xmath200 . ( b ) the corresponding order parameter @xmath57 . ( c ) the corresponding order parameter @xmath58 . figure ( a ) shows that the presence of a nearly coherent region near @xmath82 with oscillators that oscillate @xmath138 out of phase with the coherent cluster . the calculation is done with @xmath92 and @xmath80.,title="fig:",height=132 ] in a 1-cluster chimera state for @xmath90 and @xmath200 . ( b ) the corresponding order parameter @xmath57 . ( c ) the corresponding order parameter @xmath58 . figure ( a ) shows that the presence of a nearly coherent region near @xmath82 with oscillators that oscillate @xmath138 out of phase with the coherent cluster . the calculation is done with @xmath92 and @xmath80.,title="fig:",height=132 ] and @xmath37 , and ( b ) the coherent fraction @xmath115 , all as functions of @xmath24 for the chimera state shown in fig . [ fig : cosx_inho_1cluster ] . ( c ) @xmath187 and @xmath57 as functions of @xmath16 when @xmath201.,title="fig:",height=151 ] and @xmath37 , and ( b ) the coherent fraction @xmath115 , all as functions of @xmath24 for the chimera state shown in fig . [ fig : cosx_inho_1cluster ] . ( c ) @xmath187 and @xmath57 as functions of @xmath16 when @xmath201.,title="fig:",height=151 ] and @xmath37 , and ( b ) the coherent fraction @xmath115 , all as functions of @xmath24 for the chimera state shown in fig . [ fig : cosx_inho_1cluster ] . ( c ) @xmath187 and @xmath57 as functions of @xmath16 when @xmath201.,title="fig:",height=151 ] figure [ fig : cosx_inho_1cluster ] shows an example of a 1-cluster chimera state when @xmath202 ; 1-cluster chimera states of this type have not thus far been reported for @xmath90 , @xmath166 , where computations always result in 2-cluster states @xcite . to understand the origin of this unexpected 1-cluster state we use fig . [ fig : cosx_inho_1cluster](b ) to conclude that @xmath51 is of the form @xmath203 , and hence that eqs . ( [ sc - full12 ] ) and ( [ sc - full22 ] ) reduce to the single equation @xmath204 figure [ fig : romegae ] shows the result of numerical continuation of the solution of this equation as a function of @xmath24 . the figure reveals a transition at @xmath201 ( figs . [ fig : romegae](a , b ) ) . when @xmath205 , the solution is a 2-cluster chimera ; as @xmath24 increases through @xmath206 the coherent region around @xmath82 disappears , leaving a single cluster chimera state ( fig . [ fig : romegae](c ) ) . for @xmath90 and constant @xmath11 simulations always evolve into either a 2-cluster chimera state or a splay state @xcite . it is expected that when @xmath24 is small , the 2-cluster chimera is not destroyed . figures [ fig : cosx_inho_2cluster](a , b ) gives examples of this type of state with @xmath183 and @xmath185 inhomogeneities , respectively . in both cases the clusters are located at specific locations selected by the inhomogeneity . and ( a ) @xmath207 and ( b ) @xmath208 . in both cases @xmath92 and @xmath80.,title="fig:",height=132 ] and ( a ) @xmath207 and ( b ) @xmath208 . in both cases @xmath92 and @xmath80.,title="fig:",height=132 ] the order parameter profile with @xmath185 corresponding to the numerical solution in fig . [ fig : cosx_inho_2cluster](b ) has the same symmetry properties as that in figs . [ fig : cosx_bump_2cluster](b , c ) . it follows that we may set @xmath209 , @xmath210 in the order parameter representation ( [ ansatz0 ] ) , resulting once again in the self - consistency relation ( [ bump_scequation_2cluster ] ) . we have continued the solutions of this relation for @xmath185 using the simulation in fig . [ fig : cosx_inho_2cluster](b ) with @xmath211 to initialize continuation in @xmath24 . figure [ fig : chimera2 - 1 ] shows the result of numerical continuation of the order parameter for this state . no further transitions are revealed . there are three preferred locations for the coherent clusters , related by the translation symmetry @xmath212 , @xmath213 , as shown in fig . [ fig : cosx_inho_wa01_wp3_2cluster ] . ( solid line ) and @xmath139 ( dashed line ) , and ( b ) the coherent fraction @xmath115 on @xmath24 when @xmath185 and @xmath92 . , title="fig:",height=132 ] ( solid line ) and @xmath139 ( dashed line ) , and ( b ) the coherent fraction @xmath115 on @xmath24 when @xmath185 and @xmath92 . , title="fig:",height=132 ] . the simulation is done with @xmath92 and @xmath127.,title="fig:",height=132 ] . the simulation is done with @xmath92 and @xmath127.,title="fig:",height=132 ] . the simulation is done with @xmath92 and @xmath127.,title="fig:",height=132 ] we now turn to states with a spatially structured order parameter undergoing translation . with @xmath214 and constant @xmath11 the system ( [ phase_eq ] ) exhibits a fully coherent but non - splay state that travels with a constant speed @xmath215 when @xmath216 @xcite . figure [ fig : phase_cos12x](a ) shows a snapshot of this traveling coherent state while fig . [ fig : phase_cos12x](b ) shows its position @xmath155 as a function of time . the speed of travel is constant and can be obtained by solving a nonlinear eigenvalue problem @xcite . in this section , we investigate how the inhomogeneities @xmath217 and @xmath73 affect the dynamical behavior of this state . is constant . ( b ) the position @xmath155 of this state as a function of time . the simulation is done for @xmath214 with @xmath218 and @xmath127.,title="fig:",height=113 ] is constant . ( b ) the position @xmath155 of this state as a function of time . the simulation is done for @xmath214 with @xmath218 and @xmath127.,title="fig:",height=113 ] in this section we perform two types of numerical experiments . in the first , we fix @xmath122 and start with the traveling coherent solution for @xmath106 . we then gradually increase @xmath24 gradually at fixed values of @xmath25 and @xmath122 . in the second we fix @xmath220 , @xmath19 , and vary @xmath122 . for each @xmath216 we find that the coherent state continues to travel , albeit nonuniformly , until @xmath24 reaches a threshold value that depends of the values of @xmath122 and @xmath25 . the case @xmath218 provides an example . figure [ fig : hidden1](a ) shows the traveling coherent state in the homogeneous case ( @xmath106 ) while fig . [ fig : hidden1](b ) shows the corresponding state in the presence of a frequency bump @xmath221 with @xmath222 and @xmath161 . in this case the presence of inhomogeneity leads first to a periodic fluctuation in the magnitude of the drift speed followed by , as @xmath24 continues to increase , a transition to a new state in which the direction of the drift oscillates periodically ( fig . [ fig : hidden1](b ) ) . we refer to states of this type as direction - reversing waves , by analogy with similar behavior found in other systems supporting the presence of such waves @xcite . with increasing @xmath24 the reversals become localized in space ( and possibly aperiodic , fig . [ fig : hidden2](a ) ) and then cease , leading to a stationary pinned structure at @xmath223 ( fig . [ fig : hidden2](b ) ) . figure [ fig : xvst ] shows the position @xmath155 of the maximum of the local order parameter @xmath224 of the coherent state as a function of time for the cases in figs . [ fig : hidden1 ] and [ fig : hidden2 ] , showing the transition from translation to pinning as @xmath24 increases , via states that are reflection - symmetric _ on average_. the final state is a steady reflection - symmetric pinned state aligned with the imposed inhomogeneity . in fact , the dynamics of the present system may be more complicated than indicated above since a small group of oscillators located in regions where the order parameter undergoes rapid variation in space may lose coherence in a periodic fashion even when @xmath166 thereby providing a competing source of periodic oscillations in the magnitude of the drift speed . as documented in @xcite this is the case when @xmath225 . for @xmath226 , however , the coherent state drifts uniformly when @xmath166 and this is therefore the case studied in greatest detail . of the coherent state in figs . [ fig : hidden1 ] and [ fig : hidden2 ] as a function of time for ( a ) @xmath227 , ( b ) @xmath228 , ( c ) @xmath229 , and ( d ) @xmath230 . in all cases @xmath123 , @xmath218 and @xmath127.,title="fig:",height=113 ] of the coherent state in figs . [ fig : hidden1 ] and [ fig : hidden2 ] as a function of time for ( a ) @xmath227 , ( b ) @xmath228 , ( c ) @xmath229 , and ( d ) @xmath230 . in all cases @xmath123 , @xmath218 and @xmath127.,title="fig:",height=113 ] of the coherent state in figs . [ fig : hidden1 ] and [ fig : hidden2 ] as a function of time for ( a ) @xmath227 , ( b ) @xmath228 , ( c ) @xmath229 , and ( d ) @xmath230 . in all cases @xmath123 , @xmath218 and @xmath127.,title="fig:",height=113 ] of the coherent state in figs . [ fig : hidden1 ] and [ fig : hidden2 ] as a function of time for ( a ) @xmath227 , ( b ) @xmath228 , ( c ) @xmath229 , and ( d ) @xmath230 . in all cases @xmath123 , @xmath218 and @xmath127.,title="fig:",height=113 ] the profile of the pinned coherent state can also be determined from a self - consistency analysis . for @xmath231 , the local order parameter @xmath51 can be written in the form @xmath232 owing to the reflection symmetry of the solution @xmath233 ; on applying a rotation in @xmath234 we may take @xmath64 to be real . the self - consistency equation then becomes @xmath235 when @xmath236 , @xmath161 and @xmath226 the solution of these equations is @xmath237 , results that are consistent with the values obtained from numerical simulation . for these parameter values , @xmath238 for @xmath239 $ ] , indicating that all phases rotate with the same frequency @xmath37 . as we decrease @xmath24 to @xmath240 the profiles @xmath241 and @xmath242 start to touch ( fig . [ fig : romega_omega0_011 ] ) and for yet lower @xmath24 the stationary coherent state loses stability and begins to oscillate as described in the previous paragraph . we have also conducted experiments at a fixed value of @xmath79 and @xmath19 while changing @xmath122 . for example , at fixed @xmath236 , @xmath161 and @xmath226 the system is in the pinned state shown in fig . [ fig : hidden2](b ) . since the speed of the coherent state with @xmath166 gradually increases as @xmath122 decreases , we anticipate that a given inhomogeneity will find it harder and harder to pin the state as @xmath122 decreases . this is indeed the case , and we find that there is a critical value of @xmath122 at which the given inhomogeneity is no longer able to pin the structure , with depinning via back and forth oscillations of the structure @xcite . for yet smaller values of @xmath122 , this type of oscillation also loses stability and evolves into near - splay states . when we increase @xmath122 again we uncover hysteresis in each of these transitions . figure [ fig : hidden3 ] shows two distinct states at identical parameter values : @xmath236 , @xmath161 and @xmath243 generated using different protocols : fig . [ fig : hidden3](a ) shows a direction - reversing state evolved from a traveling coherent state when we increase @xmath24 from 0 to 0.12 at @xmath243 , while fig . [ fig : hidden3](b ) shows a pinned state generated from the pinned state at @xmath218 when we change @xmath122 from 0.75 to 0.76 at fixed @xmath223 . of a coherent solution when @xmath245 . ( b ) the position @xmath155 of the coherent solution as a function of time . in both cases @xmath179 , @xmath92 and @xmath127.,title="fig:",height=113 ] of a coherent solution when @xmath245 . ( b ) the position @xmath155 of the coherent solution as a function of time . in both cases @xmath179 , @xmath92 and @xmath127.,title="fig:",height=113 ] we now turn to the case @xmath246 . similar to the bump inhomogeneity case , when @xmath122 is fixed , the traveling coherent state will continue to travel until @xmath24 reaches certain threshold . however , we did not find the pinned state as in the previous subsection around @xmath218 . here we focus on the case @xmath247 for which the traveling coherent state has a reasonable speed when @xmath166 and take @xmath179 . when @xmath24 is small , the state remains coherent ( fig . [ fig : cos12x_l1_omega0_0028](a ) ) and continues to drift , albeit no longer with a uniform speed of propagation . figure [ fig : cos12x_l1_omega0_0028](b ) shows that the speed executes slow , small amplitude oscillations about a well - defined mean value @xmath248 shown in fig . [ fig : cos12x_omegav](b ) ; the corresponding time - averaged oscillation frequency @xmath249 is shown in fig . [ fig : cos12x_omegav](a ) . when @xmath24 is increased in sufficiently small increments the oscillations grow in amplitude but the solution continues to travel to the left until @xmath250 where a hysteretic transition to a near - splay state takes place . figure [ fig : cos12x_omegav](b ) shows that prior to this transition the average speed first decreases as a consequence of the inhomogeneity , but then increases abruptly just before the transition owing to the loss of coherence on the part of a group of oscillators and the resulting abrupt increase in asymmetry of the order parameter . and ( b ) the mean drift speed @xmath251 , both as functions of @xmath24 when @xmath179 , @xmath252.,title="fig:",height=132 ] and ( b ) the mean drift speed @xmath251 , both as functions of @xmath24 when @xmath179 , @xmath252.,title="fig:",height=132 ] in addition to the states discussed in the previous sections , eq . ( [ phase_eq ] ) with constant @xmath11 admits traveling one - cluster chimera states when @xmath253 $ ] and appropriate values of the the phase lag @xmath122 . this state consists of a single coherent cluster that drifts through an incoherent background as time evolves at more or less a constant speed . figure [ fig : phase_cos34x](a ) shows a snapshot of such a state when @xmath254 . the direction of motion is determined by the gradient of the phase in the coherent region : the cluster travels to the left when the gradient is positive and to the right when the gradient is negative . figure [ fig : phase_cos34x](b ) shows the position @xmath155 of the coherent cluster as a function of time and confirms that the cluster moves to the right at an almost constant speed . in @xcite , we use numerical simulations to conclude that for @xmath254 the traveling chimera is stable in the interval @xmath255 . we therefore focus on the effects of spatial inhomogeneity on the traveling chimera state when @xmath256 . in fact the traveling chimera state is more complex than suggested in fig . [ fig : phase_cos34x](a , b ) : unlike the states discussed in the previous sections , the profile of the local order parameter fluctuates in time , suggesting that the state does not drift strictly as a rigid object . for a traveling chimera state in a spatially homogeneous system . ( b ) the position @xmath155 of the coherent cluster as a function of time . the simulation is done for @xmath257 with @xmath256 and @xmath127.,title="fig:",height=113 ] for a traveling chimera state in a spatially homogeneous system . ( b ) the position @xmath155 of the coherent cluster as a function of time . the simulation is done for @xmath257 with @xmath256 and @xmath127.,title="fig:",height=113 ] in this section we investigate the effect of a bump - like inhomogeneity @xmath258 on the motion of the traveling chimera state . starting with the traveling chimera state for @xmath106 , we increase @xmath24 in steps of @xmath259 . to describe the motion of the coherent cluster , we follow the method in @xcite and determine the instantaneous position @xmath155 of the cluster by minimizing the function @xmath260 ^ 2 $ ] , where @xmath261 is a reference profile , and using the minimizer @xmath262 as a proxy for @xmath142 . we find that even small @xmath24 suffices to stop a traveling chimera from moving : fig . [ fig : kappa_omega0 ] shows that the threshold @xmath263 for @xmath264 and that it increases monotonically to @xmath265 for @xmath266 . the resulting pinned state persists to values of @xmath24 as large as @xmath267 . on @xmath25 when @xmath256.,height=151 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath228 , ( c ) @xmath229 , ( e ) @xmath230 . the average rotation frequency @xmath101 for ( b ) @xmath228 , ( d ) @xmath229 , ( f ) @xmath230 . in all cases @xmath256 and @xmath143 , @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath228 , ( c ) @xmath229 , ( e ) @xmath230 . the average rotation frequency @xmath101 for ( b ) @xmath228 , ( d ) @xmath229 , ( f ) @xmath230 . in all cases @xmath256 and @xmath143 , @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath228 , ( c ) @xmath229 , ( e ) @xmath230 . the average rotation frequency @xmath101 for ( b ) @xmath228 , ( d ) @xmath229 , ( f ) @xmath230 . in all cases @xmath256 and @xmath143 , @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath228 , ( c ) @xmath229 , ( e ) @xmath230 . the average rotation frequency @xmath101 for ( b ) @xmath228 , ( d ) @xmath229 , ( f ) @xmath230 . in all cases @xmath256 and @xmath143 , @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath228 , ( c ) @xmath229 , ( e ) @xmath230 . the average rotation frequency @xmath101 for ( b ) @xmath228 , ( d ) @xmath229 , ( f ) @xmath230 . in all cases @xmath256 and @xmath143 , @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath228 , ( c ) @xmath229 , ( e ) @xmath230 . the average rotation frequency @xmath101 for ( b ) @xmath228 , ( d ) @xmath229 , ( f ) @xmath230 . in all cases @xmath256 and @xmath143 , @xmath127.,title="fig:",height=113 ] figures [ fig : xvst_travel_bump](a , c , e ) show the position @xmath155 of the coherent cluster as a function of time obtained using the above procedure for @xmath268 , respectively , i.e. , in the pinned regime . the figures show that the equilibrium position of the coherent region is located farther from the position @xmath82 of the inhomogeneity peak as @xmath24 increases . the bump in @xmath15 thus exerts a " repelling force on the coherent cluster , whose strength increases with the height of the bump . we interpret this observation as follows . the coherent cluster can only survive when the frequency gradient is sufficiently small , and is therefore repelled by regions where @xmath15 varies rapidly . in the present case this implies that the coherent cluster finds it easiest to survive in the wings of the bump inhomogeneity , and this position moves further from @xmath82 as @xmath24 increases . this interpretation is confirmed in figs . [ fig : xvst_travel_bump](b , d , f ) showing the average rotation frequency , @xmath269 , of the oscillators . the plateau in the profile of @xmath270 indicates frequency locking and hence the location of the coherent cluster ; the fluctuations in the position of the coherent cluster are smoothed out by the time - averaging . we now turn to the effects of a periodic inhomogeneity @xmath272 . when @xmath179 and @xmath24 increases the coherent cluster initially travels with a non - constant speed but then becomes pinned in place ; as in the case of the bump inhomogeneity quite small values of @xmath24 suffice to pin the coherent cluster in place ( for @xmath273 the value @xmath274 suffices ) . as shown in figs . [ fig : xvst_travel_periodic_l1](a , c ) the position @xmath155 of the pinned cluster relative to the local maximum of the inhomogeneity ( i.e. , @xmath82 ) depends on the value of @xmath275 . as shown in figs . [ fig : xvst_travel_periodic_l1_2](a , b ) the coherent cluster travels to the right and does so with a speed that is larger when @xmath276 than when @xmath277 . this effect becomes more pronounced as @xmath24 increases . this is because the coherent structure is asymmetric , with a preferred direction of motion , and this asymmetry increases with @xmath24 . evidently , the speed of the synchronization front at the leading edge is enhanced when @xmath278 but suppressed when @xmath279 and likewise for the desynchronization front at the rear . of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath280 , ( c ) @xmath281 . the average rotation frequency @xmath101 for ( b ) @xmath280 , ( d ) @xmath281 . in all cases @xmath256 , @xmath273 and @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath280 , ( c ) @xmath281 . the average rotation frequency @xmath101 for ( b ) @xmath280 , ( d ) @xmath281 . in all cases @xmath256 , @xmath273 and @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath280 , ( c ) @xmath281 . the average rotation frequency @xmath101 for ( b ) @xmath280 , ( d ) @xmath281 . in all cases @xmath256 , @xmath273 and @xmath127.,title="fig:",height=113 ] of the pinned coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath280 , ( c ) @xmath281 . the average rotation frequency @xmath101 for ( b ) @xmath280 , ( d ) @xmath281 . in all cases @xmath256 , @xmath273 and @xmath127.,title="fig:",height=113 ] of the coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath282 , ( b ) @xmath283 . in all cases @xmath256 , @xmath273 and @xmath127.,title="fig:",height=113 ] of the coherent cluster in a traveling chimera state as a function of time when ( a ) @xmath282 , ( b ) @xmath283 . in all cases @xmath256 , @xmath273 and @xmath127.,title="fig:",height=113 ] for @xmath284 we observe similar results . the coherent cluster is pinned in space already at small values of @xmath24 . since the inhomogeneous system has the discrete translation symmetry @xmath285 the coherent cluster has @xmath21 possible preferred positions . figure [ fig : xvst_travel_periodic_l2 ] shows an example for @xmath286 . panels ( a , c ) show snapshots of the phase distribution @xmath10 for @xmath287 in the two preferred locations ( separated by @xmath288 ) , while panels ( b , d ) show the corresponding average rotation frequency @xmath101 . of pinned traveling chimera states when @xmath281 , @xmath256 and @xmath286 . ( b , d ) the corresponding average rotation frequencies @xmath101 . in both cases @xmath127.,title="fig:",height=113 ] of pinned traveling chimera states when @xmath281 , @xmath256 and @xmath286 . ( b , d ) the corresponding average rotation frequencies @xmath101 . in both cases @xmath127.,title="fig:",height=113 ] of pinned traveling chimera states when @xmath281 , @xmath256 and @xmath286 . ( b , d ) the corresponding average rotation frequencies @xmath101 . in both cases @xmath127.,title="fig:",height=113 ] of pinned traveling chimera states when @xmath281 , @xmath256 and @xmath286 . ( b , d ) the corresponding average rotation frequencies @xmath101 . in both cases in this paper we have investigated a system of non - identical phase oscillators with nonlocal coupling , focusing on the effects of weak spatial inhomogeneity in an attempt to extend earlier results on identical oscillators to more realistic situations . two types of inhomogeneity were considered , a bump inhomogeneity in the frequency distribution specified by @xmath289 and a periodic inhomogeneity specified by @xmath20 . in each case we examined the effect of the amplitude @xmath24 of the inhomogeneity and its spatial scale @xmath290 ( @xmath291 ) on the properties of states known to be present in the homogeneous case @xmath166 , viz . , splay states and stationary chimera states , traveling coherent states and traveling chimera states @xcite . we have provided a fairly complete description of the effects of inhomogeneity on these states for the coupling functions @xmath90 , @xmath292 and @xmath293 employed in @xcite . specifically , we found that as the amplitude of the inhomogeneity increased a splay state turned into a near - splay state , characterized by a nonuniform spatial phase gradient , followed by the appearance of a stationary incoherent region centered on the location of maximum inhomogeneity amplitude . with further increase in @xmath24 additional intervals of incoherence opened up , leading to states resembling the stationary multi - cluster chimera states also present in the homogeneous system . these transitions , like many of the transitions identified in this paper , could be understood with the help of a self - consistency analysis based on the ott - antonsen ansatz @xcite , as described in the appendix . the effect of inhomogeneity on multi - cluster chimera states was found to be similar : the inhomogeneity trapped the coherent clusters in particular locations , and eroded their width as its amplitude @xmath24 increased , resulting in coalescence of incoherent regions with increasing @xmath24 . more significant are the effects of inhomogeneity on traveling coherent and traveling chimera states . here the inhomogeneity predictably pins the traveling structures but the details can be complex . figures [ fig : hidden1][fig : hidden2 ] show one such complex pinning transition that proceeds via an intermediate direction - reversing traveling wave . these waves are generated directly as a consequence of the inhomogeneity and would not be present otherwise , in contrast to homogeneous systems undergoing a symmetry - breaking hopf bifurcation as described in @xcite . many of the pinning transitions described here are hysteretic as demonstrated in fig . [ fig : hidden3 ] . the traveling chimera states are particularly fragile in this respect , with small amplitude inhomogeneities sufficient to arrest the motion of these states . in all these cases the coherent regions are found in regions of least inhomogeneity , an effect that translates into an effective repulsive interaction between the coherent cluster and the inhomogeneity . in future work we propose to explore similar dynamics in systems of more realistic nonlocally coupled oscillators and compare the results with those for similar systems with a random frequency distribution . this work was supported in part by a national science foundation collaborative research grant cmmi-1232902 . we suppose that an oscillator at position @xmath16 has intrinsic frequency @xmath15 , and assume that the frequency distribution @xmath15 is continuous . the model equation is @xmath294 we next introduce the probability density function @xmath295 characterizing the state of the system . this function must satisfy the continuity equation @xmath296 where @xmath297 satisfies the relation @xmath298 and @xmath299 . we also define the local order parameter @xmath300 then @xmath301.\ ] ] the above equations can be recast in a more convenient form using the ott - antonsen ansatz @xcite @xmath302.\ ] ] here @xmath303 represents the distribution of natural frequencies at each @xmath16 . matching terms proportional to different powers of @xmath304 , we obtain @xmath305,\ ] ] where the complex order parameter @xmath306 is given by @xmath307 if we take @xmath308 set @xmath309 and perform the implied contour integration , we obtain @xmath310 in the limit @xmath311 the distribution function @xmath312 reduces to a delta function . the corresponding quantity @xmath27 satisfies @xmath313,\label{zdef}\ ] ] where @xmath306 is given by ( [ zdef ] ) . equations ( [ zdef])([zdef ] ) constitute the required self - consistency description of the nonlocally coupled phase oscillator system with an inhomogeneous frequency distribution @xmath15 .

chimera states consisting of domains of coherently and incoherently oscillating nonlocally - coupled phase oscillators in systems with spatial inhomogeneity are studied . the inhomogeneity is introduced through the dependence of the oscillator frequency on its location . two types of spatial inhomogeneity , localized and spatially periodic , are considered and their effects on the existence and properties of multi - cluster and traveling chimera states are explored . the inhomogeneity is found to break up splay states , to pin the chimera states to specific locations and to trap traveling chimeras . many of these states can be studied by constructing an evolution equation for a complex order parameter . solutions of this equation are in good agreement with the results of numerical simulations . pacs numbers : : may be entered using the ` \pacs{#1 } ` command .

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Proceed to summarize the following text: in this paper , we investigate the hausdorff young inequality for a nonlinear version of the fourier transform , and establish theorem [ thm : main ] below . before stating it precisely , we briefly discuss the linear case . given a complex - valued integrable function @xmath1 on the real line , we normalize its fourier transform as follows : @xmath2 in this way , the fourier transform is a contraction from @xmath3 to @xmath4 and it extends to a unitary operator on @xmath5 . standard interpolation tools can then be used to show that , for any @xmath6 $ ] , the fourier transform is also a contraction from @xmath7 to @xmath8 , where @xmath9 denotes the exponent conjugate to @xmath10 . this is the content of the classical hausdorff young inequality . its sharp version was first established by babenko @xcite in the case when the exponent @xmath11 is an even integer , and then by beckner @xcite for general exponents . it states that , if @xmath6 $ ] , then @xmath12 for every @xmath13 , where the optimal constant is given by @xmath14 an easy computation shows that gaussians , i.e. functions of the form @xmath15 with @xmath16 and @xmath17 , turn inequality into an equality . in other words , gaussians are _ extremizers _ for inequality . in the converse direction , lieb @xcite has shown that all extremizers for inequality are in fact gaussians . recently , christ @xcite further refined inequality by establishing the following sharpened version : given @xmath18 , there exists a constant @xmath19 such that , for every nonzero function @xmath13 , @xmath20 here , the distance from @xmath13 to the set of all gaussians , denoted @xmath21 , is naturally defined as @xmath22 we now describe the nonlinear setting of the present paper . we are interested in the simplest nonlinear model of the fourier transform , also known as the _ dirac scattering transform _ or the _ @xmath23-scattering transform_. to describe it precisely , take a measurable , bounded , and compactly supported function @xmath24 , which will often be referred to as a _ potential_. consider the initial - value problem @xmath25 for each @xmath26 , this problem has a unique solution @xmath27 , @xmath28 in the class of absolutely continuous functions . we simply write @xmath29 , @xmath30 in place of the limits @xmath31 , @xmath32 , and define the nonlinear fourier transform of the potential @xmath1 to be the function @xmath33 if @xmath1 vanishes outside some interval @xmath34 $ ] , then the initial condition in translates into @xmath35 , @xmath36 , while @xmath37 can be interpreted as @xmath38 respectively . the differential equation forces @xmath39 which in particular means that a certain size of the above vector is retained by the quantity @xmath40 alone . occasionally it is more convenient to add an extra column and turn the above vector into a @xmath41 matrix belonging to the classical lie group @xmath42 , see e.g. @xcite . sources of motivation for considering this precise instance of the nonlinear fourier transform include the eigenvalue problem for the dirac operator , the study of completely integrable systems and scattering theory , and the riemann hilbert problem ; see the expository paper @xcite for further information , and the notes @xcite for several related examples in the discrete setting . the dirac scattering transform is the simplest case of a more general transform , the akns zs nonlinear fourier transform ; see @xcite , @xcite for details . there is a strong parallel between the nonlinear and the linear fourier transforms . it is a straightforward exercise to verify the following analogues of the symmetry rules for the linear fourier transform . * _ unimodular homogeneity _ : if @xmath43 , where @xmath44 , then @xmath45 * _ modulation symmetry _ : if @xmath46 , where @xmath47 , then @xmath48 * _ translation symmetry _ : if @xmath49 , where @xmath50 , then @xmath51 * _ @xmath0-normalized dilation symmetry _ : if @xmath52 , where @xmath53 , then @xmath54 * _ conjugation symmetry _ : if @xmath55 , then @xmath56 * _ a substitute for additivity _ : if @xmath57 , where the support of @xmath58 lies to the left of that of @xmath59 , then @xmath60 we proceed to describe some nonlinear analogues of standard estimates for the linear fourier transform . the nonlinear riemann lebesgue estimate , which follows easily from grnwall s inequality , states that @xmath61 for every potential @xmath1 . the nonlinear plancherel identity , which is a well - known scattering identity that can be established via complex contour integration ( see e.g. the appendix in @xcite ) , states that @xmath62 for every potential @xmath1 as before . this equality can even be used to extend the definition of @xmath63 for functions @xmath1 that are only square - integrable , but in this case the existence of the pointwise limits @xmath31 , @xmath32 for a.e . @xmath64 is a well - known open problem ( see @xcite , @xcite ) . even if interpolation is not available in the present nonlinear setting , the work of christ and kiselev on the spectral theory of one - dimensional schrdinger operators @xcite , @xcite establishes a version of the nonlinear hausdorff young inequality which translates into the present context as follows : if @xmath6 $ ] , then there exists a constant @xmath65 , such that @xmath66 for every potential @xmath1 . an interesting question raised in @xcite is whether the constants @xmath67 can be chosen uniformly in @xmath10 , as @xmath68 . this has been confirmed in a particular toy model in @xcite , but remains an open problem in its full generality . by considering truncated gaussian potentials @xmath69}(x)\ ] ] as @xmath70 and linearizing , one can check that the constant in is at least as large as beckner s constant , i.e. @xmath71 . one may be tempted to conjecture that the optimal constant in is actually @xmath72 . while these questions are left open by the present work , we are able to provide some further supporting evidence of their validity by considering the behaviour of the nonlinear hausdorff young ratio for sufficiently small potentials . the main result of this paper is the following theorem . [ thm : main ] let @xmath18 , @xmath9 , and @xmath73 . let @xmath74 . then there exist @xmath75 and @xmath76 , depending on @xmath77 , with the following property : if @xmath78 is an interval of length @xmath79 , and @xmath80 is a measurable function such that @xmath81 and @xmath82 , then @xmath83 a few remarks may help to further orient the reader . * the parameters @xmath84 provide upper bounds for the _ height _ and the _ width _ of the function @xmath1 , respectively . they are fixed but can be arbitrarily large . * inequality implies with an optimal constant @xmath72 , but only within the restricted class of potentials considered in the theorem , and this class is allowed to depend on @xmath10 . that way theorem [ thm : main ] does not claim uniform boundedness of the constants in for any particular family of functions . it rather fixes the value of @xmath10 and shows that the nonlinear hausdorff young inequality beats the linear one in the asymptotic regime when @xmath85 . in particular , the constant @xmath86 is never attained for nonzero potentials in a sufficiently small neighborhood of the zero function ( depending on @xmath87 ) . * between the lines of the first part of the proof below ( see [ sec : far ] ) , one can easily obtain @xmath88 estimate provides a cheap version of , with @xmath89 replaced by @xmath90 , as @xmath91 , but the interest lies , of course , in obtaining the estimate with a negative sign . however , at least shows that the mere uniformity of the constants @xmath67 is trivial for potentials that are controlled in the @xmath0 norm . * it is easy to observe that is invariant under @xmath0-normalized dilations applied to @xmath1 . that way one can trade width for height or vice versa , and conclude that @xmath92 depend only on the _ product _ @xmath93 . this fact can also be checked by tracking the dependence of various parameters in the proof . let us briefly comment on the proof of theorem [ thm : main ] , which spans over the next three sections . the upshot is that for most choices of @xmath1 one can simply estimate the error arising from linearization , while for the remaining ones we verify that the nonlinear effect actually improves the estimate . to implement this strategy , we set up a case distinction , depending on whether the function @xmath1 is far or close to the set of gaussians , in an appropriate sense . the former case is the subject of [ sec : far ] , where we invoke christ s sharpened hausdorff young inequality in order to absorb the error terms coming from linearization . the latter case is the subject of [ sec : close ] , where we use a perturbative argument to expand the functional in question around a suitable gaussian that provides a good approximation for @xmath1 in _ both _ the @xmath94 and the @xmath0 senses . we are naturally led to study a certain quartic operator @xmath95 and its quadrilinear variant @xmath96 , which enjoy various symmetries . the operator @xmath95 can in turn be pointwise dominated by a power of the maximally truncated fourier transform , defined as follows : @xmath97 where the supremum is taken over all intervals @xmath98 . the classical menshov paley zygmund inequality states that , for every @xmath18 , there exists @xmath99 such that , for every @xmath13 , @xmath100 the argument makes crucial use of estimate in order to control the error terms , and to verify that the second order variation about the aforementioned gaussian has the correct sign . proofs of several technical lemmata are deferred to [ sec : proofs ] . finally , in the last section [ sec : counterexample ] , we provide a counterexample to the natural question of whether an inequality @xmath101 might hold for a general bounded compactly supported potential @xmath1 . more precisely , when @xmath102 , we exhibit an explicit linear combination of indicator functions of six contiguous intervals of unit length that disproves . because of this , we believe that theorem [ thm : main ] can not be established solely by regarding as a small perturbation of . * notation . * if @xmath103 are real numbers , we write @xmath104 or @xmath105 if there exists a finite absolute constant @xmath106 such that @xmath107 . if we want to make explicit the dependence of the constant @xmath106 on some parameter @xmath108 , we write @xmath109 or @xmath110 . we also write @xmath111 and @xmath112 . the real and imaginary parts of a complex number @xmath113 are denoted by @xmath114 and @xmath115 . the indicator function of a set @xmath116 is denoted by @xmath117 . throughout the paper it will be understood that all constants may depend on the admissible parameters @xmath77 . let @xmath80 satisfy the assumptions of theorem [ thm : main ] . our first task is to make precise what it means for @xmath1 to be far away from the set of gaussians . the following notion will be suitable for our purposes : assume that @xmath118 where @xmath119 is shorthand notation for @xmath120 here , @xmath86 denotes beckner s constant and @xmath121 is the constant promised by christ s refinement . this precise choice of @xmath119 will become clear as the proof unfolds . in particular , note that @xmath122 . going back to the defining ode , it is straightforward to check that the functions @xmath123 and @xmath124 satisfy the integral equations @xmath125 adding the two equations , the triangle inequality yields @xmath126 taking the @xmath127 norm in @xmath128 and invoking minkowski s integral inequality , we obtain @xmath129}\big\|_{\textup{l}^q(\mathbb{r } ) } + \int_{-\infty}^x |f(t)| \big\| |b(t,\xi)| + |a(t,\xi ) - 1| \big\|_{\textup{l}^q_\xi(\mathbb{r } ) } { \,{\rm d}}t.\ ] ] to estimate the quantity @xmath130}\|_{\textup{l}^q(\mathbb{r})}$ ] , we further split the analysis into two cases . _ @xmath131 in this case , for each @xmath132 we have that @xmath133}-g\|_{\textup{l}^p(\mathbb{r } ) } \geq \|f - g\|_{\textup{l}^p(\mathbb{r } ) } - \|f\mathbbm{1}_{(x,+\infty)}\|_{\textup{l}^p(\mathbb{r } ) } \geq \frac{\gamma}2 \|f\|_{\textup{l}^p(\mathbb{r})}.\ ] ] it follows that @xmath134},\mathfrak{g } ) \geq \frac{\gamma}2 \|f\mathbbm{1}_{(-\infty , x]}\|_{\textup{l}^p(\mathbb{r})},\ ] ] and christ s improved hausdorff young inequality yields @xmath135}\big\|_{\textup{l}^q(\mathbb{r } ) } \leq \big ( { \bf b}_p - c_p\big(\frac{\gamma}2\big)^2 \big ) \|f\mathbbm{1}_{(-\infty , x]}\|_{\textup{l}^p(\mathbb{r } ) } \leq \widetilde{\bf b}_p \|f\|_{\textup{l}^p(\mathbb{r})},\ ] ] where @xmath136 is defined as @xmath137 note that @xmath136 is not really a constant , since @xmath119 depends on the @xmath0 norm of @xmath1 . _ case 2 . _ @xmath138 in this case , the sharp hausdorff young inequality yields @xmath135}\big\|_{\textup{l}^q(\mathbb{r } ) } \leq { \bf b}_p \|f\mathbbm{1}_{(-\infty , x]}\|_{\textup{l}^p(\mathbb{r } ) } \leq { \bf b}_p \big(1-\big(\frac{\gamma}{2}\big)^p\big)^{\frac{1}{p } } \|f\|_{\textup{l}^p(\mathbb{r})}.\ ] ] bernoulli s inequality can then be invoked to verify that @xmath139 provided @xmath140 is chosen to be sufficiently small . here we also use that @xmath141 . in both cases , we obtain @xmath142 grnwall s lemma then implies @xmath143 letting @xmath144 and estimating @xmath145 , we finally have that @xmath146 the obtained inequality shows that , in this case , the only loss in passing from the linear to the nonlinear setting amounts to the exponential factor , which tends to @xmath147 as @xmath85 . recall the choice of @xmath119 from . it remains to choose @xmath148 small enough , so that holds , and @xmath149 thus we have verified the desired inequality in the case when the function @xmath1 is far from the gaussians . we analyse the complementary situation in the next section , where an additional smallness condition will be imposed on @xmath150 . we are now working under the assumption @xmath151 where @xmath119 was defined in . in particular , @xmath152 . thus , there exists a gaussian @xmath132 , such that @xmath153 this readily implies @xmath154 provided @xmath155 . recall our working assumptions that @xmath1 vanishes outside an interval @xmath156 of length @xmath79 , that @xmath157 is bounded from above by @xmath158 , and that it satisfies @xmath159 . under these conditions , @xmath160 the penultimate step in this chain of inequalities follows from the choice of @xmath119 and log - convexity of the @xmath94 norms . as a consequence , @xmath161 provided @xmath140 is small enough . this readily implies @xmath162 inequalities and ensure that the gaussian @xmath163 is a good approximation for the function @xmath1 on the whole real line , both in the @xmath94 and in the @xmath0 senses . we now proceed to derive the first nontrivial term in the expansion of the left - hand side of inequality for any @xmath164 . the first observation is that @xmath165 where the reflection coefficient @xmath166 satisfies riccati s differential equation @xmath167 in other words , the reflection coefficient is given by the integral equation @xmath168 from identity we obtain @xmath169 i.e. @xmath170 using the integral equation to substitute for @xmath171 on the right - hand side of , @xmath172 the first summand on the right - hand side can be recognized as @xmath173 repeating this procedure once again , i.e. substituting for @xmath174 and symmetrizing in the variables @xmath175 and @xmath176 , we conclude that @xmath177 where @xmath95 is the quartic operator defined as @xmath178 and @xmath179 is the nonlinear operator given by @xmath180 we now use the first numerical inequality established in lemma [ lm : inequalities ] below . substituting @xmath181 into the inequality from part ( a ) of lemma [ lm : inequalities ] , and then integrating in @xmath128 , we conclude from that @xmath182 where the second term is given by @xmath183 and the remainder @xmath184 is bounded by a linear combination ( with coefficients depending only on @xmath11 ) of integrals @xmath185 and , if @xmath186 , also @xmath187 approximating the function @xmath1 with the gaussian @xmath163 as discussed at the beginning of this section , we are thus reduced to showing that @xmath188 is a large enough positive quantity , that @xmath188 provides a good approximation for @xmath189 , and that the remainder term @xmath184 is appropriately small . this is accomplished via the following sequence of lemmata , whose proofs are deferred to the next section in order not to obscure the main line of reasoning . all of them hold under the ongoing assumption that @xmath1 satisfies the hypotheses of theorem [ thm : main ] , and that @xmath132 approximates @xmath1 in the sense of . the first lemma shows that the quantity @xmath188 is not too small . [ lm : hbounds ] @xmath190 the second lemma shows that @xmath188 provides a good approximation for @xmath189 . [ lm : diffhbounds ] @xmath191 the third lemma shows smallness of the remainder term @xmath184 . [ lm : rbounds ] @xmath192 we are now in a position to finish the proof of the theorem . if the parameter @xmath140 is chosen to be small enough , then so is @xmath119 , and the three lemmata combine with bounds , to yield @xmath193 if @xmath76 is small enough , then this inequality together with , , , lemma [ lm : hbounds ] , and the sharp hausdorff young inequality imply @xmath194 one last application of bernoulli s inequality finally yields @xmath195 as desired . this completes the proof of theorem [ thm : main ] modulo the verification of the lemmata , which is the content of the next section . we start with some elementary numerical inequalities . [ lm : inequalities ] given an exponent @xmath196 , the following inequalities hold . * for @xmath197 and @xmath198 one has @xmath199 with some finite constant @xmath200 depending only on @xmath11 . * for @xmath197 and @xmath198 one has @xmath201 with some finite constant @xmath202 depending only on @xmath11 . \(a ) no generality is lost in assuming that @xmath203 . we can then divide both sides of the inequality by @xmath204 , and substituting @xmath205 we are left with checking that @xmath206 if @xmath207 denotes the quotient of the two sides of the inequality , @xmath208 then we need to show boundedness from above of the function @xmath209 on @xmath210 . this is a simple consequence of the continuity of @xmath209 and finiteness of the limits : @xmath211 \(b ) this time the substitution @xmath205 turns the inequality into @xmath212 if we again denote the quotient of the two sides by @xmath207 , then the inequality follows as before from the continuity of @xmath209 and finiteness of the limits : @xmath213 start by noting that the quotient @xmath214 is invariant under arbitrary scalings , modulations , translations , and @xmath0-normalized dilations . indeed , if @xmath215 for some @xmath216 , then @xmath217 . moreover , if @xmath218 for some @xmath219 , then @xmath220 . similarly , if @xmath49 for some @xmath221 , then @xmath220 . finally , dilation invariance is easily seen from the fourier representation @xmath222 where the function @xmath223 is given by @xmath224 here we are integrating over a region in the affine hyperplane @xmath225 with respect to the @xmath226-dimensional hausdorff measure @xmath227 . the constant @xmath228 is unimportant and it is coming from the non - orthogonal choice of coordinates . it follows that , if @xmath229 , then @xmath230 on the other hand , @xmath231 this shows that the expression is invariant under @xmath0-normalized dilations , as claimed . now , going back to and writing @xmath232 one sees that any gaussian can be brought to standard form by an application of an appropriate scaling , modulation , translation , and dilation . given the symmetries of just discussed , in verifying the claim of the lemma we can assume that the gaussian approximation @xmath163 coincides with the standard gaussian , @xmath233 . in this case , we are reduced to checking that @xmath234 . again writing @xmath235 as the fourier transform of some function @xmath223 , then from formula above with @xmath236 it immediately follows that @xmath223 is nonnegative and not identically zero . also , @xmath237 is the fourier transform of another gaussian function @xmath238 , given by @xmath239 which is clearly strictly positive . it remains to invoke unitarity of the linear fourier transform and observe that @xmath240 this concludes the proof of the lemma . the proofs of the two remaining lemmata rely on the observation that @xmath241 can be pointwise controlled by the maximally truncated fourier transform @xmath242 , defined in . indeed , @xmath243 in a similar way , @xmath244 can be controlled pointwise by @xmath242 . to see this , start by noting that the reflection coefficient satisfies @xmath245 . it then follows from the integral equation that also @xmath246 . this time we get @xmath247 we start by rewriting @xmath248 as @xmath249 set @xmath250 . part ( b ) of lemma [ lm : inequalities ] allows us to bound the first integral in by a multiple of @xmath251 to bound these integrals , start by observing that estimates and imply @xmath252 hlder s inequality and the hausdorff young inequality then imply @xmath253 where the last inequality is a consequence of , , and . in particular , this term is acceptable . in a similar way , @xmath254 is likewise acceptable for @xmath255 . now we focus on the second integral from . it is useful to introduce the quadrilinear operator @xmath96 by @xmath256 so that @xmath257 . it can be estimated either by @xmath258 or ( similarly as @xmath95 ) by @xmath259 so the @xmath260 norm of this function is controlled using , , and hlder s inequality by @xmath261 invoking the menshov paley zygmund inequality , , , and then gives @xmath262 finally , yet another application of hlder s inequality combined with the linear hausdorff young inequality for @xmath163 bounds the second integral in by a constant times @xmath263 as noted before , this concludes the proof of the lemma . the proof parallels that of the previous lemma , and so we shall be brief . using hlder s inequality , the pointwise estimates and , and that fact that the maximally truncated fourier transform @xmath264 satisfies the estimate , we conclude that the five integrals in and are respectively controlled by @xmath265 since @xmath266 , the result follows . for @xmath267 , respectively.,title="fig:",width=226 ] for @xmath267 , respectively.,title="fig:",width=226 ] + for @xmath267 , respectively.,title="fig:",width=226 ] for @xmath267 , respectively.,title="fig:",width=226 ] now we disprove inequality . for computational simplicity , take @xmath102 . define : @xmath268 and consider the potential @xmath269 for a variable parameter @xmath270 . graphs of the function @xmath271 for several values of @xmath108 are depicted in figure [ fig : examples ] . let us remark that already for @xmath272 the graph looks almost identically as the one of @xmath273 , due to the effects of linearization . straightforward numerical computation of the @xmath274 norms for @xmath272 provides a direct counterexample to : @xmath275 the above decimal numbers were evaluated using the _ mathematica _ @xcite command _ nintegrate _ with options _ workingprecision @xmath276 _ and _ accuracygoal @xmath277_. the step - function [ eq : fcounter ] was discovered by numerical evaluation of the @xmath278-linear generalization of the form @xmath279 for indicator functions of arbitrary intervals @xmath280 , where @xmath281 for each @xmath282 , and subsequent numerical optimization of the coefficients . v.k . was supported in part by the croatian science foundation under the project 3526 . v.k . and j.r . were partially supported by the bilateral daad - mzo grant _ multilinear singular integrals and applications_. d.o.s . was partially supported by the hausdorff center for mathematics and dfg grant crc1060 . this work was started during a pleasant visit of the second author to the university of zagreb , whose hospitality is greatly appreciated . m. christ and a. kiselev , _ wkb asymptotic behavior of almost all generalized eigenfunctions for one - dimensional schrdinger operators with slowly decaying potentials _ , j. funct . , * 179 * ( 2001 ) , no . 2 , 426447 . v. e. zakharov , a. b. shabat , _ a refined theory of two dimensional self - focussing and one - dimensional self - modulation of waves in non - linear media _ ( russian ) , zh . . fiz . * 61 * ( 1971 ) 118134 .

the nonlinear hausdorff young inequality follows from the work of christ and kiselev . later muscalu , tao , and thiele asked if the constants can be chosen independently of the exponent . we show that the nonlinear hausdorff young quotient admits an even better upper bound than the linear one , provided that the function is sufficiently small in the @xmath0 norm . the proof combines perturbative techniques with the sharpened version of the linear hausdorff young inequality due to christ .

You are an expert at summarizing long articles. Proceed to summarize the following text: common belief holds that only polynomial interactions up to a certain degree depending on the spacetime dimension are renormalizable , in the sense that interactions of even higher order require an infinite number of subtractions in a perturbative analysis . this can be attributed to the implicit assumption that the higher - order couplings , which in general are dimensionful , set independent scales . such nonrenormalizable theories can only be defined with a cutoff scale @xmath1 , while the unknown physics beyond the cutoff is encoded in the ( thereby independent ) values of the couplings . starting from the viewpoint that the cutoff @xmath1 is the only scale in the theory , halpern and huang @xcite pointed out the existence of theories with higher - order and even nonpolynomial interactions within the conventional setting of quantum field theory . this happens because the higher - order couplings , by assumption , are proportional to a corresponding power of @xmath2 and therefore die out sufficiently fast in the limit @xmath3 ; the theories remain perturbatively renormalizable in the sense that infinitely many subtractions are not required . perhaps most important , halpern and huang so discovered nonpolynomial scalar theories which are asymptotically free , offering an escape route to the `` problem of triviality '' of standard scalar theories @xcite . to be more precise , halpern and huang analyzed the renormalization group ( rg ) trajectories for the interaction potential in the vicinity of the gaussian fixed point . the exact form of the potential was left open by using a taylor series expansion in the field as an ansatz . employing the wegner - houghton @xcite ( sharp - cutoff ) formulation of the wilsonian rg , the eigenpotentials , i.e. , tangential directions to the rg trajectories at the gaussian fixed point , were identified in linear approximation . while the standard polynomial interactions turn out to be irrelevant as expected , some nonpolynomial potentials which increase exponentially for strong fields prove to be relevant perturbations at the fixed point . for the irrelevant interactions , the gaussian fixed point is infrared ( ir ) stable , whereas the relevant ones approach this fixed point in the ultraviolet ( uv ) . possible applications of these new relevant directions are discussed in @xcite for the higgs model and in @xcite for quintessence . further nonpolynomial potentials and their applications in higgs and inflationary models have been investigated in @xcite . considering the complete rg flow of such asymptotically free theories from the uv cutoff @xmath1 down to the infrared , the halpern - huang result teaches us only something about the very beginning of the flow close to the cutoff and thereby close to the gaussian fixed point . each rg step in a coarse - graining sense `` tends to take us out of the linear region into unknown territory '' @xcite . it is the purpose of the present work to perform a first reconnaissance of this territory with the aid of the rg flow equations for the `` effective average action '' @xcite . in this framework , the standard effective action @xmath4 is considered as the zero - ir - cutoff limit of the effective average action @xmath5 $ ] which is a type of coarse - grained free energy with a variable infrared cutoff at the mass scale @xmath6 . @xmath7 satisfies an exact renormalization group equation , and interpolates between the classical action @xmath8 and the standard effective action @xmath9 . in this work , we identify the classical action @xmath10 given at the cutoff @xmath1 with a scalar o(@xmath0 ) symmetric theory defined by a standard kinetic term and a generally nonpolynomial potential of halpern - huang type . therefore , we have the following scenario in mind : at very high energy , the system is at the uv stable gaussian fixed point . as the energy decreases , the system undergoes an ( unspecified ) perturbation which carries it away from the fixed point initially into some tangential direction to one of all possible rg trajectories . we assume that this perturbation occurs at some scale @xmath1 which then sets the only dimensionful scale of the system . any other ( dimensionless ) parameter of the system should also be determined at @xmath1 ; for the halpern - huang potentials , there are two additional parameters : one labels the different rg trajectories ; the other specifies the `` distance '' scale along the trajectory . finally , the precise form of the potential at @xmath1 serves as the boundary condition for the rg flow equation which governs the behavior of the theory at all scales @xmath11 . since the rg flow equations for @xmath7 are equivalent to an infinite number of coupled differential equations of first order , a number of approximations ( truncations ) are necessary to arrive at explicit solutions . in the present work , we shall determine the rg trajectory @xmath12 for @xmath13 $ ] explicitly only in the large-@xmath0 limit which simplifies the calculations considerably . the paper is organized as follows : sec . [ hh ] , besides introducing the notation , briefly rederives the halpern - huang result in the language of the effective average action , generalizing it to a nonvanishing anomalous dimension . [ largen ] investigates the rg flow equation for the halpern - huang potentials in the large-@xmath0 limit , concentrating on @xmath14 and @xmath15 spacetime dimensions ; here , we emphasize the differences to ordinary @xmath16 theory particularly in regard to mass renormalization and symmetry - breaking properties . [ conclusions ] summarizes our conclusions and discusses open questions related to finite values of @xmath0 . as an important caveat , it should be mentioned that the results of halpern and huang have been questioned ( see @xcite and also @xcite ) , and these questions raised also affect the present work . to be honest , we have hidden the problems in the `` scenario '' described above in which an `` unspecified '' perturbation controls the shift of the system from the gaussian fixed point ( the continuum limit ) to the cutoff scale @xmath1 along a _ tangential _ direction . but since the cutoff scale @xmath1 , though large , is not at all infinitesimally separated from the gaussian fixed point , this tangential approximation is probably not sufficient to stay on the true renormalized trajectory during the shift . not only the tangent but also all ( infinitely many ) curvature moments of the trajectory had to be known in order to find an initial point right on the renormalized trajectory at @xmath1 . this point would correspond to a so - called `` perfect action '' @xcite . of course , this requires an infinite number of conditions to be imposed on the initial action at @xmath1 which we can not specify . in conventional field theories , this problem is solved by adjusting ( fine - tuning ) the initial action close to the unstable gaussian fixed point , leaving open only one a priori chosen relevant direction to the flow . but in the present case , there is an infinite number of relevant directions corresponding to the continuum of possible halpern - huang directions , and thus it seems impossible to single out only one relevant direction while frustrating the others by tuning infinitely many parameters . in other words , upon studying the flow from @xmath1 down to zero within our scenario , the continuum limit of our system remains unspecified , and therefore one important ingredient to a complete field theoretic system is missing . with these reservations in mind , we nevertheless believe that there are some lessons to be learned from the application of the rg flow equations to such potentials . concerning the investigation of the rg flow equation for the euclidean effective average action in @xmath17 dimensions , we closely follow the original work of wetterich @xcite . polynomial potentials and the large-@xmath0 limit to be discussed later have been explored in @xcite and @xcite in the effective average action approach . a comprehensive review and an extensive list of references on this subject can be found in @xcite . the effective average action can be expanded in terms of all possible o(@xmath0 ) invariants , @xmath18 = \int d^dx\left\ { u_k(\rho ) + \frac{1}{2 } z_k(\rho)\ , \partial_\mu \phi^b \partial_\mu \phi^b + \frac{1}{4 } y_k(\rho)\ , \partial_\mu \rho \partial_\mu \rho+\dots\right\ } , \label{1}\ ] ] where @xmath19 , @xmath20 labels the real components of the scalar field , and the dots represent terms involving higher derivatives ; for convenience , we shall always assume that @xmath21 during the calculation . halpern and huang derived their result in the `` local - potential approximation '' which is constituted by setting the wave function renormalization constant @xmath22 and neglecting @xmath23 and higher - derivative terms . in the present work , we shall generalize their result to a @xmath6-dependent @xmath24 which is parametrized by the anomalous dimension , @xmath25 denotes the derivative with respect to the rg `` time '' , @xmath26-\infty , 0]$ ] . ] @xmath27 . here we neglect @xmath23 and any @xmath28 dependence of @xmath24 . following @xcite , the rg flow equation for the effective average potential @xmath29 can be written as @xmath30 where the prime denotes the derivative with respect to the argument @xmath28 . the cutoff function @xmath31 is to some extent an arbitrary positive function that interpolates between @xmath32 for @xmath33 and @xmath34 for @xmath35 . it suppresses the small - momentum modes by a mass term @xmath36 acting as the ir cutoff . in eq . ( [ 3 ] ) the distinction between the @xmath37 `` goldstone modes '' and the `` radial mode '' is visible . provided that @xmath38 is given ( which we shall always assume in the this work ) , the flow of the effective potential @xmath39 ( and thus of the effective action @xmath7 in the present approximation ) is determined by eq . ( [ 3 ] ) . even if @xmath38 is neglected , eq . ( [ 3 ] ) produces qualitatively good results for polynomial effective potentials in @xmath21 @xcite . we expect similar behavior for nonpolynomial potentials . the halpern - huang result can be rederived by assuming that the system is close to the gaussian fixed point so that the effective potential and its derivatives are small . linearizing the right - hand side of eq . ( [ 3 ] ) with respect to the potential and its derivatives gives @xmath40 where we introduced the abbreviation @xmath41 , which is related to the volume of @xmath17 spheres . it is convenient to remove the explicit @xmath24 and @xmath6 dependence by using dimensionless scaling variables : @xmath42 in the same spirit , we write for the cutoff function @xmath43 where @xmath44 is a dimensionless function of a dimensionless argument , satisfying @xmath45 and @xmath46 . rewriting eq . ( [ 4 ] ) in terms of these variables and taking the rg time derivative @xmath47 on the left - hand side at fixed @xmath48 , we obtain the differential equation @xmath49 where the dot denotes a derivative with respect to the argument @xmath48 , and the complete cutoff dependence is contained in @xmath50 . \label{8}\ ] ] we are looking for eigenpotentials , i.e. , tangential directions to the rg flow of the scaling form @xmath51 , where @xmath52 classifies the possible directions and distinguishes between irrelevant ( @xmath53 ) , marginal ( @xmath54 ) and relevant ( @xmath55 ) perturbations away from the gaussian fixed point . solutions of this form can be given in terms of the kummer function @xmath56 @xcite @xmath57 . \label{9}\ ] ] for given dimension , @xmath0 , cutoff specification and anomalous dimension , the halpern - huang potential ( [ 9 ] ) depends on two dimensionless parameters : @xmath52 and @xmath58 . the latter sets a `` distance '' scale along the rg trajectories ; since it is an overall factor , the position of possible extrema of @xmath59 are independent of @xmath58 . to make contact with the literature , we note that we rediscover the results of periwal @xcite in the limit @xmath60 , where the halpern - huang result was generalized to arbitrary cutoffs within the polchinski rg approach @xcite . the results of halpern and huang are recovered by employing a sharp cutoff , for which @xmath61 is related to the volume of the @xmath62 dimensional sphere ) has to be defined carefully ; details can be found in @xcite . ] : @xmath63 various representations for the kummer function @xmath64 exist in the literature @xcite ; for further discussion , it is useful to replace the parameter @xmath52 by the combination @xmath65 then , eq . ( [ 9 ] ) reduces to standard polynomial potentials of degree @xmath66 in @xmath48 ( @xmath67 in @xmath68 ) if @xmath69 ; for all such polynomial potentials , the gaussian fixed point is ir stable . for @xmath70 , the potential vanishes , and for any other value of @xmath71 , the potential is nonpolynomial . for these cases , the asymptotic behavior for large third argument @xmath72 is given by an exponential increase @xmath73 the gaussian fixed point is uv stable ( @xmath55 ) for @xmath74 ( as long as @xmath75 ) . a particularly interesting case is given by the parameter set @xmath76 for which the eigenpotential eq . ( [ 9 ] ) is nonpolynomial and develops a minimum , inducing spontaneous symmetry breaking . to conclude our derivation of the halpern - huang results , we mention that in the particular case of @xmath77 there exist ( physically admissible ) solutions to eq . ( [ 7 ] ) which are odd under @xmath78 @xcite . the linearized flow equation has also been studied from a different perspective employing its similarity to a fokker - planck form @xcite . according to the scenario outlined in the introduction , we shall now consider the potentials found in eq . ( [ 9 ] ) taken at @xmath79 ( @xmath80 ) as the boundary condition for the complete flow equation ( [ 3 ] ) . provided that the anomalous dimension @xmath38 is only weakly dependent on @xmath6 and bounded ( as is the case , e.g. , for polynomial interactions in @xmath21 ) , some features can immediately be read off from eq . ( [ 3 ] ) : for nonpolynomial potentials with exponential asymptotics given by eq . ( [ 10 ] ) , the denominators on the right - hand side of the flow equation ( [ 3 ] ) vanish exponentially for large values of @xmath28 . therefore , @xmath81 for large @xmath28 , and the flow halts , leaving @xmath39 essentially unchanged . in particular , for symmetry - preserving potentials with a minimum at @xmath82 and @xmath83 , we may expect a rather unspectacular flow : for large @xmath28 , the above argument holds , whereas for small @xmath28 , we may always find a small region where the linearization of the flow equation is a good approximation ; there , the halpern - huang potential will still be an appropriate approximation . therefore , these potentials are expected to behave stiffly under the flow . for potentials with a minimum at nonvanishing @xmath28 ( spontaneous symmetry breaking ) with @xmath84 , the asymptotics for large @xmath28 will also stop the flow . however , the flow of @xmath39 near the nontrivial minimum can be more complicated , since @xmath85 and @xmath86 are no longer monotonic functions in this region . to the right of the minimum , these potentials may also be stiff under the flow , but the region around the origin and the minimum appear as a loose end . these heuristic arguments will be worked out and confirmed in the following section in the large-@xmath0 limit . for solving flow equations for the effective average potential of the type of eq . ( [ 3 ] ) , several techniques have been developed . of course , it is always possible to search numerically for solutions by putting the differential equation on a computer ; in fact , if one is looking for accurate results , this is the most appropriate option . however , since the potentials under consideration exhibit an exponential increase , straightforward numerics may come to its limits and a clever variable substitution has to be guessed . another possibility is to expand the potential in terms of a complete set of functions and decompose the flow equation into differential equations for the @xmath6-dependent coefficients ( generalized couplings ) . here , a choice for a useful set of functions again has to be guessed ; obviously , the polynomials as the standard choice are of no use , because the important information is contained in the nonpolynomial nature of the potential . therefore , we decide to work in the large-@xmath0 limit which puts no a priori restrictions on the form of the potential and allows for a complete integration of the flow equation . of course , the validity of the results for finite values of @xmath0 can hardly be controlled at this early stage . in the large-@xmath0 limit , the rg flow equation ( [ 3 ] ) for the potential simplifies considerably ; here we shall follow the presentation given in @xcite and @xcite . not only does the anomalous dimension @xmath38 vanish @xcite , but so does the influence of higher derivative terms ( @xmath87 ) . moreover , the goldstone modes dominate the right - hand side of eq . ( [ 3 ] ) and any contribution from the radial mode can be neglected ( this essentially changes the order of the differential equation ) . for technical reasons , one finally chooses a sharp cutoff function @xmath88 and decides to consider the flow equation for the _ derivative _ of the potential . in dimensionless variables , the large-@xmath0 limit of the flow equation reads @xmath89 of course , this equation can be obtained directly from the sharp - cutoff formulation of the rg and has already been studied by wegner and houghton @xcite ; further investigations of the wegner - houghton approach have been made in @xcite . following @xcite , this partial differential equation of first order can be solved using the standard method of characteristics and we find that the solution @xmath90 has to satisfy the equation @xmath91 where @xmath92 is defined by the integral @xmath93 this function is studied in app . a and explicit representations for @xmath14 and @xmath15 are given . the function @xmath94 is implicitly defined by the equation @xmath95 where @xmath96 represents the boundary condition for the flow equation at @xmath80 ( @xmath79 ) ; here , @xmath97 as a variable parametrizes the boundary condition and corresponds to the @xmath48 axis at @xmath79 in the @xmath98 plane . it is exactly eq . ( [ 18 ] ) that is to be inserted into eq . ( [ 16 ] ) , where the nonpolynomial potentials enter the investigation . now the route to an explicit solution is clear : ( i ) we specify the boundary condition via eq . ( [ 18 ] ) , ( ii ) insert this and an explicit representation for @xmath92 into eq . ( [ 16 ] ) , and `` solve '' ( or invert ) the resulting equation for @xmath90 . however , in practise , some complications are encountered : e.g. , inverse functions of such complicated objects as the kummer function @xmath56 are not easily obtainable . but the large-@xmath0 limit comes to the rescue once more as demonstrated in the next subsection . let us finally extract the flow of a possible minimum of the potential which is defined by @xmath99 ; from eq . ( [ 16 ] ) , we can easily extract that @xmath100 where @xmath101 denotes the minimum of the potential at @xmath80 ( t=0 ) , i.e. , the minimum of the halpern - huang potential ( in the large-@xmath0 limit ) ; by construction , it is identical to @xmath102 . the function @xmath103 can be read off from eqs . ( [ a3 ] ) and ( [ a4 ] ) of the appendix ( @xmath104 ) . reinstating dimensionful quantities ( cf . ( [ 5 ] ) ) , we find for the flow of a minimum of the potential @xmath105 here , we introduced a `` critical '' ( dimensionless ) field strength @xmath106 . of course , eq . ( [ 20 ] ) is well known in the literature @xcite and makes no particular reference to the type of potential under consideration . the only place where the potential type enters is the position of the initial minimum @xmath107 . if @xmath108 , then the classical as well as the quantum theory exhibit spontaneous symmetry breaking , since @xmath109 ; if @xmath110 the quantum theory will preserve o(@xmath0 ) symmetry . finally , if @xmath111 the classical potential @xmath112 is `` fine - tuned '' in such a way that the theory shows symmetry breaking for finite values of @xmath6 , but restores o(@xmath0 ) symmetry in the limit @xmath113 ; additionally , the potential has a vanishing mass term : @xmath114 ( by construction ) . in standard @xmath16 theory , the position of the minimum @xmath107 of @xmath112 can be chosen at will by an appropriate tuning of the negative mass term and the coupling . by contrast , for halpern - huang potentials , once the precise type of the potential is chosen by fixing @xmath52 ( or @xmath71 ) , there is no parameter left for any fine - tuning , since a possible minimum ( for theories with @xmath84 ) is independent of the last free parameter @xmath58 in eq . ( [ 9 ] ) . the question as to whether a symmetry - breaking quantum theory of the halpern - huang potentials exists has to be answered by determining the position of the initial minimum @xmath107 . this will also be investigated in the next subsection in the large-@xmath0 limit . in our scenario , the halpern - huang potential enters the flow equation as its boundary condition at the cutoff . since the flow equation is considered in the large-@xmath0 limit , it is not only useful to insert the large-@xmath0 limit of the halpern - huang potential into eq . ( [ 16 ] ) , but it is mandatory for reasons of consistency . otherwise , nonleading large-@xmath0 information would be mixed with large-@xmath0 behavior , introducing some arbitrariness into this approximation . from eq . ( [ 9 ] ) , we read off that the parameter @xmath0 occurs only in the second argument of the kummer function . unfortunately , we could not find any asymptotic expression for the kummer function with large second argument in the literature . instead of investigating this limit in terms of some appropriate series or integral representation , which might involve awkward interchanges of limiting processes , we shall use a more physically motivated approach : since the kummer function was identified with the tangential direction to the rg flow in the vicinity of the gaussian fixed point , its large-@xmath0 approximation should naturally be deducible from a large-@xmath0 study of the same subject . in other words , the desired function has to be a solution to the linearized flow equation in the large-@xmath0 limit . for technical reasons , we again turn to the derivative of the potential with respect to @xmath48 . then , the desired differential equation is obtained by linearizing eq . ( [ 15 ] ) . looking for potentials which satisfy the eigenpotential scaling condition @xmath115 , the large-@xmath0 halpern - huang equation reads @xmath116 where we again traded @xmath52 for the parameter @xmath71 as defined in eq . ( [ 12 ] ) . additionally , we made use of the `` critical '' field strength @xmath106 defined in eq . ( [ 20 ] ) . equation ( [ 21 ] ) can easily be solved for the various boundary conditions ; let us begin with a symmetry - preserving potential satisfying @xmath117 and @xmath118 : @xmath119 where the initial value @xmath120 also satisfies @xmath121 . the parameter @xmath120 is , of course , the large-@xmath0 analogue of the distance parameter @xmath58 in eq . ( [ 9 ] ) , @xmath122 . at first sight , one may doubt the validity of eq . ( [ 22 ] ) as the large-@xmath0 limit of eq . ( [ 9 ] ) , because it diverges at the critical field strength @xmath123 . nevertheless , this indeed reflects the behavior of the kummer function for large second argument , as can be checked numerically ( see fig . [ fig1](a ) ) @xcite . the critical field strength @xmath106 marks the point where the asymptotic exponential increase ( cf . ( [ 10 ] ) ) sets in ; and for larger values of @xmath0 , the slope increases without bound limit of the kummer function in the literature : it can not be defined for arbitrary values of @xmath48 . ] . of course , a potential that diverges for finite values of its argument is usually considered as inadmissible in field theory ( see , e.g. , @xcite and @xcite ) ; however , in the present case , we take the viewpoint that this potential wall at @xmath124 only symbolizes the exponential increase of the potential for finite values of @xmath0 . ( 145,52 ) ( 0,0 ) ( 0,52)(a ) ( 49,18)@xmath0=10 ( 61,13)100 ( 77,40)1000 ( 64,45)10000 ( 7,49)@xmath90 ( 80,3)@xmath125 ( 82,0 ) ( 82,52)(b ) ( 150,7)10 ( 148,40)100 ( 134,40)1000 ( 110,40)@xmath0=10000 ( 92,46)@xmath90 ( 156,20)@xmath125 let us now turn to the solution of eq . ( [ 21 ] ) for the symmetry - breaking potentials with ( @xmath84 ) . in the inner region of the potential where @xmath126 , we obtain the solution @xmath127 where the initial value this time satisfies @xmath128 . the derivative of the large-@xmath0 halpern - huang potential has a zero at @xmath124 , so that the potential itself exhibits a minimum at this position . now let us turn to the large-@xmath0 limit of the symmetry - breaking potential to the right of the minimum @xmath129 where @xmath130 . as a matter of fact , the unique solution of eq . ( [ 21 ] ) increases only very slowly , @xmath131 for @xmath132 and @xmath84 . this does certainly not reflect the expected exponential increase ; hence this solution has to be discarded . even if a more appropriate solution existed for @xmath129 , we would not be able to match them properly at @xmath124 , because the second derivative of the potential diverges at this point . in view of the results for the symmetry - preserving potential and owing to the fact that the asymptotic exponential increase for both types of potentials is the same ( see eq . ( [ 10 ] ) ) , the only possibility for the large-@xmath0 limit is to continue the potential at @xmath124 by a potential wall at @xmath133 . again , a numerical analysis of the full kummer function for large @xmath0 confirms this conjecture , as is depicted in fig . [ fig1](b ) . this concludes our large-@xmath0 analysis of the halpern - huang potential ; note that both types of the potential are formally equivalent , so that we combine them in the notation @xmath134 where @xmath135 stands for @xmath120 or @xmath136 and @xmath137 . actually , this also covers a @xmath16 potential for @xmath138 . now we are in a position to study the flow of the halpern - huang potentials from @xmath80 into the infrared regime @xmath113 in the large-@xmath0 limit . the missing piece of information to be inserted into eq . ( [ 16 ] ) is given by the inverse of eq . ( [ 24 ] ) : @xmath139 . \label{25}\ ] ] employing the representation ( [ a3 ] ) for the function @xmath92 , we find that the derivative of the potential has to satisfy the equation @xmath140 where the function @xmath141 is defined in eq . ( [ a5 ] ) . finally , eq . ( [ 26 ] ) has to be solved for @xmath90 , which we shall do in the limit @xmath142 ( @xmath143 ) for various cases in order to obtain the complete quantum effective potential . the following consideration will serve as a guide to the necessary approximations : at the cutoff @xmath80 , the dimensionful potential is of the order of the cutoff @xmath144 . for small deviations from the cutoff , @xmath145 , the potential scales according to the linearized flow equation ( halpern - huang equation ) : @xmath146 . then , the dimensionful potential scales as @xmath147 . therefore , if @xmath118 ( symmetry - preserving potentials ) , @xmath85 increases as we approach the infrared , whereas if @xmath148 ( symmetry - breaking potentials ) , @xmath85 decreases towards the infrared . of course , this argument holds strictly close to @xmath149 only , but it turns out to reproduce the unique consistent approximation schemes for extracting analytical results . let us first consider the @xmath15 potentials with @xmath118 and @xmath150 that exhibit no symmetry breaking at the cutoff . employing eq . ( [ a6 ] ) and reinstating dimensionful quantities via eq . ( [ 5 ] ) , eq . ( [ 26 ] ) reads , after neglecting terms of order @xmath151 in the limit @xmath113 ( @xmath152 ) : @xmath153 where @xmath154 denotes the mass of the theory at the cutoff . let us study eq . ( [ 27 ] ) in two limits : first , at @xmath28 close to @xmath155 , and secondly at the origin @xmath156 . at @xmath28 close to the potential wall at @xmath155 , the potential diverges , and we can approximate @xmath157 , leading us to @xmath158 in this limit , the effective potential @xmath159 remains formally identical to the large-@xmath0 halpern - huang potential ( cf . ( [ 22 ] ) ) ! this confirms our heuristic argument that the potential behaves stiffly under the flow in the region where it increases exponentially . concerning the opposite limit @xmath156 , there would be no mass renormalization at all , if eq . ( [ 28 ] ) were also correct in this limit , @xmath160 . however , in this limit , the approximation @xmath161 no longer holds , and instead we deduce from eq . ( [ 27 ] ) the transcendental equation @xmath162 therefore , the mass renormalization is governed by the only free parameter of the theory , @xmath120 : for large @xmath120 , there is effectively no renormalization , whereas the renormalized mass @xmath163 exceeds the `` classical '' mass @xmath164 for @xmath165 . typical values are @xmath166 for @xmath167 and @xmath168 ; for larger values of the rg trajectory parameter @xmath71 , the mass shift even increases : @xmath169 for @xmath167 and @xmath170 . the @xmath171 relation is plotted against @xmath120 for various @xmath71 in fig . [ fig3](a ) . by reintroducing the cutoff again via @xmath172 , ( [ 29 ] ) can be interpreted differently by writing @xmath173^a . \label{30}\ ] ] this equation tells us that the physical mass of the theory in the infrared can easily be much smaller than the cutoff by tenth of orders of magnitude , provided that @xmath120 is correspondingly small . since @xmath120 sets the distance scale on the rg trajectory , the demand for a small value of @xmath120 is consistent with our scenario : if we leave the gaussian fixed point with a very tiny perturbation @xmath174 at the high - energy scale @xmath1 , it is only _ natural _ to arrive at a low - energy theory with a similarly tiny mass compared to the cutoff . moreover , consistency of our scenario requires @xmath120 to be small in order to justify the linearization of the flow equation in deriving the halpern - huang result . to summarize , the symmetry - preserving halpern - huang potential qualitative does not change its form during the flow into the infrared ; in particular , no symmetry breaking occurs . only the slope of the potential at the origin of the theory increases for @xmath142 , which corresponds to a mass renormalization . let us begin with a dimension - independent statement referring to the position of the minimum of symmetry - breaking halpern - huang potentials with @xmath84 : in subsec . [ lnhh ] we learned that the position of the minimum of the halpern - huang potentials in the large-@xmath0 limit is independent of the parameters @xmath71 and @xmath175 : @xmath176 , or in dimensionful quantities : @xmath177 . according to the discussion following eq . ( [ 20 ] ) , the halpern - huang potentials are `` fine - tuned '' in the sense that the minimum vanishes exactly in the infrared limit @xmath113 : @xmath178 therefore , there is no symmetry breaking in the full quantum theory of halpern - huang potentials in the large-@xmath0 limit . moreover , since @xmath179 , the potential is flat at the origin and the renormalized quantum theory is massless . following the line of argument given below eq . ( [ 26 ] ) , the inner region of the potential where @xmath180 decreases towards the infrared ; hence we approximate @xmath181 in eq . ( [ 26 ] ) and obtain the transcendental equation in @xmath15 : @xmath182 here we can read off that @xmath183 is always smaller than @xmath36 . this reflects the approach to convexity of the inner part of the effective potential . to summarize , we have found , on the one hand , that the originally nontrivial minimum of the potential moves to the origin during the flow ; the inner region of the potential shrinks to a point . on the other hand , we know from the preceding subsection that the potential wall at @xmath184 does not change its position under the flow . it remains to be investigated what happens in between the minimum and the potential wall . unfortunately , we can not answer this question by the large-@xmath0 version of the flow equation , because we do not have a boundary condition for this region . at the cutoff @xmath80 , the inner region borders directly at the potential wall ; hence , there is no `` in - between '' that could serve as a boundary condition . of course , it is plausible to assume that the potential may interpolate smoothly between the origin with zero slope and the potential wall at @xmath185 with infinite slope . but alternatively , the potential can also remain flat for @xmath186 $ ] , resembling a particle - in - a - box potential . our ignorance about that part of the potential is unfortunately accompanied by our inability to predict the mass of the radial mode ; but this should not come as a surprise , since the large-@xmath0 limit neglects the radial mode anyway . ( 145,52 ) ( 0,0 ) ( 0,52)(a ) : @xmath15 ( 35,50)@xmath187 ( 61,-2)@xmath188 ( 2,10)@xmath189 ( 5,27)@xmath168 ( 12,40)@xmath170 ( 84,0 ) ( 84,52)(b ) : @xmath14 ( 147,-2)@xmath188 ( 127,50)@xmath187 ( 86,9)@xmath189 ( 87,25)@xmath168 ( 101,38)@xmath170 the investigation of the various types of potentials in @xmath14 proceeds analogously to the @xmath15 case with almost identical results . in particular , the symmetry - breaking potentials offer no new information : the inner region shrinks to a point , while the potential minimum moves to zero for @xmath113 , and the potential wall remains at @xmath190 . in between , no confirmed statement can be made within the large-@xmath0 limit , since no boundary condition governs this part of the potential . for symmetry - preserving potentials with @xmath118 , the potential again remains in the same form as at the cutoff for values of @xmath28 close to the potential wall at @xmath191 ( cf . ( [ 28 ] ) with @xmath192 replaced by @xmath1 ) . close to the origin @xmath156 , the shape of the potential is modified ; this is reflected by a mass renormalization . employing the same line of argument as given above in @xmath15 , and using eq . ( [ a7 ] ) , we find the @xmath14 analogue of eq . ( [ 29 ] ) : @xmath193 again , we find that there is no mass renormalization for large values of @xmath120 ; corrections for small values of @xmath120 are plotted in fig [ fig3](b ) . in the present paper , we have investigated the rg flow of particular nonpolynomial potentials for o(@xmath0 ) symmetric scalar theories using the effective - average - action method . these halpern - huang potentials arise from small relevant perturbations at the gaussian fixed point as tangential directions to the rg flow . apart from serious , unresolved problems with the continuum limit of these potentials , we were able to follow the flow from a given ultraviolet scale @xmath1 down to the nonperturbative infrared ; for this , a number of approximations have been made which are only under limited control . in a first step , we have neglected the influence of possible derivative couplings on the flow of the potential . secondly , assuming that the anomalous dimension is only weakly dependent on @xmath6 and bounded , the qualitative features of the flow could already be guessed from the form of the flow equation : this is because the exponential increase of the potentials essentially causes the flow to stop for large enough field values . therefore , the form of the potentials was recognized as stiff under the flow ; only the loose ends of the potential near the origin or possible extrema make room for more diversified behavior . these considerations have been verified explicitly in the large-@xmath0 limit of the system . in this limit , the exponential increase of the potentials is represented by a potential wall . the potential close to the wall and the wall itself remain unchanged even in the far infrared . those potentials with an o(@xmath0 ) symmetric ground state ( @xmath118 ) at the cutoff preserve this symmetry down to @xmath113 . our main result for such potentials is summarized in eqs . ( [ 29 ] ) , ( [ 30 ] ) and ( [ 33 ] ) , where the particular form of the mass renormalization is stated . contrary to polynomial scalar interactions where the mass varies @xmath194 during the flow , the halpern - huang potentials exhibit corrections which are governed by the rg distance parameter @xmath120 . in particular , if one demands that a renormalized ( infrared ) mass differ by several orders of magnitude from the cutoff scale @xmath1 , the bare parameters of a _ polynomial _ theory at the cutoff scale have to be fine - tuned accurately to several decimal places . by contrast , to achieve such a separation of mass scales with a _ nonpolynomial _ halpern - huang potential , an adjustment of the rg distance parameter at the cutoff to some small value is required with much less precision . additionally , the smallness of this value arises naturally , if the ( unknown ) perturbation at the gaussian fixed point is tiny . the ( symmetry - preserving ) halpern - huang potentials thus has no problem of _ naturalness_. owing to the general properties of the complete flow equation mentioned above , we believe that these properties of the symmetry - preserving potentials in the large-@xmath0 limit also hold for finite values of @xmath0 . the status of the large-@xmath0 limit is certainly different for halpern - huang potentials which offer spontaneous symmetry breaking ( @xmath84 ) . these potentials exhibit the remarkable property that the nontrivial minimum persists for any finite value of @xmath6 but vanishes in the complete quantum theory for @xmath142 in the large-@xmath0 limit ; the o(@xmath0 ) symmetry is restored and the potential becomes flat near the origin . the coincidence between the position of the minimum and the critical value of the field strength may finally be ascribed to the formal resemblance between the large-@xmath0 flow equation and its linearized version determining the halpern - huang potentials . since the complete flow equation is much more complex , it appears rather improbable that this property continues to hold for finite @xmath0 . therefore , whether or not spontaneous symmetry breaking occurs in the quantum version of the halpern - huang potential at finite @xmath0 remains an open question . the present investigation at least observes a tendency of the system to restore o(@xmath0 ) symmetry . this is in concordance with @xcite , where a one - loop calculation for the effective potential reveals a restoration of o(@xmath0 ) symmetry for potentials with ( @xmath195 ) . in this context , a possible application of the halpern - huang potentials to the higgs sector of the standard model is still questionable . even if a quantum version of the potential with spontaneous symmetry breaking exists , the naturalness of the scalar sector alone is not sufficient to solve the hierarchy problem this is because the ( standard ) yukawa coupling to the fermions leads to large scalar mass renormalizations by fermion loops . therefore , some appropriate nonpolynomial interaction has to be chosen , also in this sector . nevertheless , the price to be paid would not be too high , because not only the hierarchy problem could be circumvented without additional degrees of freedom , but also the problem of `` triviality '' would be evaded . from an intuitive point of view , the fact that the form of the potential is stable under the rg flow appears to be disappointing : since the potential remains inherently nonpolynomial , it is impossible to make contact with a would - be classical behavior that is determined by only a few ( polynomial ) terms . the latter is usually expected at large distances . for example , merely for very weak fields do the first terms in a taylor expansion of the kummer function represent a good approximation . for stronger fields , the application of the halpern - huang potentials might therefore be limited in this sense . from a technical viewpoint , our calculations hold for @xmath21 . we have given explicit results for @xmath14 and @xmath15 , and generalizations to higher dimensions are straightforward . the limiting case @xmath196 has to be treated with great care for several reasons . first of all , finite @xmath0 results may only be trusted if the flow of the anomalous dimension @xmath38 is taken into account ; at least in the case of polynomial potentials , this turned out to be obligatory @xcite in order to obtain a good picture of the kosterlitz - thouless transition . furthermore , the limit @xmath197 of the halpern - huang potentials offers several possibilities . it has already been observed variously in the literature ( see , e.g. , @xcite ) , that the sine - gordon as well as the liouville potentials solve the linearized flow equation in @xmath196 . in fact , as can be easily shown with the aid of some identities of @xcite , both types of potentials arise as limiting cases of the halpern - huang potentials for @xmath77 in combination with the @xmath198 odd solution of the linearized flow equation : to be precise , the sine - gordon potential is recovered in the limit @xmath199 for @xmath200 , whereas the liouville potential is obtained by taking the limit @xmath201 for @xmath202 . as far as the liouville theory is concerned , further similarities to the present results for the symmetry - preserving potentials are visible . in @xcite , the liouville potential has also been found to behave stiffly under the rg flow for similar reasons as in the present case . in particular , quantum liouville theory appears to equal classical liouville theory , except for a flow of the central charge by one unit and a modified mass parameter . these similarities confirm the viewpoint that the halpern - huang potentials can be regarded as higher - dimensional analogues of liouville theory . the author wishes to thank w. dittrich for helpful conversations and for carefully reading the manuscript . useful discussions with r. shaisultanov are also gratefully acknowledged . in this appendix , we present some details about the function @xmath92 appearing in the solution ( [ 16 ] ) to the flow equation ( [ 15 ] ) ; this function is defined as @xmath203 substituting @xmath204 $ ] , we arrive at the form @xmath205 where @xmath27 is always nonpositive : @xmath206-\infty , 0]$ ] . separating the zeroth - order term of a taylor expansion of the integrand , we find the convenient representation @xmath207 with the auxiliary functions @xmath208 and @xmath141 defined by @xmath209 note that @xmath210 for @xmath211 and @xmath212 . the explicit form of @xmath213 depends on the spacetime dimension . for @xmath15 , the integral can easily be evaluated by standard means , yielding @xmath214 in @xmath14 , we take care of the possibility of a nontrivial minimum ( spontaneous symmetry breaking ) and find to the right of a possible minimum @xmath215 in the `` inner '' region to the left of a possible minimum , we obtain @xmath216 where @xmath212 for reasons of consistency . 99 k. halpern and k. huang , phys . rev . lett . * 74 * , 3526 ( 1995 ) . k. halpern and k. huang , phys . d * 53 * , 3252 ( 1996 ) . a.i . larkin and d.e . khumelnitskii , sov . j. nucl . * 29 * , 1123 ( 1969 ) ; k.g . wilson , phys . * 28 * , 248 ( 1972 ) . f.j . wegner and a. houghton , rev . a * 8 * , 401 ( 1973 ) . v. branchina , hep - ph/0002013 ( 2000 ) . k. langfeld and h. reinhardt , mod . a * 13 * , 2495 ( 1998 ) ; r.f . langbein , k. langfeld , h. reinhardt , l. v. smekal , mod . a * 11 * , 631 ( 1996 ) . c. wetterich , phys . b * 301 * , 90 ( 1993 ) . morris , phys . * 77 * , 1658 ( 1996 ) ; k. halpern and k. huang , phys . rev . lett . * 77 * , 1659 ( 1996 ) . c. bagnuls and c. bervillier , hep - th/0002034 ( 2000 ) . n. tetradis and c. wetterich , nucl . b * 422 * , 541 ( 1994 ) . p. hasenfratz , in proc . _ advanced school of non - perturbative quantum field physics _ , edited by m. asorey and a. dobado , singapore , world scientific , 1998 , hep - lat/9803027 . n. tetradis and d. f. litim , nucl . b * 464 * , 492 ( 1996 ) [ hep - th/9512073 ] . j. berges , n. tetradis and c. wetterich , hd - thep-00 - 26 , hep - ph/0005122 ( 2000 ) . v. periwal , mod . a * 11 * , 2915 ( 1996 ) . j. polchinski , nucl . b * 231 * , 269 ( 1984 ) . m. abramowitz and i.a . stegun , _ handbook of mathematical functions _ , national bureau of standards , washington , ( 1964 ) . a. bonanno , phys . d * 62 * , 027701 ( 2000 ) . j. zinn - justin , _ quantum field theory and critical phenomena _ , oxford university press , ( 1989 ) . j. comellas and a. travesset , nucl . b498 , 539 ( 1997 ) . the kummer functions ( hypergeometric1f1[a , b , z ] ) can numerically as well as partly algebraically be treated by mathematica , version 4.0.1.0 , wolfram research , champaign ( 1999 ) . a. hasenfratz and p. hasenfratz , nucl . phys . * b270 * , 687 ( 1986 ) . k. halpern , phys . rev . d * 57 * , 6337 ( 1998 ) . m. reuter and c. wetterich , nucl . b * 506 * , 483 ( 1997 ) .

a class of asymptotically free scalar theories with o(@xmath0 ) symmetry , defined via the eigenpotentials of the gaussian fixed point ( halpern - huang directions ) , are investigated using renormalization group flow equations . explicit solutions for the form of the potential in the nonperturbative infrared domain are found in the large-@xmath0 limit . in this limit , potentials without symmetry breaking essentially preserve their shape and undergo a mass renormalization which is governed only by the renormalization group distance parameter ; as a consequence , these scalar theories do not have a problem of naturalness . symmetry - breaking potentials are found to be `` fine - tuned '' in the large-@xmath0 limit in the sense that the nontrivial minimum vanishes exactly in the limit of vanishing infrared cutoff : therefore , the o(@xmath0 ) symmetry is restored in the quantum theory and the potential becomes flat near the origin .

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Proceed to summarize the following text: several articles have recently focused on the electrocaloric effect ( ece ) in ferroelectrics and related materials,@xcite which bears analogy with the well known magnetocaloric effect ( mce).@xcite here we investigate the mechanisms of ece in relaxor ferroelectrics , to be referred to as _ relaxors _ and normal ferroelectrics ( or _ ferroelectrics _ ) , and discuss the specific features of these two groups of materials . in particular , we will discuss the possibility of achieving a giant ece in bulk inorganic relaxors and ferroelectric materials as well as in organic polymers . these systems offer the prospect of practical applications , such as miniaturized and energy efficient cooling devices , without the need for large electric currents commonly associated with the mce . a crucial physical quantity in ece is the change of entropy of a polar material under the application and/or removal of an external electric field . for example , when the electric field is turned on isothermally , the elementary dipolar entities in the system will become partially ordered and the entropy will be lowered . the entropy lowering of the dipolar subsystem is then compensated by an increase of the temperature of the total system , which characterizes the ece . the degree of lowering depends on the number of statistically significant configurations in the initial and final states of the system , as well as on the size of the average dipole moment and the volume density of dipolar entities . other factors may also play a role : if the system undergoes a first order phase transition under the action of external electric field , the entropy will be enhanced on crossing the borderline between the two phases , resulting in a larger ece . the line of first order transition points terminates at a critical point where the transition becomes continuous,@xcite and it will be of special interest to investigate the behavior of ece in the vicinity of the critical point . estimates of the ece can be made on the basis of thermodynamic maxwell relations using the measured heat capacity and the field and temperature dependence of the dielectric polarization . from the theoretical point of view , a central problem is how to make predictions about the temperature and field dependence of ece . as a first step , one needs to develop an appropriate phenomenological and/or mesoscopic model , which incorporates the specific physical features of the systems . here we will make use of the standard landau phenomenological model , which can be applied to both relaxors and ferroelectrics with the corresponding choice of landau coefficients . these in turn can be derived from the mesoscopic model of the material under study . in the case of relaxors , the mesoscopic model of choice is the spherical random bond random field ( srbrf ) model , which is based on the concept of reorientable polar nanoregions ( pnrs).@xcite thus we should be able to compare the ece in relaxors and ferroelectrics , and determine the parameters , which control the ece in these systems . finally , using general principles of statistical thermodynamics we will discuss the existence of a theoretical upper bound on the ece and argue that it satisfies a universal relation , which is , in principle , also applicable to mce . the temperature change of a polar system under adiabatic electric field variation from the initial value @xmath3 to final value @xmath4 can be written in the form @xcite @xmath5 which follows from the thermodynamic maxwell relation @xmath6 involving the entropy density @xmath7 and the physical dielectric polarization @xmath8 ( in units of c / m@xmath9 ) . the volume specific heat at constant field is given by @xmath10 . in deriving eq . ( [ dt1 ] ) , one tacitly assumes that the fluctuations of polarization @xmath11 can be ignored and that @xmath8 represents a thermodynamic variable given by the macroscopic average of @xmath11 . furthermore , it is implied that the system is ergodic , i.e. , its response time much shorter than the experimental time scale . if the field and temperature dependence of @xmath12 is known from experiments , the integral in eq . ( [ dt1 ] ) can be evaluated , yielding an estimate for @xmath13.@xcite in model calculations , it seems convenient to change the integration variable in eq . ( [ dt1 ] ) from @xmath14 to @xmath15 . this is readily done by applying the thermodynamic identity@xcite @xmath16 with the result @xmath17 this expression is fully equivalent to eq . ( [ dt1 ] ) , with the new integration limits given by @xmath18 and @xmath19 . the partial derivative @xmath20 can be obtained from the free energy density functional @xmath21 . ignoring fluctuations of the order parameter @xmath8 , we write @xmath22 as a power series @xmath23 this has the standard form of a mean field free energy expansion with temperature dependent coefficients @xmath24 , etc . applying the equilibrium condition @xmath25 , we obtain the equation of state @xmath26 and the temperature derivative in eq . ( [ dt2 ] ) becomes @xmath27 where @xmath28 , @xmath29 etc . it should noted be that @xmath19 in eq . ( [ dt2 ] ) is that solution of eq . ( [ es1 ] ) , which simultaneously minimizes the free energy ( [ f ] ) . the integration in eq . ( [ dt2 ] ) can now be carried out , yielding @xmath30 . \label{dt3}\ ] ] in passing , we note that @xmath31 , in general , depends on the temperature ; however , in writing down eqs . ( [ dt1 ] ) and ( [ dt2 ] ) the temperature dependence of the heat capacity had already been ignored . the expression in brackets is related to the change of the entropy density @xmath32 . using eq . ( [ f ] ) we can write @xmath33 the first term @xmath34 is the entropy at @xmath35 . it contains the configuration entropy of dipolar entities , which depends on the number of equilibrium orientations @xmath36 , say , @xmath37 for the @xmath38 equilibrium case.@xcite thus we may expect that @xmath39 , @xmath40 being the total number of dipolar entities such as pnrs in relaxors . the second term is given by @xmath41 back in eq . ( [ dt3 ] ) , @xmath42 cancels out and the ece temperature change can be rewritten in the familiar form @xcite @xmath43 with @xmath44 . it should be noted that the values of all temperature - dependent quantities @xmath45 , @xmath8 , @xmath46 , @xmath47 , etc . , on the r.h.s . ( [ dt3 ] ) are taken at the initial temperature @xmath48 , and @xmath31 at the final field value @xmath49 . the coefficients @xmath50 can be expressed in terms of linear and nonlinear susceptibilities by formally inverting the relation ( [ es1 ] ) and writing @xmath51 as a power series in @xmath49.@xcite in landau theory , close to a second order phase transition one sets @xmath52 , while @xmath53 are constants . thus , @xmath54 const . , and @xmath55 . this leaves only one nonzero term of the order @xmath56 in eq . ( [ dt3 ] ) . on the other hand , @xmath57 and the nonlinear susceptibilities are also found to diverge when @xmath58 . thus a formal inversion @xmath51 of eq . ( [ es1 ] ) in powers of @xmath49 would lead to a poorly converging series . in the following we will apply eq . ( [ dt3 ] ) in order to discuss the predictions of landau theory in two characteristic cases , namely , normal ferroelectrics and relaxors . as already mentioned , in landau theory of phase transitions in ferroelectrics , the coefficients @xmath53 in eq . ( [ f ] ) are assumed temperature independent and @xmath59 , where @xmath60 is the curie - weiss temperature . when @xmath61 and @xmath62 , a second order transition occurs at @xmath63 . for @xmath2 and @xmath64 , a first order transition appears at a temperature @xmath65 , given by the relation @xmath66 . writing @xmath67 , where @xmath68 is the curie constant , we find : @xmath69 for @xmath70 , a critical point will be located at @xmath71 , where @xcite @xmath72 turning next to relaxors , we assume that the relevant elementary dipolar entities at temperatures around the dielectric maximum are polar nanoregions or pnrs . according to the srbrf model,@xcite these pnrs are coupled through gaussian random interactions @xmath73 ( `` random bonds '' ) and are subject to gaussian random fields @xmath74 . in zero applied field , spontaneous long range order is suppressed @xmath75 . this means that for a relaxor we can still use the free energy ( [ f ] ) , however , the coefficient @xmath76 must remain positive at all temperatures . thus , for @xmath61 and @xmath62 there can be no second order phase transition . the explicit form of @xmath76 follows from eq . ( [ es1 ] ) , namely , @xmath77 where @xmath78 represents the ( quasi)static linear field - cooled dielectric susceptibility . this can be derived from the srbrf model of relaxors:@xcite @xmath79 with @xmath80 the effective curie constant @xmath68 is given explicitly in terms of the average squared dipole moments @xmath81 of pnrs , namely , @xmath82 , and the average volume @xmath83 associated with a pnr . later , we will also introduce the saturation polarization @xmath84 , where @xmath85 . for simplicity , we will henceforth neglect the difference between @xmath86 and @xmath87 , and write @xmath88 . the parameter @xmath68 can be determined experimentally from the asymptotic high temperature behavior of @xmath89.@xcite the parameter @xmath90 is defined _ via _ the average over the infinitely ranged random interaction @xmath91_{av}=j_0/n$ ] . the spherical glass order parameter , @xmath92 , is a measure of the degree of disorder . for @xmath35 it is determined by the cubic equation @xmath93 here , @xmath94 is proportional to the variance of the random bond distribution according to @xmath95_{av}-([j_{ij}]_{av})^2=j^2/n$ ] , while @xmath96 measures the correlations of quenched random fields , i.e. , @xmath97_{av}=0 $ ] and @xmath98_{av}=\delta_{ij}\delta$ ] . if @xmath99 , long range order will be suppressed ( @xmath100 ) . thus the relaxor state is characterized by three physical parameters : @xmath101 , @xmath102 , and @xmath96 . typically , in relaxors one finds @xmath103 , implying that random bonds are effectively much stronger than random fields . this allows pnrs to reorient collectively under the action of external fields and relax towards equilibrium . in the opposite case , @xmath104 , pnrs are would be trapped in a frozen static configuration of random fields and no characteristic low - frequency relaxor response due to pnr flipping could be observed . here we will consider only the case @xmath103 . from eqs . ( [ chi1 ] ) and ( [ achi ] ) we derive the coefficient @xmath76 : @xmath105 the linear susceptibility @xmath89 has been fitted to experimental data for the static field - cooled response in a variety of relaxor systems , from which the parameters of the model have been obtained.@xcite formally , the srbrf model also yields explicit expressions for nonlinear susceptibilities , from which the coefficients @xmath1 and @xmath106 in the free energy ( [ f ] ) can be determined.@xcite however , it has been shown earlier@xcite that realistic values of these coefficients can only be obtained if the coupling between pnrs and lattice strain fluctuations is included . several mechanisms for such a coupling have been investigated both at a mesoscopic @xcite and phenomenological level.@xcite it has also been shown@xcite that in real three dimensional systems strain coupling gives rise to anisotropy of the anharmonic terms in the free energy.@xcite specifically , strain coupling may change the sign of the coefficient @xmath1 and hence of the corresponding nonlinear susceptibility @xmath107 for a given direction of the applied field . in the following , we will simply consider @xmath1 and @xmath106 as free parameters and discuss separately the cases @xmath61 and @xmath2 , while keeping @xmath64 . for cubic systems , eq . ( [ f ] ) can be rewritten in a general form @xcite @xmath108 where @xmath109 are so - called dielectric stiffness coefficients with @xmath110 . for a system with orthorhombic symmetry in a field along @xmath111 $ ] we recover eq . ( [ f ] ) , where @xmath8 is the total polarization , @xmath112 , @xmath113}=(4/3)(\alpha_{11}+\alpha_{12})$ ] , and @xmath114}=(2/9)(3\alpha_{111}+6\alpha_{112}+\alpha_{123})$].@xcite experiments in pmn @xcite indicate that @xmath115 for @xmath116 $ ] , implying @xmath113}<0 $ ] . on the other hand , for tetragonal symmetry with @xmath117 $ ] one has @xmath118}=4\alpha_{11}$ ] and @xmath119 . for example , in pmn @xcite one finds in this case that @xmath120 or @xmath118}>0 $ ] , indicating that @xmath121 and @xmath122 . if @xmath123 at all temperatures , there is no first order phase transition for @xmath62 . if @xmath124 , however , a first order phase transition with a jump of polarization @xmath125 occurs for @xmath49 exceeding some threshold value @xmath126 this can readily be seen by numerical minimization of the free energy ( [ f ] ) for any pair @xmath127 . as the temperature increases , @xmath128 decreases and vanishes at an isolated critical point @xmath71,@xcite where the derivative @xmath129 diverges.@xcite the critical temperature is determined from the equation @xmath130 and @xmath131 from the second of eqs . ( [ cp ] ) . it should be noted that , generally speaking , @xmath1 and @xmath106 can also be functions of @xmath132.@xcite for @xmath133 the system is in a supercritical regime with continuous temperature and field dependence of @xmath12 . to illustrate the temperature and field dependence of the ece in relaxors and ferroelectrics , we calculate @xmath134 from eq . ( [ dt3 ] ) for a selected set of parameter values . first , we introduce rescaled , dimensionless quantities @xmath22 and @xmath8 according to @xmath135 and @xmath136 , where @xmath88 is the saturation polarization occurring at high field values and/or low temperatures . this requires a rescaling of the remaining parameters according to @xmath137 , @xmath138 , @xmath139 , etc . also , we redefine @xmath140 and @xmath141 . from eq . ( [ at1 ] ) we see that in relaxors the rescaled parameter @xmath76 behaves as @xmath142 , and @xmath143 becomes @xmath144 here and until the end of this section , the symbols @xmath50 and @xmath145 refer to dimensionless , rescaled parameters , but elsewhere in this paper the same symbols denote the true , physical values of these quantities . in ferroelectrics , @xmath146 and thus @xmath147 , and @xmath148 . in relaxors , in the high temperature limit @xmath149 , @xmath150 tends asymptotically to zero , thus @xmath151 and @xmath152 , i.e. , the same as in ferroelectrics . in fig . 1 , @xmath76 and @xmath143 are plotted for a relaxor ferroelectric with @xmath153 and @xmath154 . also shown is the behavior of a normal ferroelectric . it should be noted that the essential difference between ferroelectrics and relaxors is the behavior of the corresponding coefficients @xmath76 and @xmath143 . in the following we will choose @xmath155 const . @xmath156 and @xmath157 . the corresponding ece temperature change is obtained from eq . ( [ dt3 ] ) . using the fact that @xmath158 etc . , we find @xmath159 . \label{dt5}\ ] ] the polarization @xmath12 will be calculated numerically by simultaneously solving eq . ( [ es1 ] ) and minimizing the free energy ( [ f ] ) . we will do that separately for the two cases @xmath61 and @xmath2 , assuming the denominator @xmath160 to be a constant amplitude factor . \(i ) _ case @xmath61_. as already stated , @xmath161 in a relaxor , but in ferroelectrics @xmath162 for @xmath163 . the spontaneous polarization @xmath164 is obtained by minimization of @xmath165 . in real systems , @xmath45 may not be spatially uniform due to domains . here we assume that @xmath166 has the same value in all domains regardless of their orientation , and that the contribution of domain walls to the entropy can be neglected.@xcite in fig . 2(a ) we show the calculated values of @xmath13 for a relaxor as function of temperature for various values of @xmath167 , where @xmath131 is formally given by eq . ( [ cp ] ) , although the critical point does not exist for @xmath61 . also shown in fig . 2(b ) is @xmath13 for a ferroelectric with the same parameters @xmath168 , but with different @xmath169 and @xmath46 . @xmath13 has a peak at @xmath170 and is in general larger than in the relaxor case . at higher temperatures , however , the difference gradually disappears . \(ii ) _ case @xmath2_. eqs . ( [ chi1]-[q ] ) imply @xmath171 . thus , @xmath172 and the first order phase transition in relaxors at @xmath62 is suppressed . in a ferroelectric , however , a first order transition in zero field occurs at @xmath173 . the critical point is located at @xmath174 in relaxors , and at @xmath175 in ferroelectrics , while @xmath176 in both cases . in general , @xmath13 is found to increase with increasing field and exhibits a peak as a function of temperature . in relaxors , the peak position moves to higher temperatures with increasing field values , whereas in ferroelectrics the maximum is located at @xmath177 , where a jump of the spontaneous polarization occurs . in fig . 3 , @xmath178 is plotted for four values of the field @xmath179 , as indicated . at @xmath180 in relaxors , there is a jump of @xmath13 due to the field - induced first - order transition . on the other hand , in the case of a ferroelectric , there is a jump of @xmath13 at the zero - field first order transition temperature @xmath173 . at the critical point , @xmath13 is continuous but with an infinite slope in both cases , while at higher temperatures the difference between relaxors and ferroelectrics tends to disappear . in fig . 4 , the ece efficiency @xmath181 is plotted as a function of @xmath167 for four values of temperature close to @xmath182 . as expected , the maximum efficiency is obtained at the corresponding critical points @xmath183 and @xmath180 . larger values of @xmath13 in ferroelectrics rather than in relaxors are mainly due to the sharp decrease of @xmath143 in relaxors at @xmath184 ( cf . fig . 1 ) . however , this does not mean that ferroelectrics are better candidates for achieving giant ece . namely , one should bear in mind that there are other parameters , such as @xmath68 and @xmath31 , which also have a strong impact on the ece . moreover , the above comparison between relaxors and ferroelectrics makes only sense if the coefficients @xmath185 are indeed the same in both cases . therefore , in discussing the ece in specific systems one should carefully consider the actual physical values of all the relevant model parameters , as discussed in the following section . it has been observed experimentally in a variety of systems that the entropy change @xmath186 in eq . ( [ dt4 ] ) is proportional to @xmath187,@xcite suggesting that the terms of order @xmath188 and higher in the expansion ( [ s1 ] ) make no contribution . assuming for simplicity that @xmath100 , we recover from eq . ( [ s1 ] ) the empirical quadratic relation @xmath189 where the coefficient @xmath190 can be expressed through eqs . ( [ s1 ] ) and ( [ at1 ] ) , i.e. , @xmath191 according to the landau model , in ferroelectrics the partial derivative is equal to @xmath192 or @xmath193 at all temperatures , whereas in relaxors it approaches the value @xmath194 at high temperatures , but is in general a function of temperature . thus , in relaxors , @xmath190 is expected to be a function of temperature with @xmath195 . using the relations ( [ dt4 ] ) and ( [ ds ] ) , the ece temperature change @xmath13 can be written as @xmath196 a quadratic relation of the same form is predicted by eq . ( [ dt5 ] ) if the terms @xmath47 , @xmath197 , ... etc . are neglected . it is exactly true in the case @xmath55 discussed in section iii . clearly , the above relation is applicable only in the _ quadratic _ regime where the empirical eq . ( [ ds ] ) is valid . the parameter @xmath190 can be determined directly from the measured ece temperature change @xmath13 , with @xmath12 extracted from dielectric experiments . alternatively , and especially in ferroelectrics or relaxors at temperatures above the freezing temperature , we can obtain both @xmath12 and the curie constant @xmath68 from the dielectric data , and then deduce @xmath190 from eq . ( [ beta ] ) . we can then predict @xmath13 from eq . ( [ dt6 ] ) . it turns out , however , that the values of @xmath13 measured directly may differ from the ones deduced from dielectric data . for example , in the case of relaxor ferroelectric terpolymer p(vdf - trfe - cfe ) at @xmath198 k and @xmath199 mv / m , the ece measured directly is @xmath200 k , leading to @xmath201 v m c@xmath202k@xmath202,@xcite whereas the value deduced from dielectric data is @xmath203 k.@xcite a tentative explanation of this discrepancy is that even far above the freezing temperature @xmath204 k the system may still be nonergodic , so that the maxwell relations , based on equilibrium thermodynamics , are not applicable.@xcite as a consequence , the value of @xmath12 measured on a short time scale is smaller than its thermodynamic long - time limit . another possibility is that the empirical relation eq . ( [ ds ] ) is an effective quasi - linear relation observed in a broad range of large field values , while its derivation based on the landau expansion is by assumption restricted to small fields . we can obtain an estimate for the maximum ece temperature change @xmath205 by assuming that in a sufficiently strong electric field the polarization reaches its saturation value @xmath88 . relation ( [ ds ] ) is not expected to be valid in this _ saturation _ regime , and we must return to the general expression ( [ dt4 ] ) . obviously , the expansion ( [ s1 ] ) can not be applied due to convergence problems . on the other hand , it is well known @xcite that in the saturation regime the excess entropy of the dipolar subsystem tends to zero . therefore , according to eq . ( [ s ] ) , @xmath206 should approach the negative value of the configuration entropy @xmath42 , i.e. , @xmath207 . thus , in the saturation regime eq . ( [ dt4 ] ) leads to @xmath208 this relation gives the theoretical upper bound on ece in terms of just three physical quantities , @xmath209 , @xmath31 , and the configuration number @xmath36 . interestingly , this result does not depend explicitly on the dipole moment @xmath87 . moreover , it does not contain any information about possible phase transition occurring in the quadratic regime . for a given value of electric field @xmath49 , the borderline between the two regimes is expected to occur at some temperature @xmath210 where the dipolar energy becomes equal to the thermal fluctuation energy , i.e. , @xmath211 . for @xmath212 the system is in the quadratic , and for @xmath213 in the saturation regime . for the above terpolymer p(vdf - trfe - cfe ) , we can estimate the dipole moment @xmath87 from the relations @xmath214 c / m@xmath9 and @xmath215 . using the value of @xmath190 determined directly from @xmath13 , we have @xmath216 c m , and for @xmath217 mv / m we thus find @xmath218 k. the average volume associated with a pnr in relaxors , @xmath83 , in eq . ( [ dtm1 ] ) is not _ a priory _ known and depends on the total number of pnrs . we can estimate @xmath209 from the measured values of @xmath68 and @xmath219 using the relation @xmath220 , and rewrite eq . ( [ dtm1 ] ) in the form @xmath221 the value of @xmath219 can be extracted , for example , from hysteresis loops in the saturation regime , and @xmath68 from the asymptotic behavior of @xmath89 . as already discussed above , this value of @xmath68 may differ from the one derived from the experimental value of parameter @xmath222 , observed in ece experiments in the effective quadratic regime . in table i , the predicted values of @xmath205 for a set of selected systems are listed using the values of @xmath68 deduced from dielectric experiments . it should be noted that eqs . ( [ dtm1 ] ) and ( [ dtm2 ] ) provide just a theoretical upper bound for the ece in the systems listed . in practice , the limit of a fully polarized dipolar subsystem might not be accessible because dielectric breakdown could occur before complete saturation is reached . nonetheless , the predicted values of @xmath205 permit a comparison between various sytems and might be useful in the search for a giant ece . ( [ dtm1 ] ) indicates that a giant ece is expected to occur in systems with a small value of @xmath209 , or equivalently a large number @xmath40 of dipolar entities at fixed volume @xmath223 . for illustration , let us consider a specific example , i.e. , the ferroelectric copolymer p(vdf - trfe ) . this system consists of microscopic crystalline layers of polarized material embedded in an amorphous environment.@xcite electron irradiation breaks up the layered structure into smaller dipolar units , and turns the polymer into a relaxor . since the number of these new entities is now larger than the number of the original microcrystallites , and assuming the same value of saturation polarization , one expects a stronger ece to occur in the irradiated relaxor copolymer than in the original ferroelectric copolymer . this is corroborated by the experimental values of the coefficient @xmath190 for the irradiated relaxor copolymer p(vdf - trfe),@xcite @xmath224 v m c@xmath202k@xmath202 , and for the original ferroelectric copolymer in the paraelectric phase,@xcite @xmath225 v m c@xmath202k@xmath202 . in the case of inorganic relaxor 8/65/35 plzt thin films , the value of the coefficient @xmath190 is@xcite @xmath226 v m c@xmath202k@xmath202 , whereas for the ferroelectric pzt one finds@xcite @xmath227 v m c@xmath202k@xmath202 . again , @xmath190 is larger for the relaxor ; however , this comparison seems less conclusive since the difference in composition between the two systems is much greater than in the above organic case . relation ( [ dtm1 ] ) has been derived here in the framework of landau theory , however , its validity is essentially based on thermodynamic and statistical principles and is hence quite general . in particular , it is independent of any mesoscopic models such as the srbrf model . moreover , it can be easily generalized to magnetic systems , where @xmath13 represents the mce temperature change . the smallest physical limit for @xmath209 in ferroelectrics is the volume of the unit cell @xmath228 , yielding the ultimate upper bound on @xmath205 in eq . ( [ dtm1 ] ) for ferroelectric materials . to emphasize this point , let us multiply the numerator and denominator by the avogadro number @xmath229 to obtain @xmath230 here , @xmath231 is the gas constant and @xmath232 the molar specific heat . this simple result does not explicitly contain any information on the microscopic nature of the system , i.e. , whether it is dielectric , magnetic , etc . , since the corresponding electric or magnetic dipole moment does not appear in eq . ( [ dtm3 ] ) . therefore , we can regard the above relation as a universal law for the theoretical upper bound on the ece and/or mce in electrically or magnetically polarizable solids . of course , eq . ( [ dtm3 ] ) implies that the polarization or magnetization must have reached complete saturation . in the dielectric case , this requires large electric fields with the already mentioned possibility of dielectric breakdown . in the magnetic case , however , extremely large fields of the order of @xmath233 t may be necessary . ( [ dtm3 ] ) is valid in the saturation regime @xmath234 , typically at low temperatures , where @xmath235 is expected to be temperature dependent . in ferroelectrics , the formation of domains should be taken into account , and the entropy limit @xmath236 could only be reached in very high fields . in relaxors , there is an additional difficulty that the relaxation of pnrs at low temperatures is extremely slow and the above limit can only be reached after very long times . at high temperatures , @xmath235 normally approaches a certain limit . in metals one has @xmath237 according to the dulong - petit law , which is generally not valid for complex solids . for example , in perovskites , @xmath238.@xcite thus , the maximum ece in perovskites at high temperatures is of the order @xmath239 . this limit applies to a large group of perovskites , i.e. , paraelectrics , ferroelectrics , relaxors , etc . specifically , for @xmath37 ( see table i ) , we find @xmath240 . for example , if the system can support large electric fields such that @xmath241 k , this would lead to a giant ece temperature change @xmath242 k at room temperature . this exceeds roughly by a factor of @xmath243 the estimated values for perovskites in table i , which were derived from the physical values of the parameters @xmath219 , @xmath68 , and @xmath31 . in this paper , we studied the mechanism of ece in relaxor ferroelectrics ( or _ relaxors _ ) and in normal ferroelectrics ( referred to as _ ferroelectrics _ ) . starting from the widely accepted result for the ece temperature change @xmath134 , based on the thermodynamic maxwell relation , we derived an alternative expression which could directly be applied in theoretical model calculations . the results for @xmath134 have been obtained in two physical regimes of field - temperature variables @xmath127 . the first of these is the so - called _ quadratic _ regime , where @xmath13 is proportional to the square of the dielectric polarization @xmath12 , and the second the _ saturation _ regime , where @xmath12 is allowed to reach its maximum value @xmath219 . in the quadratic regime , the system can be described by the landau - type free energy model , in which the harmonic landau coefficient @xmath76 depends on the physical nature of the system , i.e. , the behavior of @xmath76 in relaxors differs from that of ferroelectrics . the anharmonic coefficients @xmath53 , which are common to both cases , then determine the critical behavior of the system in a field @xmath49 . the case @xmath61 does not show any pronounced anomalies . for @xmath2 , the polarization @xmath8 and hence @xmath13 rises steeply in a relaxor near the isolated critical point @xmath244 , whereas in a ferroelectric , the largest effect occurs near the first order phase transition in zero field , which is absent in the relaxor case . the ece efficiency @xmath181 shows a similar behavior in both cases , and at higher fields the same asymptotic values are found . of course , these conclusions only apply provided that all the remaining physical features , such as the number of equilibrium orientations @xmath36 of the elementary dipolar entities , etc . , are the same in both cases . experimentally , the ece entropy change in the _ quadratic _ regime is found to behave as @xmath245 , which is trivially reproduced by the landau model . the coefficient @xmath190 tends to have larger values in relaxors , leading to a stronger ece . in irradiated organic polymer relaxors , this can be explained by the larger number of polar nanoregions ( pnrs ) and thus a smaller average pnr volume @xmath209 . in the _ saturation _ regime , the entropy of the dipolar subsystem generally approaches the negative value of the configuration entropy , @xmath246 , and the maximum ece value @xmath205 thus crucially depends on the orientational degeneracy @xmath36 . this work was supported by the slovenian research agency through grants p1 - 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the electrocaloric effect ( ece ) in normal and relaxor ferroelectrics is investigated in the framework of a thermodynamic approach based on the maxwell relation and a landau - type free energy model . the static dielectric response of relaxors is described by the spherical random bond random field model , yielding the first landau coefficient @xmath0 , which differs from the usual expression for ferroelectrics . the fourth - order coefficient @xmath1 is treated as a phenomenological parameter , which is either positive or negative due to the anisotropy of the stress - mediated coupling between the polar nanoregions . when @xmath2 , the maximum ece in a relaxor is predicted near the critical point in the temperature - field phase diagram , whereas in a ferroelectric it occurs at the first order phase transition . the theoretical upper bound on the ece temperature change is estimated from the values of saturated polarization , effective curie constant , and specific heat of the material .

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Proceed to summarize the following text: m _ spitzer_/irac image of ngc 5907 . spectra from @xmath9 10 to 37 @xmath0mwere taken at the nucleus and at distances of 5 , 10 , and 15 kpc from the nucleus along the galaxy s major axis.,width=377 ] the physical conditions and excitation mechanisms of atomic and molecular gas in the outer disks of nearby spiral galaxies are only beginning to be explored . most of this gas is thought to be neutral atomic hydrogen ( ) , or cold ( t @xmath10 50 k ) molecular hydrogen ( @xmath2 ) . the presence of cold @xmath2 is usually inferred only by indirect means via observations of carbon monoxide ( co ) , and quantified by assuming an ( uncertain ) empirical conversion factor between the two molecules . direct detection of @xmath2 is preferable . however , since @xmath2 has no allowed dipole radiative transitions , it has to be heated above @xmath9 100 k to radiate significantly via quadrupole pure - rotational transitions in the mid - infrared ( mid - ir ) or through ro - vibrational transitions from even warmer gas emerging in the near - infrared . furthermore , the other direct observational window the detection through the absorption of uv radiation in the electronic lyman werner bands is challenging , and only under rare conditions has it been possible to detect the presence of cold @xmath2 in the galaxy through fuv absorption ( e.g. , * ? ? ? * ; * ? ? ? * ) . the _ infrared space observatory ( iso ) _ provided the first opportunity to directly observe warm extragalactic molecular hydrogen in nearby galaxies , unhampered by the atmosphere ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? more recently the _ spitzer space telescope _ @xcite has provided a wealth of new data on rotational @xmath2 emission lines in dozens of nearby galaxies , ranging from normal galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ) to ultraluminous infrared galaxies ( ulirgs ; * ? ? ? unusually strong intergroup @xmath2 emission associated with a large - scale ( @xmath9 30 kpc ) x - ray emitting shock has recently been found associated with the compact stephan s quintet galaxy group @xcite and the _ taffy galaxy _ bridge ( b. w. peterson et al . 2010 , in preparation ) . similarly large @xmath2 line fluxes have been found in 17 galaxies in a sample of 55 low - luminosity radio galaxies @xcite . in stephan s quintet and in the low - luminosity radio galaxies , very weak thermal continua are detected , suggesting shock excitation of @xmath2 , rather than excitation via photodissociation regions ( pdrs ) associated with star formation ( e.g. , * ? ? ? other sources of @xmath2 heating , for example cosmic ray heating , have also been suggested to explain the strong @xmath2 emission in the orion bar @xcite . it is therefore of interest to examine the strength of @xmath2 emission in the outer regions of nearby galaxies , where star formation and cosmic ray heating are much reduced , and where the dominant gas component is usually assumed to be neutral atomic hydrogen rather than molecular gas . by outer regions in this paper we mean the radii 10 kpc and beyond from the nucleus . _ iso _ observations of the nearby edge - on galaxy ngc 891 directly detected the abundant warm @xmath2 out to a distance of 11 kpc from the galaxy center @xcite , with warm @xmath2 mass surface densities of @xmath9 3000 m@xmath6 pc@xmath7 . this suggested a dominant contribution of molecular hydrogen to the mass - density of the disk , and perhaps that molecular hydrogen could contribute a significant part of the `` missing mass '' in this galaxy . intrigued by these early _ iso _ results , we pursued _ spitzer _ infrared spectrograph ( irs ; * ? ? ? * ) observations of two local , nearly edge - on galaxies , ngc 4565 and ngc 5907 , to explore the possibility of massive reservoirs of warm molecular gas far from the nuclei . these early - mission irs high resolution spectra cover infrared wavelengths from 10 @xmath0 m to 37 @xmath0 m , and target the 00 s(0 ) and 00 s(1 ) @xmath2 lines , which are known to contain the strongest emission from the mass in warm molecular gas in nearby galaxies ( e.g. , * ? ? ? the spectral range also covered several other mid - ir lines which assisted us in exploring the importance of star formation as an excitation mechanism in these regions . although the current observations are not extremely sensitive , they provide interesting constraints on the nature of @xmath2 emission in the outer disks of galaxies . to assist in our analysis , we also utilized _ spitzer _ infrared array camera ( irac ) 8 @xmath0 m images of ngc 4565 ( figure [ fig1 ] ) and ngc 5907 ( figure [ fig2 ] ) taken in _ program pid 3 ( p.i . giovanni fazio ; m. l. n. ashby 2009 , private communication ) . finally , we utilized archival _ spitzer _ multiband imaging photometer ( mips ) images of ngc 4565 and ngc 5907 at 24 , 70 , and 160 @xmath0 m ( figure [ fig3 ] shows these maps for ngc 4565 ) . ngc 4565 is a nearby ( we adopted a distance of 10 mpc for our observations ) , sb - type nearly edge - on ( inclination 88 ; @xcite @xcite ) large ( d@xmath11=158 ) disk galaxy with a nucleus classified as sy1.9 @xcite . a sharp dust lane delineates the disk plane of the galaxy , and there is significant obscuration caused by dust within the galactic plane . @xcite found that this galaxy has a nuclear molecular disk as well as a molecular gas ring at a distance of @xmath9 12 ( 36 kpc ) from the nucleus , and weaker extended molecular gas emission . the molecular gas ring has an associated dust ring , which is seen in the _ spitzer _ 8 @xmath0 m image shown in figure [ fig1 ] . the distribution is asymmetric along the disk plane , with substantially more emission coming from the northwestern side , and there is a strong , continuous warp in the emission starting at @xmath9 7 on both sides of the nucleus @xcite . @xcite showed that at a radius of @xmath9 10 kpc the interstellar medium ( ism ) transitions from being dominated by molecular gas to being dominated by atomic gas . ngc 5907 is a similarly large ( d@xmath11=126 ) , nearby ( adopted distance 11 mpc ) , almost edge - on ( inclination 87 ; @xcite @xcite ) , disk galaxy . co observations show a fast - rotating nuclear molecular disk with bar - like non - circular motions beyond the nucleus @xcite . the distribution shows a warp at both the southeastern and northwestern sides of the nucleus , starting at @xmath9 5 radius @xcite . cccccc + position & [ neii]12.81@xmath0 m & [ neiii]15.55@xmath0 m & [ siii]18.71@xmath0 m & [ siii]33.48@xmath0 m & [ siii]34.81@xmath0 m + + se 5 kpc ex & 187@xmath126 & 44@xmath125 & 65@xmath126 & 335@xmath1213 & 397@xmath1215 + se 5 kpc pt & 283@xmath129 & 73@xmath129 & 118@xmath1210 & 531@xmath1221 & 650@xmath1215 + se 10 kpc ex & 95@xmath125 & 28@xmath1212 & 48@xmath124 & 284@xmath126 & 234@xmath1213 + se 10 kpc pt & 144@xmath127 & 47@xmath1220 & 88@xmath128 & 386@xmath129 & 383@xmath1213 + se 15 kpc ex & 30@xmath127 & @xmath10 14 & 13@xmath123 & & + se 15 kpc pt & 45@xmath1211 & @xmath10 23 & 24@xmath125 & & + nw 5 kpc ex & 207@xmath126 & 40@xmath125 & 87@xmath123 & 411@xmath128 & 487@xmath127 + nw 5 kpc pt & 314@xmath129 & 67@xmath129 & 158@xmath126 & 652@xmath1213 & 799@xmath1211 + nw 10 kpc ex & 69@xmath123 & 41@xmath125 & 36@xmath123 & 199@xmath126 & 144@xmath125 + nw 10 kpc pt & 104@xmath125 & 68@xmath127 & 65@xmath125 & 316@xmath1210 & 236@xmath128 + nw 15 kpc ex & 28@xmath128 & 28@xmath128 & @xmath10 10 & 40@xmath125 & 69@xmath128 + nw 15 kpc pt & 43@xmath1212 & 47@xmath1213 & @xmath10 19 & 64@xmath128 & 113@xmath1213 + cccccccccc pt & 55@xmath129 & 322@xmath126 & 47@xmath123 & 354@xmath128 & 238@xmath1213 & 52@xmath127 & 431@xmath1211 & 726@xmath1213 & 867@xmath1221 + ex & 39@xmath126 & 213@xmath124 & 30@xmath122 & 212@xmath125 & 131@xmath127 & 37@xmath125 & 302@xmath128 & 457@xmath128 & 529@xmath1213 + lccc nucleus ex & 189@xmath125 & 170@xmath127 & 29@xmath126 + nucleus pt & 281@xmath128 & 298@xmath1214 & 43@xmath1210 + se 5 kpc ex & 176@xmath123 & 121@xmath124 & @xmath10 12 + se 5 kpc pt & 261@xmath125 & 213@xmath127 & @xmath10 17 + se 10 kpc ex & 88@xmath121 & 52 @xmath122 & @xmath10 17 + se 10 kpc pt & 130@xmath122 & 92@xmath124 & @xmath10 25 + se 15 kpc ex & & @xmath10 14 & @xmath10 16 + se 15 kpc pt & & @xmath10 25 & @xmath10 23 + nw 5 kpc ex & 221@xmath124 & 83@xmath123 & @xmath10 13 + nw 5 kpc pt & 327@xmath126 & 145@xmath125 & @xmath10 19 + nw 10 kpc ex & 89@xmath125 & 36@xmath123 & @xmath10 19 + nw 10 kpc pt & 132@xmath127 & 64@xmath125 & @xmath10 27 + nw 15 kpc ex & 36@xmath122 & @xmath10 13 & @xmath10 13 + nw 15 kpc pt & 53@xmath123 & @xmath10 22 & @xmath10 18 + we observed ngc 4565 and ngc 5907 with both high resolution modules of _ spitzer s _ irs instrument on 2005 january 10 and 2005 june 6 , respectively ( pid 3319 ; and ) . the short - high ( sh ) module covers wavelengths from 9.9 to 19.6 @xmath0 m and has a slit size of 47 @xmath13 113 , while the long - high ( lh ) module brackets the 18.7 to 37.2 @xmath0 m wavelength range with a slit size of 111 @xmath13 223 . we took one cycle of `` staring mode '' observations with a 120 s ramp time with the sh module and a 240 s ramp time with the lh module . the effective integration times were approximately doubled to @xmath9 240 s ( sh ) and @xmath9 480 s ( lh ) because each cycle takes two spectra , moving the target to positions 1/3 and 2/3 slit lengths away from the end of the slit along the slit long axis . we observed three positions along the galaxy major axes on both sides of the nuclei at distances of 5 , 10 , and 15 kpc from the nucleus . we also observed the nucleus of ngc 4565 . projections of the sh and lh slits on the 8 @xmath0 m irac galaxy images are shown in figures [ fig1 ] and [ fig2 ] . by overlaying our observed positions on visible light and maps we confirmed that the gaseous and stellar warps start beyond the outermost observed locations in these two galaxies . only in the northwest 15 kpc pointing in ngc 5907 could a very small amount of @xmath2 have been missed if it strictly follows the distribution . however , even in that position the majority of emission comes from along the major axis of the galaxy . cccccc se 5 kpc ex & 187@xmath125 & 33@xmath122 & 70@xmath124 & 386@xmath1221 & 553@xmath129 + se 5 kpc pt & 284@xmath127 & 55@xmath124 & 127@xmath127 & 613@xmath1233 & 906@xmath1214 + se 10 kpc ex & 48@xmath122 & 21@xmath123 & 30@xmath122 & 197@xmath1214 & 201@xmath127 + se 10 kpc pt & 72@xmath123 & 35@xmath125 & 54@xmath124 & 312@xmath1222 & 329@xmath1211 + se 15 kpc ex & @xmath10 12 & @xmath10 13 & @xmath10 13 & @xmath10 21 & @xmath10 31 + se 15 kpc pt & @xmath10 18 & @xmath10 22 & @xmath10 24 & @xmath10 34 & @xmath10 51 + nw 5 kpc ex & 168@xmath128 & 31@xmath125 & 50@xmath122 & 379@xmath1212 & 481@xmath124 + nw 5 kpc pt & 254@xmath1212 & 52@xmath128 & 90@xmath124 & 602@xmath1219 & 789@xmath126 + nw 10 kpc ex & 28@xmath122 & [email protected] & 13@xmath122 & 115@xmath1210 & 146@xmath124 + nw 10 kpc pt & 43@xmath123 & [email protected] & 24@xmath123 & 182@xmath1216 & 239@xmath127 + nw 15 kpc ex & @xmath10 14 & @xmath10 7 & @xmath10 18 & @xmath10 22 & @xmath10 32 + nw 15 kpc pt & @xmath10 21 & @xmath10 12 & @xmath10 33 & @xmath10 35 & @xmath10 53 + the background brightnesses ( due to ecliptic emission ) were @xmath9 29 mjy sr@xmath8 for ngc 4565 , and 1718 mjy sr@xmath8 for ngc 5907 . no separate background spectra were taken , since the recommended observing strategy during the first cycle of _ spitzer _ observations was still evolving and no clear recommendations existed at the time . this considerably complicated the removal of bad pixels from the spectra , discussed below . we did not use the `` peak - up '' option since our targets are extended , but the intrinsic irs pointing accuracy of @xmath9 1@xmath14@xmath14 was sufficient for our purposes . we used spectra that were processed through the standard _ spitzer _ irs pipeline ( version s13.2.0 ) . we first edited the basic calibrated data frames to remove bad or `` rogue '' pixels , using a custom - made software script that allows interactive removal of isolated bad pixels from these data . we then ran the spectra through the s17 version of the custom spectral extraction software spice provided by the _ spitzer _ science center , using the whole slit width extractions and initially the standard point source calibration for flux calibration . corrections were later made to line fluxes for both point and extended source calibration using the slit - loss factors provided in spice , and these two extreme limits bracket the true ( unknown ) distribution of gas in the slits . a comparison of the @xmath2 28 @xmath0 m line fluxes for the two nod positions in both galaxies shows differences of @xmath1010% , suggesting the gas distribution is relatively smooth on the size scale of the nods ( five and nine arcseconds , respectively , for sh and lh slits ) . extended emission is also suggested by structure along the slit in individual images . we consider the distribution of the emission - line gas to be close to flat and extended , and apply the corresponding calibration , but also quote , for reference , the very unlikely values derived by applying the point source calibration , when interpreting the properties of the observed galaxies . for the lh spectra of ngc 4565 , we encountered a low - level `` fringing '' effect not seen in the spectra of ngc 5907 . this effect would normally have been removed had we obtained dedicated `` off '' observations ( not obtained during our cycle-1 observations ) , and appears to be the result of incomplete `` jail bar '' removal in the pipeline ( an effect in which parts of the detector array show a patterning , which in this case was brighter than usual ) . to remove this effect , we decided to use the observations taken at 15 kpc southeast ( se ) of the nucleus of ngc 4565 as a reference , and subtracted this lh spectrum from all the others . this led to a significant improvement in the spectra . nonetheless , such a procedure runs the risk that there may have been faint emission at that reference position which would be removed from all other lh points ( this primarily affects the @xmath2 00 s(0 ) line which lies in the lh module . however , unlike the 15 kpc northwest ( nw ) point , which clearly shows 28 @xmath0 m s(0 ) emission even before we performed the subtraction , the se point appears devoid of emission as can be seen in figure [ fig4 ] . we feel confident , therefore , that the use of the se 15 kpc spectrum as a reference has not adversely affected our conclusions . although faint emission might have been present in this reference spectrum , based on measurements of the raw spectrum of ngc 4565 at the 15 kpc se position , we believe that an rms upper limit to such emission is 1.2 @xmath13 10@xmath15 w m@xmath7 hz@xmath8 at the position of the 00 s(0 ) line , corresponding to less than 10@xmath16 of the faintest @xmath2 emission detected at the 15 kpc nw point . in other words , the subtraction of the reference spectrum from the observations introduces a systematic error ( not to be confused with a random error ) which is estimated to be less than 10% of the faintest emission detected , a result that does not affect the conclusions of this paper . none of the sh spectra were affected by this instrumental effect . finally , we combined ( by averaging ) the spectra obtained at the two nod positions at each separate radial distance . the final spectra are shown in figures [ fig5 ] and [ fig6 ] . the 00 s(0 ) and 00 s(1 ) transitions of @xmath2 were detected at the 5 and 10 kpc distances from the nucleus in both galaxies . @xmath2 was also detected at the northwest 15 kpc location in ngc 4565 . on this side of the galaxy there is also substantially more emission @xcite . the 00 s(2 ) transition of @xmath2 was also detected in the seyfert nucleus of ngc 4565 . a variety of forbidden lines ( [ ] 12.81 @xmath0 m , [ ] 15.55 @xmath0 m , [ ] 18.71/33.48 @xmath0 m , and [ ] 34.82 @xmath0 m ) were detected in most locations of both galaxies . the continuum emission from the nucleus of ngc 4565 appears relatively flat , although it shows the broad pah emission feature around 17 @xmath0 m . because the irs apertures cover several hundred parsecs , most of this pah emission is likely emitted by the disk of this galaxy . in addition to the forbidden lines detected elsewhere in this galaxy , [ ] 10.51 @xmath0 m , [ ] 14.32/24.31 @xmath0 m , and [ ] 25.89 @xmath0 m were detected in the nucleus . the indicator lines of active galactic nuclei ( agns ) , [ ] 14.32 @xmath0 m and [ ] 24.31 @xmath0 m ( e.g. , * ? ? ? * ) , were both detected at a signal - to - noise ratio of @xmath910 . we extracted line fluxes by fitting gaussians to the lines using the smart software package @xcite . we fitted the broad aromatic features ( indicated in figures [ fig5 ] and [ fig6 ] ) with lorentzian profiles ( cf . the lorentzian method used by * ? ? ? * ) . the extracted fluxes are given in tables [ table1a ] , [ table1b ] , [ table1c ] , [ table2a ] , and [ table2b ] , where `` pt '' indicates point source calibrated spectra and `` ex '' indicates fluxes corresponding to a flat , infinitely extended distribution , as explained in detail in section 2 . the uncertainties were estimated by taking into account the quality of the profile fit . generally high signal - to - noise ratios ( @xmath17 10 ) were obtained for most of the lines except in the outer regions of the disks . upper limits were estimated as 4 @xmath18 , where @xmath19 is the rms noise in the region of the expected line and @xmath20@xmath21 is the width of an unresolved line at the corresponding wavelength ( which corresponds essentially to the width of the bandpass for the high resolution modules , @xmath21/600 ) . to assist in diagnosing the gas excitation conditions , we also estimated the flux densities in the irac 8 @xmath0 m images of ngc 4565 and ngc 5907 , under the areas covered by the irs slits in our observations . we used the same irac 8 @xmath0 m filter width as @xcite to convert the flux densities from jy into fluxes in w m@xmath7 , but we have not attempted to subtract the stellar emission from the irac image as it is generally only a few per cent of the total emission at 8 @xmath0 m . cccc se 5 kpc ex & 180@xmath127 & 92@xmath125 & @xmath10 16 + se 5 kpc pt & 267@xmath1210 & 162@xmath128 & @xmath10 24 + se 10 kpc ex & 51@xmath122 & 18@xmath122 & @xmath10 12 + se 10 kpc pt & 75@xmath123 & 32@xmath123 & @xmath10 18 + se 15 kpc ex & @xmath10 17 & @xmath10 9 & @xmath10 9 + se 15 kpc pt & @xmath10 25 & @xmath10 16 & @xmath10 14 + nw 5 kpc ex & 188@xmath123 & 104@xmath123 & @xmath10 13 + nw 5 kpc pt & 278@xmath124 & 182@xmath126 & @xmath10 20 + nw 10 kpc ex & 51@xmath123 & 15@xmath122 & @xmath10 15 + nw 10 kpc pt & 76@xmath124 & 27@xmath123 & @xmath10 22 + nw 15 kpc ex & @xmath10 18 & @xmath10 13 & @xmath10 16 + nw 15 kpc pt & @xmath10 26 & @xmath10 23 & @xmath10 24 + to investigate the gas excitation conditions we compared the strengths of various emission lines by forming line ratios . when comparing line ratios formed from lines taken with two different modules ( lh and sh ) , we scaled the fluxes by a factor of 4.66 , the ratio of the areas of the two module apertures . we also applied the extended source calibration correction to the line fluxes before taking the ratio to be consistent with this approach . we show the line ratios in figures [ fig7 ] and [ fig8 ] . in ngc 4565 the [ ] 33.48 @xmath0m/ [ ] 18.71 @xmath0 m ratios are close to 1 , typical for extranuclear regions seen in the sings sample of nearby galaxies @xcite , except at 10 kpc se where the ratio drops below 0.4 . this would imply a drop in the electron density by factors of a few hundreds @xcite . this ratio is slightly higher , between 1 and 2 , in ngc 5907 , covering very well the region in which most of the extranuclear areas studied by @xcite fall . the [ ] 34.81 @xmath0m/ [ ] 33.48 @xmath0 m ratio in ngc 4565 has a surprisingly low value of just above 1 at the nucleus , which is at the lower end of values seen in the nuclei of agn galaxies in the sample of @xcite . lcccc nuc & 164 ( 122 ) & 2.6 ( 2.0 ) & 4.1 ( 59 ) & 6.6 ( 95 ) + se 5 & 149 ( 112 ) & 2.5 ( 1.9 ) & 4.8 ( 74 ) & 7.7 ( 119 ) + se 10 & 146 ( 110 ) & 2.4 ( 1.8 ) & 2.4 ( 40 ) & 3.8 ( 64 ) + nw 5 & 135 ( 106 ) & 2.3 ( 1.7 ) & 7.5 ( 113 ) & 12 . ( 182 ) + nw 10 & 132 ( 104 ) & 2.3 ( 1.7 ) & 3.3 ( 51 ) & 5.2 ( 82 ) + nw 15 & @xmath10136 ( 103 ) & 2.3 ( 1.6 ) & 1.1 ( 21 ) & 1.8 ( 34 ) + lcccc se 5 & 139 ( 107 ) & 2.4 ( 1.8 ) & 5.8 ( 90 ) & 9.3 ( 145 ) + se 10 & 129 ( 101 ) & 2.2 ( 1.6 ) & 2.0 ( 32 ) & 3.3 ( 51 ) + nw 5 & 142 ( 109 ) & 2.4 ( 1.8 ) & 5.7 ( 89 ) & 9.2 ( 144 ) + nw 10 & 124 ( 100 ) & 2.1 ( 1.6 ) & 2.3 ( 33 ) & 3.6 ( 53 ) + the [ ] 15.55 @xmath0m/ [ ] 12.81 @xmath0 m ratio behaves as expected in both galaxies . it is higher in low - metallicity regions ( towards larger radii in both galaxies ) , and lower towards the center in regions that are expected to have a higher metallicity . it achieves a high value in the nucleus of ngc 4565 , consistent with what was seen by @xcite in the sings sample , presumably due to the higher excitation conditions near an agn . the @xmath2 s(0 ) to @xmath2 s(1 ) ratio hovers around 0.5 in both galaxies and is seen to increase towards the outer disk in both galaxies on both sides of the disk . this may primarily be an effect of the temperature , and it will be discussed in more detail in section [ excisection ] . we also show the @xmath2 s(0 ) to [ ] 33.48 @xmath0 m ratio . [ ] 33.48 @xmath0 m is mostly excited by star formation , and thus this ratio can be used as a rough indicator of the significance of star formation induced excitation of the @xmath2 molecule . we see that the ratio stays fairly constant at around 0.5 in both galaxies , but goes up at the 15 kpc nw point in ngc 4565 . this is consistent with the ionization level of molecules dropping in the outermost disk , as discussed in section [ excitdiscussion ] below . figure [ fig9 ] shows the ratios of the fluxes in the 11.3 and 7.7 @xmath0 m pah features versus the distance from the nucleus on the nw side of ngc 4565 . the irac 8 @xmath0 m fluxes measured in the irac image within the sh aperture and at the same spatial locations as the spectra were used as a proxy for the 7.7 @xmath0 m pah flux . the ratio is increasing towards 15 kpc nw , which most likely implies that the ism is becoming less ionized towards the outer disk , as the 7.7 @xmath0 m pah feature consists of more ionized dust material than the 11.3 @xmath0 m pah feature ( e.g. , * ? ? ? * ) . we constructed excitation diagrams ( figures [ fig10 ] and [ fig11 ] ) from the @xmath2data in order to place constraints on the molecular gas properties . these diagrams plot the column density ( n@xmath22 ) of @xmath2 in the upper level of each transition , normalized by its statistical weight , versus the upper level energy e@xmath22 ( e.g. , * ? ? ? * ) , which we derived from the measured fluxes assuming local thermodynamic equilibrium for each position observed . both extended and point source flux distributions are shown for the detected lines . the extended source results include a wavelength - dependent slit - loss correction which makes the extended flux calibration differ from the point source flux calibration typically by a factor of 1.5 , and has a further geometrical correction of 4.66 for the different areas of the sh and lh slits . the grey area between the two limiting cases ( point and extended ) is the parameter space that most likely encompasses the actual case . however , we stress that the point source assumption is very unrealistic , as discussed in section [ observ ] and given the small slit aperture and the thickness of the disks . as we will see , this assumption also leads to unlikely low gas temperatures close to 100 k , and therefore high implied @xmath2 gas surface densities . for this reason we prefer to consider the extended source limit to be much closer to the actual situation , but the point source provides a useful ( although unrealistic ) boundary . the uncertainty in the @xmath2 properties is governed largely by this uncertainty in the slit loss corrections rather than the formal errors , which are quite small because the lines were all detected with quite high signal to noise ratios ( snr ; they vary from 10 to 50 in most cases ) . the solid lines indicate the best fits to the s(0 ) , s(1 ) , and s(2 ) data points assuming @xmath2 in thermal equilibrium with a single - temperature component ( we will discuss the consequences of relaxing this assumption below ) . the fits also assume a thermal equilibrium ratio for the ortho - to - para species ( o / p ) , as is reasonable if the density of the @xmath2 is above the critical density ( which , for the low j transitions , is typically @xmath9 100 mol @xmath23 ) , a condition probably satisfied in most cases . for temperatures less than @xmath9 300 k , this leads to o / p ratios @xmath10 3 . for example , at t = 115120 k , o / p = 2 for thermal equilibrium . however , it is far from clear that thermal equilibrium is appropriate in all cases . for example , if the excitation mechanism were a shock , then the passage of the shock could leave the @xmath2 molecules in a state where they do not have enough time to equilibrate . this is another source of uncertainty . for example , if o / p was 3 instead of 2 ( a case where the gas has not had time to come into thermal equilibrium ) , then this would change the calculated temperature from t = 120 k to 113 k with a corresponding increase in the total @xmath2 mass surface density . this uncertainty in the o / p ratio is comparable with the uncertainty in the clumpiness in the @xmath2 distribution which leads to the broad range of possible temperatures as shown in figures [ fig10 ] and [ fig11 ] . the single - temperature fits to these data are shown in each panel of the figures . outside the nucleus only an upper limit is available for the s(2 ) line . therefore , fitting more than one thermal component is not statistically justifiable the fits are the formal solutions . we note that the assumption that the source of @xmath2 is a point source always yields very low @xmath2 temperatures , bordering on becoming physically unreasonable . thus we believe that the warmer temperatures implied by the extended source calibration are more physically reasonable for the case of these edge - on galaxies . one exception is the nucleus of ngc 4565 , where a point - source assumption may be reasonable , as it contains a seyfert nucleus . based on these assumptions , tables [ excipars ] and [ excipars2 ] summarize the derived @xmath2 physical parameters for ngc 4565 and ngc 5907 , respectively : temperature ( k ) , equilibrium ortho / para ratio , column density of @xmath2 ( mol cm@xmath7 ) , and the mass surface density of @xmath2 ( m@xmath6 pc@xmath7 ) . the temperatures and mass surface densities for ngc 4565 and ngc 5907 are also shown in figures [ fig12 ] and [ fig13 ] . the gas is colder in the outer disk where the mass surface density is also lower , creating the apparent impression of a correlation between temperature and mass surface density . the derived ( extended source ) mass surface densities are more than 100 times smaller than those found in ngc 891 by @xcite . see section [ darkmatter ] for more discussion about the implication of the implied warm @xmath2 mass surface densities . if one adopts the extended source assumption , and excludes the nucleus of ngc 4565 which is significantly warmer , there is no obvious change in the fitted temperature with radius within the uncertainty from @xmath1 = 5 kpc to @xmath1 = 10 kpc on both sides of this galaxy ( t = 146149 k on the southeastern side and t = 132134 k on the northwestern side ) . even at @xmath1 = 15 kpc on the northwestern side , the upper limit to the s(1 ) flux provides a temperature limit which is at least consistent with a flat temperature distribution . the situation is different in ngc 5907 , where the outermost 10 kpc points seem more than 10 k cooler than those measured at 5 kpc . for this galaxy the radial temperature profile is also symmetric , unlike that in ngc 4565 . in the previous discussion we have made an assumption that a single - temperature model is reasonable . this is clearly not the case for the nucleus of ngc 4565 , where the 00 s(2 ) line was detected . the first panel of figure [ fig10 ] shows that the single - temperature fits do not pass through the s(2 ) point . indeed , in general , extragalactic sources almost always show a range of allowable temperatures and often a multiple - component fit is required . one consequence of fitting a multiple - temperature model is that the warmer component softens the slope of the fit in the excitation diagram . this means that the lower temperature component becomes even cooler , once a warm component is subtracted . to illustrate this we have fitted a two - component model to the nucleus of ngc 4565 and derived the following temperatures and column densities . instead of a single ( in this case point - like ) nuclear source with t = 122@xmath124 k and a column density n@xmath25 of 5.9 @xmath13 10@xmath26 mol cm@xmath7 , we obtain t(1 ) = 115@xmath123 k , n(1)@xmath25 = 7.4 @xmath13 10@xmath26 mol cm@xmath7 , and t(2 ) = 450550 k , n(2)@xmath25 = 34 @xmath13 10@xmath27 mol cm@xmath7 . the warmer component is less constrained because the error bar on the s(2 ) line is larger than that of the s(0 ) and s(1 ) lines . note that the effect in this case of relaxing the single - temperature model is to increase the cold component column density by 25% . the warmer component adds a negligible amount to the final column density . it is very likely that the nucleus of ngc 4565 is different from the disk because it contains a seyfert component which may contribute additional heating to the @xmath2emission . this is reflected in the generally higher single - temperature fits shown in figure [ fig10 ] . it may seem odd that the two - component fit gives a temperature for the cold component in the nuclear pointing that is colder than elsewhere in the disk . however , the thermodynamics of the @xmath2 molecule is likely to be very complex , involving a multi - phase medium with unknown heating and cooling conditions . it is also possible that the density distribution of the clouds near the nucleus is very different from elsewhere in the disk , and there may be more very dense cold clouds near the nucleus . with the spatial resolution afforded by the irs , we can not resolve this question . furthermore , magnetohydrodynamic shocks may be driven into a clumpy medium near the nucleus , which will lead to a range of temperatures . it should also be noted that it is impossible to estimate what the separate contributions of the disk and the nucleus are to the observed line fluxes in the nuclear pointings . thus , depending on the strength of a second or third component , the temperature of the coolest component is always lowered relative to a single - temperature fit - case . because we used only single - temperature fits , it is possible that we underestimated the total @xmath2 column density if warmer components were present , because a cooler @xmath2 temperature implies a larger total @xmath2 column density . since we have , in general , no information about a warmer component , we can not do more than fit a single - temperature component and accept a degree of uncertainty in the final @xmath2 column densities and masses . the line fluxes measured in the seyfert nucleus of ngc 4565 are listed in table [ table1b ] . the continuum appears relatively flat , although it shows a signature of the broad pah emission feature around 17 @xmath0 m . since the apertures are relatively large ( covering several hundreds of pc ) , a lot of this pah emission is likely to come from the disk of ngc 4565 . the 11.3 @xmath0 m and 12.9 @xmath0 m pah features are also strong , but weaker than in the spectra taken at 5 kpc from the nucleus . the detected emission lines come from @xmath2 , o , ne , s , and si . the agn indicator lines of [ ] 14.32 @xmath0 m and [ ] 24.31 @xmath0 m ( e.g. , * ? ? ? * ) are both detected at s / n @xmath9 10 . the @xmath2 s(0)/@xmath2 s(1 ) ratio reaches its minimum at the nuclear position ( see figure [ fig7 ] ) , implying the highest gas temperatures , as can also be seen in figure [ fig10 ] . the [ ] 15.55 @xmath0m/ [ ] 12.81 @xmath0 m ratio reaches a peak in the nucleus . the value of @xmath9 1 for this ratio indicates a moderate nuclear starburst @xcite . it is also consistent with the classification of ngc 4565 as a sy1.9 galaxy @xcite . in both galaxies , ngc 4565 and ngc 5907 , we see that the emission line intensities and the derived mass surface densities of @xmath2 emission ( as well as the intensity of the forbidden lines ) decrease with increasing radius , while the temperature decreases only slightly . also , the 20-cm radio continuum , for ngc 4565 shown in figure [ fig14 ] @xcite , decreases strongly towards the 15 kpc radius ( which in ngc 4565 is actually outside the detected radio continuum emission on the nw side ) . we calculated the ratio of the @xmath2 luminosity surface density over the total infrared ( tir ) emission luminosity surface density ( figure [ fig15 ] ) . tir was calculated as in equation ( 9 ) of @xcite over the lh slit area , measuring surface brightness values in the 8 , 24 , 70 , and 160 @xmath0 m spitzer irac and mips maps that were all smoothed to the resolution of the 160 @xmath0 m map , at the positions of the observed irs slits . this ratio is relatively constant with the radius at about 0.2%0.4% . this value is somewhat higher than the 0.05%0.1% typically seen in the sings sample , but since we could not match the resolution and aperture of the broad - band images , from which tir was estimated , to the single slit observations taken in the staring mode , such a bias is expected . when plotting the @xmath2 emission power over the irac 8 @xmath0 m power ( figure [ fig16 ] ) we see that the points in ngc 4565 and ngc 5907 lie generally above the star formation region points in @xcite . specifically , we see an increase in the ratio towards the outer 15 kpc nw point in ngc 4565 . we also see no change in the @xmath2 s(0)/11.3 @xmath0 m pah ratio ( figure [ fig17 ] ) on the northwestern side of the disk of ngc 4565 , but we see an increase in the 11.3 @xmath0m/7.7 @xmath0 m pah feature ratio ( figure [ fig9 ] ) from 10 kpc to 15 kpc . one explanation is that the pahs become more neutral in the lower uv excitation environment of the outer disk at 15 kpc nw in ngc 4565 . the possible change in pah excitation from ionized to neutral changes the relative strengths of the 11.3 @xmath0 m with respect to the 7.7 @xmath0 m pahs because the 11.3 @xmath0 m pah feature becomes more dominant as the pahs become more neutral . this might naturally explain why the 15 kpc nw point in figure [ fig15 ] stands out . it is not due to the @xmath2 emission becoming relatively stronger at 15 kpc , but due to the 7.7 @xmath0 m pah feature becoming weaker . we also measured the 24 @xmath0 m flux densities in the areas covered by the irs slits in ngc 4565 , and noticed that the 24 @xmath0 m flux density decreases with radius . since the 24 @xmath0 m emission is a relatively good proxy of the star formation intensity ( e.g. , * ? ? ? * ) , this implies that the uv flux intensity is decreasing with radius , therefore producing a more neutral ism at larger radii , consistent with our results derived from the pah flux ratio . @xcite have shown that there is an apparently strong coupling between the surface densities of neutral hydrogen and dark matter in spiral galaxies , with a significant and pronounced peak in @xmath28/@xmath29 @xmath9 9 . to see whether this holds in ngc 5907 , we used the multicomponent dynamical model of ngc 5907 by @xcite . for ease of calculation we approximated the dark matter distribution by a singular isothermal sphere . at a radial distance of 10 kpc from the nucleus we calculate a @xmath30 = 188 m@xmath6 pc@xmath7 . @xmath31 @xmath9 17.4 m@xmath6 pc@xmath7 @xcite and the ratio of the two is 10.8 . including the warm @xmath2 gas only reduces the dark matter to gas ratio to 9 . thus ngc 5907 appears to follow the relationship found by @xcite . at a radial distance of 15 kpc from the nucleus the warm @xmath2 is undetected . assuming , as suggested by the data , an extended , smooth emission distribution , @xmath32 @xmath33 3 m@xmath6 pc@xmath7 . using our approximation to the barnaby thronson model , @xmath30/@xmath32 @xmath34 42 . at 15 kpc , @xmath35 = 10.3 m@xmath6 pc@xmath7 and @xmath30/@xmath35 = 12.2 . it is therefore clear that the mass of the ism in ngc 5907 , including the mass of the warm molecular gas , is too small by more than an order of magnitude to account for the requisite dark matter . unfortunately there is no model of the mass distribution and dynamics of ngc 4565 similar to that of @xcite for ngc 5907 . the neutral atomic hydrogen properties were studied by @xcite , and the cold molecular gas properties by @xcite while visible light surface photometry of ngc 4565 was performed by @xcite . we assumed that the stars and the ism are confined to a thin disk and the dark halo can be described , again , by a singular isothermal sphere . the rotation curve is given by @xcite . @xcite showed that the optical disk is truncated at a radius of 24.9 kpc , comparable to where @xcite sees a warp . using a @xmath36-band luminosity of the old disk of 1.4@xmath1310@xmath37 l@xmath6 and a median value of 7.5 ( corrected for hubble constant h@xmath38 = 75 km s@xmath8 mpc@xmath8 ) for the mass to luminosity ratio of sab sb galaxies from @xcite , the mass of the luminous stellar disk is 10.5@xmath1310@xmath37 m@xmath6 . this may be an overestimate as some fraction of the mass quoted by @xcite is dark . the total mass of the neutral atomic hydrogen is 5.96@xmath1310@xmath39 m@xmath6 @xcite and that of the cold molecular hydrogen 2.4@xmath1310@xmath39 m@xmath6 @xcite . we assumed that the ism and the stars are confined to a thin disk and all components are truncated at 25 kpc . @xcite suggests that the velocity at the truncation radius is @xmath40 where @xmath41 is 25 kpc for ngc 4565 and @xmath42 is the total mass . beyond @xmath41=25 kpc the velocity is assumed to decline in a keplerian fashion . at a radial distance of 35 kpc , the observed circular velocity of the galaxy is about 214 km s@xmath8 @xcite . from the mass and velocity components of our model we find that the halo contributes a velocity of 150 km s@xmath8 to the system . we reflect these values back to a radius of 15 kpc at which we observed the most distant emission from warm @xmath2 , and we recalculate the mass surface densities . assuming a singular isothermal sphere for the dark matter , the mass surface density of dark matter is 86 m@xmath6 pc@xmath7 , the ratio of the mass surface densities of dark matter and warm @xmath2 ( with the much likelier smoothly distributed extended source emission calibration ) is @xmath34 86/1.8 = 48 , and that of dark matter to all of the ism components ( neutral atomic hydrogen , warm molecular hydrogen , and cold molecular hydrogen ) is @xmath34 15 . from our analysis it is clear that the mass surface densities of the warm molecular gas can not produce the observed rotation velocities at large radii in ngc 4565 and ngc 5907 . the `` missing mass '' in these two galaxies can not be accounted for by warm @xmath2 gas . the molecular gas at the 15 kpc nw point in the outer disk of ngc 4565 , as probed by the @xmath2 rotational lines , has roughly the same temperature and the same ratio of the @xmath2 to far - ir power as in the inner disk . however , the star formation rate at the 15 kpc nw point , traced by the mid - ir emission , has substantially decreased , compared to the inner disk . in other words , although the intensity of the exciting radiation field has been reduced ( as seen also in the change in the ratio of the ionized to neutral pah molecules and in a reduction in the tir intensity when comparing the 10 kpc nw and 15 kpc nw points ) , the molecular gas is heated to a similar temperature throughout the disk . therefore , something other than star formation may be heating the gas at the 15 kpc nw point . cosmic ray ( cr ) heating of the @xmath2 does not appear viable at the 15 kpc nw point because we see a dramatic decrease in the strength of the synchrotron radio continuum emission , which is a tracer of cosmic rays accelerated in the magnetic field of the galaxy , between the 10 and 15 kpc nw points , as shown in figure [ fig14 ] . the 15 kpc nw point lies outside the detectable signal in the radio continuum maps of @xcite . the upper limit of the 20-cm radio continuum ( 150 @xmath0jy / beam ; 3@xmath19 ) at the 15 kpc nw point suggests a difference of a factor of @xmath17 64 in the 20-cm radio continuum flux density between the 10 and 15 kpc points . this change is not reflected in the decrease in the @xmath2 line luminosity which is only a factor of 7 . however , despite the lack of detected radio continuum at the 15 kpc nw point , we can not completely rule out cr excitation . if we assume an equipartition of energy between crs in the disk and the magnetic energy density , the upper limit to the radio continuum flux density corresponds to an upper limit for the equipartition magnetic field strength of @xmath44 @xmath33 1 @xmath0 g , following the assumptions discussed in @xcite . this corresponds to a magnetic energy density ( and a comparable cr energy density ) of @xmath9 9.6 @xmath13 10@xmath45 ergs @xmath23 . for a canonical synchrotron lifetime in the mid - plane of 10@xmath46 yrs , the crs could potentially provide @xmath47 @xmath33 4.7 @xmath13 10@xmath48 w / kpc@xmath49 of power if such a population of crs existed below the detection limit of the radio continuum observations . interestingly , this is only a factor of two lower than the @xmath2line luminosity in the s(0 ) line at 15 kpc nw ( 7 @xmath13 10@xmath48 w / kpc@xmath49 ) , and therefore cr heating , although unlikely ( it would require very rapid deposition timescales of much less than 10@xmath46 yrs and high heating efficiency ) , can not be completely ruled out . we note that the equivalent @xmath50 and @xmath51 for the 10 kpc nw point in ngc 4565 are 3.3 @xmath0 g and 5.0 @xmath13 10@xmath52 ergs @xmath23 , respectively = 10 kpc and 5 kpc at @xmath1 = 15 kpc . ] , and the ratio of @xmath51/@xmath53(@xmath2 ) @xmath9 0.5 at that point . this suggests that within the radio continuum emitting disk of ngc 4565 , trapped cosmic rays can , in principle , provide energy to heat the @xmath2 in the disk ( again high heating efficiency would be needed ) . in those same regions , star formation , through pdr heating , provides a more likely channel for heating the @xmath2 gas ( see * ? ? ? the feasibility of cosmic ray heating at @xmath1 = 15 kpc can also be estimated using an independent ionization argument @xcite . if we assume that the cosmic rays heat the gas through partial ionization of the @xmath2 to h@xmath54@xmath55 , then the rate of ionization through cosmic ray heating must balance the rate of cooling of the @xmath2 per molecule . for a column density @xmath56 @xmath9 1.1 @xmath13 10@xmath57 mol cm@xmath7 and the observed luminosity in the s(0 ) line , we estimate the @xmath2 cooling per molecule to be 6.7 @xmath13 10@xmath58 w mol@xmath8 . @xcite estimate the cr heating rate per molecule to be 8 @xmath13 10@xmath59 @xmath60@xmath61 w , where @xmath60@xmath61 is the @xmath2ionization rate . under these assumptions , for cr heating to balance the @xmath2 cooling would require an ionization rate of @xmath9 10@xmath62 s@xmath8 . this value is comparable to that measured in the galactic center ( see * ? ? ? * ) , but is unlikely to be realized in the outer disk of ngc 4565 . this again suggests that cr excitation is an unlikely source of heating for the s(0 ) line at @xmath1 = 15 kpc unless conditions there are very unusual . the two remaining options for @xmath2 excitation in the outer disk are heating within extended pdr regions , or shock heating . we have already shown that the ratio of the @xmath2 power to the far - ir emission power is consistent with pdr heating ( in comparison with the models of * ? ? ? * ) , but it is not clear if this process works in the outer disk . indeed , it is very likely ( and observations with the _ herschel space observatory _ will help to resolve this issue ) that a large component of the tir flux we see in the outer disk of ngc 4565 comes from cirrus clouds heated by the general radiation field of the outer disk , and not from young stars . thus one is left with the puzzling result that the @xmath2 excitation remains constant to within a factor of two in the outer disk which does not contain a high concentration of young stars . widely distributed pdr regions around a smoothly distributed set of faint young stars may be responsible for the excitation , but this can not be demonstrated with our observations . another possible way of heating the @xmath2 is by shocks , perhaps through a recent passage of a disturbance through the outer disk of ngc 4565 . a recent model of how @xmath2 can be excited in a powerful shock propagating through a multi - phase medium in stephan s quintet has been presented by @xcite . however , this model was tuned to the specific problem of how to generate large amounts of power in the @xmath2lines in a 1000 km sec@xmath8 shock moving through a clumpy medium . to explain the emission in the outer regions of ngc 4565 via shocks would require considerably less energy input , but there is no obvious source of energy to drive the shocks . curiously , recent _ spitzer _ observations of a high - latitude cirrus cloud within the galaxy @xcite have revealed unusually strong @xmath2emission from regions that are clearly not associated with pdrs . shock heating is one possible explanation . these observations suggest that the excitation of @xmath2 in galaxy disks is not yet well understood . we have examined the excitation of gas , dust , and pahs and the physical conditions of the ism in two nearby normal edge - on disk galaxies , ngc 4565 and ngc 5907 , out to 15 kpc from the nucleus of each galaxy . our most important conclusions can be summarized as follows . 1 . we have detected the rotational 17 @xmath0 m s(0 ) and 28 @xmath0 m s(1 ) @xmath2 line transitions at 5 and 10 kpc , and most interestingly , the s(0 ) @xmath2 line at 15 kpc nw from the nucleus of ngc 4565 . we have discovered that in these two edge - on galaxies , ngc 4565 and ngc 5907 , the warm molecular gas temperature ( although uncertain ) and the ratio of the @xmath2 line luminosity surface density to the total infrared luminosity surface density are rather flat with radius . however , the active star formation rate , as measured by , e.g. , the 24 @xmath0 m emission , falls rapidly with radius . this result is potentially inconsistent with excitation of the @xmath2 emission by photodissociation regions in the outer disks of these galaxies . alternatives to the @xmath2 excitation in the outer disk are cosmic ray heating and shocks . based on the midplane radio continuum emission intensities , excitation by cosmic rays and photodissociation regions are both viable in the inner disk . however , in the outer disk the non - detection of radio continuum implies that cosmic rays are less important there . therefore , extended photodissociation regions or shocks can excite the emission at the outermost disk , as seen in ngc 4565 at the 15 kpc nw point . we see an increase of the 11.3 @xmath0m/7.7 @xmath0 m pah feature strength ratio ( where we used the irac 8 @xmath0 m band to be a proxy of the 7.7 @xmath0 m emission ) at the 15 kpc nw position in ngc 4565 . we also see that the summed @xmath2 line intensity over the 8 @xmath0memission intensity ratio increases at the same position . our interpretation is that the @xmath2 s(0 ) 28 @xmath0 m emission at the 15 kpc nw position may still be excited by ( weaker ) emission from photodissociation regions , coming from a more neutral medium at this large distance from the nucleus , as the strength of the 7.7 @xmath0 m pah feature , which traces more highly ionized dust , decreases with respect to the strength of the 11.3 @xmath0 m pah feature , which traces more neutral dust . the observations strongly suggest that the warm molecular gas is smoothly distributed . assuming such an extended distribution , the detected mass surface densities of warm molecular hydrogen are very low at large radii in both galaxies . it is very unlikely that this component of the ism contributes at any significant level to the `` missing mass '' in the outer regions of these two edge - on disk galaxies . the seyfert 1.9 nucleus in ngc 4565 revealed [ ] 14.32 @xmath0 m , [ ] 24.31 @xmath0 m , [ ] 10.51 @xmath0 m , and [ ] 25.89 @xmath0mlines , as well as the 12.28 @xmath0 m @xmath2 s(2 ) line . the higher excitation forbidden lines are expected to be seen in seyfert nuclei . we are grateful to tom jarrett at ipac for helping us to correct the rogue pixels with his custom - made iraf script . we thank the anonymous referee for very helpful and detailed comments . we acknowledge stimulating discussions with eric murphy on the mid - ir and far - ir properties of nearby galaxies . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . smart was developed by the irs team at cornell university and is available through the _ spitzer _ science center at caltech . the irs was a collaborative venture between cornell university and ball aerospace corporation funded by nasa through the jet propulsion laboratory and ames research center . this work is based on observations made with the _ spitzer space telescope _ , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . support for this work was provided by nasa through an award issued by jpl / caltech .

we have observed warm molecular hydrogen in two nearby edge - on disk galaxies , ngc 4565 and ngc 5907 , using the _ spitzer _ high - resolution infrared spectrograph . the 00 s(0 ) 28.2 @xmath0 m and 00 s(1 ) 17.0 @xmath0 m pure rotational lines were detected out to 10 kpc from the center of each galaxy on both sides of the major axis , and in ngc 4565 the s(0 ) line was detected at @xmath1 = 15 kpc on one side . this location is beyond the transition zone where diffuse neutral atomic hydrogen starts to dominate over cold molecular gas , and marks a transition from a disk dominated by high surface - brightness far - ir emission to that of a more quiescent disk . it also lies beyond a steep drop in the radio continuum emission from cosmic rays in the disk . despite indications that star formation activity decreases with radius , the @xmath2 excitation temperature and the ratio of the @xmath2 line and the far - ir luminosity surface densities , @xmath3(l@xmath4)/@xmath3(l@xmath5 ) , change very little as a function of radius , even into the diffuse outer region of the disk of ngc 4565 . this suggests that the source of excitation of the @xmath2 operates over a large range of radii , and is broadly independent of the strength and relative location of uv emission from young stars . although excitation in photodissociation regions is the most common explanation for the widespread @xmath2 emission , cosmic ray heating or shocks can not be ruled out . at @xmath1 = 15 kpc in ngc 4565 , outside the main uv and radio continuum - dominated disk , we derived a higher than normal @xmath2 to 7.7 @xmath0 m pah emission ratio , but this is likely due to a transition from mainly ionized pah molecules in the inner disk to mainly neutral pah molecules in the outer disk . the inferred mass surface densities of warm molecular hydrogen in both edge - on galaxies differ substantially , being 4(60 ) m@xmath6 pc@xmath7 and 3(50 ) m@xmath6 pc@xmath7 at @xmath1 = 10 kpc for ngc 4565 and ngc 5907 , respectively . the higher values represent very unlikely point - source upper limits . the point source case is not supported by the observed emission distribution in the spectral slits . these mass surface densities can not support the observed rotation velocities in excess of 200 km s@xmath8 . therefore , warm molecular hydrogen can not account for dark matter in these disk galaxies , contrary to what was implied by a previous _ iso _ study of the nearby edge - on galaxy ngc 891 .

You are an expert at summarizing long articles. Proceed to summarize the following text: the authors of ref . @xcite investigated the effects of the variation of the mass parameter @xmath1 on the thick branes . they used a real scalar field , which has a potential of the @xmath4 model , as the background field of the thick branes . it was found that the number of the bound states ( in the case without gravity ) or the resonant states ( in the case with gravity ) increases with the parameter @xmath1 . that work considered the simplest yukawa coupling @xmath2 , where @xmath3 is the coupling constant . the authors stated that as the value of @xmath1 is increasing , the maximum of the matter energy density splits into two new maxima , and the distance of the new maxima increases and the brane gets thicker . the authors also stated that the brane with a big value of @xmath1 would trap fermions more efficiently . in this paper , we reinvestigated the effect of the variation of the mass parameter @xmath1 on the thick branes , because the above investigation does not analyze the zero mode in details and contains some misconceptions . we only focus attention in the case with gravity . we find that the variation of @xmath1 on the thick brane is associated to the phenomenon of brane splitting . from the static equation of motion , we analyze the asymptotic behavior of @xmath5 and find that the zero mode for left - handed fermions can be localized on the brane depending on the value for the coupling constant @xmath3 and the mass parameter @xmath1 . we also show that as the value of @xmath1 is increasing the simplest yukawa coupling does not support the localization of fermions on the brane , as incompletely argued in ref . @xcite . the action for our system is described by @xcite @xmath6,\ ] ] where @xmath7 , @xmath8 is the 5d bulk cosmological constant and the scalar potential @xmath9 is given by @xcite @xmath10 where @xmath11 . there are three minima for @xmath9 , one is at @xmath12 ( local minima ) corresponding to a disordered bulk phase and the other two are at @xmath13 ( global minima ) with @xmath14 they are degenerated and correspond to ordered bulk phases . as @xmath15 ( @xmath16 ) , @xmath17 , @xmath9 has three degenerated global minima . for the case with gravity , the critical value of @xmath1 is not @xmath18 but a smaller effective critical value @xmath19 . in this case , @xmath20 @xcite . the line element in this model is considered as @xmath21 where @xmath22 , @xmath23 , and @xmath24 is the so - called warp factor . we suppose that @xmath25 and @xmath26 . for this model , the equations of motion are @xmath27 @xmath28 @xmath29 it is possible to rewrite ( [ em2b ] ) and ( [ em3b ] ) as @xmath30 the boundary conditions can be read as follows @xmath31 @xmath32 the matter energy density has the form @xmath33.\ ] ] at this point , it is also instructive to analyze the matter energy of the toy model @xmath34 substituting ( [ de ] ) in ( [ ephi ] ) , we get @xmath35\,,\ ] ] using ( [ em3b ] ) and ( [ em3c ] ) , we obtain the value of the matter energy given by @xmath36-\lambda\int^{\infty}_{-\infty}dy% \mathrm{e}^{2a(y)}.\ ] ] as @xmath37 , the value of the matter energy depends on the asymptotic behavior of the warp factor . if @xmath38 then @xmath39 and by the analysis to eq . ( [ em2b ] ) , we can see that @xmath40 . therefore , @xmath41 and the value of the matter energy is zero . this fact is the same to the case of branes with generalized dynamics @xcite . the scalar curvature ( or ricci scalar ) is given by @xmath42 the profiles of the matter energy density is shown in fig . ( [ fde ] ) for some values of @xmath1 . figure ( [ fde ] ) clearly shows that for @xmath43 the matter energy density has not a single - peak around @xmath44 . the core of the brane is localized at @xmath44 for @xmath43 , because this region has a positive matter energy density . on the other hand , as the value of @xmath1 is increasing , we can see that the single brane splits into two sub - branes and as @xmath45 each sub - brane is a thick brane . this phenomenon is so - called of brane splitting @xcite . from the peak of the matter energy density is evident know where the core of the branes are located . therefore , the brane does not get thicker with the increases of the value of the mass parameter @xmath1 , as argued in ref . the profiles of the matter energy density and the ricci scalar are shown in fig . ( [ desc ] ) for @xmath46 . note that the presence of regions with positive ricci scalar is connected to the capability to trap matter near to the core of the brane @xcite and it reinforces the conclusion of the analyzes from the matter energy density . also note that far from the brane , @xmath47 tends to a negative constant , characterizing the @xmath48 limit from the bulk . , @xmath49 , @xmath43 ( thin line ) , @xmath46 ( dashed line ) and @xmath50 ( dotted line).,width=264 ] , @xmath49 and @xmath46.,width=264 ] the action for a dirac spinor field coupled with the scalar fields by a general yukawa coupling is @xmath51\,,\ ] ] where @xmath3 is the positive coupling constant between fermions and the scalar field . moreover , we are considering the covariant derivative @xmath52 , where @xmath53 and @xmath54 , denote the local lorentz indices and @xmath55 is the spin connection . here we consider the field @xmath56 as a background field . the equation of motion is obtained as @xmath57 at this stage , it is useful to consider the fermionic current . the conservation law for @xmath58 follows from the standard procedure and it becomes @xmath59 where @xmath60 . thus , if @xmath61 then four - current will be conserved . the condition ( [ cj0 ] ) is the purely geometrical assertion that the curved - space gamma matrices are covariantly constant . using the same line element ( [ metric ] ) and the representation for gamma matrices @xmath62 , the condition ( [ cj0 ] ) is trivially satisfied and therefore the current is conserved . the equation of motion ( [ dkp ] ) becomes @xmath63\psi=0.\ ] ] now , we use the general chiral decomposition @xmath64 with @xmath65 and @xmath66 . with this decomposition @xmath67 and @xmath68 are the left - handed and right - handed components of the four - dimensional spinor field , respectively . after applying ( [ dchiral ] ) in ( [ em ] ) , and demanding that @xmath69 and @xmath70 , we obtain two equations for @xmath71 and @xmath72 @xmath73\alpha_{l_{n}}=m_{n}\mathrm{e}^{-a}\alpha_{r_{n}}\,,\ ] ] @xmath74\alpha_{r_{n}}=-m_{n}\mathrm{e}^{-a}\alpha_{l_{n}}\,.\ ] ] inserting the general chiral decomposition ( [ dchiral ] ) into the action ( [ ad ] ) , using ( [ ea1 ] ) and ( [ ea2 ] ) and also requiring that the result take the form of the standard four - dimensional action for the massive chiral fermions @xmath75 where @xmath76 and @xmath77 , the functions @xmath71 and @xmath72 must obey the following orthonormality conditions @xmath78 implementing the change of variables @xmath79 @xmath80 and @xmath81 , we get @xmath82 @xmath83 where @xmath84 using the expressions @xmath85 and @xmath86 , we can recast the potentials ( [ vefa ] ) and ( [ vefb ] ) as a function of @xmath87 @xcite-@xcite @xmath88,\label{vya } \\ v_{r}(z(y ) ) & = & v_{l}(z(y))|_{\eta\rightarrow-\eta}\,.\label{vyb}\end{aligned}\ ] ] it is worthwhile to note that we can construct the schrdinger potentials @xmath89 and @xmath90 from eqs . ( [ vya ] ) and ( [ vyb ] ) . at this stage , it is instructive to state that with the change of variable ( [ cv ] ) we get a geometry to be conformally flat @xmath91 now we focus attention on the condition ( [ cj0 ] ) for the line element ( [ cm ] ) . in this case we obtain @xmath92 therefore , the current is no longer conserved for the line element ( [ cm ] ) @xcite . it is known that , in general , the reformulation of the theory in a new conformal frame leads to a different , physically inequivalent theory . this issue has already a precedent in cosmological models @xcite . under this arguments , we only use the change of variable ( [ cv ] ) to have a qualitative analysis of the potential profiles ( [ vya ] ) and ( [ vyb ] ) , which is a fundamental ingredient for the fermion localization on the brane . now we focus attention on the calculation of the zero mode . substituting @xmath93 in ( [ ea1 ] ) and ( [ ea2 ] ) and using @xmath80 and @xmath81 , respectively , we get @xmath94,\ ] ] @xmath95.\ ] ] this fact is the same to the case of two - dimensional dirac equation @xcite . at this point is worthwhile to mention that the normalization of the zero mode and the existence of a minimum for the effective potential at the localization on the brane are essential conditions for the problem of fermion localization on the brane . this fact was already reported in @xcite . in order to guarantee the normalization condition ( [ orto ] ) for the left - handed fermion zero mode ( [ mzl ] ) , the integral must be convergent , _ i.e _ @xmath96<\infty.\ ] ] this result clearly shows that the normalization of the zero mode is decided by the asymptotic behavior of @xmath97 . furthermore , from ( [ vya ] ) and ( [ vyb ] ) , it can be observed that the effective potential profile depends on the @xmath97 choice . this fact implies that the existence of a minimum for the effective potential @xmath98 or @xmath99 at the localization on the brane is decided by @xmath97 . this point will be more clear when it is considered a specific yukawa coupling . therefore , the behavior of @xmath97 plays a leading role for the fermion localization on the brane @xcite . having set up the two essential conditions for the problem of fermion localization on the brane , we are now in a position to choice some specific forms for yukawa couplings . from now on , we mainly consider the simplest case @xmath100 . first , we consider the normalizable problem of the solution . in this case , we only need to consider the asymptotic behavior of the integrand in ( [ cono ] ) . it becomes @xmath101\,.\ ] ] by the analysis from eq . ( [ em2b ] ) , we obtain that @xmath102 . for the integral @xmath103 , we only need to consider the asymptotic behavior of @xmath56 for @xmath38 @xcite and as @xmath104 the equation ( [ in ] ) becomes @xmath105\,.\ ] ] this result clearly shows that the zero mode of the left - handed fermions is normalized only for @xmath106 . now , under the change @xmath107 ( @xmath108 ) we obtain that the right - handed fermions can not be a normalizable zero mode . the shape of the potentials @xmath89 and @xmath90 are shown in fig . ( [ pe ] ) for some values of @xmath1 . figure [ pe](a ) shows that the effective potential @xmath89 , is indeed a volcano - like potential for @xmath43 . as @xmath1 increases the well structure of @xmath89 gets a double well . figure [ pe](b ) shows that the potential @xmath90 has also a well structure , but the minimum of @xmath90 is always positive , therefore the potential does not support a zero mode . the shapes of the matter energy density , @xmath89 potential and @xmath109 are shown in fig . [ a](a ) ( @xmath43 ) shows that the zero mode is localized on the brane . on the other hand , fig . [ a](b ) ( @xmath50 ) clearly shows that the normalizable zero mode is localized between the two sub - branes , as a consequence the zero mode is not localized on the brane . therefore , we can conclude that the zero mode of the left - handed fermions is localized on the brane only as @xmath110 . + + + + we have reinvestigated the effects of the variation of the mass parameter @xmath1 on the thick branes as well as the localization of fermions . we showed that the variation of @xmath1 is associated to the phenomenon of brane splitting , therefore the brane does not get thicker with the increases of the value of @xmath1 , as argued in ref . we can conclude that the appearance of two sub - branes is associated to phase transition for @xmath111 ( a disordered phase between two ordered phases ) . also , we showed that the value of the matter energy depends on the asymptotic behavior of the warp factor . from the static equation of motion we have analyzed the asymptotic behavior of @xmath5 and showed that the zero mode of the left - handed fermions for the simplest yukawa coupling @xmath2 is normalizable under the condition @xmath106 and it can be trapped on the brane only for @xmath110 , because the zero mode has a single - peak at the localization of the brane . we also showed that as @xmath45 the zero mode has a single - peak between the two sub - branes and as a consequence the normalizable zero mode is not localized on the brane . therefore , the brane with a big value of @xmath1 would not trap fermions more efficiently , in opposition to what was adverted in ref . @xcite . this work completes and revises the analyzing of the research in ref . @xcite , because in that work does not analyze the zero mode in full detail and contain some misconceptions . additionally , we showed that the change of variable @xmath112 leads to a non conserved current , because the curved - space gamma matrices are not covariantly constant . an interesting issue will be investigate the effects of non - conserved current on resonances modes and bear out the main conclusion of ref . @xcite . a.s . de castro and m.b . hott , phys . a * 351 * , 379 ( 2006 ) ; l.b . castro and a.s . de castro , j. phys . a : math . theor . * 40 * , 263 ( 2007 ) ; l.b . castro and a.s . de castro , phys . scr . * 75 * , 170 ( 2007 ) ; l.b . castro and a.s . de castro , int . e * 16 * , 2998 ( 2007 ) ; l.b . castro , a.s . de castro and m.b . hott , int . j. mod e * 16 * , 3002 ( 2007 ) ; l.b . castro and a.s . de castro , phys . scr . * 77 * , 045007 ( 2008 ) .

in a recent paper published in this journal , zhao and collaborators [ phys . rev . d * 82 * , 084030 ( 2010 ) ] analyze a toy model of thick branes generated by a real scalar field with the potential @xmath0 , and investigate the variation of the mass parameter @xmath1 on the branes as well as the localization and resonances of fermions . in that research the simplest yukawa coupling @xmath2 was considered . in that work does not analyze the zero mode in details and also contains some misconceptions . in this paper , the effect of the variation of the mass parameter @xmath1 on the brane is reinvestigated and it is associated to the phenomenon of brane splitting . furthermore , it is shown that the zero mode for the left - handed fermions can be localized on the brane depending on the values for the coupling constant @xmath3 and the mass parameter @xmath1 .

You are an expert at summarizing long articles. Proceed to summarize the following text: in deriving the partition function for a desired ensemble , the most common approach is to maximize an entropy function with constraints appropriate to the thermodynamic condition . while equivalent to the approach proposed below , such a method ( called the traditional approach hereafter ) does not make clear to students the explicit role of the assumption of equal _ a priori _ states and the corresponding role of the entropy in the thermodynamic potential for the microcanonical ensemble . indeed , @xmath2 is often taken as a postulate@xcite and its connection to the statistical formula @xmath4 ( appearing on boltzmann s tombstone ) is not obvious . further , in the traditional approach , the role of the entropy in understanding equilibrium in non - isolated , open ensembles can be confusing . we note in passing that concerns over the rigor of the method of most probable distribution prompted darwin and fowler to develop a derivation of the partition function based upon complex analysis.@xcite also , infrequently stressed is the gibbs entropy , @xmath1 , where @xmath5 is the probability of finding a system in a given state , which can be invoked for any equilibrium ensemble and associated state probabilities.@xcite it is a direct consequence of the statistical entropy formula , @xmath4 , in conjunction with the gibbs construction of an ensemble that contains a large number of macroscopic subsystems , each consistent with the desired thermodynamic variables ; @xmath6 gives the number of possible realizations within the gibbs construction for the ensemble under consideration . the gibbs entropy also permits the derivation of the connection between the characteristic thermodynamic function and the partition function for a given ensemble without further appeal to thermodynamic expressions , as is required in the traditional approach . in the present approach , first , the connections between the statistical formula @xmath7 , the gibbs entropy , @xmath1 , and the microcanonical entropy expression @xmath2 are clarified . the condition for microcanonical equilibrium , and the associated role of the entropy in the thermodynamic potential then arises from the postulate of equal _ a priori _ states . the derivation of the canonical partition function follows by invoking the gibbs construction and the first and second law of thermodynamics _ via _ the fundamental equation , @xmath8 , that incorporates the conditions of conservation of energy and composition without the needs for explicit constraints . the role of the temperature ( coming from the constraint of total energy and an appeal to appropriate thermodynamic relationships in the traditional approach ) is immediately apparent and also introduced _ via _ the fundamental equation . the need for explicit maximization of any function is thus also avoided . legendre transforming a particular thermodynamic function to include desired thermodynamic control variables for an ensemble of interest and invoking equilibrium leads to the corresponding partition function . using the resulting probabilities in the gibbs entropy expression directly connects the partition function to the thermodynamic potential . the central role of the entropy in establishing equilibrium for a given ensemble emerges naturally from the current approach . connections to generalized ensemble theory also arise and are presented in this context . the present approach is novel in providing clarity as to the roles played by the different formulas and physical quantities of interest . further , it makes explicit the assumptions inherent in deriving the partition function for an ensemble and provides its direct connection to the relevant thermodynamic potential in a systematic fashion . this approach also makes deriving the partition function for a given ensemble a simplified , straight - forward process , even for more challenging examples such as the isothermal - isobaric ensemble . using this approach in the classroom has led to better retention and understanding of the foundations of statistical mechanics and an ability for students to confidently apply the machinery to problems that arise in their subsequent work . we begin by introducing the concept of an ensemble of replicas that describe the molecular states corresponding to a given macrostate ; this picture is referred to as the gibbs construction " herein , due to it s original introduction by gibbs@xcite who addressed many of the subtleties inherent@xcite in the formulation of statistical mechanics . consider a collection of macroscopic molecular subsystems " of @xmath9 molecules within a volume @xmath10 , each of which is part of the larger gibbs construction , the totality of which is known as the system " . no other constraints have yet been imposed , _ i.e. _ the system s macrostate is otherwise unspecified . it is desirable to define the microscopic statistics of this system as thoroughly as possible and then apply any other constraints at the end . let the total number of subsystems in our collection be known as @xmath11 . then let @xmath12 , the occupation number , denote the number of subsystems from this collection that are in the same thermodynamic state . these occupations will thus take on a large value in the thermodynamic limit and they obey a sum rule , @xmath13 . note , technically the energy is course - grained , _ i.e. _ specified to within a small but otherwise arbitrary range ( these arguments are presented in detail elsewhere@xcite ) and the results are insensitive to this choice . first , consider the following combinatoric formula : @xmath14 @xmath15 is the number of ways in which the set of occupations @xmath16 may be arranged consistent with the given macrostate . first , it is to be shown that when evaluated at fixed energy , this quantity @xmath17 may be identified with the thermodynamic entropy of the ensemble of systems at equilibrium , with each systems entropy given by s = @xmath18 . note , the expression necessarily involves the logarithm of the combinatoric expression to make the entropy an extensive property ; for two independent systems the possible number of arrangements is the product of those for the individual systems , @xmath19 . next the connection between the combinatoric formula @xmath20 and the gibbs entropy is presented ; the details of this have been given elsewhere.@xcite applying the sterling approximation@xcite to the factorial function gives the entropy : @xmath21 further substituting @xmath22 where @xmath23 is identified as the probability of a particular state and using the property of the natural log gives a system s entropy as:@xcite @xmath24 this is the gibbs entropy in an as yet unspecified ensemble with its associated probabilities ; the gibbs entropy is an entirely general definition that , for any equilibrium ensemble , specifies the relationship between the partition function and the associated characteristic thermodynamic function . now , specializing to a set of microcanonical subsystems , and invoking the equilibrium principle of equal _ a priori _ states , _ i.e. _ @xmath25 , gives the well known result : @xmath26 it is also simple and useful to show that the gibbs entropy , and thus the thermodynamic entropy , is maximized microcanonically@xcite by the state - independent probabilities @xmath27 . proceeding , taking the derivative of equation [ eq : ge ] and setting it to zero as @xmath28 gives @xmath29 , a constant value independent of the summation index . thus , normalizing the probabilities as , @xmath30 immediately yields @xmath31 . thus , for an isolated system , the assumption of equal _ a priori _ states leads to a probability @xmath5 that is independent of index , _ i.e. _ every subsystem has energy @xmath32 by construction . further , the characteristic maximum entropy in the microcanonical equilibrium ensemble also follows . then applying the gibbs entropy expression leads to the identification of the thermodynamic entropy as the characteristic function of the microcanonical ensemble and gives its relationship to the @xmath33 partition function , @xmath34 , which can also be interpreted as the density of states@xcite at that energy specializing the gibbs construction from the previous section to include temperature , we have a collection of subsystems all possessing the same @xmath35 . this can be thought of by placing the subsystems in contact with a large heat bath of temperature @xmath36.@xcite we now imagine that each subsystem ( after having achieved equilibrium with the heat bath by definition ) is to be insulated and the energy of the @xmath37 subsystem is measured as @xmath38 , and for which there is also an associated macroscopic entropy @xmath39 . of great importance , we also note that the thermodynamic energy @xmath38 is exactly equal to the microscopic configurational energy of the subsystem upon insulation . furthermore , the details and/or rates involved in the insulation process are irrelevant for an equilibrium ensemble . using the earlier result , the entropy for a collection of subsystems with a specified energy @xmath38 is @xmath40 , where @xmath11 is the number of subsystems with energy @xmath38 in the ensemble . consider the ratio of the density at energies @xmath41 : @xmath42 the fundamental equation of thermodynamics@xcite is now invoked : @xmath43 the canonical ensemble is given by a state with well defined thermodynamic variables , @xmath44 . so the energy function , @xmath45 is legendre transformed to a new thermodynamic function , the helmholtz free energy , @xmath46 _ via _ : @xmath47 where the condition for canonical equilibrium is that @xmath48 and @xmath49 are constant , giving : @xmath50 integrating between two state points gives : @xmath51 note , the constraints of fixed temperature , particle number and volume have been explicitly enforced by using the fundamental equation , equation [ eq : fe ] and dropping differential terms that are fixed canonically . substituting equation [ eq : ds ] into equation [ eq : ratio ] gives : @xmath52 where @xmath53 and @xmath54 ( _ i.e. _ the probability of choosing the @xmath37 state from the entire ensemble at equilibrium with the heat bath ) . the normalization factor , @xmath55 , may be readily recognized as the canonical partition function . most importantly , we note that the insulation procedure applied to each subsystem has allowed us to identify the macroscopic energy ( and entropy ) of that microcanonical system , with the microscopic energy of the molecular configuration present at the time of insulation . we can now proceed to use the gibbs entropy expression , equation [ eq : ge ] , substituting the canonical expression for @xmath5 to obtain : @xmath56 above , @xmath57 represents the canonical average energy that is identified with the thermodynamic energy , @xmath32.@xcite thus , the relationship @xmath58 is obtained directly from the gibbs entropy . note , the gibbs entropy is defined for any set of probabilities and , as was shown above , is simply a consequence of the combinatoric formula @xmath59 interpreted in the context of the gibbs construction . thus , an entropy can be associated even with nonequilibrium probabilities . however , in that case , the entropy does not play the role of being the constrained maximized quantity that it does at equilibrium and its utility , in such circumstances , is unclear . further note , the role of temperature is introduced _ via _ the fundamental equation without further appeal to thermodynamic relationships . this emphasizes the role of temperature as the system is in contact with a heat bath different energy ranges are now accessible with canonical probabilities . the ability of a diathermal system to exchange energy with its surroundings also clarifies how the concepts of work and entropy make sense for an open system and provides their relationship to the temperature . gibbs construction for the canonical ensemble . the subsystems of identical @xmath60 are in thermal equilibrium with a large bath at temperature @xmath36.,width=3 ] using the gibbs construction from the previously derived canonical ensemble , the constraint that all subsystems possess identical @xmath9 can now be relaxed . we now consider the ratio of subsystem , having chosen a particular system from the energy level @xmath38 with @xmath61 molecules : @xmath62 as before , the fundamental equation of thermodynamics can be relied upon to relate the change in entropy to the other variables of our ensemble . in this case , the probabilities that will generate the macrostate corresponding to constant @xmath63 are desired . after doing so , the entropy _ via the boltzmann law _ is used to determine the microscopic states . legendre transforming the canonical thermodynamic equation to substitute @xmath64 for @xmath9 : @xmath65 where at equilibrium @xmath66 and with constant @xmath63 : @xmath67 upon integrating , the difference equation for the entropy is found : @xmath68\end{aligned}\ ] ] the ratio of microstates then becomes : @xmath69 and since @xmath70 @xmath71 where the normalization factor @xmath72 is the _ grand canonical partition function _ : @xmath73 it may be noted that in this case the constraint of constant @xmath9 was merely relaxed , legendre transformed to the corresponding macrostate , and the partition function then followed quite naturally and simply . the resulting probabilities can now be substituted into the gibbs entropy and the relationship between the thermodynamic potential and partition function is thus directly established . gibbs construction for the grand canonical ensemble . @xmath64 has been legendre transformed to replace @xmath9 as the macroscopic constant , and so the set of subsystems includes those of differing @xmath9 values.,width=3 ] having successfully derived the grand canonical partition function by relaxing the constant @xmath9 constraint , it can now be shown that the isothermal - isobaric ensemble is generated by starting with constant @xmath74 and relaxing the condition of constant @xmath10 . consider the number of subsystems with both volume @xmath75 and energy @xmath76 : @xmath77 legendre transforming our desired variables to a new characteristic function @xmath78 , and then applying the condition of equilibrium @xmath79 and our constant differential terms : @xmath80 and so the ratio of observable subsystems becomes : @xmath81 } \nonumber \\ = \frac{e^{\beta e_i}e^{\beta pv_j}}{e^{\beta e_{i+1}}e^{\beta pv_{j+1}}}\end{aligned}\ ] ] and since @xmath70 @xmath82 where the normalization factor , @xmath83 is the _ isothermal - isobaric partition function_. again , the associated probabilities can now be substituted into the gibbs entropy and the relationship between the thermodynamic potential and partition function is thus directly established . gibbs construction for the isothermal - isobaric ensemble . @xmath84 has been legendre transformed to replace @xmath10 as the macroscopic constant , and so the set of subsystems includes those of differing @xmath10 values.,width=3 ] based on the previous sections , it may be noted that if a partition function for _ any _ macrostate in thermal equilibrium is to be derived , one shall _ always _ arrive at an expression that involves an exponential function ; this follows as a consequence of the boltzmann law . furthermore , the partition function being the normalization factor to express this as a probability means that one will always have a discrete sum . therefore , our partition functions will always be some variation on a theme amounting to a sum of exponentials . " in the continuum limit , it can be shown that the canonical partition function @xmath85 ( written as a sum over energy levels ) transforms as : @xmath86 or , in other words , the canonical partition function is the laplace transform of the microcanonical partition function @xmath11 . how does it work the other way ? let s apply the inverse laplace transform to @xmath85 : @xmath87 where the phase space differential form @xmath88 . now , @xmath89 and because no singularity is present in the right - half of the complex plane , the contour may be taken vertically through @xmath90 . since @xmath91 along the integration , the substitution @xmath92 can be made : @xmath93 where indeed equation [ eq : delta_function ] can be identified as the microcanonical partition function . thus , any constant energy shell ensemble may be laplace transformed to an ensemble of a new intensive variable . as the partition functions are related to one another through the laplace transform , this is isomorphic to the thermodynamic potentials ( to each of which may be associated a particular partition function ) being related through the legendre transform.@xcite the quantum harmonic oscillator is an illustrative example of how the canonical partition function may be transformed to the microcanonical case : @xmath94 \label{eq : nve_qho}\end{aligned}\ ] ] the reason that the aforementioned recipe " for generating the partition function ( as outlined in sections [ sec : canonical],[sec : grand_canonical ] and [ sec : isobaric ] ) in an arbitrary ensemble works is due to the thermodynamic relations and the boltzmann law . the underlying mathematical structure that allows this has also been previously formulated as generalized ensemble theory.@xcite an approach is presented for deriving partition functions that is an alternative to more common methods . it emphasizes the central role that ( maximizing ) the boltzmann entropy plays in connecting the molecular states of the system to the observable thermodynamics . using this technique in a classroom setting for a beginning graduate class in statistical mechanics has led to systemization and demystification of the derivation for useful ensembles . also , the role of legendre transforms to introduce thermodynamic control variables appears naturally and is tied directly to both the derivation of the ensemble and corresponding partition function . within this formalism , students are clear on how the thermodynamic potential relates to a given ensemble and the role of equal _ a priori _ states . further , relating the partition function to the thermodynamic potential using the gibbs entropy is straightforward and no further appeal to thermodynamic expressions is required as the relevant thermodynamic connection was included from the start of the derivation . finally , the similarities between the derivation method demonstrated and the relations known from generalized ensemble theory have been noted . it is our hope that the formulaic approach presented here will be of utility in both research and pedagogy . the authors acknowledge funding from the u.s . department of energy , basic energy sciences ( grant no . de0gg02 - 07er46470 ) . lawrence livermore national laboratory is operated by lawrence livermore national security , llc , for the u.s . department of energy , national nuclear security administration under contract de - ac52 - 07na27344 .

a pedagogical approach for deriving the statistical mechanical partition function , in a manner that emphasizes the key role of entropy in connecting the microscopic states to thermodynamics , is introduced . the connections between the combinatoric formula @xmath0 applied to the gibbs construction , the gibbs entropy , @xmath1 , and the microcanonical entropy expression @xmath2 are clarified . the condition for microcanonical equilibrium , and the associated role of the entropy in the thermodynamic potential is shown to arise naturally from the postulate of equal _ a priori _ states . the derivation of the canonical partition function follows simply by invoking the gibbs ensemble construction at constant temperature and using the first and second law of thermodynamics ( _ via _ the fundamental equation @xmath3 ) that incorporate the conditions of conservation of energy and composition without the needs for explicit constraints ; other ensemble follow easily . the central role of the entropy in establishing equilibrium for a given ensemble emerges naturally from the current approach . connections to generalized ensemble theory also arise and are presented in this context .

You are an expert at summarizing long articles. Proceed to summarize the following text: strongly correlated electron systems exhibit the highest attained superconducting transition temperatures currently known , and a rich variety of complex electronic phases@xcite . many compounds among this family of mott insulators , such as the cuprates , are quasi - two - dimensional layered materials . this renders them ideal candidates for bilayer exciton condensation , which is the topic of this publication . the effort to achieve the condensation of excitons has a long history starting just after the discovery of bcs theory@xcite . an exciton is the bound state of an electron and a hole and as such it can bose condense . the obvious advantage of considering excitons above cooper pairs is the strong coulomb attraction between the electron and the hole ; allowing in principle for a much higher critical temperature . to reduce the exciton lifetime problems caused by electron - hole recombination , it has been suggested to spatially separate the electrons and holes in their own subsequent layers@xcite . this indeed has resulted in the experimental realization of exciton condensates , first in the so - called quantum hall bilayers@xcite and more recently without an externally applied magnetic field in electrically gated , optically pumped semiconductor quantum wells@xcite . the successes of exciton condensation in semiconductor 2deg bilayer systems have led to many proposals for exciton condensation in alternative bilayer materials , such as gated topological insulators@xcite or double layer graphene@xcite . however , these proposals are limited to the bcs paradigm of weak coupling . on the other hand , mott insulators provide a completely different route to exciton condensation@xcite . naively one would expect that the localization of the electrons and holes leads to a higher critical temperature , since @xmath2 is determined by the competition between the electronic kinetic energy and the electron - hole attraction . but the physics of exciton condensation in mott insulators is in fact much richer . instead of the picture that the electron - hole pair lives in a conduction and valence band , an exciton now consists of a double occupied and vacant site bound together on an interlayer rung , see figure [ lattice ] . to estimate the binding energy , consider the in - plane charge - transfer excitons which are known to have a binding energy of the order of 1 - 2 ev@xcite . due to the small interlayer distances of order 1 nm we expect that a similar energy scale will set the binding of the interlayer exciton . as such , excitons in a mott bilayer are most likely in the strongly coupled regime . side view of a strongly correlated electron bilayer with an exciton present . the red arrows denote the spin of the localized electrons , and the exciton is a bound state of a double occupied and an empty site . ] furthermore , a single doublon - holon pair inserted into a mott insulator leads to dynamical frustration effects@xcite , even stronger than seen for a single hole in the @xmath0 model@xcite . the study of excitons in strongly correlated materials thus catches the complexity of doped mott insulators . as we discussed elsewhere @xcite the bosonic nature of the excitons actually falls short to completely eliminate all `` fermion - like '' signs : there are still left - over signs of the phase - string type@xcite . however , it is easy to demonstrate that collinear spin order is a sufficient condition for these signs to cancel out , leaving a truly bosonic dynamics controlling the ground state and long wavelength physics . the problem thereby reduces to that of hard - core bosons ( the excitons ) in a sign - free spin background . this is very similar to the `` spin - orbital '' physics described by kugel - khomskii type models@xcite , which can be viewed after all as describing d - d excitons interacting with spins . also the lattice implementations@xcite of the so(5 ) model@xcite for ( cuprate ) superconductivity are in this family . such bosonic problems can be handled with standard ( semi - classical ) mean field theory , and therefore the regime of finite exciton density can be addressed in an a - priori controlled manner . in most bilayer exciton set - ups , such as the quantum hall bilayers or the pumped systems , there is no controllable equilibrium exciton density . in these cases one can hardly speak of the exciton density as a conserved quantity , and exciton condensation in the sense of spontaneously broken @xmath3 symmetry is impossible@xcite . however , in mott insulators the dopant density per layer could be fixed by , for example , chemical doping . the effective exciton chemical potential is then by definition large compared to the recombination rate . effectively , the excitons are at finite density in equilibrium and hence spontaneous @xmath3 symmetry breaking is possible in the mott insulating bilayer . besides the exciton superfluid phase one anticipates a plethora of competing orders , as is customary in strongly correlated materials . at zero exciton density the bilayer heisenberg system exhibits already interesting magnetism , in the form of the antiferromagnet for small rung coupling turning via an @xmath4-qnls quantum phase transition into an `` incompressible quantum spin liquid '' for larger rung couplings that can be viewed as a continuation of pair singlets ( `` valence bonds '' ) stacked on the rungs @xcite . the natural competitor of the exciton superfluid at finite density is the exciton crystal and one anticipates that due to the strong lattice potential this will tend to lock in at commensurate densities forming exciton `` mott insulators '' . we will wire this in by taking also the exciton - exciton dipolar interaction into account that surely promotes such orderings . in principle there is the interesting possibility that all these orders may coexist microscopically forming an `` antiferromagnetic supersolid '' @xcite . in this bosonic setting we can address it in a quite controlled manner , but we find that at least for the strongly coupled `` small '' excitons assumed here this does not happen . the reason is interesting . we already alluded to the dynamical `` frustration '' associated with the exciton delocalizing in the anti - ferromagnetic spin background , which is qualitatively of the same kind as for the standard `` electron '' t - j model . at finite densities this turns into a tendency to just phase separate on a macroscopic scale , involving antiferromagnets , exciton crystalline states and high density diamagnetic exciton superfluids , respectively . even though the exciton dipolar repulsion is long - ranged , there is no possibility of frustrated phase separation as suggested for the electronic order in cuprates@xcite because the @xmath5 interaction falls off too quickly . however , if one correctly incorporates the full exciton dipolar interaction , a variety of different exciton ordered phase may arise@xcite . here we restrict ourselves to nearest neighbor repulsion only , which allows for the formation of a checkerboard ordered exciton crystalline state . it is disappointing that apparently in this system only conventional ground states occur . however , this is actually to a degree deceptive . the hamiltonian describing the physics at the lattice scale describes a physics where the exciton- and spin motions are `` entangled '' : the way in which these subsystems communicate gets beyond the notion of just being strongly coupled , since the motions of the exciton motions and the spin dynamics can not be separated . by coarse graining this all the way to the static order parameters ( the mean fields ) an effective decoupling eventually results as demonstrated by the pure ground states . however , upon going `` off - shell '' this spin - exciton entanglement becomes directly manifest in the form of unexpected and rather counterintuitive effects on the excitation spectrum . a simple example is the zero exciton density antiferromagnet . from the rather controlled linear spin wave self - consistent born approximation ( lsw - scba ) treatment of the one exciton problem @xcite we already know that the resulting exciton spectrum can be completely different from that in a simple semiconductor . we compute here the linearized excitations around the pure antiferromagnet , recovering the lsw - scba result in the `` adiabatic limit '' where the exciton hopping is small compared to the exchange energy of the spin system , which leads to a strong enhancement of the exciton mass . in the opposite limit of fast excitons , the energy scale is recovered but the `` ising - confinement '' ladder spectrum revealed by the lsw - scba treatment is absent . the reason is clear : in the language of this paper , the couplings between the exciton- and spin - wave modes become very big and these need to be re - summed in order to arrive at an accurate description of the exciton propagator , while our mean - field treatment corresponds with a complete neglect of these exciton - spin interactions . the real novelty in this regard is revealed in the high density exciton superfluid phase . the spin system forms here a ground state that is a product state of pair - singlets living on the rungs . besides the superfluid phase modes one expects in addition also the usual massive spin - triplet excitations associated with the ( incompressible ) singlet vacuum . the surprise is that these are characterized by a dispersion which is in part determined by the _ superfluid density of the exciton condensate _ , as we already announced elsewhere@xcite for which we present here the details . counterintuitively , by measuring the spin fluctuations one can in principle determine whether the excitons are condensed in a superfluid . let us complete this introduction by specifying the point of departure : the hamiltonian describing strongly bound excitons propagating through a bilayer heisenberg spin 1/2 system . this model is derived and discussed at length in our earlier papers @xcite and here we just summarize the outcome . due to the strong electron - electron interactions the electronic degrees of freedom are , at electronic half - filling , reduced to spin operators @xmath6 governed by the bilayer heisenberg model@xcite @xmath7 the subscript denotes spin operators on site @xmath8 in layer @xmath9 . the heisenberg @xmath10 is antiferromagnetic with @xmath11 and @xmath12 . the interlayer exciton can hop around , thereby interchanging places with the spin background . in the strong - coupling limit of exciton binding energies the exciton hopping process is described by the hamiltonian @xmath13 where @xmath14 is the exciton state on an interlayer rung , and @xmath15 represent the rung spin states . whenever an exciton hops , it effectively exchanges the spin configuration on its neighboring site . this exciton @xmath0 model was derived earlier in refs . @xcite , where the optical absorption was computed in the limit of vanishing exciton density @xmath16 . in order to study the system with a finite density of excitons , we need to enrich the current @xmath0 model with two extra terms : a chemical potential and an exciton - exciton interaction . the chemical potential is straightforwardly @xmath17 the exciton - exciton interaction requires more thought . the bare interaction between two interlayer excitons results from their electric dipole moment . since all interlayer exciton dipole moments are pointing in the same direction the full exciton - exciton interaction is described by a repulsive @xmath5 interaction . hence the interaction strength decays sufficiently fast to avoid the coulomb catastrophe responsible for frustrated phase separation@xcite . we consider it reasonable to only include the nearest - neighbor repulsion , @xmath18 here @xmath19 is the energy scale associated with nearest neighbor exciton repulsion . this number can get quite high : given a typical interlayer distance@xcite of @xmath20 and an intersite distance of @xmath21 the bare dipole interaction energy is 14 ev . in reality , we expect this energy to be lower due to quantum corrections and screening effects . however , the exciton - exciton interaction scale remains on the order of electronvolts and thus larger than the estimated heisenberg @xmath22 and hopping @xmath23 . let us finally consider the effects of interlayer hopping of electrons , which leads to the annihilation of excitons , @xmath24 this term explicitly breaks the @xmath3 symmetry associated with the conservation of excitons . while this term is almost certainly present in any realistic system , it is a matter of numbers whether it is relevant . in the present case of cuprates , where each layer can be doped by means of chemical substitution , we expect the chemical potential @xmath25 to be significantly larger than the interlayer tunneling @xmath26 . consequently , the interlayer hopping is barely relevant . throughout this publication we will discuss the effects that the inclusion of a small @xmath26 will have . the full model hamiltonian describing a finite density of excitons in a strongly correlated bilayer is thus @xmath27 let us now summarize the layout of our paper . most of the physics of hard - core excitons on a lattice can be captured using an effective @xmath1 model , which is studied in section [ secxxz ] . the ground state phase diagram of the full exciton @xmath0 model is derived in section [ secmft ] , using both numerical simulations and analytical mean field theory . the excitations and the corresponding susceptibilities are discussed in section [ seceex ] . we conclude this paper with a discussion on possible further lines of theoretical and experimental research in section [ secconc ] . the hamiltonian , equation ( [ fullh ] ) , has five model parameters : @xmath22 , @xmath28 , @xmath23 , @xmath19 and @xmath25 . however , most properties of the excitons can be understood by considering the problem of hard - core bosons on a lattice . in this section we will argue that the exciton degrees of freedom can be described by an effective @xmath1 model . based on some reflections on the mathematical symmetries of the full exciton @xmath0 model , we will describe the properties of this effective @xmath1 model in subsection [ subsecxxz1 ] . we will conclude this section with an outline of the method used to obtain the excitation spectrum of the model . before characterizing different phases of the model we need to assess the algebraic structure of the exciton @xmath0 model . the set of all operators that act on the local hilbert space form the _ dynamical algebra _ , whereas the symmetries of the system are grouped together in the _ symmetry algebra_. to derive the dynamical algebra , it is instructive to start with the bilayer heisenberg model which has , on each interlayer rung , a @xmath29 dynamical algebra@xcite . upon inclusion of the exciton hopping term we need more operators , since now the local hilbert space on an interlayer rung is five - dimensional ( four spin states and the exciton ) . consider the spin - to - exciton operator @xmath30 and its conjugate @xmath31 . their commutator reads @xmath32 = |e \rangle \langle e | - | s\ ; m \rangle \langle s \ ; m | \equiv 2 e^z_{sm}\ ] ] where we have introduced the operator @xmath33 to complete a @xmath34 algebraic structure . we could set up such a construction for each of the four spin states @xmath15 . under these definitions the exciton hopping term , equation ( [ excitonhop ] ) , can be rewritten in terms of an @xmath35-model for each spin state , @xmath36 where the sum over @xmath37 runs over the singlet and the three triplets . note that the exciton chemical potential , equation ( [ hmu ] ) , acts as an externally applied magnetic field to this @xmath35-model , and that the exciton - exciton repulsion , equation ( [ hv ] ) , can be rewritten as an antiferromagnetic ising term in the @xmath33 operators . the dynamical algebra therefore contains four @xmath34 algebras in addition to the @xmath38 from the bilayer heisenberg part . the closure of such an algebra is necessarily @xmath39 , which is the largest algebra possible acting on the five - dimensional hilbert space . hence we need a full @xmath39 dynamical algebra to describe the exciton @xmath0 model at finite density . the operators that compose this algebra are enumerated in appendix [ appendixa ] . from the @xmath35-representation of the hopping term one can already deduce that we have four distinct @xmath3 symmetries associated with spin - exciton exchange . the bilayer heisenberg model contains two separate @xmath34 symmetries , associated with in - phase and out - phase interlayer magnetic order . therefore the full symmetry algebra of the model is @xmath40 ^ 2 \times [ u(1)]^4 $ ] . breaking of the @xmath34 symmetry amounts to magnetic ordering , which is most likely antiferromagnetic ( and therefore also amounts to a breaking of the lattice symmetry ) . each of the @xmath3 algebras can be broken leading to exciton condensation . note that next to possible broken continuous symmetries , there also might exist phases with broken translation symmetry . the checkerboard phase , already anticipated in the introduction , is an example of a phase where the lattice symmetry is broken into two sublattices . when discussing the dynamical algebra of the exciton @xmath0 model we found that the exciton hopping terms are similar to an @xmath35-model . the main reason is that the excitons are , in fact , hard - core bosons and thus allow for a mapping onto pseudospin degrees of freedom . viewed as such , the exciton - exciton interaction equation ( [ hv ] ) is similar to an antiferromagnetic ising term and the exciton chemical potential equation ( [ hmu ] ) amounts to an external magnetic field in the @xmath41-direction . together they form an @xmath1-model in the presence of an external field , which has been investigated in quite some detail elsewhere@xcite as well as in the context of exciton dynamics in cold atom gases@xcite . in order to understand the basic competition between the checkerboard phase and the superfluid phase of the excitons , it is worthwhile to neglect the magnetic degrees of freedom and study first this effective @xmath1-model for the excitons only . the transition between the checkerboard and superfluid phases is known as the ` spin flop'-transition@xcite . keeping the identification of the exciton degrees of freedom as @xmath1 pseudospin degrees of freedom in mind , let us review the basics of the @xmath1 hamiltonian @xmath42 where @xmath43 creates a hard - core bosonic particle @xmath44 out of the vacuum @xmath45 . this model has a built - in competition between @xmath46 , which favors a superfluid state , and @xmath47 , which favors a crystalline state where all particles are on one sublattice and the other sublattice is empty . the external field or chemical potential @xmath25 tunes the total particle density . the ground state can now be found using mean field theory . it is known that for pseudospin @xmath48 models in @xmath49d the quantum fluctuations are not strong enough to defeat classical order and therefore we can rely on mean field theory , as supported by exact diagonalization studies@xcite . to find the ground state we introduce a variational wavefunction describing a condensate of excitons , @xmath50 the mean - field approximation amounts to choosing @xmath51 constant and @xmath52 only differing between the two sublattices . we find the following mean - field energy @xmath53 let s rewrite this in terms of @xmath54 and @xmath55 , @xmath56 when @xmath57 the ground state is fully polarized in the @xmath41-direction . this means either zero particle density for negative @xmath25 , or a @xmath58 for the positive @xmath25 case . starting from the empty side , increasing @xmath25 introduces a smooth distribution of particles . this phase amounts to the superfluid phase of the excitons . the particle density on the two sublattices is equal and the total density is given by @xmath59 at the critical value of the chemical potential @xmath60 a first order transition occurs towards the checkerboard phase : the spin flop transition . in the resulting phase , which goes under various names such as the antiferromagnetic model . to avoid confusion , from now on we will use the term antiferromagnetism only when referring to the spin degrees of freedom in the full exciton @xmath0 model . ] , solid , checkerboard or wigner crystalline phase , the sublattice symmetry is broken . the resulting ground state phase diagram is shown in figure [ figxxz]a , where we also show the dependence of the particle density on @xmath25 . * a. * the ground state phase diagram of the @xmath1 model , equation ( [ xxzmodel ] ) . the graph shows the mean field particle density @xmath61 as a function of @xmath25 , with model parameters @xmath62 and @xmath63 . one clearly distinguishes the fully polarized phases for large @xmath25 , the superfluid phase with a linear @xmath61 vs @xmath25 dependence and the crystalline checkerboard phase with @xmath64 . in between the checkerboard and the superfluid phase a non - trivial first order transition exists , with a variety of coexistence ground states with the same ground state energy . the insets show how the @xmath65-vectors look like in the different phases . * b. * finite temperature phase diagram of the @xmath1 model with the same parameters . the background coloring corresponds to a semiclassical monte carlo computation of @xmath61 , the solid lines are analytical mean field results for the phase boundaries . we indeed see the checkerboard phase and the superfluid phase , as well as a high - temperature non - ordered ` normal ' phase . ] at finite temperatures in @xmath49d there can be algebraic long - range order . at some critical temperature a kosterlitz - thouless phase transition@xcite will destroy this long - range order . the topology of the phase diagram however can be obtained using the finite temperature mean field theory for which we need to minimize the mean field thermodynamic potential@xcite @xmath66 \nonumber \\ & & - \frac{\mu}{2 } \tanh \left ( \frac{\beta m}{2 } \right ) \cos \delta \theta \cos \overline{\theta}.\end{aligned}\ ] ] expectation values are @xmath67 and the parameter @xmath68 needs to be determined self - consistently . the resulting phase diagram is shown in figure [ figxxz]b , which is of the form discussed by fisher and nelson@xcite . the first order quantum phase transition at @xmath69 turns out to be non - trivial , a point which is usually overlooked in the literature . a trivial first order transition occurs when there are two distinct phases with exactly the same energy . in the case presented here , there is a infinite set of mean field order parameters all yielding different phases yet still having the same energy . a simple analytic calculation shows that the energy of the ground state at the critical point is @xmath70 . now rewrite the mean field parameters @xmath71 and @xmath72 into a sum and difference parameter @xmath73 for each value of @xmath74 with @xmath75 we can find a value of @xmath76 such that the mean field energy is exactly @xmath77 . this has interesting consequences . if one can control the density instead of the chemical potential around a first order transition , in general phase separation would occur between the two competing phases . from the mean field considerations above it is unclear what would happen in a system described by the @xmath1 hamiltonian , equation ( [ xxzmodel ] ) . all phases would be equally stable , at least on the mean field level , and every phase may occur in regions of any size . such a highly degenerate state may be very sensible to small perturbations . we consider it an interesting open problem to study the dynamics of such a highly degenerate system , and whether this degeneracy may survive the inclusion of quantum corrections . in the introduction we mentioned the existence of interlayer hopping , equation ( [ interlayerhopping ] ) . qualitatively the @xmath26 is irrelevant , which can be seen in the @xmath1 pseudospin language where it takes the form of a tilt of the magnetic field in the @xmath78-direction , @xmath79 as a result the phase diagram is shifted but not qualitatively changed . the effect of the @xmath26 on the excitation spectrum is briefly discussed in section [ sfphasesection ] . of direct experimental relevance are the elementary excitations of a phase . the dispersion of these excitations can be computed using the ` equations of motion'-method based on the work of zubarev@xcite . we present the formalities of this method in appendix [ appendixc ] . in this subsection we briefly show the essence of this technique , applied to the @xmath1 model . later , in section [ seceex ] , we will compute the excitations for the full exciton @xmath0 model . the key ingredients of this zubarev - approach are the heisenberg equations of motion , @xmath80 where @xmath81 runs over all nearest neighbors . these equations can not be solved exactly , and one relies on the approximation controlled by the mean field vacua . that is , we neglect fluctuations of the order parameters , so that products of operators on different sites are replaced by@xcite @xmath82 where @xmath83 denotes the mean field expectation value . by such a decoupling the heisenberg equations of motion become a coupled set of linear equations which can be solved easily . in the homogeneous phase we thus obtain , after fourier transforming , @xmath84 we find an analytical expression for the excitations in the superfluid phase , @xmath85 where @xmath86 . for small momenta this excitation has a linear dispersion , conform to the goldstone theorem requiring a massless excitation as a result of the spontaneously broken @xmath3 symmetry . exactly at @xmath87 the dispersion reduces to @xmath88 , hence the gap at @xmath89 closes thus signaling a transition towards the checkerboard phase . at the critical point and in the checkerboard phase , we need to take into account the fact that expectation values of operators differ on the two sublattices . the heisenberg equations of motion now reduce to six ( instead of three ) linear equations , which can be straightforwardly solved . for now we postpone the discussion on the dispersion of elementary excitations to section [ seceex ] , where the full exciton @xmath0 model will be considered using the technique discussed here . in the previous section we have seen that the effective @xmath1 model predicts the existence of both an exciton superfluid phase and a checkerboard phase , separated by a first order transition . now we derive the ground state phase diagram for the full exciton @xmath0 model given by equation ( [ fullh ] ) . we will proceed along the same lines as in the previous section , starting with a variational wavefunction . numerical simulation of this wavefunction creates an unbiased view on the possible inhomogeneous and homogeneous ground state phases . this serves as a basis to further analyze the phase diagram with analytical methods . the analytical mean field theory also allows us to characterize the three homogeneous phases : the antiferromagnet , the superfluid and the checkerboard crystal . finally , combining the numerical and analytical mean field results we obtain the ground state phase diagram , see figure [ finalphasediagramfig ] . recall that the local hilbert space consists of four spin states @xmath15 and the exciton state @xmath14 . we therefore propose a variational wavefunction consisting of a product state of a superposition of all five states on each rung . for the spin states we take the @xmath38 coherent state@xcite @xmath90 which needs to be superposed with the exciton state , @xmath91 to obtain the total variational ( product state ) wavefunction @xmath92 this full wavefunction acts as ansatz for the numerical simulations . note that the homogeneous phases can be described by this wavefunction with the parameters @xmath93 and @xmath76 only depending on the sublattice . given this wavefunction , the expectation value of a product of operators on different sites decouples , @xmath94 . the only nonzero expectation values of spin operators are for @xmath95 and it equals @xmath96 where @xmath97 is the unit vector described by the angles @xmath98 and @xmath99 . this variational wavefunction therefore assumes interlayer nel order of magnitude @xmath100 , which enables us to correctly interpolate between the perfect nel order at @xmath101 and the singlet phase @xmath102 present in the bilayer heisenberg model . the exciton density at a rung @xmath8 is trivially given by @xmath103 . given the variational wavefunction , we can use simulated annealing to develop an unbiased view on the possible mean field ground state phases . therefore we start out with a lattice with on each lattice site the variables @xmath52 , @xmath104 , @xmath105 , @xmath51 and @xmath103 and with periodic boundary conditions . the energy of a configuration is @xmath106 we performed standard metropolis monte carlo updates of the lattice with fixed total exciton density . the fixed total exciton density is imposed as follows : if during an update the exciton density @xmath103 is changed , the exciton density on one of the neighboring sites is corrected such that the total exciton density remains constant . the main results of the simulation are shown in figure [ mcresults ] , for various values of the hopping parameter @xmath23 and exciton density @xmath76 . we performed the computations on a @xmath107 lattice . notice that even though true long - range order does not exist in two dimensions , the correlation length of possible ordered phases is larger than the size of our simulated lattice . the other parameters are fixed at @xmath108 mev , @xmath109 and @xmath110 ev . the heisenberg couplings @xmath108 mev and @xmath109 are obtained from measurements of undoped ybco - samples@xcite , which we consider to be qualitatively indicative of all strongly correlated electron bilayers . the dipolar coupling is estimated at 2 ev , following our discussion in the introduction . for each value of @xmath76 and @xmath23 we started at a high temperature @xmath111 ev , to slowly reduce the temperature to @xmath112 ev while performing a full update of the whole lattice 10 million times . we expect that by such a slow annealing process we obtain the true ground state of equation ( [ scenergy ] ) , devoid of topological defects . once we arrive at the low temperature state , we performed measurements employing 200.000 full updates of the system . results from the semi - classical monte carlo simulations . here shown are color plots , with on the horizontal axes the exciton density @xmath76 and on the vertical axes the hopping parameter @xmath23 ( in ev ) . other parameters are fixed at @xmath108 mev , @xmath109 and @xmath110 ev . the five measurements shown here are the nel order parameter , equation ( [ firstmcformula ] ) , the checkerboard order parameter , equation ( [ cbmcformula ] ) , the superfluid density , equation ( [ sfdensityformula ] ) , the phase coherence , equation ( [ phasemcf ] ) , and the ratio signaling phase separation according to equation ( [ lastmcformula ] ) , 0 means complete phase separation , 1 means no phase separation . notice that the prominent line at @xmath113 signals the checkerboard phase . ] we measured six different order parameter averages : * the nel order parameter defined by @xmath114 where we first sum over all spin vectors and then take the norm . * the checkerboard order , defined as the difference in exciton density between the sublattices divided by the maximal difference possible . the maximal difference possible equals @xmath115 , so @xmath116 * the superfluid density is given by the expectation value of the exciton operator . here we do not make a distinction between singlet exciton condensation or triplet exciton condensation . therefore @xmath117 * now the superfluid density is not the only measure of the condensate , we can also probe the rigidity of the phase @xmath118 . therefore we sum up all the phase factors on all sites , @xmath119 if the phase is disordered , this sum tends to zero . on the other hand , complete phase coherence in the condensate phase implies that this quantity equals unity . * finally , we considered a measure of phase separation between the checkerboard phase and the superfluid phase . if the exciton condensate and the checkerboard phase are truly coexisting , then the maximal superfluid density attainable would be @xmath120 where @xmath121 . if there is phase separation however , the actual superfluid density is less than this maximal density . therefore we also measured the ratio @xmath122 to quantify the extent of phase separation . when this ratio is less than @xmath123 this indicates phase separation . typical configurations for the exciton density per site , obtained in the monte carlo simulation on a @xmath124 square lattice . the color scale indicates the exciton density . all five figures have model parameters @xmath108 mev , @xmath109 and @xmath110 ev . * a : * separation between the antiferromagnetic phase ( without excitons , hence shown black ) and the exciton condensate with smooth exciton density ( @xmath125 , @xmath126 ev ) . * b : * separation between checkerboard - like localized excitons and an antiferromagnetic background ( @xmath127 , @xmath128 ev ) . * c : * separation between the checkerboard phase and a low density exciton condensate ( @xmath129 , @xmath126 ev ) . * d : * separation between the checkerboard phase and a high density exciton condensate ( @xmath130 , @xmath131 ev ) . * e : * the region where antiferromagnetic order , checkerboard order and the exciton condensate are all present ( @xmath132 , @xmath133 ) . ] the results for a full scan for the range @xmath134 and @xmath135 ev are shown in figure [ mcresults ] . in figures [ mcdensityplots ] and [ phasesepfig ] we have displayed typical exciton density configurations for various points in the phase diagram . in combination these results suggest that there are three homogeneous phases present in the system : the antiferromagnet at low exciton densities , the exciton superfluid at high exciton hopping energies and the checkerboard crystal at half - filling of excitons . however , for most parts of the phase diagram the competition between the three phases appears to result in phase separation . let us investigate the phase separation in somewhat more detail . in our earlier work we found that the motion of an exciton in an antiferromagnetic background leads to dynamical frustration@xcite . in other words : excitons do not want to coexist with antiferromagnetism . the introduction of a finite density of excitons will therefore induce phase separation . for large @xmath23 , we find macroscopic phase separation between the antiferromagnet and the exciton superfluid , see figure [ mcdensityplots]a . at low exciton kinetic energy the excitons will crystallize in a checkerboard pattern as can be seen in figure [ mcdensityplots]b . close to half - filling the role of the dipole repulsion @xmath19 becomes increasingly relevant . the first order ` spin flop ' transition we discussed in section [ subsecxxz1 ] implies that there will be phase separation between the superfluid and the checkerboard order . figures [ mcdensityplots]c and d show this phase separation . finally there is a regime where the condensate , the checkerboard order and the nel order are all present . however , given the dynamical frustration on the one hand and the spin - flop transition on the other hand , we again predict phase separation . a typical exciton configuration in this parameter regime is shown in figure [ mcdensityplots]e . different exciton configurations with their respective energies on a @xmath136 lattice , to show whether there is macroscopic phase separation . the model parameters are @xmath137 ev , @xmath108 mev , @xmath109 , @xmath110 ev and @xmath138 . yellow indicates the presence of excitons , and in the black regions there is antiferromagnetic order . * a : * the lowest energy state is the one with complete macroscopic phase separation . * b : * more complicated phase separation , such as the halter form depicted here , are higher in energy . * c : * starting at high temperatures with the configuration a , we slowly lowered the temperature . the resulting configuration shown here is a local minimum . * d : * using the same slow annealing as for c starting from configuration b. the local energy minimum obtained this way is lower in energy than the configuration c. we conclude that even though macroscopic phase separation has the lowest energy , there are many local energy minima without macroscopic phase separation . ] these simulated annealing results suggest that phase separation dominates the physics of this exciton system . to check whether the numerics are reliable we inspected directly the energies of the various homogeneous mean field solutions , using the maxwell construction for phase separated states . the constructed phase separated configurations and their energies are shown in figure [ phasesepfig ] . the lowest energy configuration ( [ phasesepfig]a ) has macroscopic phase separation between the checkerboard and the antiferromagnetic phase . intermediate states with one blob of excitons ( [ phasesepfig]c ) are slightly higher in energy than states with two blobs of excitons ( [ phasesepfig]d ) . however , even though macroscopic phase separation has the lowest energy , configurations with more blobs have more entropy . consequently for any nonzero temperatures complete macroscopic phase separation is not the most favorable solution . this is indeed seen in the numerical simulations : annealing leads to high - entropy states such as figure [ phasesepfig]d rather than to the lowest energy configuration . we thus conclude that the dominant phases are the antiferromagnet , the superfluid and the checkerboard . the competition between these three phases leads to phase separation in most parts of the phase diagram . the unbiased monte carlo simulations show the direction in which further analytical research should be directed : we will use mean field theory to characterize the three homogeneous phases . given the fact that we are dealing with a hard - core boson problem , we know that mean field theory is qualitatively correct . a remaining issue is whether one can tune the exciton chemical potential rather than the exciton density in realistic experiments . since we are prescient about the many first - order phase transitions in this system , we will perform the analysis with a fixed exciton density ( the canonical ensemble ) . using the maxwell construction and the explicit @xmath25 vs. @xmath76 relations , we can transform back to the grand - canonical ensemble . the numerical simulations suggest that the only solutions breaking translational symmetry invoke two sublattices , @xmath139 and so forth for @xmath140 , @xmath98 , @xmath118 and @xmath99 . this broken translational symmetry allows for the antiferromagnetic and exciton checkerboard order . evaluation of the energy @xmath141 of the variational wavefunction , equation ( [ variationalwavefunction ] ) , directly suggests that we can set @xmath142 on all sites . we restrict the spin vectors to be pointing in the @xmath143 direction only . since we anticipate magnetic ordering we have the freedom to choose the direction of the ordering . similar arguments hold for the choice @xmath144 ; when breaking the @xmath3 symmetry associated with exciton condensation we are free to choose the phase direction . ] we are left with four parameters @xmath145 and @xmath146 , and as it turns out it will be more instructive to rewrite these in terms of sum and difference variables , @xmath147 the mean field energy per site is now given by @xmath148 \nonumber \\ & & - \frac{1}{4 } zt \sqrt{((1 - \overline{\rho})^2 - \delta_\rho^2)(\overline{\rho}^2 - \delta_\rho^2 ) } \cos \delta_\chi \nonumber \\ & & - \mu \overline{\rho } + \frac{1}{2 } z v ( \overline{\rho}^2 - \delta_\rho^2 ) \label{meanfielde}\end{aligned}\ ] ] which has to be minimized for a fixed average exciton density @xmath149 with the constraint @xmath150 . the resulting mean field phase diagram for typical values of @xmath151 and @xmath19 , and for various @xmath152 , is shown in figure [ canonicaldiagram ] . the canonical mean - field phase diagram for typical values of @xmath108 mev , @xmath109 and @xmath110 ev whilst varying @xmath23 and the exciton density @xmath76 . in the absence of exciton , at @xmath153 , we have the pure antiferromagnetic nel phase ( af ) . exactly at half - filling of excitons ( @xmath154 ) and small hoping energy @xmath155 we find the checkerboard phase ( cb ) where one sublattice is filled with excitons and the other sublattice is filled with singlets . for large values of @xmath23 we find the singlet exciton condensate ( ec ) , given by the wavefunction @xmath156 . the coexistence of antiferromagnetism and superfluidity for small @xmath76 and @xmath23 is an artifact of the mean field theory . conform the monte carlo results of figure [ mcresults ] , for most parts of the phase diagram phase separation ( ps ) is found . ] as long as the exciton density is set to zero , the mean field ground state is given by the ground state of the bilayer heisenberg model , @xmath157 the nel order is given by @xmath158 and the energy of the antiferromagnetic state is @xmath159 the introduction of excitons in an antiferromagnetic background leads to dynamical frustration effects which disfavors the coexistence of excitons and antiferromagnetic order@xcite . in fact , the numerical simulations already ruled out coexistence of superfluidity and antiferromagnetism . for large exciton hopping energy @xmath23 it becomes more favorable to mix delocalized excitons into the ground state . due to the bosonic nature of the problem this automatically leads to exciton condensation . the delocalized excitons completely destroy the antiferromagnetic order and the exciton condensate is described by a superposition of excitons and a singlet background , @xmath160 here we wish to emphasize the ubiquitous coupling to light of the superfluid . the dipole matrix element allows only spin zero transitions , and since the exciton itself is @xmath161 the dipole matrix element is directly related to the superfluid density , @xmath162 the dipole matrix element thus acts as the order parameter associated with the superfluid phase . in most bilayer exciton condensates , such as the one in the quantum hall regime@xcite , this order parameter is also nonzero in the normal phase because of interlayer tunneling of electrons . one can therefore not speak strictly about spontaneous breaking of @xmath3 symmetry in such systems ; there is already explicit symmetry breaking due to the interlayer tunneling . in strongly correlated electron systems the finite @xmath26 is small compared to the chemical potential @xmath25 . as discussed in the introduction , the mott insulating bilayers now effectively allow for spontaneous @xmath3 symmetry breaking , and the above dipole matrix element acts as a true order parameter . note that the irrelevance of interlayer hopping @xmath26 implies that this order parameter is , unfortunately , not reflected in photon emission or interlayer tunneling measurements . the exciton condensate is a standard two - dimensional bose condensate . the @xmath3 symmetry present in the @xmath35-type exciton hopping terms is spontaneously broken and we expect a linearly dispersing goldstone mode in the excitation spectrum , reflecting the rigidity of the condensate . we will get back to the full excitation spectrum in section [ seceex ] . the energy of the singlet exciton condensate is @xmath163 and the exciton density is given by @xmath164 whenever the exciton hopping is small , the introduction of excitons into the system leads to the ` spin flop ' transition towards the checkerboard crystalline phase . as shown in the context of the @xmath1 model , this phase implies that one sublattice is completely filled with excitons and the other sublattice is completely empty . on the empty sublattice , any nonzero @xmath28 will guarantee that the singlet spin state has the lowest energy . hence the average exciton density is here @xmath165 and the energy of the checkerboard phase is given by @xmath166 it is interesting to note that the checkerboard phase is in fact similar to a bose mott insulator : with the new doubled unit cell we have one exciton per unit cell . the nearest neighbor dipole repulsion now acts as the ` on - site ' energy preventing extra excitons per unit cell . within the analytical mean field theory set by equation ( [ meanfielde ] ) there exists a small region where antiferromagnetism and the exciton condensate coexist . there the energy of the homogenous coexistence phase is lower than the energy of macroscopic phase separation of the antiferromagnet and the condensate , as obtained using the maxwell construction . however , within numerical simulations we found no evidence of coexistence . instead , we found microscopic phase separation , which hints at a possible complex inhomogeneous phase . we therefore conclude that the homogeneous mean field theory discussed here is insufficient to find the true ground state . finally , when the exciton density is unity we have a system composed of excitons only . in the parlance of hard - core bosons this amounts to a exciton mott insulator . this rather featureless phase is adiabatically connected to a standard electronic band insulator : the system is now composed of two layers where each layer has an even number of electrons per unit cell . the energy of the exciton mott insulator is , trivially @xmath167 in this mean field theory most of the phase transitions are first order , with the exciton density varying discontinuously along the transition . the critical values of @xmath25 or @xmath168 for the first order transitions are @xmath169 the transitions towards the coexistence region from the antiferromagnet or the condensate are second order . additionally , the transition from the condensate to the exciton mott insulator is second order . the critical values of @xmath168 or @xmath25 at these second order transitions are @xmath170 the subscripts indicate the phases : antiferromagnetic phase ( af ) , coexistence phase ( co ) , exciton condensate ( ec ) , exciton mott insulator ( ei ) , checkerboard phase ( cb ) . for any nonzero @xmath171 the first order transitions from the antiferromagnetic or coexistence phase towards the checkerboard phase are ` standard ' in the sense that at the critical value of @xmath25 there are only two mean field states with equal energy . this is also true for the transitions from the antiferromagnet to the exciton condensate except at a single point . at the tricritical point @xmath172 separating the coexistence phase , the antiferromagnetic phase and the exciton condensate , we can set the parameters @xmath173 , @xmath174 and @xmath175 given by the value in the coexistence phase . now the energy becomes independent of the exciton density @xmath149 . similarly , at the critical value of @xmath176 describing the transition between the checkerboard phase to the singlet exciton condensate , we can choose the mean field parameters @xmath173 , @xmath177 and @xmath178 with these parameters , the energy becomes independent of @xmath76 . this implies that the mean field theory predicts highly degenerate states at the critical values of @xmath25 , similar to the one we found in the @xmath1 model . the phase separation that thus occurs can be between an infinite set of possible ground states that have all a different exciton density . coincidentally , the numerical simulations indicate that around the two ` degenerate ' critical points indeed all the three phases are present . while the macroscopic phase separated state might have the lowest energy , figure [ phasesepfig ] suggests that more complicated patterns of phase separation are likely to occur . the degeneracy of the critical points on the level of mean fields theory might be responsible for richer physics in these special regions of the phase diagram . the canonical ground state phase diagram of the exciton @xmath0 model , which is a combination of the semi - classical monte carlo result and the mean field computations . in the background we have put the mean field phase diagram of figure [ canonicaldiagram ] , whilst the lines show the phase diagram as obtained from the monte carlo simulations . the dotted area represents phase separation between the condensate , antiferromagnetic and checkerboard order . furthermore : ec means exciton condensate , cb means checkerboard phase , af means antiferromagnetism and ps stands for phase separation . ] finite temperature graph of the phase coherence in the exciton condensate region of the phase diagram . here @xmath179 ev and @xmath180 and the other parameters are the same as in a. a clear transition is observed at around 0.06 ev , which amounts to a transition temperature of about 700 kelvin . ] combining the simulated annealing results of figure [ mcresults ] with the analytical mean field results of figure [ canonicaldiagram ] we arrive at the definitive mean field phase diagram of the exciton @xmath0 model in figure [ finalphasediagramfig ] . there are three main phases : the antiferromagnet at zero exciton density , the checkerboard crystal at exciton density @xmath154 and the superfluid at high hopping energy @xmath23 . for most parts of the phase diagram , phase separation between these three phases occurs in any possible combination . the competition between these three phases leads generally to macroscopic phase separation . finally , within the limitations of the semi - classical monte carlo approach we deduce an estimate of the transition temperature towards the superfluid state . given a typical point in the phase diagram where the exciton condensate exists , at @xmath179 ev and @xmath181 , we find a kosterlitz - thouless transition temperature of approximately 700 kelvin , see figure [ mcresults]c . this number should be taken not too seriously , as the exciton @xmath0 model might not be applicable at such high temperatures given possible exciton dissociation . additionally , at high temperatures the electron - phonon coupling becomes increasingly important , which we neglect in our exciton @xmath0 model . nonetheless , our estimate suggests that exciton superfluidity may extend to quite high finite temperatures . each phase of the excitons in the strongly correlated bilayer has distinct collective modes , that are in principle measurable by experiment . in order to obtain the dispersions of the collective modes we employ the technique of the heisenberg equations of motion , introduced in the context of the @xmath1 model in section [ xxzexc ] and further formalized in appendix [ appendixc ] . in the case of the exciton @xmath0 model the set of equations is larger and analytical solutions can in general not be obtained . whenever this is the case we compute the dispersions numerically . quantities of direct experimental relevance are the dynamical susceptibilities . we are for instance interested in the absorptive part of the dynamical magnetic susceptibility , defined by @xmath182 here @xmath183 is the ground state of the system and @xmath184 are the excited states with energy @xmath185 . it appears unlikely that bilayer exciton systems can be manufactured in bulk form which is required for neutron scattering , while there is a real potential to grow these using thin layer techniques . therefore the detection of the dynamical spin susceptibility forms a realistic challenge for resonant inelastic x - ray scattering ( rixs)@xcite measurements with its claimed sensitivity for interface physics@xcite . furthermore we are interested in the charge dynamical susceptibility @xmath186 which is directly related to the polarization propagator . we use the operator @xmath187 because this amounts to the interlayer dipole matrix element . therefore , this charge dynamical susceptibility expresses the excitonic excitations . it can be observed by optical absorption experiments@xcite at @xmath188 . finite wavelength measurements may be obtained using the aforementioned rixs@xcite technique , or using electron energy loss spectroscopy ( eels)@xcite . the method we use to compute the susceptibilities , based on the heisenberg equations of motion method , is also described in appendix [ appendixc ] . the three dominant phases we encountered in our mean field analysis will have distinct magnetic and optical responses . let us briefly summarize our main findings with respect to the collective excitations . the results for the antiferromagnetic phase are shown in figures [ afspinwaves ] to [ afantiadiabatic ] . this limit of vanishing exciton density has been studied with in far greater rigor than our current zubarev method is capable of@xcite . we can therefore compare the results of the zubarev method with a full resummation of spin - exciton interactions using the self - consistent born approximation ( lsw - scba ) . it turns out that for small exciton kinetic hopping @xmath23 the non - interacting equations - of - motion method yields reliable results . for large @xmath23 one needs the full scba code to correctly reproduce the dynamical frustration effects of excitons in the antiferromagnetic background . the collective modes of the exciton condensate are shown in figures [ ec - exciton ] and [ ec - spin ] . due to the absence of dynamical frustration and the presence of a spin - gap we expect that these results survive in a fully interacting computation . in fact , here the modes of the simple hard - core boson system discussed in section [ secxxz ] can be used as a template . just as for the phase diagram , the qualitative features of @xmath1 model are still of relevance for the more complicated @xmath0 model . nonetheless , in this condensate phase the interplay between excitonic and magnetic degrees of freedom gives rise to a rather counterintuitive effect . we find that the _ exciton superfluid density _ can be detected directly in a measurement of the _ magnetic excitations _ , as we already announced elsewhere@xcite . in contrast , in the checkerboard crystalline phase the spin and exciton degrees of freedom are once again decoupled . in the remainder of this section we will elaborate further on these results for each phase separately . throughout the following discussion , the model parameters are @xmath108 mev , @xmath109 , @xmath189 ev and a varying @xmath23 and @xmath76 . in order to visualize the susceptibilities we have convoluted @xmath190 with a lorentzian of width @xmath191 ev . the color scale of the susceptibility plots is in arbitrary units . the spin wave dispersions ( * a. * ) and the dynamical magnetic susceptibility ( * b. * ) in the antiferromagnetic phase . in this phase , the spin wave dispersions are not influenced by exciton dynamics . as is known from previous studies , there are two transversal spin waves and two longitudinal spin waves@xcite . the transversal spin waves are gapless around either @xmath192 ( solid red line ) or the @xmath193 point ( dotted blue line ) . the longitudinal spin waves , which are associated with interlayer fluctuations ( solid green line ) , are nearly flat and have a gap of order @xmath194 . the dynamic magnetic susceptibility ( * b. * ) only shows one transversal spin wave . these results and all subsequent figures are obtained using @xmath195 mev and @xmath109 , as is expected for the undoped bilayer cuprate ybco @xcite . ] the exciton modes in the antiferromagnetic phase in the adiabatic regime @xmath196 . here we have chosen @xmath197 ev , @xmath195 mev and @xmath198 . within the equations of motion picture there are four exciton modes ( * a. * ) , which come in pairs of two with a small interlayer splitting . due to the antiferromagnetic order the exciton bands are renormalized with respect to a free hard - core boson ( * b. * ) . the susceptibility corresponding to the free exciton motion ( * c. * ) is verified by the fully interacting lsw - scba results ( * d. * ) . this is to be expected : in the adiabatic regime spins react much faster than the exciton motion and the exciton still moves freely dressed by a spin polaron , reducing its bandwidth to order @xmath199 . ] the exciton modes in the antiferromagnetic phase in the antiadiabatic regime @xmath200 . here we have chosen @xmath201 ev , @xmath195 mev and @xmath198 .. just like in figure [ afadiabatic ] we find four exciton bands ( * a. * ) , renormalized with respect to the free hard - core boson results ( * b. * ) . however , upon inclusion of the interaction the free susceptibility ( * c. * ) gets extremely renormalized ( * d. * ) . the large exciton kinetic energy together with the relatively spin dynamics create an effective potential for the exciton : the exciton becomes localized and the confinement generates a ladder spectrum . note that thus in the antiadiabatic regime the free results ( * a. * , * c. * ) can not be trusted . ] in the limit of zero exciton density we recover the well - known bilayer heisenberg physics@xcite . as discussed in section [ ssgrandcanonical ] , the spins tend to order antiferromagnetically . the excitations spectrum thus contains a goldstone spin wave with linear dispersion around @xmath192 and a similar mode centered around @xmath202 . in addition , the bilayer nature is reflected in the presence of two longitudinal spin waves with a gap of order @xmath194 and a narrow bandwidth of order @xmath28 . the excitation spectrum and the corresponding magnetic dynamical susceptibility is shown in figure [ afspinwaves ] . since the spin modes of the bilayer antiferromagnet are independent of any exciton degrees of freedom , we will not discuss these any further . the dynamics of an isolated exciton in an antiferromagnetic background has been studied extensively by means of a linear spin - wave self - consistent born approximation technique ( lsw - scba)@xcite . the non - interacting equations of motion method used in this paper , amounts to the complete neglect of exciton - spin interactions , while these are on the foreground of the ( resummed ) lsw - scba computation . however , the mere existence of lsw - scba results allows us to compare it with our current non - interacting calculations . let us therefore first go through the lsw - scba results . there we need to distinguish between two limits : the adiabatic limit with @xmath196 shown in figure [ afadiabatic ] , and the anti - adiabatic limit where @xmath200 shown in figure [ afantiadiabatic ] . consider a single exciton in an antiferromagnetic background . now if this exciton hops to a neighboring site , it will leave behind two spins that are ferromagnetically aligned with their neighbors . this process is called dynamical frustration and limits severely the motion of an exciton . in the adiabatic limit ( @xmath196 ) this causes the exciton bandwidth to be drastically reduced to an order @xmath199 . in addition , the magnetic background acts as a confining potential leading to small but detectable ladder states at higher energies . at the other hand , in the anti - adiabatic regime @xmath200 exciton hopping will destroy the antiferromagnetic order as it will be surrounded by a cloud of frustrated spins . the quasiparticle picture completely breaks down and the spectral weight of the exciton is redistributed to a wide incoherent spectral bump . the ladder spectrum arising from the effective confinement will still be visible , though smeared out . the equations - of - motion method however ignores the effects of spin - exciton interactions such as dynamical frustration . it treats the excitons as well - defined quasiparticles . as such we can already guess beforehand that the non - interacting results will be reliable in the adiabatic regime . indeed , in the equations - of - motion method we find four exciton modes corresponding to either the singlet @xmath203 or @xmath204 triplet exciton @xmath205 operator , just as in the lsw - scba . when @xmath206 we can write out an analytical expression for the non - interacting dispersions , @xmath207 where each branch is twofold degenerate . this degeneracy is lifted when @xmath208 , leading to a splitting of order @xmath171 which is largest around @xmath192 and @xmath193 . in the limit of @xmath196 the dispersions , equation ( [ excdispersion ] ) , indeed result in an effective exciton bandwidth of order @xmath199 , conform the fully interacting theory as can be seen in figure [ afadiabatic ] . the natural question then arises : how is it possible that in the present non - interacting theory the exciton bandwidth depends on the spin parameter @xmath22 ? for sure , the effective exciton model introduced in section [ secxxz ] has no such renormalization as is shown in figure [ afadiabatic ] . there the exciton bandwidth fully depends on @xmath209 . however , it is important to realize that the exciton operators @xmath210 do not commute with the antiferromagnetic order parameter operator @xmath211 . as a result the mean field energy of exciting an exciton is shifted either up or down ( depending on the sublattice ) yielding a gap between the two exciton branches of @xmath212 . now for small @xmath23 , propagation of the exciton requires that one has to pay the energy shift @xmath194 to move through both sublattices . as a result the effective hopping is reduced by a factor @xmath168 . therefore the exciton bandwidth renormalization , seen in the full lsw - scba , is already present at the mean field level . for large @xmath168 however we will pay a price for the convenience of the non - interacting equations of motion method . at the mean field level one still expects the dispersions to be described by equation ( [ excdispersion ] ) . however , upon inclusion of the interaction corrections this picture breaks down completely . the bandwidth of the non - interacting exciton is of order @xmath209 , whereas in the interacting theory an incoherent ladder spectrum of the same width arises . thus for large @xmath168 the non - interacting results can not be trusted . however , this only applies to the antiferromagnetic phase due to the presence of dynamical frustration . in general , it appears that the non - interacting results are qualitatively correct in the absence of gapless modes that need to be excited in order for an exciton to move . this condition is naturally met for the other two phases . we therefore expect that exciton - spin interactions only lead to qualitative changes in the antiferromagnetic phase . by simple selection rules one can already conclude that the singlet exciton mode couples to light . as a consequence this is the mode that is visible in the charge dynamical susceptibility , which is related to the polarization propagator . the exciton excitations are shown in figures [ afadiabatic]d ( for @xmath213 ) and [ afantiadiabatic]d ( for @xmath214 ) . finally , note that at the transition from the antiferromagnetic phase to the checkerboard phase the gap in the exciton spectrum vanishes at @xmath202 . dispersions and susceptibilities of the goldstone mode associated with the exciton condensate . we have set @xmath215 ev , @xmath195 mev and @xmath198 , and the exciton density is either @xmath216 ( left column ) or @xmath217 ( right column ) . * a , b. * in the simple hard - core boson model the condensate phase clearly show the superfluid phase mode , linear at small momenta . * c , d . * in the full @xmath0 model the goldstone mode has a similar dispersion as in the @xmath1 model . the speed of the mode scales with the superfluid density . at higher densities the mode softens around @xmath202 , and when this gap closes a first order transition to the checkerboard phase sets in . * e , f . * the absorptive part of the charge susceptibility , which can be measured with for example eels or rixs . ] dispersions and magnetic susceptibilities of the exciton condensate . we have set @xmath215 ev , @xmath195 mev and @xmath198 , and the exciton density is either @xmath216 ( left column ) or @xmath217 ( right column ) . * as the exciton condensate is spin singlet , we assume that the excitation spectrum is governed by propagating triplet modes . these modes have a gap of order @xmath28 and a bandwidth of order @xmath194 . * c , d . * in contrast to the simple heisenberg results , the actual triplet modes have enhanced kinetics@xcite . the modes are split in a spin - dominated branch with small gap and large bandwidth proportional to the superfluid density ( * e , f . * ) ; and an exciton - dominated branch with a large gap and a small bandwidth ( * g , h . * ) . ] the mode spectrum of superfluid phase , as shown in figures [ ec - exciton ] and [ ec - spin ] , is characterized by a linearly dispersing goldstone mode associated with the broken @xmath3 symmetry . this superfluid phase mode has vanishing energy at the @xmath192 point , where we find the inescapable linear dispersion relation @xmath218 the speed of the superfluid phase mode is the same as for the @xmath1 model in equation ( [ xxzsuperfluidspeed ] ) up to a rescaling of the @xmath23 and @xmath19 parameters . indeed , this speed is proportional to the superfluid density @xmath219 . this mode can be seen in the charge susceptibility , figures [ ec - exciton]e and f. the goldstone mode has a gap at @xmath202 which decreases monotonically with increasing exciton density . precisely at the first order transition towards the checkerboard phase this gap closes . this mode softening at @xmath202 is reminiscent of the roton in superfluid helium : the wavelength of the roton is the same as the lattice constant of solid helium . next to the goldstone mode there are two triplet excitations , shown in figure [ ec - spin ] , each one three - fold degenerate . the degeneracy obviously arises from the standard triplet degeneracy @xmath220 . the two branches however distinguish between _ exciton - dominated _ modes and _ spin - dominated _ modes , let us discuss them separately . the spin - dominated modes have a gap of order @xmath221 , which is similar to the triplet gap in the bilayer heisenberg model for large @xmath171 . however , the bandwidth of these excitations scales with @xmath23 rather than with @xmath22 , as would be customary in a system without exciton condensation ( see figures [ ec - spin]a and b ) . we discussed this in great detail in recent work@xcite , so let us briefly review these results . in the absence of a excitons the motion of triplets is governed by the heisenberg superexchange yielding a bandwidth of order @xmath22 . now introduce fock operators @xmath222 and @xmath223 , so that the exciton - triplet exchange equation ( [ excitonhop ] ) reads @xmath224 this is an interaction term , thus seemingly irrelevant to the bandwidth of the triplet . however , when the exciton condensation sets in the operator @xmath225 obtains an expectation value , in fact @xmath226 where @xmath227 is the condensate density . therefore the higher order exchange term yields a quadratic triplet hopping term @xmath228 and the bandwidth of the triplet excitations becomes of order @xmath229 . now remember that the exciton hopping energy @xmath23 resulted , in second order perturbation theory , from the ratio @xmath230 where @xmath231 is the electron hopping energy and @xmath232 is the nearest neighbor coulomb repulsion@xcite . the heisenberg coupling however was given by @xmath233 where @xmath234 is the onsite coulomb repulsion . since for obvious reasons @xmath235 , we find that the triplet bandwidth is enhanced whenever exciton condensation sets is . this enhancement is clearly visible in the spin susceptibility @xmath236 , which allows for an experimental probe of the exciton superfluid density . the other branch of triplet excitations is dominated by triplet excitons , and is therefore barely visible in the spin susceptibility and not visible in the exciton susceptibility ( which only shows singlet excitons ) . that it is indeed dominated by triplet excitons can be inferred from computing the matrix elements of the operator @xmath237 , which are shown in figures [ ec - spin]g and h. furthermore , the gap @xmath238 is a function of exciton model parameters only . the bandwidth of this mode is of order @xmath239 , relatively independent of the exciton density . as a result , for large superfluid densities the exciton - dominated modes cross the spin - dominated triplet modes . one can directly see this in the excitation spectrum for @xmath217 as shown in figure [ ec - spin]d . we can compare the triplet spectrum to the mode spectrum of the singlet phase of the bilayer heisenberg model . when @xmath240 the ground state consists of only rung singlets . the excitation towards a triplet state , shown in figures [ ec - spin]a and b , has a gap @xmath241 and a bandwidth of order @xmath194 , which is considerably smaller than the @xmath239 bandwidth in the condensate . however , because the topology of the triplet mode is the same we expect that the effect of the spin - exciton interactions is the same in the bilayer heisenberg model as for the superfluid . since earlier lsw - scba showed no changes in the spectrum due to interactions , we infer that the non - interacting results for the superfluid are reliable . to conclude our discussion of the excitations of the superfluid phase let us consider the influence of the interlayer tunneling . in the context of the @xmath1 model we noticed that interlayer tunneling has no qualitative influence on the phase diagram itself . however , the presence of a weak interlayer tunneling may act as potential pinningthe phase@xcite opening a gap in the superfluid phase mode spectrum of order @xmath242 . persistent currents can still exist , but one needs to overcome this gap in order to get the exciton supercurrent flowing . the excitation spectrum of the checkerboard phase . * a. * in the simple hard - core boson model there are two exciton modes associated with the doublon and the holon excitation . * b. * the spin modes are decoupled from the exciton modes in the full @xmath0 model . there is only one possible spin excitation : changing the singlet groundstate into a non - propagating triplet . * c. * the exciton modes , on the other hand , can still propagate . the excitation of removing an exciton can propagate through the checkerboard . * d. * the propagating mode that changes an exciton into a singlet is detectable by optical means and thus shows up in the charge susceptibility . ] the third homogeneous phase of the exciton @xmath0 model is the checkerboard phase . in this phase the unit cell is effectively doubled with one exciton per unit cell . this state is analogous to a bose mott insulator . the trivial excitations are then the doublon and the holon : create two bosons per unit cell which costs an energy @xmath243 or to remove the boson . the latter will generate a propagating exciton mode , with dispersion @xmath244 there are two such propagating modes : one associated with the singlet exciton and one with the triplet exciton . precisely at the transition towards the superfluid phase , one of these exciton waves becomes gapless . note that the arguments that lead to the bandwidth renormalization in the antiferromagnetic phase also apply here , leading to an exciton bandwidth of order @xmath245 . the dispersions and the corresponding charge dynamical susceptibility can be seen in figure [ dispersionfigcb ] . in the spin sector one can excite a localized spin triplet on the empty sublattice . the triplet gap is set by the interlayer energy @xmath28 , and the dispersion is flat because this triplet can not propagate , as can be seen in figure [ dispersionfigcb]b . we have studied the possibility of exciton condensation in strongly correlated electron bilayers . starting from the description of the mott state , with localized electrons , an exciton is defined as an interlayer bound state of a double occupied and vacant site . in the strong coupling limit , as of relevance to laboratory systems based on mott insulators , the physics of such a system is described by the exciton @xmath0 model , equation ( [ fullh ] ) . we constructed the ground state phase diagram ( figure [ finalphasediagramfig ] ) , based on both numerical simulations and analytical mean field theory . three distinct phases are dominant : the antiferromagnetic phase , the checkerboard phase and the exciton condensate . for most parts of the phase diagram however , macroscopic phase separation will occur between these three phases . measurements of the spin and charge susceptibilities may discern in which one of the three main phases a specific system is in . the antiferromagnetic phase is characterized by a spin wave centered at @xmath202 , whilst in the exciton condensate the triplet bandwidth acts as a probe for the superfluid density ( see figure [ ec - spin ] and ref . @xcite ) . in the checkerboard phase the spin degrees of freedom are reflected only in a localized triplet excitation at low energy . the charge dynamic susceptibility shows distinct qualitative behavior depending on the phase . in the antiferromagnet the spin - exciton interactions play an important role @xcite . the superfluid phase is characterized by the visibility of the condensate goldstone mode , whereas the checkerboard phase has propagating exciton waves with bandwidth @xmath246 . note however that since we expect phase separation to occur for most model parameters , realistic samples will likely display features from all phases in its susceptibilities . our theoretical work presented here is largely based on the assumption of strong coupling . in this limit , the excitons behave as local hard - core bosons . if the exciton binding energy is less dominant , the exciton will extend over more lattice sites and thus probably enable coexistence phases . on the other hand we expect that spin - exciton interactions destabilize the coexistence phases , since these interactions generally lead to frustration effects . one could also wonder what happens if one includes longer - ranged interactions for the excitons , with the possibility of exciton stripes and incommensurate charge ordered phases@xcite . next , we are dealing with first order phase transitions where small changes may have severe consequences . combining all these effects may lead to significant changes in the phase diagram , most notably in the regime where we predict phase separation . within the context of the strongly coupled exciton @xmath0 model , a weaker exciton binding energy can be incorporated via interaction and hopping terms for the next nearest neighbors , next next nearest neighbors , etcetera . this might lead to complex ordered phases such as stripes@xcite . such phases are found in many strongly correlated electron systems@xcite , and studying these in the context of the simple bosonic exciton @xmath0 model might shed new light on the more troublesome fermionic @xmath0 model . in addition to stripy behavior other non - trivial exciton density profiles may occur when one considers a density imbalance between the electrons and holes . semiconductor imbalanced systems are predicted to exhibit fulde - ferrell - larkin - ovchinnikov density modulated phases@xcite . it is worthwhile to investigate whether such phase can exist in strongly correlated electron systems . next to an improvement of the phase diagram , we can also improve the susceptibilities by including the effect of exciton - spin interactions . similar to our earlier work @xcite on the interaction between excitons and spins in the limit of a single exciton , one could perform a diagrammatic expansion of these interactions . we expect that , apart from our earlier results in the antiferromagnetic phase , inclusion of spin - exciton interactions will not qualitatively alter the excitation spectra . experimentally , the close coupling of p- and n - doped mott insulators is still relatively ill explored . however , important advances in complex oxide thin film growth , by techniques such as molecular beam epitaxy ( mbe ) and pulsed laser deposition ( pld ) equipped with in situ monitoring tools such as reflective high energy electron diffraction ( rheed ) are making it possible now to grow multilayers of perovskite oxides - of which many are mott insulators - with unit cell precision . a complicating factor in fabricating multilayers of p- and n - doped perovskites , like the cuprate family from which also the high - tc superconductors are derived , are the oftentimes conflicting ( de)-oxygenation requirements . optimized deposition and post - anneal procedures have made it possible however to make thin film contacts between n- and p - doped superconducting cuprates @xcite , which is now further being explored in our labs to create and study the parallel n - p combinations resembling the theoretical model . an interesting additional system that can be included in this endeavor is the 2-dimensional electron gas that is formed at the interfaces between selected oxide band - insulators such as srtio@xmath247 and laalo@xmath247 . in this respect it is noteworthy that in specific configurations , in particular a 1 unit cell srtio@xmath247 layer on top of a 2 unit cell laalo@xmath247 layer grown on tio@xmath248-terminated srtio@xmath249 , a system of a closely coupled 2-dimensional electron gas and a 2-dimensional hole gas has been realized @xcite . finally we note that some cuprate high - tc materials appear to have an intrinsic stacking of electron - doped and hole - doped cuo@xmath248 layers , such as ba@xmath248ca@xmath247cu@xmath250o@xmath251f@xmath248 @xcite , where one could look for excitonic effects . this research was supported by the dutch nwo foundation through a vici grant . the authors thank sergei mukhin and kai wu for helpful discussions . in this appendix we will define the operators that compose the @xmath39 dynamical lie algebra , as described in the beginning of section [ algebrasec ] . from the bilayer heisenberg we already have the @xmath38 spin subalgebra @xmath252 where we use the obvious short - handed notation for the singlet and triplet kets and bras . the commutation relation between these operators read @xmath253 & = & i \epsilon^{abc } s^c \\ \left [ s^a , \widetilde{s}^b \right ] & = & \left [ \widetilde{s}^a , s^b \right ] = i \epsilon^{abc } \widetilde{s}^c \\ \left [ \widetilde{s}^a , \widetilde{s}^b \right ] & = & i \epsilon^{abc } s^c.\end{aligned}\ ] ] there are 12 exciton operators in the @xmath35-like part of the hamiltonian , which we denote by @xmath254 where @xmath255 and @xmath256 , with commutation relations @xmath257 & = & 2e^z_{sm } \\ \left [ e^z_{sm},e^+_{sm } \right ] & = & e^+_{sm}.\end{aligned}\ ] ] if we consider commutators between @xmath258-operators with different @xmath37 then we obtain operators that are fully spin - dependent . we find that the only nonzero commutators between @xmath259 operators with @xmath260 , @xmath261 & = & | s \ ; m \rangle \langle s ' \ ; m ' | \\ \left [ e^z_{sm},e^+_{s'm ' } \right ] & = & \frac{1}{2 } e^+_{s'm'}.\end{aligned}\ ] ] we need some more operators to close the spin @xmath262 subalgebra , therefore define @xmath263 operators by ( note that @xmath264 ) , @xmath265 to complete the spin @xmath262 subalgebra we define @xmath266 and the corresponding @xmath267 . finally , notice that we have one operator too much in our listing , since there are only 24 operators in @xmath39 . thus there exists a linear dependency relation between some operators , which is @xmath268 so when constructing the heisenberg equations of motion , we will exclude one of these three from our formalism . the most logical step is to throw out the combination @xmath269 and leave the sum @xmath270 the 24 operators of our dynamical @xmath39 algebra are the three @xmath271- , three @xmath272- , two @xmath273- , three @xmath274- , two @xmath193- and eleven @xmath258-operators . in this appendix we elaborate a bit further on the heisenberg equations of motion method , as introduced in paragraph [ xxzexc ] . the aim of this method is to find the spectrum of excitations , building on the foundations given by the mean field approximation . given a full set of local operators @xmath275 , we can construct the heisenberg equations of motion @xmath276\ ] ] which is in general impossible to solve . we employ the notation that @xmath8 indicates the lattice site , and @xmath277 is the index denoting the type of operator . the right hand side of this equation contains products of operators at different lattice sites . such products can be decoupled within the mean field approximation as @xcite @xmath278 where @xmath8 and @xmath279 are different lattice sites . upon fourier transforming lattice position into momentum and time into energy , we thus obtain a set of linear equations for the operators , @xmath280 the spectrum of excitations is simply found by solving this eigenvalue equation for the matrix @xmath281 . in order to find the matrix elements @xmath282 that enter in susceptibilities we need will introduce the following scheme . assume that the hamiltonian is of the form @xmath283 where the sum over @xmath284 runs over momenta , and @xmath285 indicates the different excited states . now @xmath286 is a creation operator , and irrespective of whether we are dealing with fermions or bosons we have the following equations of motion @xmath287 that is : every eigenvector of @xmath288 corresponding to a negative eigenvalue can be identified as a creation operator for one of the elementary excitations . however , the eigenvalue equation itself is not enough because it does not yield the proper normalization of @xmath289 . since we have the eigenvector solution @xmath290 we can write out the ( anti)commutation relation for @xmath286 in terms of the ( anti)commutation relations for the @xmath291 . upon requiring that on the mean field level the operators @xmath286 obey canonical commutation relations , that is for bosons @xmath292 \rangle = \delta_{nn'},\ ] ] we obtain a proper normalization for the new creation operators . we can invert the normalized matrix @xmath293 to express @xmath291 in terms of the creation operators @xmath286 . finally , using @xmath294 we can compute the wanted matrix element for @xmath291 . as an example of this technique we can compute the matrix element @xmath295 for the antiferromagnetic heisenberg model on a square lattice . the mean field ground state is the nel state , which leads to the following equations of motion , @xmath296 where the subscript @xmath297 and @xmath298 denote the two different sublattices , and @xmath299 . we quite easily infer that the eigenvalues are @xmath300 and thus we have one eigenvector corresponding to a creation operator , and one to an annihilation operator . if we define @xmath301 then the commutation relations tell us that the eigenvector matrix @xmath302 must satisfy @xmath303 \rangle = - 2 u_{11}^2 \langle s^z_{a } \rangle - 2 u_{12}^2 \langle s^z_{b } \rangle = - u_{11}^2 + u_{12}^2.\ ] ] the initial @xmath304 operator , which enters in the spin susceptibility , can be expressed in terms of the eigenvector matrix as @xmath305 some straightforward algebra now yields @xmath306 which is the same susceptibility one can obtain by using the holstein - primakoff linear spin wave approximation . the approximation scheme we introduced here can therefore be viewed as a generalization of the linear spin wave approximation .

we studied the possibility of exciton condensation in mott insulating bilayers . in these strongly correlated systems an exciton is the bound state of a double occupied and empty site . in the strong coupling limit the exciton acts as a hard - core boson . its physics are captured by the exciton @xmath0 model , containing an effective @xmath1 model describing the exciton dynamics only . using numerical simulations and analytical mean field theory we constructed the ground state phase diagram . three homogeneous phases can be distinguished : the antiferromagnet , the exciton checkerboard crystal and the exciton superfluid . for most model parameters , however , we predict macroscopic phase separation between these phases . the exciton superfluid exists only for large exciton hopping energy . additionally we studied the collective modes and susceptibilities of the three phases . in the superfluid phase we find the striking feature that the bandwidth of the spin - triplet excitations , potentially detectable by resonant inelastic x - ray scattering ( rixs ) , is proportional to the superfluid density . the superfluid phase mode is visible in the charge susceptibility , measurable by rixs or electron energy loss spectroscopy ( eels ) .

You are an expert at summarizing long articles. Proceed to summarize the following text: the late time acceleration of the universe can be explained by modifying general relativity ( gr ) on cosmological scales , avoiding the need of invoking a cosmological constant @xmath11 or an exotic repulsive fluid ( a.k.a . , dark energy ) . many popular modified gravity ( mg ) theories rely on an extra scalar field @xmath12 to mediate a fifth force , making the distributions and motions of galaxies different from those predicted by gr ( see * ? ? ? * and references therein ) . in particular , the coherent infall of galaxies onto massive clusters will exhibit systematic deviations due to the enhanced gravitational forces . @xcite ( 2013 , hereafter zw13 ) demonstrated that in @xmath13+gr simulations the velocity distribution of galaxies in the virial and infall regions of clusters ( hereafter abbreviated gik for galaxy infall kinematics ) is well described by a 2component velocity distribution model , which can be reconstructed from measurements of the redshift space cluster galaxy cross correlation function , @xmath10 . in this paper , we apply gik modelling to two suites of different mg simulations and investigate the possible signals of mg imprinted on the redshift space distribution of galaxies around clusters , using dark matter particles and halos as proxies for galaxies . ( for more general discussions of clusters as tests of cosmic acceleration theories we refer readers to @xcite , and for a succinct discussion of distinguishing mg from dark energy to @xcite 2009 . ) while deviations from gr may be welcomed on cosmological scales , a `` screening '' mechanism must be invoked in mg theories to recover gr in high density regions like the solar system , where gr has passed numerous stringent tests ( e.g. , * ? ? ? * ; * ? ? ? current viable screening mechanisms generally fall into two classes : * the chameleon like mechanism , in which the self interactions of the scalar field are regulated by a potential @xmath14 @xcite . objects are screened when their gravitational potential @xmath15 is larger than @xmath16 , where @xmath17 is the cosmic mean of @xmath12 and @xmath18 is the coupling between matter and @xmath12 . in other words , the effective scalar charge @xmath19 that responds to @xmath12 is reduced by the ambient gravitational potential . this type of screening operates in @xmath0 @xcite , symmetron @xcite ) , and dilaton @xcite theories . in this paper we will focus on the chameleon mechanism within @xmath0 , where the ricci scalar , @xmath20 , in the einstein hilbert action is replaced by @xmath21 where @xmath0 is an arbitrary function of @xmath20 . is tightly constrained by observations @xcite . * the vainshtein mechanism , in which the self interactions of the scalar field are determined by the derivatives of @xmath12 , which suppress the scalar field and fifth force in high density regions @xcite . scalar fields that exhibit vainshtein screening are generally called ` galileons ' because of an internal galilean symmetry @xcite . for an isolated spherical source , the force transition happens at a characteristic radius @xmath22 ( called the vainshtein radius ) , where @xmath23 is the schwarzschild radius of the source and @xmath24 in models of interest is on the order of the hubble radius @xmath25 . within @xmath26 the scalar field is suppressed ( @xmath27 ) , forming a `` sphere '' of screened region around the source . this mechanism is at play in the dvali porrati ( dgp , * ? ? ? * ) and massive gravity @xcite theories . for our purpose , we simplify this class of model as a theory with a @xmath13 background cosmology and an extra galileon type scalar field that manifests vainshtein screening @xcite . in both the @xmath0 and galileon models , the maximum force enhancement is @xmath28 times the normal gravity , but the `` fifth force '' that produces this enhancement has different ranges in the two models . since the chameleon scalar field becomes yukawa screened , the fifth force does not have infinite range , i.e. , it can not reach to cosmological scales . galileons , however , are never massive , so their force has an infinite range , thus having a much larger impact on linear perturbation theory than chameleons do . in the local universe , however , the chameleon screening predicts a richer set of observational signatures that are detectable with astrophysical tests , because it is possible to have order unity violation of the macroscopic weak equivalence principle ( wep ) , i.e. , extended objects do not fall at the same rate as in gr @xcite . in environments of low background @xmath15 , objects with deep gravitational potential can self screen , while those with shallow potential remain unscreened . for example , @xcite estimated that there could be up to @xmath29 separation of the stellar disk ( composed of self screened objects ) from the dark matter and gas ( both unscreened ) inside unscreened dwarf galaxies , using orbital simulations under @xmath0 . in contrast , there is no analogous order one violation in the vainshtein case , but the vainshtein `` spheres '' of individual objects interfere with each other . for example , in a two body system where the separation is @xmath30 , the interference reduces the infall acceleration , and this reduction becomes most significant for two objects with equal masses @xcite . the infall zone around clusters lies at the transition between the linear scale , where gravity is universally enhanced , and the local universe where gr is frequently recovered , providing a unique avenue for distinguishing mg from gr . however , in both screening mechanisms the scalar @xmath12 is coupled to density fluctuations via a nonlinear field equation , which can only be solved jointly with the matter field using numerical simulations . @xcite proposed a halo model - based approach to model the line - of - sight ( los ) velocity dispersions of galaxies in the infall region under both modified and normal gravities . in this study we hope to provide a complete picture of the coherent motions and distributions of galaxies around clusters in the two mg theories and their systematic deviations from gr , fully taking into account the nonlinearities that are intrinsic to the chameleon and vainshtein mechanisms . the gik model of zw13 describes the average galaxy infall in cluster centric coordinates in terms of a 2d velocity distribution at each radius , comprising a virialized component with an isotropic gaussian velocity distribution and an infall component described by a skewed 2d _ t_-distribution with a characteristic infall velocity @xmath31 and separate radial and tangential dispersions . the virialized component is confined within a `` shock radius '' that is close to the virial radius of the cluster , so in the infall region ( several to @xmath32 ) ] the gik model reduces to a single infall component . zw13 demonstrated that gik profiles can be robustly recovered from the measurement of @xmath10 within the millennium simulation @xcite , and they applied the method to rich galaxy groups in the sloan digital sky survey ( sdss , * ? ? ? in particular , the inferred @xmath31 profile provides a promising way of estimating the average dynamical mass of clusters and is likely insensitive to baryonic physics and galaxy bias ( zu et al . in prep ) . although the zw13 method is calibrated against gr simulations , we will show that the gik model is also an excellent description of the infall behavior in mg simulations . studies have shown that the peculiar velocities are more distinctively affected by modifications to gravity than the matter density field alone @xcite , and we expect the gik to be a particularly acute test of mg theories and their screening mechanisms . a virtue of using galaxy dynamics in the outskirts of clusters is that independent information on the average cluster mass profiles can be robustly extracted from stacked weak lensing ( wl ) experiments @xcite . since photons do _ not _ respond to the extra scalar field , lensing mass estimates will be different from dynamical mass estimates if gravity is modified from gr on relevant scales @xcite . however , implementing this test requires measuring mass profiles on scales where screening is inefficient , i.e. , the cluster infall region rather than within the virial radius . for any given cluster sample detected by imaging , x - ray , or sunyaev-zeldovich ( sz ) experiments , one can either directly compare the average lensing mass and the gik estimated dynamical mass or search for inconsistency between the measured @xmath10 and the gik predicted @xmath10 using lensing mass estimates and assuming @xmath13+gr . alternatively , a simplified test can be performed when wl measurements are unavailable . since any volume limited cluster sample is thresholded by some mass observable ( i.e. , galaxy richness , x - ray luminosity , or sz decrement ) that correlates with the true mass ( with some scatter ) , we can estimate the corresponding threshold in true mass using the abundance matching ( am ) technique . however , the uncertainties of am may be large when the scatter in the mass observable true mass relation is large or / and the completeness of the cluster sample is low . for the sake of simplicity , in this paper we concentrate on the am - based approach , and focus on cluster samples selected to have the same rank order in mass in the gr and mg simulations . we present the results of gik modelling of the chameleon and galileon simulations separately in the paper , as the two simulation sets were run with different initial conditions , cosmic expansion histories , and box sizes and resolution . we first introduce the chameleon simulations in [ sec : frsim ] , including the specific @xmath0 model , and halo statistics and kinematics . [ sec : gik ] presents the results of gik modelling and the measurements of @xmath10 for the chameleon clusters and compares that to gr . in [ sec : vain ] , we presents a similar set of results for the galileon clusters . we summarize and discuss the future prospects of our method in [ sec : con ] . for the chameleon modified gravity , we use a suite of large volume @xmath0 simulations ( @xmath33 box and @xmath34 particles ) evolved with the n - body code ecosmog @xcite . the same set of simulations has been used to study the non - linear matter and velocity power spectra @xcite , redshift space distortions @xcite , and halo and void properties @xcite in @xmath0 gravity . the large volume of the simulations allows us to derive robust statistics from a large number of massive clusters ( @xmath35 ) . we do not make distinctions between the main halos and sub halos , but include all the bound groups of particles identified by the spherical density halo finder ahf ( amiga s halo finder , * ? ? ? the halo mass is defined by @xmath36 , where @xmath37 is the overdensity for virialized halos ( @xmath38 at @xmath5 for typical @xmath13 cosmology ; * ? ? ? * ) and @xmath39 is the mean density of the universe . we will briefly describe the @xmath0 models here and refer the readers to @xcite for more details on the simulations . the simulations adopt a specific @xmath0 gravity model introduced by @xcite , with the functional form of @xmath0 as @xmath40 where the mass scale @xmath41 , @xmath42 and @xmath43 are dimensionless parameters , and @xmath44 controls the sharpness of the transition of @xmath0 from @xmath45 in high curvature limit ( @xmath20@xmath46@xmath45 ) to @xmath47 in low curvature limit ( @xmath20@xmath46@xmath48 ) . the corresponding scalar field in @xmath0 theory is @xmath49 , commonly denoted as @xmath50 ( i.e. , the scalaron ) . matching to the expansion history of flat @xmath13 universes ( e.g. , @xmath51 and @xmath52 ) requires a @xmath53 and a field value @xmath54 , where the subscript @xmath55 represents the present day values . therefore , the hu & sawicki model is effectively described by two parameters : @xmath44 and @xmath56 . models with @xmath57 are capable of evading solar system tests . for cosmological tests of chameleon theories , models with @xmath58 are ruled out by cluster abundance constraints from @xcite , and models with @xmath59 below @xmath60 are nearly indistinguishable from @xmath13 universes . we shall study two cosmologically interesting @xmath0 models with @xmath61 and @xmath62 , @xmath63 , which will hereafter be referred to as f5 and f4 , respectively . @xcite reported constraints of @xmath64 from other astrophysical tests , so these models may no longer be observationally viable , but they nonetheless provide useful illustrations of mg effects and a natural comparison for the vainshtein screening model discussed later . the expansion history of the two @xmath0 simulations is matched to one flat @xmath13 simulation that evolves from the same initial conditions under normal gravity with @xmath51 , @xmath52 , @xmath65 , and @xmath66 ( referred to as the `` gr '' simulation ) . the evolution of structure is the same in the three universes up to epochs around @xmath67 , which is the starting time of the simulations , since the fifth force in @xmath0 gravity is vastly suppressed until then . by studying the time evolution of the matter and velocity divergence power spectra in @xmath0 , @xcite showed that different @xmath0 models are in different stages of the same evolutionary path at any given time , and that varying the model parameter @xmath56 mainly varies the epoch marking the onset of the fifth force . the exact epoch of onset in each model depends on the scale of interest , i.e. , at higher redshift for smaller scales ( see figure 8 in * ? ? ? we choose to focus on the @xmath5 output of the simulations , mimicking the portion of the universe most observed by existing and near term future redshift surveys that contain large samples of clusters . figure [ fig : mf ] compares the halo mass functions from the gr , f5 , and f4 simulations . the bottom panel shows the fractional enhancement of the halo mass function in the @xmath0 simulations relative to the gr one . the shaded region ( @xmath68 ) indicates the mass range where the halo catalogs are incomplete and the bumps in the shaded area of the bottom panel are likely due to numerical effects . although sub halos were included in the halo catalogs , at @xmath69 the halo mass functions are mostly contributed by main halos . the two @xmath0 models predict very similar halo abundances below @xmath70 , enhanced by @xmath71 over the gr abundances . for halos in this mass range , the fifth force was activated early enough that the structure formation has somehow converged in the two @xmath0 models . however , on the high mass end , the halo mass function in the f4 model shows even stronger enhancement over gr , while in the f5 model the number of halos becomes closer to the @xmath13 prediction with increasing mass . the divergence of the halo mass functions predicted by the two @xmath0 models beyond @xmath72 indicates that the fifth force only started affecting the formation of massive clusters recently in the f5 model , producing smaller enhancement in the halo abundance compared to f4 . for some very massive clusters in the f5 model ( e.g. , see figure 3 in * ? ? ? * ) , the gravitational potential has begun to dominate the background scalaron , activating the chameleon mechanism to recover gr in the infall region , while in the f4 model the chameleon screening likely never activates anywhere in the universe . figure [ fig : mf ] is in good agreement with the study of @xcite , where a series of smaller but higher resolution @xmath0 simulations are employed . @xcite modeled the halo mass functions measured from the same sets of @xmath0 simulations using environment and mass dependent spherical collapse model in combination with excursion set theory . with neither well resolved sub halos nor simulated galaxies in the simulations , the common prescription for constructing mock galaxy catalogs is through halo occupation distributions ( hods , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . however , the minimum required mass threshold for a complete halo sample in the chameleon simulations , @xmath73 , is overly high for any meaningful hod galaxies similar to the milky way and m31 would be absent . therefore , we simply use particles and halos as our proxies for galaxies in the paper , and the behavior of hod galaxies should be intermediate between that of the `` particle '' galaxies and `` halo '' galaxies . since we are focused on the relative behavior of mg and gr simulations , the impact of choices for galaxy proxy on our conclusions should be small . we will usually refer simply to `` galaxies '' , when it is clear from context whether we are using particles or halos as our proxies . before going into the statistical properties of galaxy infall , we hope to gain some intuitive understanding of the differences in infall between mg and gr by looking at figure [ fig : vflow ] . taking advantage of the same initial condition shared by the @xmath0 and gr simulations , we locate at the frame center two primary clusters ( @xmath74 ) at @xmath5 that formed from the same seed in the initial density fluctuation field in the f4 ( red , for which the fifth force is stronger ) and gr ( gray ) simulations , respectively , and plot all halos with mass above @xmath75 as circles , with their relative velocities to the primary cluster indicated by the arrows . the dark matter density field in the gr simulation is illustrated by the grayscale background , highlighting the three filamentary structures that funnel the infalling halos . the radius of each circle is proportional to the halo mass , with the thick ones representing halos more massive than @xmath76 . the length of each arrow is @xmath77 , corresponding to the redshift space displacement ( in units of @xmath78 ) that would be seen by a distant observer aligned with the velocity vector . figure [ fig : vflow ] shows that the gik around _ individual _ clusters is highly anisotropic , and while the difference in the spatial distribution between gr and f4 halos is irregular and fairly mild , it is somewhat enhanced in redshift space . to avoid clutter , we do not show halos from the f5 model , which should display smaller differences from gr because of its smaller impact from the fifth force . in the next section , we will show that by stacking individual frames like figure [ fig : vflow ] for clusters of similar mass , the anisotropy goes away and the average infall kinematics can be well described by the gik model proposed in zw13 , for both the @xmath0 and @xmath13 models . more importantly , the enhanced difference in the redshift space between mg and gr models can be captured by systematic differences in the parameters of the gik model , namely , the characteristic infall velocity @xmath31 , the radial velocity dispersion @xmath79 , and the tangential velocity dispersion @xmath80 . the halo mass functions in figure [ fig : mf ] suggest that the fifth force became unscreened earlier in f4 than in f5 , which gives it more time to affect the velocity field in the former . as a result , at @xmath5 clusters in both the f4 and f5 models should exhibit more enhanced galaxy infall compared to gr , but we expect the enhancement to be more substantial in the f4 model . for the small number of screened clusters in the f5 model , the infalling galaxies around chameleon clusters should feel similar instantaneous accelerations as their counterparts around similar clusters in gr . however , the peculiar velocities of galaxies were enhanced by the fifth force when they were further away from the clusters . by the time they reached to the screened region , the peculiar velocities were already higher , so the infall stays stronger even though the underlying gravity recovers to gr . our goal here is to quantify this modification to galaxy infall induced by @xmath0 gravities as function of @xmath56 within the framework of gik modelling . to compare the average gik among the three simulations , we select dark matter halos with @xmath81 in the @xmath13 simulation as our fiducial gr cluster sample , and those with the same mass range in the @xmath0 simulations as the `` equal mass '' ( em ) cluster samples . as mentioned in the introduction , we also select specific halo samples in the @xmath0 simulations to have the same rank order in mass as the fiducial gr clusters , i.e. , the `` equal rank '' ( er ) cluster samples . the er clusters generally have slightly larger masses than the em ones , with @xmath82 and @xmath83 in f5 and f4 models , respectively . the er sample thus resembles the set of clusters that formed from the same initial density peaks as the fiducial gr ones . in the limit of very rare , highly biased peaks , the large scale cluster bias @xmath84 , defined by the ratio between the cluster matter correlation function and the matter auto correlation @xmath85 , is @xmath86 , yielding @xmath87 on large scales . operationally , the em comparison would be most relevant to an observational study of clusters whose virial masses are calibrated by wl ( and thus accurate in both gr and mg ) . alternatively , if one ranks clusters by a mass proxy such as galaxy richness , x ray luminosity , or sz signal , then selects clusters above a threshold ( i.e. , am method ) , the er comparison is more relevant . hereafter we simply denote the fiducial gr sample as `` gr '' while comparing it to the `` em '' and `` er '' samples in mg simulations . as mentioned in the introduction , the two component gik model is an excellent analytic description of the joint 2d distribution of radial and tangential velocities of galaxies in the cluster centric frame , @xmath88 . here we will present the results of gik modelling for the cluster samples defined above , and refer the readers to zw13 for details on the gik parameterization and fitting procedures . figure [ fig : vblob ] shows the best fitting @xmath88 for the gr ( top ) , f5em ( middle ) , and f4em ( bottom ) cluster samples at five different radial bins , using dark matter particles as proxy for galaxies . negative @xmath89 indicates falling toward clusters . following zw13 , we define @xmath90 as the tangential velocity component that is projected in the plane of los axis and galaxy position vector in the cluster centric frame ( see the 3d diagram in the figure 2 of zw13 ) . since the _ average _ galaxy motion around the cluster center is isotropic , the probability distribution of @xmath90 is symmetric about zero . hereafter we refer to @xmath90 simply as the `` tangential velocity '' . note that the hubble flow is subtracted when defining @xmath89 , but it will be incorporated when modelling @xmath10 . the gik of the three cluster samples in figure [ fig : vblob ] show some generic trends with radius : 1 ) the distribution has two distinct components on very small scales ( leftmost column ) but only shows a single infall component on large scales ; 2 ) the infall component is symmetric about the mean @xmath89 near the turn around radius , where infall velocity is comparable to hubble flow ( middle column ) ; 3 ) the @xmath89 distribution of the infall component is skewed toward positive velocities beyond the turn around radius ( two right columns ) , but is negatively skewed below that radius ( two left columns ) . the impacts of modified gravity on @xmath88 are subtle but nonetheless visually apparent in figure [ fig : vblob ] when comparing the three models at the same radial bin ( i.e. , within the same column ) . the solid vertical line in each panel indicates the most probable radial velocity , which shifts to more negative @xmath89 with increasing @xmath59 ( i.e. , from top to bottom ) at each radius . simultaneously , the dispersions of the infall component in the radial and tangential directions also increase as function of @xmath56 the joint distributions in the f4 model are more extended than in gr . this increased width results in the decreased peak amplitude of the distributions ( which are normalized to unity by definition ) . to quantify the differences in gik among the three simulations , we will focus on the impacts of modified gravity on three of the gik parameters ( @xmath31 , @xmath79 , and @xmath80 ) . we ignore the other gik parameters defined by zw13 ( seven in total ) , including two parameters describing the virialized component that are irrelevant to this paper , and two others that describe the skewness and the kurtosis of the infall component , which we found to be insensitive to modified gravity . note that the characteristic radial velocity @xmath31 is not the mean , median , or mode of the radial velocity distribution , but is a characteristic velocity naturally associated with the definition of the skewed _ t_-distribution used for describing the gik infall component ( see zw13 , equation 6 ) . figure [ fig : v6p ] presents the best fit gik parameters as functions of radius for the f5 ( left ) and the f4 ( right ) models , respectively . in each panel we show the parameter profiles for all three types of cluster samples ( gr , em , and er ) . the shaded region in each panel indicates the radial bins where gik is dominated by the virialized component and the fit to the infall component is less robust . for the f5 model , the halo mass function is close to that of @xmath13 at high masses , so the em and er samples have little difference in mass and their gik profiles look very similar . however , as expected from figure [ fig : vblob ] , the two f5 cluster samples in the @xmath0 simulation show stronger infall ( top left ) and larger velocity dispersions ( center and bottom left ) compared to the gr sample . for the f4 model , the difference between the em and er samples is larger , especially in the @xmath31 profiles ( top right ) , where the difference between the er and gr profiles is almost double that between the em and gr ones . for both samples , the f4 results are more easily distinguished from gr than the f5 results , as expected . since we anticipate that er comparisons will be more observationally relevant in most cases , we will focus henceforth on the gik of er cluster samples in the mg simulations . we highlight the differences in gik profiles between er and gr clusters for both chameleon models in figure [ fig : vdiff ] . the characteristic infall velocity profiles exhibit significant effects from the chameleon gravity on scales below @xmath91 , showing @xmath92 and @xmath93 enhancement at @xmath94 for the f5 and f4 models , respectively . beyond @xmath91 the @xmath31 profiles converge to the gr prediction . the dispersion profiles in @xmath0 models deviate from gr on all distance scales , with the differences almost constant and decreasing with radius for @xmath79 and @xmath80 , respectively . the magnitude of the deviations we see here is very encouraging @xmath92 difference in both the @xmath31 and the dispersions is already detectable within @xmath95 in zw13 , where a preliminary gik constraint is obtained using two samples of sdss rich groups ( with group number @xmath96 and @xmath97 , respectively ) and the sdss dr7 main galaxy sample . using dark matter particles as proxy for galaxies effectively assumes that galaxies have the same density profile and velocity distribution as dark matter particles within halos . in reality , we expect central galaxies to have low peculiar velocities relative to the halo center of mass , and the spatial distribution may be less concentrated than the matter ( see , e.g. , * ? ? ? * ) . to bracket the expectations for constraining gik using realistic galaxy samples , we repeat the above experiment using halos instead of particles as proxy for galaxies . this mimics the scenario where a luminous red galaxy ( lrg , * ? ? ? * ) galaxy sample is employed , implying approximately one galaxy per halo as one extreme of the hod @xcite . figure [ fig : vdiff_h ] summarizes the result of this experiment . we denote the curves correspondingly as `` ` gr_h ` '' , `` ` f5_h ` '' , and `` ` f4_h ` '' in the figure . the gik profiles are much noisier because of the rarity of halos , but the differences among the three samples are similar to those seen in figure [ fig : vdiff ] , but smaller in magnitude by about a factor of two . we infer that the difference in gik seen in figure [ fig : vdiff ] using `` particle '' galaxies has approximately equal contributions from two sources , chameleon modifications to the random motions within halos ( i.e. , ` 1-halo' ) and the impact of chameleon gravity on the bulk inflow of halos ( i.e. , ` 2-halo' ) . quantitative predictions for a particular galaxy sample will require simulations that resolve the host halos and thus allow a full hod model of the population , incorporating both 1-halo and 2-halo effects with appropriate weight . the redshift space cluster galaxy cross correlation function , @xmath10 , is a comprehensive characterization of the statistical relation between clusters and galaxies , influenced by both the real space cross correlation @xmath98 and the peculiar velocities induced by the cluster gravitational potential . mathematically , @xmath99 , a function of projected cluster galaxy separation @xmath100 and line - of - sight redshift separation @xmath101 , can be derived by convolving the real space @xmath102 with the hubble flow corrected los velocity distribution , which can be straightforwardly predicted from the gik model ( see equation 11 in zw13 ) . zw13 demonstrated that the gik can be extracted from the observed @xmath10 , by taking advantage of the non degenerate imprint of each gik element on the 2d pattern of @xmath10 . in this section we will examine the impact of chameleon gravity on the real and redshift space cluster galaxy correlation functions in the @xmath0 simulations , using particles as proxy for galaxies . for measuring @xmath98 , we count the numbers of particles around clusters in spherical shells of successive radii , ranging from @xmath103 to @xmath104 with logarithmic intervals , then average over all clusters in each bin and normalize by the particle numbers expected in a randomly located shell of equal volume . we measure @xmath10 in a similar way , counting galaxies in cylindrical rings of successive los distance @xmath101 for each projected separation @xmath100 ( assuming a distant observer approximation so that the los is an axis of the box ) . uncertainties in both measurements are estimated by jackknife re sampling the octants of each simulation box . we start by showing the real space cluster galaxy correlation function @xmath98 for the gr , f4er , and f5er cluster samples in the left panels of figure [ fig : xi3d_ucurve ] . in the top left panel , all three correlation functions exhibit a break at @xmath105 , marking the transition from the nfw like density distribution @xcite within halos to a biased version of the matter auto correlation function on large scales @xcite . the bottom left panel shows the ratio of @xmath98 between the er samples in @xmath0 simulations and the gr sample . on scales below the break radius , @xmath98 of the er clusters show enhancement of @xmath106 and @xmath107 in the f5 and f4 models , respectively , because they are intrinsically more massive than their counterparts in the @xmath13 simulation . on scales larger than @xmath108 , the er clusters have nearly the same large scale clustering as the gr sample . since we are measuring the cross correlation with dark matter particles , these @xmath98 profiles also determine the cluster galaxy wl profile ( see , e.g. , eq . 13 of @xcite 2012 ) . on intermediate scales , there is a bump at @xmath109 in the ratio between @xmath98 of the f4er and the gr samples , but not in the ratio curve for f5er . in the f4 model , the fifth force became unscreened from quite early time , so that the velocity field has been enhanced for a long time by @xmath5 . this means that the peculiar velocities are enhanced by roughly the same factor @xmath110 as the gravitational force , where @xmath110 is between unity and @xmath28 , and the kinetic energy of particles in f4 is thus @xmath111 times that in gr ; meanwhile , the gravitational potential in f4 is @xmath112 times as deep as in gr . the net result is that the mg effect on the particle kinetic energy dominates over its effect on the cluster potential , so that as a compromise the particles tend to move toward the outer parts of clusters . the situation is different in the f5 model , where the fifth force became unscreened quite late . by @xmath5 , the fifth force has become unscreened but only for a short period , so that particle velocities have not been significantly affected by it . on the other hand , due to the disappearance of the screening , the potential of the cluster suddenly became deeper . the result is a stronger mg effect on the potential than on the kinetic energy of particles , and particles tend to move toward the inner parts of clusters . these features have been observed in the halo density profiles of @xmath0 simulations before ( e.g. , * ? ? ? * ) , and similar ones have been found in coupled quintessence simulations . using cluster wl measurements , @xcite exploited this small scale enhancement of cluster density profiles in chameleon gravity and obtained a constraint of @xmath113 at @xmath114 confidence level . figure [ fig : xirs ] presents the redshift space cluster galaxy correlation functions for the three cluster samples , showing a stronger small scale fingers - of - god ( fog , * ? ? ? * ) effect with increasing @xmath59 , but with similar los squashing effect on large scales ( a.k.a . , kaiser effect , * ? ? ? however , the most easily visible features of @xmath10 in figure [ fig : xirs ] are driven by the radial gradient of cluster density profiles , which are fairly insensitive to the influence of chameleon modifications to gravity according to the left panel of figure [ fig : xi3d_ucurve ] . modified gravity also changes the shape of @xmath10 at fixed @xmath100 via its effects on the gik , re distributing matter / galaxies along the @xmath101 axis . to reveal this los distortion of @xmath10 by @xmath0 , following zw13 , we compute the characteristic los distance @xmath115 by fitting a powered exponential function to @xmath10 at each fixed @xmath100 , @xmath116 where @xmath8 is the characteristic length scale at which @xmath10 drops to @xmath117 of its maximum value at @xmath118 . the shape parameter @xmath119 yields a gaussian cutoff for @xmath120 and simple exponential for @xmath121 , though any value is allowed in the fit . the results of this fitting are shown in the right panel of figure [ fig : xi3d_ucurve ] . the @xmath8 vs. @xmath100 curves exhibit the characteristic u - shape discovered in zw13 fog stretching at small @xmath100 gives way to kaiser compression at intermediate @xmath100 which gives way to hubble flow expansion at large @xmath100 . clearly , the @xmath10 distribution along the los is a sensitive probe of @xmath0 models at @xmath9 , becoming more extended with increasing @xmath59 ( e.g. , @xmath122 , @xmath123 , and @xmath124 at @xmath125 for gr , f5 , and f4 models , respectively ) . the detailed shape of @xmath126 reflects a complex interplay among the four elements of the galaxy kinematics around clusters , including the three gik profiles of the infall component ( @xmath31 , @xmath79 , and @xmath80 ) and the virial component ( see figure 10 of zw13 for an illustrative experiment ) . for the same reason , the increase of @xmath8 with @xmath56 has different origins at different projected distances . below @xmath127 , the response of @xmath8 to @xmath56 is uniform with @xmath100 , caused by the uniform increase of dispersion in virial motions . for @xmath128 , the @xmath56 dependence of @xmath8 has two contributing sources , one being the increase of tangential velocity dispersions , the other the increase of maximum infall velocities , which transport high speed matter / galaxies from one side of the cluster in real space to the opposite side in redshift space ( i.e. , the portion of the fog effect caused by infall ) . for @xmath129 , @xmath56 influences the @xmath8 profile mainly via the increase of tangential velocity dispersions . the radial velocity dispersions only enter into play at large @xmath100 , where the diagnostic power of @xmath10 is diminishing . although chameleon theories like the hu & sawicki @xmath0 model include the phenomenology of @xmath13 without a _ true _ cosmological constant , @xcite proved that the theories that invoke a chameleon like scalar to explain cosmic acceleration essentially rely on a form of dark energy rather than a genuine mg effect , even if they are initially described in terms of an altered gravitational action . conversely , the galileon class of theories is capable of accelerating the cosmic expansion even in the absence of any form of dark energy ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , i.e. , they have so called `` self accelerating '' solutions . here we study the gik for a simplified version of such kind of galileon theories , where an extra galileon type scalar field that manifests the vainshtein mechanism permeates a universe with the @xmath13 background cosmology . we employ a suite of galileon simulations with @xmath130 particles on a @xmath130 grid of @xmath131 , evolved using the particle mesh code of @xcite , which was updated by @xcite ( for other galileon / dgp simulations , see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the simulations were first used by @xcite for studying the statistics of matter clustering in real and redshift spaces . we will briefly introduce the galileon implementation and parameterization here and refer the readers to @xcite for details . as described in the introduction , the vainshtein mechanism has only one parameter , @xmath24 , which is interpreted as the compton wavelength associated with the graviton mass in massive gravity theories , so that the vainshtein radius of a point mass with mass @xmath132 is @xmath133 for extended objects of the same mass @xmath132 , the vainshtein radii are generally several times larger than @xmath134 e.g. , a nfw halo with @xmath135 has @xmath136 , and galaxies generally have @xmath137 . therefore , whereas the cluster interior below the virial radius belongs to the strongly vainshtein screened regime , the cluster infall region is weakly screened , displaying complex interference among galileon fields sourced by the primary cluster and the infalling galaxies and galaxy groups . we make use of three simulations with the same expansion history and initial condition , one of them a flat @xmath13 universe evolved under normal gravity with @xmath51 , @xmath52 , @xmath65 , and @xmath138 , and the other two immersed in galileon scalar fields with @xmath139 and @xmath140 . in the figures we refer to them simply as `` gr '' ( i.e. , @xmath141 ) , @xmath139 , and @xmath142 , respectively , and a smaller @xmath24 implies an earlier onset of the fifth force and an effectively stronger fifth force in the late universe ( see , e.g. , figure 5 of * ? ? ? * for the change of linear growth rate as function of @xmath24 ) . note that the `` gr '' simulation is different from what we used for comparing to the @xmath0 simulations , albeit with similar cosmology . dark matter halos are identified via a spherical overdensity finder with the halo mass defined by @xmath143 , different from the @xmath144 used for the chameleon simulations . since @xmath145 , a halo would have a higher @xmath146 than @xmath144 ( e.g. , @xmath147 for a @xmath76 cluster at @xmath5 in @xmath13 ) . the halo mass functions , halo bias functions , and matter power spectra of the three simulations can be found in @xcite . targeting the same redshift range as in [ sec : frsim ] , we use the @xmath148 output of the simulations ( not @xmath5 due to the different sets of recorded epochs in the two suites of simulations ) . since the volume of the galileon simulations is only @xmath149 of that of the chameleon simulations , we have to select samples with a wider mass bin size for robust gik measurements . for the fiducial cluster sample in the gr simulation we include clusters with @xmath150 , and similar to [ sec : frsim ] we also select two er cluster samples from respective galileon simulations , with @xmath151 and @xmath152 in the @xmath139 and @xmath142 models , respectively . unlike [ sec : frsim ] , we do not show the results from the em cluster samples for the gr vs. galileon comparisons , as the relative difference between the em and er samples is similar to what we see in the chameleon simulations . because of the smaller volume , we can only afford to use dark matter particles as proxy for galaxies . note that the galileon cluster sample here comprises halos that are intrinsically smaller than the one used in [ sec : frsim ] , due to different overdensity thresholds used in mass definitions . figure [ fig : vdiff_mark ] compares the gik profiles of the two er samples in the galileon simulations to that of the fiducial gr cluster sample . as intrinsically less massive systems , the fiducial gr clusters show weaker infall velocities and velocity dispersions than the gr clusters used in the chameleon comparison at all distances ( e.g. , comparing the black curves in the top panels of figure [ fig : vdiff_mark ] to the black curves in figure [ fig : v6p ] ) . however , the relative difference between the mg and gr samples overall looks very similar to what we see in the chameleon comparison ( e.g. , comparing the bottom panels of figure [ fig : vdiff_mark ] to figure [ fig : vdiff ] ) . as expected , @xmath31 , @xmath79 , and @xmath80 all become stronger with decreasing @xmath24 ( i.e. , stronger fifth force ) . specifically , @xmath31 shows @xmath153 and @xmath154 enhancement at @xmath2 for the @xmath139 and @xmath142 models , respectively , which are comparable to the fractional enhancements in the f5 and f4 chameleon models , respectively . the only major difference between figure [ fig : vdiff_mark ] and figure [ fig : vdiff ] appears in the bottom middle panel : the deviation of @xmath79 profiles from the gr prediction decreases as function of distance , from @xmath155/@xmath156 at @xmath157 to @xmath158/@xmath159 at @xmath2 , whereas in the chameleon comparison the deviation of @xmath79 stays more or less constant with distance this difference in @xmath79 may be reflecting the different ranges of fifth force in the two models : the galileon has infinite range , so the force is enhanced further way from clusters , whereas the chameleon force is yukawa suppressed on large scales . however , the pattern of @xmath10 is insensitive to @xmath79 at @xmath9 where the @xmath10 measurement is the most robust , making it an unpromising tool for distinguishing the two mechanisms . the gik profiles of the em cluster samples in the galileon simulations closely follow those of their corresponding er counterparts , albeit with slightly weaker amplitudes . figure [ fig : xi3d_ucurve_mark ] compares the real space cluster galaxy cross correlation function @xmath98 ( left ) and the characteristic los distance @xmath8 ( right ) measured for the three cluster samples . the shaded region indicates the scales below the force softening length of the simulations , where the correlation function and velocity dispersions are artificially suppressed . on small scales , the ratios between @xmath98 of the galileon er samples and fiducial gr sample ( bottom left ) display similar features to these in figure [ fig : xi3d_ucurve ] , including an enhancement interior to the virial radius because the er clusters are more massive , and a bump around @xmath160 , though this is only @xmath161 away from the force resolution limit to check whether the bump is a numerical artefact , we repeated the same measurements of @xmath98 using a suite of higher resolution ( but smaller volume ) galileon simulations and verified that the bump is physical . similar to the @xmath0 case in the f4 model , the fast transition from galileon force outside to normal gravity inside causes a sudden change of the depth of the potential well , but not the galaxy velocity ( or kinetic energy ) , which has been experiencing enhancement well before infall . as the galaxies have excessive kinetic energy , they tend to move to the outer parts of halos , making the density profile lower in the central region of clusters and higher near the edges . on large scales , the er samples in the galileon simulations exhibit stronger clustering than the gr clusters . this enhancement in @xmath98 can be understood by starting from the findings of @xcite , which suggest that the halo mass function , the halo bias function , and the matter power spectrum of the galileon simulations are like those of @xmath13 universes with higher @xmath162 . by examining the kaiser effect , where changing @xmath162 can not mimic the large scale boost in the redshift space clustering due to mg . see @xcite for details . ] for example , at @xmath163 the @xmath139 and the @xmath142 simulations resemble the @xmath13 universes with @xmath164 and @xmath165 , respectively . since the large scale @xmath98 of er clusters is almost linearly proportional to the effective @xmath162 , we observe @xmath166 and @xmath167 enhancement in @xmath98 at @xmath168 for clusters in the @xmath139 and @xmath142 models , respectively . for the redshift space cluster galaxy cross correlation function , because the impact of the galileon field on gik is similar to that in the chameleon models , the @xmath8 curves in galileon simulations also exhibit similar deviations from the gr curves , as shown in the right panel of figure [ fig : xi3d_ucurve_mark ] . the lack of upturn of @xmath8 at small @xmath100 is a consequence of force resolution suppressing velocity dispersions ; we expect that simulations with higher force resolution would show the characteristic u - shape for all three models . the overall lower amplitude of @xmath8 compared to figure [ fig : xi3d_ucurve ] is again the result of selecting intrinsically less massive halos . when we select the galileon clusters to have equal mass to the gr clusters , the enhancement in large scale clustering ( left panel ) disappears because the large scale bias is steep function of mass , but the differences in gik and @xmath8 remain . we have investigated the impact of modified gravity on the galaxy infall motion around massive clusters by applying the gik model developed in zw13 to two suites of @xmath0 and galileon n - body simulations . both mg theories seek to explain cosmic acceleration by modifying gr on cosmological scales , but they recover gr in dense regions via two distinct `` screening '' effects : the potential driven chameleon mechanism in @xmath0 and the density driven vainshtein mechanism in galileon . however , within the range of parameter space probed by our simulations ( i.e. , @xmath169 for @xmath0 and @xmath170 for galileon ) , despite having quite different cosmic growth histories , the two theories exhibit strikingly similar gik deviations from gr , with @xmath1 enhancement in the characteristic infall velocity at @xmath2 , and @xmath171 broadening in the radial and tangential velocity dispersions across the infall region , for clusters with mass @xmath172 at @xmath5 . these deviations are detectable through gik modelling of the redshift space cluster galaxy correlation function @xmath10 , especially when combined with cluster wl measurements . we highlight the imprint of mg on @xmath10 using the characteristic u - shaped curve of @xmath8 , which increases by @xmath7 at @xmath9 from the gr prediction . we find little difference between the gik profiles predicted by the two screening mechanisms , except for slightly different trends of the radial velocity dispersion with distance . it is unclear whether the similar signature of these two distinct modified gravity theories on gik is a coincidence , or a generic result for any typical scalar tensor theory that recovers the observed @xmath13like expansion history and reduces to normal gravity in the solar system and binary pulsars . in either case , our findings imply that , in combination with wl , galaxy infall kinematics offer a powerful non parametric cosmological test of modified gravity . ongoing galaxy redshift surveys will provide large samples of clusters with good statistics for measuring @xmath10 and inferring gik out to large scales . the main systematic uncertainty arises from the imperfect understanding of the impact of galaxy formation physics on gik . within the context of gik modelling and calibration , the infall behavior of realistic galaxies could differ from that of tracers in cosmological simulations ( e.g. , halos / sub halos in n - body simulations , post processed galaxies in semi analytical galaxy formation models , and simulated galaxies in hydrodynamic simulations , etc ; see @xcite 2013 ) . however , we expect minimal impact on the characteristic infall velocity , which is our main tool of estimating the dynamical mass profiles of clusters , as any physical process that modifies galaxy kinematics within halos likely only adds scatter to the velocity dispersions rather than changing the mean . within the context of testing gravity , the effects of galaxy formation physics could be partly degenerate with those of modified gravity . the observed properties of galaxies , including luminosity , morphology , colour , star formation , and clustering , are known to correlate with the environment ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , and mg theories also rely on the environment to mediate the strength of the fifth force . for example , a sample of preferentially blue , star forming galaxies may show enhanced infall velocities compared to a galaxy sample that is unbiased in colour in @xmath13 universes , and the enhancement could be mistaken as signal of modified gravity were the selection bias not properly accounted for . we will investigate the potential systematic uncertainties induced by galaxy formation physics in a future paper , using mock galaxy samples constructed from different hod and semi - analytical model prescriptions . redshift surveys that probe a range of galaxy types are especially valuable for those cosmological tests because one can check that different classes of galaxies lead to the same cosmological conclusions even though the galaxy samples themselves have different clustering and kinematics . our gik modelling of @xmath10 is complementary to other semi analytical approaches based on the halo model @xcite , both seeking to model the velocity distribution around massive clusters for testing gravity ( also see * ? ? ? * ; * ? ? ? * for an alternative method of modelling galaxy redshift space distortion based on hod ) . the semi analytical velocity model adopted in @xcite has three components : the empirical infall velocity from the spherical collapse model , the halo - halo pairwise velocity distribution , and the intra - halo velocities ( assumed maxwellian with constant scatter ) . while the model itself is highly informative , the accuracy is slightly lacking compared to simulation predictions . we instead use the simulations as emulators for gik , trading more computer time for better accuracy in the prediction of our model . in terms of observational applications , our method differs from @xcite in two significant aspects . first , they use the stacked redshift differences as the observable , but the model predicts the los velocity dispersion , so they are affected by the systematics in the subtraction of hubble flow in the 2-halo term ; in our method hubble flow is naturally incorporated in the calculation of @xmath10 . second , they consider the velocity distributions up to the second moment , while we are able to model the entire @xmath88 including all higher moments . established as one of the most powerful probes of dark energy , stacked wl analysis of clusters requires deep imaging surveys that can simultaneously yield lensed background galaxies and foreground cluster sample . forecasts for stage iii and stage iv dark energy experiments predict cluster wl constraints that are competitive with supernovae , baryon acoustic oscillations , and cosmic shear ( see * ? ? ? * and 8.4 ) . to complement wl as a cosmological test of gravity , gik modelling of galaxy clusters requires overlap with a large galaxy redshift survey , such as the ongoing baryon oscillation spectroscopic survey ( boss , * * ) , its higher redshift successor eboss ( see * ? ? ? * ) , and the deeper surveys planned for future facilities such as bigboss @xcite , despec @xcite , the subaru prime focus spectrograph @xcite , _ euclid _ @xcite , and _ wfirst _ @xcite . we expect that , in combination with the stacked cluster wl analysis , the redshift space cluster galaxy cross correlations can reveal an accurate and complete picture of the average galaxy infall around clusters , allowing stringent tests of modified gravity theories for the origin of the accelerating expansion of the universe . we thank lam hui and bhuvnesh jain for helpful discussions . y.z . acknowledges the hospitality of the dept . of physics and astronomy at the university of pennsylvania where he enjoyed a fruitful discussion with participants of the `` novel probes of gravity and dark energy '' workshop . d.h.w . and y.z . are supported by the nsf grant ast-1009505 . is also supported by the ohio state university through the distinguished university fellowship . is supported by the royal astronomical society and durham university . e.j . acknowledges the support of a grant from the simons foundation , award number 184549 . m.w . and e.j . were partially supported by , and some of the numerical simulations reported here were performed on a cluster supported in part by , the kavli institute for cosmological physics at the university of chicago through grants through grants nsf phy-0114422 and nsf phy-0551142 and an endowment from the kavli foundation and its founder fred kavli . we also acknowledge resources provided by the university of chicago research computing center . was additionally supported by u.s . dept . of energy contract de - fg02 - 90er-40560 .

infrared modifications of general relativity ( gr ) can be revealed by comparing the mass of galaxy clusters estimated from weak lensing to that from infall kinematics . we measure the 2d galaxy velocity distribution in the cluster infall region by applying the galaxy infall kinematics ( gik ) model developed by @xcite to two suites of @xmath0 and galileon modified gravity simulations . despite having distinct screening mechanisms , namely , the chameleon and the vainshtein effects , the @xmath0 and galileon clusters exhibit very similar deviations in their gik profiles from gr , with @xmath1 enhancement in the characteristic infall velocity at @xmath2 and @xmath3 broadening in the radial and tangential velocity dispersions across the entire infall region , for clusters with mass @xmath4 at @xmath5 . these deviations are detectable via the gik reconstruction of the redshift space cluster galaxy cross correlation function , @xmath6 , which shows @xmath7 increase in the characteristic line - of - sight distance @xmath8 at @xmath9 from gr predictions . with overlapping deep imaging and large redshift surveys in the future , we expect that the gik modelling of @xmath10 , in combination with the stacked weak lensing measurements , will provide powerful diagnostics of modified gravity theories and the origin of cosmic acceleration . galaxy : clusters : general galaxies : kinematics and dynamics cosmology : large - scale structure of universe

You are an expert at summarizing long articles. Proceed to summarize the following text: the work of wyner @xcite led to the development of the notion of secrecy capacity , which quantifies the maximum rate at which a transmitter can reliably send a secret message to a receiver , without an eavesdropper being able to decode it . more recently , researchers have considered secrecy for the two - user broadcast channel , where each receiver acts as an eavesdropper for the independent message transmitted to the other . this problem was addressed in @xcite , where inner and outer bounds for the secrecy capacity region were established . further work in @xcite studied the multiple - input single - output ( miso ) gaussian case , and @xcite considered the general mimo gaussian case . it was shown in @xcite that , under an input covariance constraint , both confidential messages can be simultaneously communicated at their respective maximum secrecy rates , where the achievablity is obtained using secret dirty - paper coding ( s - dpc ) . however , under an average power constraint , a computable secrecy capacity expression for the general mimo case has not yet been derived . in principle , the secrecy capacity for this case could be found by an exhaustive search over the set of all input covariance matrices that satisfy the average power constraint @xcite . clearly , the complexity associated with such a search and the implementation of dirty - paper encoding and decoding make such an approach prohibitive except for very simple scenarios , and motivates the study of simpler techniques based on linear precoding . while low - complexity linear transmission techniques have been extensively investigated for the broadcast channel ( bc ) without secrecy constraints , e.g. , @xcite-@xcite , there has been relatively little work on considering secrecy in the design of linear precoders for the bc case . in @xcite , we considered linear precoders for the mimo gaussian broadcast channel with confidential messages based on the generalized singular value decomposition ( gsvd ) @xcite . it was shown numerically in @xcite that , with an optimal allocation of power for the gsvd - based precoder , the achievable secrecy rate is very close to the secrecy capacity region . in this paper , we show that for a two - user mimo gaussian bc with arbitrary numbers of antennas at each node and under an input covariance constraint , linear precoding is optimal and achieves the same secrecy rate region as s - dpc for certain input covariance constraints , and we derive an expression for the optimal precoders in these scenarios . we then use this result to develop a sub - optimal closed - form algorithm for calculating linear precoders for the case of average power constraints . our numerical results indicate that the secrecy rate region achieved by this algorithm is close to that obtained by the optimal s - dpc approach with a search over all suitable input covariance matrices . in section [ secii ] , we describe the model for the mimo gaussian broadcast channel with confidential messages and the optimal s - dpc scheme , proposed in @xcite . in section [ seciii ] , we consider a general mimo broadcast channel under a matrix covariance constraint , we derive the conditions under which linear precoding is optimal and achieves the same secrecy rate region as s - dpc , and we find the corresponding optimal precoders . we then present our sub - optimal algorithm for designing linear precoders for the case of an average power constraint in section [ seciv ] , followed by numerical examples in section [ secv ] . section [ secvi ] concludes the paper . * notation : * vector - valued random variables are written with non - boldface uppercase letters ( _ e.g. , _ @xmath0 ) , while the corresponding non - boldface lowercase letter ( @xmath1 ) denotes a specific realization of the random variable . scalar variables are written with non - boldface ( lowercase or uppercase ) letters . the hermitian ( i.e. , conjugate ) transpose is denoted by @xmath2 , the matrix trace by tr ( . ) , and * i * indicates an identity matrix . the inequality @xmath3 ( @xmath4 ) means that @xmath5 is hermitian positive ( semi-)definite . mutual information between the random variables @xmath6 and @xmath7 is denoted by @xmath8 , @xmath9 is the expectation operator , and @xmath10 represents the complex circularly symmetric gaussian distribution with zero mean and variance @xmath11 . we consider a two - receiver multiple - antenna gaussian broadcast channel with confidential messages , where the transmitter , receiver 1 and receiver 2 possess @xmath12 , @xmath13 , and @xmath14 antennas , respectively . the transmitter has two independent confidential messages , @xmath15 and @xmath16 , where @xmath15 is intended for receiver 1 but needs to be kept secret from receiver 2 , and @xmath16 is intended for receiver 2 but needs to be kept secret from receiver 1 @xcite . the signals at each receiver can be written as : @xmath17 where @xmath1 is the @xmath18 transmitted signal , and @xmath19 is white gaussian noise at receiver @xmath20 with independent and identically distributed entries drawn from @xmath21 . the channel matrices @xmath22 and @xmath23 are assumed to be unrelated to each other , and known at all three nodes . the transmitted signal is subject to an average power constraint when @xmath24 for some scalar @xmath25 , or it is subject to a matrix power constraint when @xcite : @xmath26 where @xmath27 is the transmit covariance matrix , and @xmath28 . compared with the average power constraint , ( [ lin3 ] ) is rather precise and inflexible , although for example it does allow for the incorporation of per - antenna power constraints as a special case . it was shown in @xcite that for any jointly distributed @xmath29 such that @xmath30 forms a markov chain and the power constraint over @xmath0 is satisfied , the secrecy rate pair @xmath31 given by @xmath32 is achievable for the mimo gaussian broadcast channel given by ( [ lin1 ] ) , where the auxiliary variables @xmath33 and @xmath34 represent the precoding signals for the confidential messages @xmath15 and @xmath16 , respectively @xcite . in @xcite , the achievablity of the rate pair ( [ lin4 ] ) was proved . liu _ et al . _ @xcite analyzed the above secret communication problem under the matrix power - covariance constraint ( [ lin3 ] ) . they showed that the secrecy capacity region @xmath35 is rectangular . this interesting result implies that under the matrix power constraint , both confidential messages @xmath15 and @xmath16 can be _ simultaneously _ transmitted at their respective maximal secrecy rates , as if over two separate mimo gaussian wiretap channels . to prove this result , liu _ _ showed that the secrecy capacity of the mimo gaussian wiretap channel can also be achieved via a coding scheme that uses artificial noise and random binning ( * ? ? ? * theorem 2 ) . under the matrix power constraint ( [ lin3 ] ) , the achievablity of the optimal corner point @xmath36 given by ( * ? ? ? * theorem 1 ) @xmath37 is obtained using dirty - paper coding based on double binning , or as referred to in @xcite , secret dirty paper coding ( s - dpc ) . more precisely , let @xmath38 maximize ( [ lin5 ] ) , and let @xmath39 where @xmath40 and @xmath41 are two independent gaussian vectors with zero means and covariance matrices @xmath42 and @xmath43 , respectively , and the precoding matrix @xmath44 is defined as @xmath45 . one can easily confirm the achievablity of the corner point @xmath36 by evaluating ( [ lin4 ] ) for the above random variables and noting that in ( [ lin1 ] ) , @xmath46 . note that under the matrix power constraint @xmath47 , the input covariance matrix that achieves the corner point in the secrecy capacity region satisfies @xmath48 @xcite . the matrix @xmath49 that maximizes ( [ lin5 ] ) is given by @xcite @xmath50\bc^h \bs^{\frac{1}{2}}\end{aligned}\ ] ] where @xmath51 $ ] is an invertible is invertible since both components of the pencil ( [ lin9 ] ) are positive definite . ] generalized eigenvector matrix of the pencil @xmath52 satisfying @xcite @xmath53\bc=\mathbf{\lambda}\\ & \bc^h\left[\bs^{\frac{1}{2}}\bg^h\bg\bs^{\frac{1}{2}}+\bi\right]\bc=\bi \ ; , \end{split}\end{aligned}\ ] ] where @xmath54 contains the generalized eigenvalues sorted without loss of generality such that @xmath55 the quantity @xmath56 denotes the number of generalized eigenvalues greater than one @xmath57 , and defines the following matrix partitions : @xmath58 \qquad \bc = [ \bc_1\ ; \,\bc_2 ] \ ; , \label{lin11}\ ] ] where @xmath59 , @xmath60 , @xmath61 contains the @xmath56 generalized eigenvectors corresponding to @xmath62 and @xmath63 the @xmath64 generalized eigenvectors corresponding to @xmath65 . now , by applying ( [ lin8 ] ) in ( [ lin5 ] ) , the corner rate pair @xmath36 can be calculated as ( ( * ? ? ? * theorem 3 ) ) @xmath66 for the average power constraint in ( [ lin2 ] ) , there is no computable secrecy capacity expression for the general mimo case . in principle the secrecy capacity region for the average power constraint , @xmath67 , could be found through an exhaustive search over all suitable matrix power constraints @xcite , ( * ? ? ? * lemma 1 ) : @xmath68 for any given semidefinite @xmath47 , @xmath35 can be computed as given by ( [ lin13 ] ) . then , the secrecy capacity region @xmath67 is the convex hull of all of the obtained corner points using ( [ lin13 ] ) . the complexity associated with such a search , as well as that required to implement dirty - paper encoding and decoding , are the main drawbacks of using s - dpc to find the secrecy capacity region @xmath67 for the average power constraint . this makes linear precoding ( beamforming ) techniques an attractive alternative because of their simplicity . to address the performance achievable with linear precoding , we first describe the conditions under which linear precoding is optimal in attaining the same secrecy rate region that is achievable via s - dpc , when the broadcast channel is under an input covariance constraint . in particular , in the next section we show that this equivalence holds for matrix power constraints that satisfy a certain property , and we derive the linear precoders that achieve optimal performance . section [ seciv ] then uses these results to derive a sub - optimal algorithm for the case of the average power constraint . in this section we answer the following questions : a. for a given general mimo gaussian bc described by ( [ lin1 ] ) , where each node has an arbitrary number of antennas and the channel input is under the covariance constraint ( [ lin3 ] ) , is there any @xmath69 for which linear precoding can attain the secrecy capacity region ? b. if yes , how can such @xmath47 be described ? c. for such @xmath47 , what is the optimal linear precoder that allows the rectangular s - dpc capacity region given by ( [ lin13 ] ) to be achieved ? d. if @xmath47 does not satisfy the condition for optimal linear precoding in ( a ) , what is the worst - case loss in secrecy capacity incurred by using the linear precoding approach described in ( b ) anyway ? to begin , we give the following theorem as an answer to questions ( a ) and ( b ) above . [ lin_thm1 ] suppose the matrix power constraint @xmath69 on the input covariance @xmath27 in ( [ lin3 ] ) leads to generalized eigenvectors in ( [ lin10 ] ) that satisfy @xmath70 , i.e. @xmath71 . then the secrecy capacity region @xmath35 can be achieved with @xmath72 , where @xmath33 and @xmath34 are _ independent _ gaussian precoders respectively corresponding to @xmath15 and @xmath16 , with zero means and covariance matrices @xmath73 and @xmath74 , with @xmath73 defined in ( [ lin8 ] ) . recall that for any @xmath69 , the secrecy capacity region @xmath35 is rectangular , so we only need to show that when @xmath75 , the linear precoders @xmath33 and @xmath34 characterized in this theorem are capable of achieving the corner point @xmath76 given by ( [ lin13 ] ) . from ( [ lin4 ] ) , the achievable secrecy rate @xmath77 is given by @xmath78 where ( [ lin16 ] ) and the second part of ( [ lin17 ] ) come from the fact that @xmath33 and @xmath34 are independent . equation ( [ lin18 ] ) is proved in appendix a. one can similarly show that @xmath79 is achievable to complete the proof . theorem [ lin_thm1 ] shows that the secrecy capacity region corresponding to any @xmath47 with orthogonal @xmath61 and @xmath63 can be achieved using either linear independent precoders @xmath33 and @xmath34 , as defined in theorem [ lin_thm1 ] , or using the s - dpc approach , as given by ( [ lin6 ] ) . the next theorem expands on the answer to question ( b ) above , and also addresses ( c ) . first however we present the following lemma which holds for any @xmath69 . [ lin_lem1 ] for a given bc under the matrix power constraint ( [ lin3 ] ) , for any @xmath69 we have @xmath80 , where @xmath81 is the number of positive eigenvalues of the matrix @xmath82 . please see appendix b. the following theorem presents a more specific condition on @xmath47 that results in generalized eigenvectors that satisfy @xmath70 . [ lin_thm2 ] for any @xmath69 , the generalized eigenvectors @xmath61 and @xmath63 in ( [ lin10 ] ) are orthogonal _ iff _ there exists a matrix @xmath83 such that @xmath84 and @xmath85 simultaneously block diagonalizes @xmath86 and @xmath87 : @xmath88 \qquad \bt^h\bg^h\bg\bt = \left[\begin{array}{ccc}\bk_{\bg1 } & \b0\\ \b0 & \bk_{\bg2}\end{array}\right ] \;,\ ] ] where the @xmath89 matrices @xmath90 and @xmath91 satisfy @xmath92 and @xmath93 . the proof begins by noting that if @xmath84 , then the pencil in ( [ lin9 ] ) and @xmath94 have exactly the same generalized eigenvalue matrix @xmath95 , and thus the same secrecy capacity regions . the remainder of the proof can be found in appendix c. while algorithms exist to find @xmath85 that jointly block diagonalizes @xmath86 and @xmath87 ( see for example @xcite and references therein ) , as mentioned in appendix c only those @xmath85 that lead to @xmath96 and @xmath97 are acceptable . later , we will demonstrate that for any bc there are an infinite number of matrix constraints @xmath47 that can achieve such a block diagonalization and hence allow for an optimal linear precoding solution . to conclude this section , we now answer question ( d ) posed above . define the projection matrices @xmath98 and @xmath99 , and note that in general , equation ( [ lin8 ] ) is equivalent to @xmath100 and @xmath101 . when @xmath102 , the optimal covariance matrices for @xmath33 and @xmath34 also satisfy @xmath103 the following theorem explains the loss in secrecy that results when linear precoding with these covariances is used for a matrix constraint @xmath47 that does _ not _ satisfy @xmath102 . [ lin_lem2 ] assume a linear precoding scheme @xmath72 for independent gaussian precoders @xmath33 and @xmath34 with zero means and covariance matrices @xmath104 and @xmath105 , respectively . also define @xmath106 . the loss in secrecy capacity that results from using this approach in the two - user bc is _ at most _ @xmath107 for each user . in particular , the following secrecy rate pair is achievable : @xmath108 see appendix d. [ lin_rem2 ] note that if @xmath61 and @xmath63 are orthogonal , then @xmath109 and @xmath110 , @xmath111 , is achievable , as discussed in theorem 1 . so far we have shown that if the broadcast channel ( [ lin1 ] ) is under the matrix power constraint @xmath47 ( [ lin3 ] ) , then linear precoding as defined by theorem [ lin_thm1 ] is an optimal solution when @xmath47 satisfies the condition described in theorem [ lin_thm2 ] . in the following we propose a suboptimal closed - form linear precoding scheme for the general mimo gaussian bc under the _ average _ power constraint ( [ lin2 ] ) , where as mentioned earlier there exists no optimal closed - form solution that characterizes the secrecy capacity region . we begin with some preliminary results , then we develop the algorithm for the general mimo case , and finally we present an alternative algorithm specifically for the miso case since it offers additional insight . [ lin_remetx1 ] suppose that the input covariance matrix @xmath27 leads to a point on the pareto boundary of the secrecy capacity region given by ( [ lin14 ] ) under the average power constraint ( [ lin2 ] ) . then @xmath112 and @xmath27 can not have any component in the nullspace of @xmath113 , and thus @xmath114 , where @xmath115 , @xmath116 and @xmath117 contains the singular vectors corresponding to the non - zero singular values of @xmath113 . according to remark [ lin_remetx1 ] , we can assume without loss of generality that @xmath113 is full - rank ; otherwise , we could replace @xmath118 with @xmath119 and have an equivalent problem where @xmath120 is full - rank and the secrecy capacity region is the same ( in such a case , @xmath12 would then represent the number of transmitted data streams rather than the number of antennas ) . with this result , we have the following lemma . [ lin_lem3 ] define @xmath121 then @xmath122 and @xmath123 commute and hence share the same set of eigenvectors : @xmath124 where @xmath125 is the ( unitary ) matrix of eigenvectors and @xmath126 the corresponding eigenvalues . see appendix e. without loss of generality , we assume that the columns of @xmath125 are sorted such that the first @xmath127 diagonal elements of @xmath128 are greater than the first @xmath127 diagonal elements of @xmath129 , and the last @xmath130 diagonal elements of @xmath128 are less than or equal to those of @xmath129 . recall from lemma [ lin_lem1 ] that @xmath131 , where @xmath81 is the number of positive eigenvalues of @xmath82 . thus , @xmath132 \qquad \bsig_2 = \left[\begin{array}{ccc}\bsig_{2\rho } & \b0\\\b0 & \bsig_{2\bar{\rho}}\end{array}\right]\ ] ] where @xmath133 is @xmath134 , @xmath135 is @xmath136 , @xmath137 and @xmath138 . now define @xmath139 where @xmath140 and @xmath125 are given in lemma [ lin_lem3 ] and @xmath141 is any block - diagonal matrix partitioned in the same way as @xmath128 and @xmath129 . with these definitions , we see from ( [ lin21 ] ) that @xmath142 and @xmath143 are block diagonal . thus , from theorem [ lin_thm2 ] , a bc with the matrix power constraint @xmath144 leads to a matrix pencil @xmath145 with generalized eigenvectors @xmath146 $ ] that satisfy @xmath147 , where @xmath148 correspond to generalized eigenvalues that are larger or less - than - or - equal - to one , respectively . [ lin_remetx2 ] since the above result holds for any block - diagonal @xmath149 with appropriate dimensions , then for every bc there are an infinite number of matrix power constraints @xmath150 that achieve a block diagonalization and hence allow for an _ optimal _ linear precoding solution . in the following , we restrict our attention to diagonal rather than block - diagonal matrices @xmath151 , for which a closed form solution can be derived . from theorem [ lin_thm1 ] , we have the following result . [ lin_lem4 ] for any diagonal @xmath152 , the secrecy capacity of the broadcast channel in ( [ lin1 ] ) under the matrix power constraint @xmath153 defined in ( [ lin20])-([lin23 ] ) can be obtained by linear precoding . in particular , @xmath154 = v_1 + v_2\end{aligned}\ ] ] where @xmath155 and @xmath156 are independent gaussian random vectors with zero means and covariance matrices @xmath157 and @xmath158 such that @xmath159 \ ; , \end{aligned}\ ] ] and as before @xmath160 represent independently encoded gaussian codebook symbols corresponding to the confidential messages @xmath15 and @xmath16 , with zero means and covariances @xmath161 and @xmath162 respectively given by @xmath163 \bphi_{\bw}^h \bw \label{lin25a } \\ \bs_\bw - \bk_{t\bw}^ * & = \bw\bphi_{\bw}\left [ \begin{array}{cc } \b0 & \b0 \\ \b0 & \bp_2 \end{array } \right ] \bphi_{\bw}^h \bw \ ; . \label{lin25b}\end{aligned}\ ] ] the matrix @xmath164 simultaneously block diagonalizes @xmath86 and @xmath87 , so by theorems [ lin_thm1 ] and [ lin_thm2 ] we know that linear precoding can achieve the secrecy capacity region . the proof is completed in appendix f by showing the equality in ( [ lin25 ] ) , and showing that ( [ lin25a ] ) corresponds to the optimal covariance in ( [ lin8 ] ) . from the proof in appendix f and ( [ lin21])-([lin23 ] ) , we see that under the matrix power constraint @xmath150 given by ( [ lin24 ] ) with diagonal @xmath151 , the general bc is transformed to an equivalent bc with a set of parallel independent subchannels between the transmitter and the receivers , and it suffices for the transmitter to use independent gaussian codebooks across these subchannels . in particular , the diagonal entries of @xmath157 and @xmath158 represent the power assigned to these independent subchannels prior to application of the precoder @xmath165 in ( [ lin23 ] ) do not represent the actual transmitted power , since the columns of @xmath165 are not unit - norm . ] . from ( [ lin25 ] ) , the signals at the two receivers are given by @xmath166 + \bz_1 \\ & = & \bgam_1 \bsig_1 \left [ \begin{array}{c } \bv'_1 \\ \bv'_2 \end{array } \right ] + \bz_1 \\ & = & \bgam_1 \left [ \begin{array}{c } \bsig_{1\rho } \bv'_1 \\ \bsig_{1\bar{\rho } } \bv'_2 \end{array } \right ] + \bz_1 \\ \by_2 & = & \bgam_2 \left [ \begin{array}{c } \bsig_{2\rho } \bv'_1 \\ \bsig_{2\bar{\rho } } \bv'_2 \end{array } \right ] + \bz_2 \ ; , \end{aligned}\ ] ] where @xmath167 are unitary . the confidential message for receiver 1 is thus transmitted with power loading @xmath157 over those subchannels which are degraded for receiver 2 ( @xmath168 ) , while receiver 2 s confidential message has power loading @xmath158 over subchannels which are degraded for receiver 1 ( @xmath169 ) . any subchannels for which the diagonal elements of @xmath170 are equal to those of @xmath171 are useless from the viewpoint of secret communication , but could be used to send common non - confidential messages . from theorem [ lin_thm1 ] , the rectangular secrecy capacity region of the mimo gaussian bc ( [ lin1 ] ) under the matrix power constraint @xmath150 ( [ lin24 ] ) is defined by the corner points @xmath172 where @xmath173 is given by ( [ linap35 ] ) in appendix f. note that we have explicitly written @xmath174 as a function of the diagonal matrix @xmath175 to emphasize that @xmath157 contains the only parameters that can be optimized for @xmath174 . more precisely , since for a given matrix power constraint @xmath150 , @xmath176 and @xmath177 are channel dependent and thus fixed , as shown in ( [ lin21])-([lin22 ] ) . a similar description is also true for @xmath178 . here we propose our sub - optimal closed form solution based on linear precoding for the broadcast channel under the _ average _ power constraint ( [ lin2 ] ) . the goal is to find the diagonal matrix @xmath151 in ( [ lin24 ] ) that maximizes @xmath179 in ( [ lin27 ] ) for a given allocation of the transmit power to message @xmath180 , and that satisfies the average power constraint in ( [ lin28 ] ) . ] @xmath181 noting that @xmath125 can be written as @xmath182 $ ] , where @xmath183 is a @xmath184 submatrix corresponding to the eigenvalues in @xmath176 , ( [ lin28 ] ) can be rewritten as @xmath185 where we defined positive definite matrices @xmath186 , @xmath111 . our sub - optimal closed - form solution for the bc under the average power constraint ( [ lin2 ] ) is not optimal , since instead of doing an exhaustive search over all @xmath69 with @xmath187 as indicated in ( [ lin14 ] ) , we will only consider specific @xmath47 matrices of the form given for @xmath150 in ( [ lin24 ] ) with diagonal @xmath151 . since @xmath188 is only a function of @xmath189 , @xmath190 and @xmath191 can be optimized separately for any power fraction @xmath192 ( @xmath193 ) under the constraints @xmath194 and @xmath195 , respectively . [ lin_lem5 ] for any @xmath192 , @xmath193 , the diagonal elements of the optimal @xmath196 and @xmath197 are given by @xmath198 where @xmath199 , @xmath200 , and @xmath201 are the @xmath202 diagonal elements of @xmath176 , @xmath177 , and @xmath203 , respectively , where @xmath204 . also @xmath205 , @xmath206 , and @xmath207 are the @xmath202 diagonal elements of @xmath171 , @xmath170 , and @xmath208 , respectively , where @xmath209 . the lagrange parameters @xmath210 and @xmath211 are chosen to satisfy the average power constraints @xmath194 and @xmath195 , respectively . we want to optimize diagonal matrices @xmath157 and @xmath158 so that the secrecy rates @xmath190 and @xmath191 , given by ( [ lin27 ] ) , are maximized for a given @xmath192 , @xmath193 . since @xmath212 only depends on @xmath189 , the two terms in ( [ lin27 ] ) can be maximized independently . we show the result for @xmath213 ; the procedure for @xmath214 is identical . from ( [ lin27 ] ) , the lagrangian associated with @xmath215 is @xmath216 -\mu_1 \sum_{i } a_{1i } \,p_{1i } \;,\end{aligned}\ ] ] where @xmath210 is the lagrange multiplier . since @xmath217 , eq . ( [ eq15 ] ) represents a convex optimization problem . the optimal @xmath218 with diagonal elements given by ( [ lin30 ] ) is simply obtained by applying the kkt conditions to ( [ eq15 ] ) . [ lin_cor1 ] for any @xmath192 , @xmath193 , let @xmath219 and @xmath220 represent the corner points given by ( [ lin27 ] ) for the optimal @xmath218 and @xmath221 , given by ( [ lin30 ] ) and ( [ lin31 ] ) . the achievable secrecy rate region of the above approach under the average power constraint ( [ lin2 ] ) is the convex hull of all obtained corner points and is given by @xmath222 it is interesting to note that , unlike the conventional broadcast channel without secrecy constraints where uniform power allocation is optimal in maximizing the sum - rate in the high snr regime @xcite , the high snr power allocation for the bc with confidential messages is a special form of waterfilling as described in the following lemma . [ lin_lemext1 ] for high snr @xmath223 , the asymptotic optimal power allocations given by ( [ lin30])-([lin31 ] ) are @xmath224 to show ( [ linext1 ] ) we note that @xmath225 when @xmath226 . thus ( [ lin30 ] ) can be written as @xmath227 ( [ linext2 ] ) is proved similarly . it is also worth noting that the solution in ( [ lin30])-([lin31 ] ) approaches the standard point - to - point mimo waterfilling solution when one of the channels is dominant . for example , let @xmath228 . we will show that the optimal input covariance simplifies to the waterfilling solution for @xmath229 , given by @xmath230 , where unitary @xmath231 and diagonal @xmath232 are obtained from the eigenvalue decomposition @xmath233 . the capacity of the point - to - point mimo gaussian link is @xmath234 when @xmath235 , we note from ( [ lin20 ] ) and ( [ lin21 ] ) that @xmath236 , @xmath237 , @xmath238 , and @xmath239 . consequently , @xmath240 and @xmath241 , where @xmath242 is a diagonal matrix with diagonal elements given by ( [ lin30 ] ) . the average power constraint in ( [ lin29 ] ) becomes @xmath243 , where @xmath244 when @xmath235 . thus the @xmath202 diagonal element of @xmath245 converges to the @xmath202 diagonal element of @xmath246 . starting from ( [ lin30 ] ) and applying lhpital s rule , when @xmath228 and hence @xmath247 , we have @xmath248 , and consequently , @xmath249 here we focus on the bc in ( [ lin1 ] ) for the miso case under an average power constraint , where both receivers have a single antenna and the transmitter has @xmath250 antennas : @xmath251 where the channels are represented by the @xmath252 vectors @xmath253 and @xmath254 . the miso case is the only bc scenario whose secrecy capacity region under the _ average _ power constraint ( [ lin2 ] ) is characterized in closed - form . in particular , it was shown in @xcite that @xmath255 where @xmath256 is the secrecy rate pair on the pareto boundary of the secrecy capacity region for the power fraction @xmath192 , @xmath193 , where power @xmath257 is allocated to receiver 1 s message and @xmath258 is allocated to receiver 2 s message . furthermore , we have @xcite @xmath259 where @xmath260 @xmath261 is the unit length principal generalized eigenvector of @xmath262 , @xmath263 is the largest generalized eigenvalue of latexmath:[\[(\bi+\frac{(1-\alpha)p_t}{1+\alpha p_t length generalized eigenvector corresponding to @xmath263 . note that the achievablity of ( [ lin34 ] ) is still based on s - dpc . while we could have just used the results of section [ sec : mimo ] for the miso case , we will see that the advantage of considering a different approach here is that we obtain a more succinct expression for the achievable secrecy rate region for linear precoding , and we are able to quantify the loss in secrecy rate incurred by linear precoding under the average power constraint compared with @xmath265 . this was not possible in the mimo case . referring to ( [ lin6 ] ) , it was shown in @xcite that for the secrecy rate pair given by ( [ lin34 ] ) , @xmath40 and @xmath41 have covariance matrices @xmath266 and @xmath267 , respectively . thus , the specific input covariance matrix that attains ( [ lin34 ] ) is given by @xmath268 where @xmath269 and @xmath270 . equivalently , one can say that under the _ matrix _ power constraint @xmath271 , the corner point of the corresponding rectangular secrecy capacity region is given by ( [ lin34 ] ) . the union of these corner points constructs the pareto boundary of the secrecy capacity region under the _ average _ power constraint , where any point on the boundary is given by ( [ lin34 ] ) for a different @xmath192 and is achieved under the matrix power constraint @xmath271 given by ( [ lin35 ] ) . using the above fact , we now present a different linear precoding scheme as an alternative to corollary [ lin_cor1 ] for the miso bc under the average power constraint ( [ lin2 ] ) . [ lin_cor2 ] using the linear precoding scheme proposed in theorem [ lin_lem2 ] for the miso bc under an average power constraint , the following secrecy rate region is achievable : @xmath272 where @xmath273 @xmath274 and @xmath275 are given by ( [ lin34 ] ) , @xmath276 and where @xmath277 is the unit length principal generalized eigenvector of @xmath278 . from remark [ lin_remetx1 ] , and by noting that for any miso bc , @xmath279 has at most 2 non - zero eigenvalues , any miso bc can be modeled with a scenario involving just two transmit antennas . thus , without loss of generality , we assume that @xmath280 . from theorem [ lin_lem2 ] , we only need to characterize @xmath281 and @xmath282 , where @xmath281 ( @xmath282 ) is the generalized eigenvector of the pencil @xmath283 corresponding to the generalized eigenvalue larger ( less ) than 1 , @xmath284 ( @xmath285 ) . from ( [ lin6 ] ) and ( [ lin8 ] ) , the covariance matrix of @xmath40 can be rewritten as @xmath286\left [ \begin{array}{ccc } ( \bc_1^h\bc_1)^{-1 } & 0\\ 0 & 0 \end{array } \right][\bc_1\ ; \bc_2]^h \bs_q^{\frac{1}{2 } } = \frac{1}{\bc_1^h\bc_1}\,\bs_q^{\frac{1}{2 } } \bc_1\;\bc_1^h \bs_q^{\frac{1}{2 } } \ ; .\end{aligned}\ ] ] comparing ( [ lin38 ] ) with the covariance matrix of @xmath40 reported in @xcite , we have @xmath287 . this results in is required for a precise equaltiy , but since this term disappears in the final result , we simply ignore it . ] @xmath288 on the other hand , from the definition of @xmath281 and @xmath282 ( see ( [ lin10])-([lin11 ] ) for example ) , we have @xmath289^h\left[\bs_q^{\frac{1}{2}}\bh\bh^h\bs_q^{\frac{1}{2}}+\bi\right][\bc_1\ ; \bc_2]=\left[\begin{array}{ccc}\lambda_1 & 0\\ 0 & \lambda_2\end{array}\right ] = \left[\begin{array}{ccc}\gamma_1(\alpha ) & 0\\ 0 & \gamma^{-1}_2(\alpha)\end{array}\right]\\ & [ \bc_1\ ; \bc_2]^h\left[\bs_q^{\frac{1}{2}}\bg\bg^h\bs_q^{\frac{1}{2}}+\bi\right][\bc_1\ ; \bc_2]=\bi \end{split}\end{aligned}\ ] ] , @xmath290.,width=288,height=288 ] where @xmath291 and @xmath263 are defined after ( [ lin34 ] ) , and the fact that @xmath292 and @xmath293 comes from the argument after ( [ lin35 ] ) and by comparing ( [ lin13 ] ) and ( [ lin34 ] ) . substituting ( [ lin39 ] ) in ( [ lin40 ] ) , after some simple calculations , @xmath281 can be explicitly written as in ( [ lin41 ] ) . recalling that @xmath281 is the principal generalized eigenvector of ( [ lin37 ] ) and @xmath282 , which corresponds to the smallest generalized eigenvalue of the pencil ( [ lin37 ] ) , is the principal generalized eigenvector of the pencil @xmath294 we obtain ( [ lin42 ] ) . the proof is completed by using ( [ lin41 ] ) and ( [ lin42 ] ) in ( [ lin19 ] ) . , @xmath295.,width=288,height=288 ] in this section , we provide numerical examples to illustrate the achievable secrecy rate region of the mimo gaussian bc under the average power constraint ( [ lin2 ] ) . in the first example , we have @xmath296 , @xmath297 $ ] and @xmath298 $ ] , which is identical to the case studied in ( * ? ? ? 3 ( d ) ) . fig . [ lin_exm1 ] compares the achievable secrecy rate region of the proposed linear precoding scheme in section iv - a with the secrecy capacity region obtained by the optimal s - dpc approach together with an exhaustive search over suitable matrix constraints , as described in section ii . we see that in this example , the performance of the proposed linear precoding approach is essentially identical to that of the optimal s - dpc scheme . in the next example , we study the miso bc for @xmath299 . fig . [ lin_exm41 ] shows the average secrecy rate regions for s - dpc and the suboptimal linear precoding algorithms described in corollary 1 and 2 . this plot is based on an average of over 30000 channel realizations , where the channel coefficients were generated as independent @xmath300 random variables . we see that corollary 2 provides near optimal performance when @xmath301 or @xmath302 , while corollary 1 is better for in - between values of @xmath192 . the degradation of using linear precoding with corollary 1 is never above 15% for any @xmath192 . we have shown that for a two - user gaussian bc with an arbitrary number of antennas at each node , when the channel input is under the matrix power constraint , linear precoding is optimal and achieves the secrecy capacity region attained by the optimal s - dpc approach if the matrix constraint satisfies a specific condition . we characterized the form of the linear precoding that achieves the secrecy capacity region in such cases , and we quantified the maximum loss in secrecy rate that occurs if the matrix power constraint does not satisfy the given condition . based on these observations , we then formulated a sub - optimal approach for the general mimo scenario based on linear precoding for the case of an average power constraint , for which no known characterization of the secrecy capacity region exists . we also studied the miso case in detail . numerical results indicate that the proposed linear precoding approaches yield secrecy rate regions that are close to the secrecy capacity achieved by s - dpc . from ( [ lin17 ] ) , we have @xmath303 the covariance @xmath42 , given by ( [ lin8 ] ) , can be rewritten as @xmath304 \left[\begin{array}{ccc}(\bc_1^h\bc_1)^{-1 } & \b0\\ \b0 & \b0\end{array}\right]\left[\begin{array}{ccc}\bc^h_1\\\bc^h_2\end{array}\right ] \bs^{\frac{1}{2 } } \nonumber\\ & = \bs^{\frac{1}{2}}\,\bc_1 ( \bc_1^h\bc_1)^{-1 } \bc^h_1\ , \bs^{\frac{1}{2 } } = \bs^{\frac{1}{2}}\,\bp_{\bc_1}\,\bs^{\frac{1}{2 } } \ ; , \end{aligned}\ ] ] where @xmath305 is the projection matrix onto the column space of @xmath61 . moreover , let @xmath306 be the projection onto the space orthogonal to @xmath61 . consequently , we have @xmath307\bc^h \bs^{\frac{1}{2 } } \ ; , \label{linap7}\end{aligned}\ ] ] where in ( [ linap6 ] ) , @xmath308 comes from the fact that @xmath70 , and @xmath309 $ ] is full - rank . following the same steps as in the proof of ( * ? ? ? * lemma 2 ) or ( * ? ? ? b ) , we can convert the case when @xmath69 , @xmath310 , to the case where @xmath311 with the same secrecy capacity region . from ( [ lin10 ] ) and ( [ lin11 ] ) we have @xmath312\bc^{-1}-\bi\right ] \bs^{-1/2 } \\ & \bg^h\bg= \bs^{-1/2}\left[\bc^{-h}\bc^{-1}-\bi\right ] \bs^{-1/2 } \ ; . \end{split}\end{aligned}\ ] ] using ( [ linap7 ] ) and ( [ linap8 ] ) , we have : @xmath313 \bc^h\cdot\left[\bc^{-h}\left[\begin{array}{ccc}\mathbf{\lambda}_1 & 0\\ 0 & \mathbf{\lambda}_2\end{array}\right]\bc^{-1}-\bi\right ] \bs^{-1/2}\right|\nonumber \\ & = \left|\bi+ \left[\begin{array}{ccc } \b0 & \b0\\\b0 & ( \bc_2^h\bc_2)^{-1}\end{array}\right ] \cdot\left[\left[\begin{array}{ccc}\mathbf{\lambda}_1 & 0\\ 0 & \mathbf{\lambda}_2\end{array}\right]-\bc^h\bc\right ] \right| \label{linap9 } \\ & = \left|\left[\begin{array}{ccc}\bi & \b0\\\b0 & ( \bc_2^h\bc_2)^{-1}\mathbf{\lambda}_2\end{array } \right]\right| \label{linap10 } \\ & = \left|(\bc_2^h\bc_2)^{-1}\mathbf{\lambda}_2\right| = \left|(\bc_2^h\bc_2)^{-1}\right|\cdot\left|\mathbf{\lambda}_2\right| \ ; , \label{linap11}\end{aligned}\ ] ] where ( [ linap9 ] ) comes from the fact that @xmath314 . finally , ( [ linap10 ] ) holds since @xmath71 and @xmath315 is block diagonal . similarly , one can show that @xmath316 and @xmath317\bc^{-1}\right| = \left|(\bc^h\bc)^{-1}\right|\cdot\left|\mathbf{\lambda}_1\right| \cdot\left|\mathbf{\lambda}_2\right| \nonumber\\ & = \left|(\bc_1^h\bc_1)^{-1}\right|\cdot\left|(\bc_2^h\bc_2)^{-1}\right|\cdot\left|\mathbf{\lambda}_1\right| \cdot\left|\mathbf{\lambda}_2\right| \;. \label{linap14}\end{aligned}\ ] ] substituting ( [ linap11 ] ) , ( [ linap13 ] ) and ( [ linap14 ] ) in ( [ linap4 ] ) , we have @xmath318 , and this completes the proof . from ( [ lin10])-([lin11 ] ) , we know that @xmath319 , where @xmath56 represents number of generalized eigenvalues of the pencil ( [ lin9 ] ) that are greater than 1 . from ( [ lin10])-([lin11 ] ) , we have @xmath320\bc_1=\mathbf{\lambda}_1 \label{linap1 } \\ & \bc_1^h\left[\bs^{\frac{1}{2}}\bg^h\bg\bs^{\frac{1}{2}}+\bi\right]\bc_1=\bi \;.\label{linap2}\end{aligned}\ ] ] subtracting ( [ linap1 ] ) from ( [ linap2 ] ) , a straightforward computation yields @xmath321\bs^{\frac{1}{2}}\bc_1= \mathbf{\lambda}_1-\bi \succ\b0 \;.\end{aligned}\ ] ] from ( [ linap3 ] ) , we have @xmath322\bs^{\frac{1}{2}}\bc_1\succ\b0 $ ] , from which it follows that @xmath323 . similarly one can show that @xmath324 , where @xmath325 corresponds to the generalized eigenvalues of the pencil ( [ lin9 ] ) which are less than 1 , and @xmath326 represents number of negative eigenvalues of @xmath82 . we want to show that @xmath122 and @xmath123 commute , where @xmath389 let the invertible matrix @xmath390 and diagonal matrix @xmath391 respectively represent the generalized eigenvectors and eigenvalues of @xmath392 , so that @xmath393\widehat{\bc}=\widehat{\mathbf{\lambda } } \label{linap26}\\ & \widehat{\bc}^h\left[\bw\bg^h\bg\bw+\bi\right]\widehat{\bc}=\bi \;. \label{linap27}\end{aligned}\ ] ] adding ( [ linap26 ] ) and ( [ linap27 ] ) , we have @xmath394\widehat{\bc}= 3\,\widehat{\bc}^h\widehat{\bc}=(\widehat{\mathbf{\lambda}}+\bi ) \;,\end{aligned}\ ] ] from which it results that @xmath390 must be of the form @xcite @xmath395 where @xmath125 is an unknown unitary matrix . in the following , as we continue the proof , @xmath125 is characterized too . substituting ( [ linap28 ] ) in ( [ linap26 ] ) and ( [ linap27 ] ) , it is revealed that the unitary matrix @xmath125 represents the common set of eigenvectors for the matrices @xmath396 and @xmath397 , and thus both matrices commute . in particular , @xmath398\bphi_{\bw } & = 3\,\widehat{\mathbf{\lambda } } ( \widehat{\mathbf{\lambda}}+\bi)^{-1}=3\,(\widehat{\mathbf{\lambda}}^{-1}+\bi)^{-1 } \\ \bphi_{\bw}^h\left[\bw\bg^h\bg\bw+\bi\right]\bphi_{\bw}&=3\ , ( \widehat{\mathbf{\lambda}}+\bi)^{-1}\;. \end{split}\end{aligned}\ ] ] consequently , @xmath128 and @xmath129 are diagonal : @xmath399 it is interesting to note that , since @xmath400 and @xmath401 , we have @xmath402 . we first consider the generalized eigenvalue decomposition for @xmath403 where @xmath164 is given by ( [ lin23 ] ) and @xmath404\overline{\bc}_{\bw}=\mathbf{\lambda}_{\bw}\\ & \overline{\bc}_{\bw}^h\left[\bt_{\bw}^h\bg^h\bg\bt_{\bw}+\bi\right]\overline{\bc}_{\bw}=\bi \ ; . \end{split}\end{aligned}\ ] ] using ( [ lin21 ] ) , and noting that @xmath125 is unitary and @xmath151 is diagonal , a straightforward calculation yields @xmath405\overline{\bc}_{\bw}=\mathbf{\lambda}_{\bw}\\ & \overline{\bc}_{\bw}^h\left[\bsig_2\bp+\bi\right]\overline{\bc}_{\bw}=\bi\ ; , \end{split}\end{aligned}\ ] ] where @xmath128 and @xmath129 are respectively ( diagonal ) eigenvalue matrices of @xmath122 and @xmath123 , as given by ( [ lin21 ] ) . thus , @xmath406 is diagonal and is given by @xmath407 consequently , we have @xmath408 . let @xmath409 , @xmath410 and @xmath411 represent the @xmath202 diagonal elements of @xmath128 , @xmath129 and @xmath151 , respectively . we note that for any @xmath411 , @xmath412 iff @xmath413 . thus , based on the argument that we made after lemma [ lin_lem3 ] , the first @xmath127 diagonal elements of @xmath414 represent generalized eigenvalues greater than 1 . letting @xmath415\end{aligned}\ ] ] where @xmath157 is @xmath416 and @xmath158 is @xmath417 , we have : @xmath418 = \left[\begin{array}{ccc}\mathbf{\lambda}_{1\bw } & \b0\\\b0 & \mathbf{\lambda}_{2\bw}\end{array } \right ] \ ; , \end{aligned}\ ] ] where @xmath133 and @xmath135 ( @xmath111 ) are given by ( [ lin22 ] ) . consequently , ( [ linap32 ] ) can be rewritten as @xmath419= \left[\begin{array}{ccc}\left(\bsig_{2\rho}\bp_1+\bi\right)^{-\frac{1}{2 } } & \b0\\\b0 & \left(\bsig_{2\bar{\rho}}\bp_2+\bi\right)^{-\frac{1}{2}}\end{array}\right ] \;.\end{aligned}\ ] ] @xmath141 , linear precoding is an optimal solution for the bc under the matrix power constraint @xmath420 , where @xmath164 is given by ( [ lin23 ] ) . more precisely , from theorem [ lin_thm1 ] , @xmath72 is optimal , where @xmath33 and @xmath34 are independent gaussian precoders , respectively corresponding to @xmath15 and @xmath16 with zero means and covariance matrices @xmath421 and @xmath422 , where @xmath421 is given by @xmath423\bc_{\bw}^h \bs_{\bw}^{\frac{1}{2}}\end{aligned}\ ] ] and @xmath424 is the generalized eigenvector matrix for @xmath425 we note that there exists a unitary matrix @xmath331 for which @xmath426 @xcite , where @xmath427 . we also note that , from remark [ lin_remap2 ] , @xmath428 and @xmath429 . thus , @xmath421 can be rewritten as @xmath430\overline{\bc}_{\bw}^h \bt_{\bw}^h = \bt_{\bw}\left[\begin{array}{ccc}\bi & \b0\\\b0 & \b0\end{array } \right]\bt_{\bw}^h \label{linap39 } \\ & = \bw\bphi_{\bw}\bp^{\frac{1}{2}}\left[\begin{array}{ccc}\bi & \b0\\\b0 & \b0\end{array } \right]\bp^{\frac{1}{2}}\bphi^h_{\bw}\bw = \bw\bphi_{\bw}\left[\begin{array}{ccc}\bp_1 & \b0\\\b0 & \b0\end{array } \right]\bphi^h_{\bw}\bw \ ; , \label{linap40}\end{aligned}\ ] ] where ( [ linap39 ] ) comes from ( [ linap36 ] ) , and ( [ linap40 ] ) comes from ( [ linap34 ] ) . consequently , @xmath422 can be written as @xmath431\bphi^h_{\bw}\bw \ ; . \label{linap41}\end{aligned}\ ] ] from ( [ linap40 ] ) and ( [ linap41 ] ) , under the matrix power constraint @xmath150 given by ( [ lin24 ] ) , the optimal linear precoding is @xmath72 , where precoding signals @xmath33 and @xmath34 are independent gaussian vectors with zero means and covariance matrices given by ( [ linap40 ] ) and ( [ linap41 ] ) , respectively . alternatively , the optimal precoder can be represented as @xmath432 $ ] , where precoding signals @xmath433 and @xmath434 are independent gaussian vectors with zero means and diagonal covariance matrices respectively given by @xmath157 and @xmath158 . in both cases @xmath435 , and the same secrecy rate region is achieved . 1 a. wyner , `` the wire - tap channel , '' _ bell . j. _ , vol . 54 , no . 8 , pp . 1355 - 1387 , jan . 1975 . r. liu , i. maric , p. spasojevic , and r. d. yates , discrete memoryless interference and broadcast channels with confidential messages : secrecy rate regions , " _ ieee trans . inf . theory _ 54 , no . 6 , pp . 2493 - 2512 , june 2008 . r. liu and h. v. poor , secrecy capacity region of a multiple - antenna gaussian broadcast channel with confidential messages , " _ ieee trans . inf . theory _ 1235 - 1249 , mar . r. liu , t. liu , h. v. poor , and s. shamai , multiple - input multiple - output gaussian broadcast channels with confidential messages , " _ ieee trans . inf . theory _ , 4215 - 4227 , 2010 . q. h. spencer , a. l. swindlehurst , and m. haardt , zero - forcing methods for downlink spatial multiplexing in multiuser mimo channels , " _ ieee trans . signal processing _ , vol . 461 - 471 , feb . 2004 . t. yoo and a. goldsmith , on the optimality of multi - antenna broadcast scheduling using zero - forcing beamforming , " _ ieee j. select . areas commun . _ , special issue on 4 g wireless systems , vol . 3 , pp.528 - 541 , mar . a. wiesel , y. eldar , and s. shamai , linear precoding via conic optimization for fixed mimo receivers , " _ ieee trans . signal processing _ , vol . 161 - 176 , jan . 2006 . a. fakoorian and a. l. swindlehurst , `` dirty paper coding versus linear gsvd - based precoding in mimo broadcast channel with confidential messages , '' in _ proc . ieee globecom _ , dec a. khisti and g. wornell , `` secure transmission with multiple antennas ii : the mimome wiretap channel , '' _ ieee trans . 56 , no . 11 , pp . 5515 - 5532 , 2010 . a. fakoorian and a. l. swindlehurst , `` optimal power allocation for the gsvd based mimo gaussian wiretap channel , '' in _ isit _ , july 2012 . r. bustin , r. liu , h. v. poor , and s. shamai ( shitz ) , `` a mmse approach to the secrecy capacity of the mimo gaussian wiretap channel , '' _ eurasip journal on wireless comm . and net . 2009 , article i d 370970 , 8 pages , 2009 . r. a. horn and c. r. johnson , _ matrix analysis _ , university press , cambridge , uk , 1985 . h. weingarten , y. steinberg , and s. shamai ( shitz ) , `` the capacity region of the gaussian multiple - input multiple - output broadcast channel , '' _ ieee trans . inf . 9 , pp . 3936 - 3964 , 2006 . d. nion , `` a tensor framework for nonunitary joint block diagonalization , '' _ ieee trans . signal processing _ , vol . 4585 - 4594 , oct . s. a. a. fakoorian and a. l. swindlehurst , `` mimo interference channel with confidential messages : achievable secrecy rates and beamforming design , '' _ ieee trans . on inf . forensics and security _ , vol . j. lee , and n. jindal , `` high snr analysis for mimo broadcast channels : dirty paper coding versus linear precoding , '' _ ieee trans . inf . theory _ , 4787 - 4792 , dec . we want to characterize the matrices @xmath69 for which @xmath327 has generalized eigenvectors with orthogonal @xmath61 and @xmath63 . for any positive semidefinite matrix @xmath328 , there exists a matrix @xmath329 such that @xmath84 @xcite . more precisely , @xmath330 , where @xmath331 can be any @xmath332 unitary matrix ; thus @xmath85 is not unique . [ lin_remap2 ] let the invertible matrix @xmath333 and the diagonal matrix @xmath334 respectively represent the generalized eigenvectors and eigenvalues of @xmath335 so that @xmath336\overline{\bc}=\overline{\mathbf{\lambda}}\\ & \overline{\bc}^h\left[\bt^h\bg^h\bg\bt+\bi\right]\overline{\bc}=\bi \ ; , \end{split}\end{aligned}\ ] ] where @xmath330 for a given unitary matrix @xmath331 . by comparing ( [ lin10 ] ) and ( [ linap_ex1 ] ) , one can confirm that @xmath337 and @xmath338 , where @xmath339 and @xmath95 are respectively the generalized eigenvectors and eigenvalues of ( [ linap22 ] ) , as given by ( [ lin10 ] ) . also note that , for any unitary @xmath331 , @xmath340 . thus , finding a @xmath69 such that ( [ linap22 ] ) has orthogonal @xmath61 and @xmath63 ( block diagonal @xmath315 ) is equivalent to finding a @xmath85 , @xmath84 , such that ( [ linap23 ] ) has orthogonal @xmath341 and @xmath342 ( block diagonal @xmath343 ) . the _ if _ part of theorem [ lin_thm2 ] is easy to show . we want to show that if @xmath84 and @xmath85 simultaneously block diagonalizes @xmath86 and @xmath87 , as given by ( [ linshrink1 ] ) such that @xmath344 and @xmath345 , then @xmath71 . from the definition of the generalized eigenvalue decomposition , we have @xmath346\overline{\bc}= \overline{\bc}^h \left[\begin{array}{ccc}\bi+\bk_{\bh1 } & \b0\\ \b0 & \bi+\bk_{\bh2}\end{array}\right]\overline{\bc}= \left[\begin{array}{ccc}\bd_1 & \b0\\ \b0 & \bd_2 \end{array}\right]\\ & \overline{\bc}^h\left[\bt^h\bg^h\bg\bt+\bi\right]\overline{\bc}=\overline{\bc}^h \left[\begin{array}{ccc}\bi+\bk_{\bg1 } & \b0\\ \b0 & \bi+\bk_{\bg2}\end{array}\right]\overline{\bc}= \left[\begin{array}{ccc}\bi & \b0\\ \b0 & \bi \end{array}\right ] \ ; , \\ \end{split}\end{aligned}\ ] ] from which we have @xmath347=\left[\begin{array}{ccc}\overline{\bc}_{11 } & \b0\\ \b0 & \overline{\bc}_{22}\end{array}\right ] \;,\ ] ] where the invertible matrix @xmath348 and diagonal matrix @xmath349 are respectively the generalized eigenvectors and eigenvalues of @xmath350 . since @xmath351 , then @xmath352 , which shows that @xmath341 corresponds to generalized eigenvalues that are bigger than or equal to one . we have a similar definition for @xmath353 and diagonal matrix @xmath354 , corresponding to @xmath355 . finally , since @xmath343 is block diagonal , then @xmath315 , where @xmath339 is the generalized eigenvector matrix of ( [ linap22 ] ) , is block diagonal as well . this completes the _ if _ part of the theorem . in the following , we prove the _ only if _ part of theorem [ lin_thm2 ] ; _ i.e. , _ we show that if @xmath69 results in ( [ linap22 ] ) having orthogonal @xmath61 and @xmath63 , then there must exist a square matrix @xmath85 such that @xmath84 and @xmath356 and @xmath357 are simultaneously block diagonalized as in ( [ linshrink1 ] ) with @xmath344 and @xmath345 . let @xmath358 have the eigenvalue decomposition @xmath359 , where @xmath360 is unitary and @xmath361 is a positive semidefinite diagonal matrix . also let @xmath362 have the eigenvalue decomposition @xmath363 , where @xmath364 is unitary and @xmath365 is a positive definite diagonal matrix . one can easily confirm that @xcite @xmath366 and @xmath367 , where @xmath339 and @xmath95 are respectively the generalized eigenvectors and eigenvalues of ( [ linap22 ] ) . also let @xmath339 be ordered such that @xmath309 $ ] , where @xmath61 corresponds to the generalized eigenvalues bigger than ( or equal to ) 1 . we have @xmath368 from ( [ linap25 ] ) , @xmath315 is block diagonal iff the unitary matrix @xmath364 is block diagonal . recalling that @xmath364 is the eigenvector matrix of @xmath362 , a block diagonal @xmath364 leads to @xmath362 , and consequently @xmath369 must be block diagonal . thus , if @xmath315 is block diagonal , _ @xmath71 , there must exist a unitary matrix @xmath360 such that @xmath369 and @xmath370 are simultaneously block diagonal . is actually diagonal , and hence also block diagonal . ] letting @xmath371 results in ( [ linap24 ] ) , for which we must have @xmath344 and @xmath345 , otherwise it contradicts the ordering of @xmath309 $ ] . this completes the proof . [ lin_remap1 ] by applying the schur complement lemma @xcite on @xmath372^h\left[\bc_1\quad \bc_2\right]=\left [ \begin{array}{ccc}\bc_1^h\bc_1 & \bc_1^h\bc_2\\ \bc_2^h\bc_1 & \bc_2^h\bc_2\end{array}\right]\ ] ] and recalling the fact that @xmath339 is full - rank , we have that @xmath373 is full rank . similarly , one can show that @xmath374 exists . also , we have @xmath375 @xmath376 . define @xmath377 $ ] , so that @xmath378^h\left[\bp^\perp_{\bc_2}\bc_1\quad \bc_2\right]=\left [ \begin{array}{ccc}\bc_1^h\bp^\perp_{\bc_2}\bc_1 & \b0\\ \b0 & \bc_2^h\bc_2\end{array}\right]\;.\ ] ] consequently , we can write @xmath379\widehat{\bc}^h\end{aligned}\ ] ] and @xmath380\widehat{\bc}^h \ ; .\end{aligned}\ ] ] in the following we show the achievablity of @xmath77 in ( [ lin19 ] ) . the achievablity of @xmath381 is obtained in a similar manner . since @xmath33 and @xmath34 in theorem [ lin_lem2 ] are independent , from ( [ lin17 ] ) we have @xmath382 recalling ( [ linap8 ] ) , we have @xmath383 where we used remark [ lin_remap1 ] to obtain ( [ linap18 ] ) . from ( [ linap15 ] ) , we have @xmath384 \widehat{\bc}^h\cdot\left[\bc^{-h}\left[\begin{array}{ccc}\mathbf{\lambda}_1 & 0\\ 0 & \mathbf{\lambda}_2\end{array}\right]\bc^{-1}-\bi\right ] \right|\nonumber \\ & = \left|\bi+\left[\begin{array}{ccc } \b0 & \b0\\\b0 & ( \bc_2^h\bc_2)^{-1}\end{array}\right ] \cdot\left[\widehat{\bc}^h\bc^{-h}\left[\begin{array}{ccc}\mathbf{\lambda}_1 & 0\\ 0 & \mathbf{\lambda}_2\end{array}\right]\bc^{-1}\widehat{\bc}-\widehat{\bc}^h\widehat{\bc } \right ] \right|\nonumber \\ & = \left|\bi+\left[\begin{array}{ccc } \b0 & \b0\\\b0 & ( \bc_2^h\bc_2)^{-1}\end{array}\right ] \cdot\left[\left[\begin{array}{ccc}\bi & \bn^h\\ \b0 & \bi\end{array}\right ] \left[\begin{array}{ccc}\mathbf{\lambda}_1 & 0\\ 0 & \mathbf{\lambda}_2\end{array}\right]\left[\begin{array}{ccc}\bi & \b0\\ \bn & \bi\end{array}\right ] -\widehat{\bc}^h\widehat{\bc } \right ] \right| \label{linap19}\\ & = \left|\bi+\left[\begin{array}{ccc } \b0 & \b0\\\b0 & ( \bc_2^h\bc_2)^{-1}\end{array}\right ] \cdot\left[\left[\begin{array}{ccc}\mathbf{\lambda}_1+\bn^h\mathbf{\lambda}_2\bn & \bn^h\mathbf{\lambda}_2\\ \mathbf{\lambda}_2\bn & \mathbf{\lambda}_2\end{array}\right ] -\widehat{\bc}^h\widehat{\bc } \right ] \right| \nonumber\\ & = \left|\left[\begin{array}{ccc } \bi & \b0 \\ ( \bc_2^h\bc_2)^{-1}\mathbf{\lambda}_2\bn & ( \bc_2^h\bc_2)^{-1}\mathbf{\lambda}_2\end{array}\right ] \right| \nonumber\\ & = \left|(\bc_2^h\bc_2)^{-1}\mathbf{\lambda}_2\right| = \left|(\bc_2^h\bc_2)^{-1}\right|\cdot\left|\mathbf{\lambda}_2\right| \ ; , \label{linap20}\end{aligned}\ ] ] where in ( [ linap19 ] ) , @xmath385 , and we used the fact that @xmath386\ , \left[\bp^\perp_{\bc_2}\bc_1 \quad \bc_2\right ] = \left[\begin{array}{ccc}\bi & \b0\\ \bn & \bi\end{array}\right ] \;.\end{aligned}\ ] ] similarly , we have @xmath387 \cdot\left[\left[\begin{array}{ccc}\bi & \bn^h\\ \b0 & \bi\end{array}\right ] \left[\begin{array}{ccc}\bi & \b0\\ \bn & \bi\end{array}\right ] -\widehat{\bc}^h\widehat{\bc } \right ] \right| \nonumber\\ & = \left|\bi+\left[\begin{array}{ccc } ( \bc_1^h\bp^\perp_{\bc_2}\bc_1)^{-1 } & \b0\\\b0 & \b0\end{array}\right ] \cdot\left[\left[\begin{array}{ccc}\bi+\bn^h\bn & \bn^h\\ \bn & \bi\end{array}\right ] -\widehat{\bc}^h\widehat{\bc } \right ] \right| \nonumber\\ & = \left|\left[\begin{array}{ccc } ( \bc_1^h\bp^\perp_{\bc_2}\bc_1)^{-1 } ( \bi+\bn^h\bn ) & ( \bc_1^h\bp^\perp_{\bc_2}\bc_1)^{-1}\bn^h \\ \b0 & \bi\end{array}\right ] \right| \nonumber\\ & = \left|(\bc_1^h\bp^\perp_{\bc_2}\bc_1)^{-1}\right|\cdot\left|\bi+\bn^h\bn\right| \;. \label{linap21}\end{aligned}\ ] ] subsituting ( [ linap18 ] ) , ( [ linap20 ] ) and ( [ linap21 ] ) in ( [ linap17 ] ) , we have @xmath388 , which completes the proof .

we study the optimality of linear precoding for the two - receiver multiple - input multiple - output ( mimo ) gaussian broadcast channel ( bc ) with confidential messages . secret dirty - paper coding ( s - dpc ) is optimal under an input covariance constraint , but there is no computable secrecy capacity expression for the general mimo case under an average power constraint . in principle , for this case , the secrecy capacity region could be found through an exhaustive search over the set of all possible matrix power constraints . clearly , this search , coupled with the complexity of dirty - paper encoding and decoding , motivates the consideration of low complexity linear precoding as an alternative . we prove that for a two - user mimo gaussian bc under an input covariance constraint , linear precoding is optimal and achieves the same secrecy rate region as s - dpc if the input covariance constraint satisfies a specific condition , and we characterize the corresponding optimal linear precoders . we then use this result to derive a closed - form sub - optimal algorithm based on linear precoding for an average power constraint . numerical results indicate that the secrecy rate region achieved by this algorithm is close to that obtained by the optimal s - dpc approach with a search over all suitable input covariance matrices .

You are an expert at summarizing long articles. Proceed to summarize the following text: the realization of bose - einstein condensates ( bec)@xcite , achieved by taking dilute alkali gases to ultra low temperatures @xcite is certainly among the most exciting recent experimental achievements in physics . since then , investigations dedicated to the comprehension of new phenomena associated to this state of matter as well as its properties have flourished , either in the experimental or theoretical domains . the fast increasing of the control and production of bose - einstein condensates ( becs ) in different geometries has permitted the study of these systems in different physical situations . the fragmentation of a bec to produce a josephson junction @xcite using becs opened the possibility to study the quantum tunnelling of the atoms across a barrier between the condensates @xcite . using superposition of light in different direction is possible to create any arbitrary trapping configuration as for example a ring or a superconducting quantum interference devices ( squid ) with an atom bec @xcite . another experimental realization of fragmentation of becs is the two - legs bosonic ladder to study chiral current and meissner effect @xcite . there is also the possibility to produce optical lattices in one - dimension ( 1d ) , two - dimensions ( 2d ) and three - dimensions ( 3d ) using one , two or three orthogonal standing waves @xcite . these experimental realisations have opened the possibility to introduce new models that permit the study of the tunnelling of the atoms between the becs . some of these models are exactly solvable by the algebraic bethe ansatz method @xcite and this open the possibility to take in account quantum fluctuations that allows us go beyond the results obtained by mean field approximations . this fruitful approach can furnish some new insights in this area , and contribute as well to the increasingly interesting field of integrable systems itself @xcite . the algebraic formulation of the bethe ansatz , and the associated quantum inverse scattering method ( qism ) , was primarily developed in @xcite . the qism has been used to unveil properties of a considerable number of solvable systems , such as , one - dimensional spin chains , quantum field theory of one - dimensional interacting bosons @xcite and fermions @xcite , two - dimensional lattice models @xcite , systems of strongly correlated electrons @xcite , conformal field theory @xcite , integrable systems in high energy physics @xcite and quantum algebras ( deformations of universal enveloping algebras of lie algebras ) @xcite . for a pedagogical and historical review see @xcite . more recently solvable models have also showed up in relation to string theories ( see for instance @xcite ) . remarkably it is important to mention that exactly solvable models are recently finding their way into the lab , mainly in the context of ultracold atoms @xcite but also in nuclear magnetic resonance ( nmr ) experiments@xcite turning its study as well as the derivation of new models an even more fascinating field . i am considering here the bosonic multi - states lax operator introduced by the author in @xcite that permits to solve a family of models of fragmented becs coupled by josephson tunnelling . this lax operator is a generalization of the bosonic lax operator in @xcite , where a lax operator is defined for a single canonical boson operator , but instead of a single operator we choose a linear combination of independent canonical boson operators . the paper is organized as follows . in section 2 , i will review briefly the algebraic bethe ansatz method and present the lax operators and the transfer matrix for both models . in section 3 , i present a generalized model with a family of models of fragmented becs coupled by josephson tunnelling and show their solutions . in section 4 , i summarize the results . in this section we will shortly review the algebraic bethe ansatz method and present the transfer matrix used to get the solution of the models @xcite . we begin with the @xmath0-invariant @xmath1-matrix , depending on the spectral parameter @xmath2 , @xmath3 with @xmath4 , @xmath5 and @xmath6 . above , @xmath7 is an arbitrary parameter , to be chosen later . it is easy to check that @xmath8 satisfies the yang - baxter equation @xmath9 where @xmath10 denotes the matrix acting non - trivially on the @xmath11-th and the @xmath12-th spaces and as the identity on the remaining space . next we define the monodromy matrix @xmath13 , @xmath14 such that the yang - baxter algebra is satisfied @xmath15 in what follows we will choose a realization for the monodromy matrix @xmath16 to obtain solutions of a family of models for multilevel two - well bose - einstein condensates . in this construction , the lax operators @xmath17 have to satisfy the relation @xmath18 then , defining the transfer matrix , as usual , through @xmath19 it follows from ( [ rtt ] ) that the transfer matrix commutes for different values of the spectral parameter ; i. e. , @xmath20=0 , \;\;\;\;\;\;\ ; \forall \;u,\;v.\ ] ] consequently , the models derived from this transfer matrix will be integrable . another consequence is that the coefficients @xmath21 in the transfer matrix @xmath22 , @xmath23 are conserved quantities or simply @xmath24-numbers , with @xmath25 = 0 , \;\;\;\;\;\;\ ; \forall \;j,\;k.\ ] ] if the transfer matrix @xmath22 is a polynomial function in @xmath2 , with @xmath26 , it is easy to see that , @xmath27 for the standard bosonic operators satisfying the canonical commutation relations @xmath28 = [ \hat{p}_{i},\hat{q}_{j } ] = 0 , \qquad [ \hat{p}_{i},\hat{q}_{j}^{\dagger } ] = \delta_{pq}\delta_{ij}\hat{i},\ ] ] @xmath29= + \hat{p}_{j}^{\dagger}\delta_{pq}\delta_{ij } , \qquad [ \hat{n}_{pi},\hat{q}_{j}]= -\hat{p}_{j}\delta_{pq}\delta_{ij},\ ] ] with @xmath30 , @xmath31 and @xmath32 , we have the following lax operators , @xmath33 and @xmath34 if the conditions , @xmath35 and @xmath36 , are satisfied . the above lax operators satisfy the equation ( [ rll ] ) . in this section i present some applications of the lax operators ( [ l2 ] ) and ( [ l3 ] ) for models with different number of becs @xmath37 and @xmath38 . the generalized hamiltonian is , @xmath39 the parameters , @xmath40 , describe the atom - atom @xmath41-wave scattering between the atoms in the respective becs , the @xmath42 parameters are the relative external potentials between the becs and @xmath43 are the energies in the becs . the parameters @xmath44 are the tunnelling amplitudes . the operators @xmath45 are the number of atoms operators . the labels @xmath46 and @xmath47 stand for the becs @xmath37 and @xmath38 with @xmath48 and @xmath49 . we just remark that @xmath50 . the becs are coupled by josephson tunnelling and the total number of atoms , @xmath51 , is a conserved quantity , @xmath52 = 0 $ ] . the state space is spanned by the base @xmath53 and we can write each vector state as @xmath54 where @xmath55 is the vacuum vector state in the fock space . we can use the states ( [ state1 ] ) to write the matrix representation of the hamiltonian ( [ h1 ] ) . the dimension of the space increase very fast when we increase @xmath56 , @xmath57 where @xmath58 is the total number of becs in the system and @xmath56 is a constant @xmath24-number , @xmath59 . in the case where we have only two becs @xcite ( one @xmath60 and one @xmath61 ) the dimension is @xmath62 . in the figs . ( [ gf1 ] ) and ( [ gf2 ] ) we show some graphs for different values of @xmath63 and @xmath64 . the balls with their respective labels are representing the condensates and the tubes are representing the tunnelling of the atoms between the respective condensates . [ cols="^,^ " , ] now we use the co - multiplication property of the lax operators to write , @xmath65 following the monodromy matrix ( [ monod ] ) we can write the operators , @xmath66 taking the trace of the operator ( [ lh2 ] ) we get the transfer matrix @xmath67 from ( [ c14b ] ) we identify the conserved quantities of the transfer matrix ( [ tu2 ] ) , @xmath68 @xmath69 @xmath70 we can rewrite the hamiltonian ( [ h1 ] ) using these conserved quantities @xmath71 with the following identification for the parameters @xmath72 @xmath73 @xmath74 @xmath75 @xmath76 the hamiltonian ( [ h1 ] ) is related with the transfer matrix ( [ tu2 ] ) by the equation , @xmath77 we use as pseudo - vacuum the product state , @xmath78 with @xmath79 denoting the fock vacuum state for the becs @xmath80 and @xmath81 denoting the fock vacuum state for the becs @xmath82 , for @xmath83 and @xmath84 . for this pseudo - vacuum we can apply the algebraic bethe ansatz method in order to find the bethe ansatz equations ( baes ) , @xmath85 the eigenvectors @xcite @xmath86 of the hamiltonian ( [ h1 ] ) or ( [ h4 ] ) and of the transfer matrix ( [ tu2 ] ) are @xmath87 and the eigenvalues of the hamiltonian ( [ h1 ] ) or ( [ h4 ] ) are , @xmath88 where the @xmath89 are solutions of the baes ( [ bae2 ] ) and @xmath56 is the total number of atoms . we can choose arbitrarily the spectral parameter @xmath2 . choosing @xmath90 , for example , we can write the baes ( [ bae2 ] ) in the limit @xmath91 as just one equation @xmath92 with @xmath93 if the bethe roots @xmath89 are real numbers , the bae ( [ bae3 ] ) is the equation of a @xmath56-dimensional sphere of radii @xmath94 and center in @xmath95 in this limit and with @xmath96 we can write the eigenvalues as @xmath97 i have solved a family of fragmented bose - hubbard models using the multi - states boson lax operators introduced by the author in @xcite . these models can be considered as graphs , with the becs in the vertices and the edge representing the tunnelling between the respective becs . the graphs can appear in one - dimension ( 1d ) , two - dimension ( 2d ) and in three - dimension ( 3d ) . when we increase the number of becs we get a ring of becs @xmath37 and a chain of becs @xmath38 in the center of the ring . we can consider the becs identical or different . i have showed that in the limit @xmath91 , if the bethe roots are all real numbers , they are on a @xmath56-dimensional sphere . the author acknowledge capes / faperj ( coordenao de aperfeioamento de pessoal de nvel superior / fundao de amparo pesquisa do estado do rio de janeiro ) for financial support . 10 kulish p p and sklyanin e k , _ integrable quantum field theories : proceedings of the symposium held at tvrminne , finland - lecture notes in physics _ editor : j. hietarinta and c. montonen , * 151 * , springer - verlag , berlin , ( 1982 ) 61 . takhtajan l a , _ quantum groups : proceedings of the 8th international workshop on mathematical physics held at the arnold sommerfeld institute , clausthal , frg - lecture notes in physics _ , editor : h. -d . doebner and j. -d . hennig , * 370 * , springer - verlag , berlin , ( 1990 ) 3 . essler f h l and korepin v e , _ exactly solvable models of strongly correlated electrons _ , world scientific , singapore , ( 1994 ) . essler f h l , frahm h , ghmann f , klmper a and korepin v e , _ the one - dimensional hubbard model _ , cambridge university press , cambridge , ( 2005 ) . bazhanov v , lukyanov s and zamolodchikov a b , _ commun . _ * 177 * ( 1996 ) 381 . lipatov l , _ jetp lett . _ * 59 * ( 1994 ) 596 . jimbo m , _ field theory , quantum gravity and strings : proceedings of a seminar series held at daphe , observatoire de meudon , and lpthe , universit pierre et marie curie , paris - lecture notes in physics _ , editor : h. j. de vega and n. snchez , * 246 * , springer - verlag , berlin , ( 1986 ) 335 .

i present the exact solution of a family of fragmented bose - hubbard models and represent the models as graphs in one - dimension , two - dimensions and three - dimensions with the condensates in the vertices . the models are solved by the algebraic bethe ansatz method .

You are an expert at summarizing long articles. Proceed to summarize the following text: bayesian approaches have inherent advantages in solving inference and decision problems , but practical applications pose challenges for computation . as these challenges have been met bayesian approaches have proliferated and contributed to the solution of applied problems . mcgrayne ( 2011 ) has recently conveyed these facts to a wide audience . the evolution of bayesian computation over the past half - century has conformed with exponential increases in speed and decreases in the cost of computing . the influence of computational considerations on algorithms , models , and the way that substantive problems are formulated for statistical inference can be subtle but is hard to over - state . successful and innovative basic and applied research recognizes the characteristics of the tools of implementation from the outset and tailors approaches to those tools . recent innovations in hardware ( graphics processing units , or gpus ) provide individual investigators with massively parallel desktop processing at reasonable cost . corresponding developments in software ( extensions of the c programming language and mathematical applications software ) make these attractive platforms for scientific computing . this paper extends and applies existing sequential monte carlo ( smc ) methods for posterior simulation in this context . the extensions fill gaps in the existing theory to provide a thorough and practical foundation for sequential posterior simulation , using approaches suited to massively parallel computing environments . the application produces generic posterior simulators that make substantially fewer analytical and programming demands on investigators implementing new models and provides faster , more reliable and more complete posterior simulation than do existing methods inspired by conventional predominantly serial computational methods . sequential posterior simulation grows out of smc methods developed over the past 20 years and applied primarily to state space models ( particle filters " ) . seminal contributions include baker ( 1985 , 1987 ) , gordon et al . ( 1993 ) , kong et al . ( 1994 ) , liu and chen ( 1995 , 1998 ) , chopin ( 2002 , 2004 ) , del moral et al . ( 2006 ) , andrieu et al . ( 2010 ) , chopin and jacob ( 2010 ) and del moral et al . ( 2011 ) . the posterior simulator proposed in this paper , which builds largely on chopin ( 2002 , 2004 ) , has attractive properties . * it is highly generic ( easily adaptable to new models ) . all that is required of the user is code to generate from the prior and evaluate the prior and data densities . * it is computationally efficient relative to available alternatives . this is in large part due to the specifics of our implementation of the simulator , which makes effective use of low - cost massively parallel computing hardware . * since the simulator provides a sample from the posterior density at each observation date conditional on information available at that time , it is straightforward to compute moments of arbitrary functions of interest conditioning on relevant information sets . marginal likelihood and predictive scores , which are key elements of bayesian analysis , are immediately available . also immediately available is the probability integral transform at each observation date , which provides a powerful diagnostic tool for model exploration ( generalized residuals " ; e.g. , diebold et al . , 1998 ) . * estimates of numerical standard error and relative numerical efficiency are provided as an intrinsic part of the simulator . this is important but often neglected information for the researcher interested in generating dependable and reliable results . * the simulator is robust to irregular posteriors ( e.g. , multimodality ) , as has been pointed out by jasra et al . ( 2007 ) and others . * the simulator has well - founded convergence properties . but , although the basic idea of sequential posterior simulation goes back at least to chopin ( 2002 , 2004 ) , and despite its considerable appeal , the idea has seen essentially no penetration in mainstream applied econometrics . applications have been limited to relatively simple illustrative examples ( although this has begun to change very recently ; see herbst and schorfheide , 2012 ; fulop and li , 2012 ; chopin et al . , 2012 ) . this is in stark contrast to applications of smc methods to state space filtering ( particle filters " ) , which have seen widespread use . relative to markov chain monte carlo ( mcmc ) , which has become a mainstay of applied work , sequential posterior simulators are computationally costly when applied in a conventional serial computing environment . our interest in sequential posterior simulation is largely motivated by the recent availability of low cost hardware supporting massively parallel computation . smc methods are much better suited to this environment than is mcmc . the massively parallel hardware device used in this paper is a commodity graphical processing unit ( gpu ) , which provides hundreds of cores at a cost of well under one dollar ( us ) each . but in order to realize the huge potential gains in computing speed made possible by such hardware , algorithms that conform to the single - instruction multiple - data ( simd ) paradigm are needed . the simulators presented in this paper conform to the simd paradigm by design and realize the attendant increases in computing speed in practical application . but there are also some central issues regarding properties of sequential posterior simulators that have not been resolved in the literature . since reliable applied work depends on the existence of solid theoretical underpinnings , we address these as well . our main contributions are as follows . 1 . _ theoretical basis . _ whereas the theory for sequential posterior simulation as originally formulated by chopin ( 2002 , 2004 ) assumes that key elements of the algorithm are fixed and known in advance , practical applications demand algorithms that are adaptive , with these elements constructed based on the information provided by particles generated in the course of running the algorithm . + while some progress has been made in developing a theory that applies to such adaptive algorithms , ( e.g. , douc and moulines , 2008 ) , we expect that a solid theory that applies to the kind of algorithms that are used and needed in practice is likely to remain unattainable in the near future . + in section [ sec : algorithms ] we provide an approach that addresses this problem in a different way , providing a posterior simulator that is highly adapted to the models and data at hand , while satisfying the relatively straightforward conditions elaborated in the original work of chopin ( 2002 , 2004 ) . numerical accuracy . _ while chopin ( 2004 ) provides a critical central limit theorem , the extant literature does not provide any means of estimating the variance , which is essential for assessing the numerical accuracy and relative numerical efficiency of moment approximations ( geweke , 1989 ) . this problem has proved difficult in the case of mcmc ( flegal and jones , 2010 ) and appears to have been largely ignored in the smc literature . in section [ sec : parpostsims ] we propose an approach to resolving this issue which entails no additional computational effort and is natural in the context of the massively parallel environment as well as key in making efficient use of it . the idea relies critically upon the theoretical contribution noted above . marginal likelihood . _ while the approach to assessing the asymptotic variance of moment approximations noted in the previous point is useful , it does not apply directly to marginal likelihood , which is a critical element of bayesian analysis . we address this issue in section [ sec : pml ] . parallel implementation . _ it has been noted that sequential posterior simulation is highly amenable to parallelization going back to at least chopin ( 2002 ) , and there has been some work toward exploring parallel implementations ( e.g. , lee et al . , 2010 ; fulop and li , 2012 ) . building on this work , we provide a software package that includes a full gpu implementation of the simulator , with specific details of the algorithm tuned to the massively parallel environment as outlined in section [ sec : algorithms ] and elsewhere in this paper . the software is fully object - oriented , highly modular , and easily extensible . new models are easily added , providing the full benefits of gpu computing with little effort on the part of the user . geweke et al . ( 2013 ) provides an example of such an implementation for the logit model . in future work , we intend to build a library of models and illustrative applications utilizing this framework which will be freely available . section [ ss : software ] provides details about this software . 5 . _ specific recommendations . _ the real test of the simulator is in its application to problems that are characteristic of the scale and complexity of serious disciplinary work . in section [ sec : application ] , we provide one such application to illustrate key ideas . in other ongoing work , including geweke et al . ( 2013 ) and a longer working paper written in the process of this research ( durham and geweke , 2011 ) , we provide several additional substantive applications . as part of this work , we have come up with some specific recommendations for aspects of the algorithm that we have found to work well in a wide range of practical applications . these are described in section [ ss : details ] and implemented fully in the software package we are making available with this paper . the sequential simulator proposed in this paper is based on ideas that go back to chopin ( 2002 , 2004 ) , with even earlier antecedents including gilkes and berzuini ( 2001 ) , and fearnhead ( 1998 ) . the simulator begins with a sample of parameter vectors ( particles " ) from the prior distribution . data is introduced in batches , with the steps involved in processing a single batch of data referred to as a cycle . at the beginning of each cycle , the particles represent a sample from the posterior conditional on information available up to that observation date . as data is introduced sequentially , the posterior is updated using importance sampling ( kloek and van dijk , 1978 ) , the appropriately weighted particles representing an importance sample from the posterior at each step . as more data is introduced , the importance weights tend to become unbalanced " ( a few particles have most of the weight , while many others have little weight ) , and importance sampling becomes increasingly inefficient . when some threshold is reached , importance sampling stops and the cycle comes to an end . at this point , a resampling step is undertaken , wherein particles are independently resampled in proportion to their importance weights . after this step , there will be many copies of particles with high weights , while particles with low weights will tend to drop out of the sample . finally , a sequence of metropolis steps is undertaken in order to rejuvenate the diversity of the particle sample . at the conclusion of the cycle , the collection of particles once again represents a sample from the posterior , now incorporating the information accrued from the data newly introduced . this sequential introduction of data is natural in a time - series setting , but also applicable to cross - sectional data . in the latter case , the sequential ordering of the data is arbitrary , though some orderings may be more useful than others . much of the appeal of this simulator is due to its ammenability to implementation using massively parallel hardware . each particle can be handled in a distinct thread , with all threads updated concurrently at each step . communication costs are low . in applications where computational cost is an important factor , nearly all of the cost is incurred in evaluating densities of data conditional on a candidate parameter vector and thus isolated to individual threads . communication costs in this case are a nearly negligible fraction of the total computational burden . the key to efficient utilization of the massively parallel hardware is that the workload be divided over many simd threads . for the gpu hardware used in this paper optimal usage involves tens of thousands of threads . in our implementation , particles are organized in a relatively small number of groups each with a relatively large number of particles ( in the application in section [ sec : application ] there are @xmath0 groups of @xmath1 particles each ) . this organization of particles in groups is fundamental to the supporting theory . estimates of numerical error and relative numerical efficiency are generated as an intrinsic part of the algorithm with no additional computational cost , while the reliability of these estimates is supported by a sound theoretical foundation . this structure is natural in a massively parallel computing environment , as well critical in making efficient use of it . the remainder of this section provides a more detailed discussion of key issues involved in posterior simulation in a massively parallel environment . it begins in section [ subsec : compenv ] with a discussion of the relevant features of the hardware and software used . section [ subsec : notation_assumption ] sets up a generic model for bayesian inference along with conditions used in deriving the analytical properties of various sequential posterior simulators in section [ sec : algorithms ] . section [ sec : parpostsims ] stipulates a key convergence condition for posterior simulators , and then shows that if this condition is met there are attractive generic methods for approximating the standard error of numerical approximation in massively parallel computing environments . section [ sec : algorithms ] then develops sequential posterior simulators that satisfy this condition . the particular device that motivates this work is the graphics processing unit ( gpu ) . as a practical matter several gpu s can be incorporated in a single server ( the host " ) with no significant complications , and desktop computers that can accommodate up to eight gpu s are readily available . the single- and multiple - gpu environments are equivalent for our purposes . a single gpu consists of several multiprocessors , each with several cores . the gpu has global memory shared by its multiprocessors , typically one to several gigabytes ( gb ) in size , and local memory specific to each multiprocessor , typically on the order of 50 to 100 kilobytes ( kb ) per multiprocessor . ( for example , this research uses a single nvidia gtx 570 gpu with 15 multiprocessors , each with 32 cores . the gpu has 1.36 gb of local memory , and each multiprocessor has 49 kb of memory and 32 kb of registers shared by its cores . ) the bus that transfers data between gpu global and local memory is significantly faster than the bus that transfers data between host and device , and accessing local memory on the multiprocessor is faster yet . for greater technical detail on gpu s , see hendeby et al . ( 2010 ) , lee et al . ( 2010 ) and souchard et al . ( 2010 ) . this hardware has become attractive for scientific computing with the extension of scientific programming languages to allocate the execution of instructions between host and device and facilitate data transfer between them . of these the most significant has been the compute unified device architecture ( cuda ) extension of the c programming language ( nvidia , 2013 ) . cuda abstracts the host - device communication in a way that is convenient to the programmer yet faithful to the aspects of the hardware important for writing efficient code . code executed on the device is contained in special functions called kernels that are invoked by the host code . specific cuda instructions move the data on which the code operates from host to device memory and instruct the device to organize the execution of the code into a certain number of blocks with a certain number of threads each . the allocation into blocks and threads is the virtual analogue of the organization of a gpu into multiprocessors and cores . while the most flexible way to develop applications that make use of gpu parallelization is through c / c++ code with direct calls to the vendor - supplied interface functions , it is also possible to work at a higher level of abstraction . for example , a growing number of mathematical libraries have been ported to gpu hardware ( e.g. , innovative computing laboratory , 2013 ) . such libraries are easily called from standard scientific programming languages and can yield substantial increases in performance for some applications . in addition , matlab ( 2013 ) provides a library of kernels , interfaces for calling user - written kernels , and functions for host - device data transfer from within the matlab workspace . we augment standard notation for data , parameters and models . the relevant observable random vectors are @xmath2 @xmath3 and @xmath4 denotes the collection @xmath5 . the observation of @xmath2 is @xmath6 , @xmath7 denotes the collection @xmath8 , and therefore @xmath9 denotes the data . this notation assumes ordered observations , which is natural for time series . if @xmath10 is independent and identically distributed the ordering is arbitrary . a model for bayesian inference specifies a @xmath11 unobservable parameter vector @xmath12 and a conditional density @xmath13 with respect to an appropriate measure for @xmath14 . the model also specifies a prior density @xmath15 with respect to a measure @xmath16 on @xmath17 . the posterior density @xmath18 follows in the usual way from and @xmath19 . the objects of bayesian inference can often be written as posterior moments of the form @xmath20 $ ] , and going forward we use @xmath21 to refer to such a generic function of interest . evaluation of @xmath21 may require simulation , e.g. @xmath22 $ ] , and conventional approaches based on the posterior simulation sample of parameters ( e.g. geweke , 2005 , section 1.4 ) apply in this case . the marginal likelihood @xmath23 famously does not take this form , and section [ sec : pml ] takes up its approximation with sequential posterior simulators and , more specifically , in the context of the methods developed in this paper . several conditions come into play in the balance of the paper . [ cond : prior_evaluate](prior distribution ) . the model specifies a proper prior distribution . the prior density kernel can be evaluated with simd - compatible code . simulation from the prior distribution must be practical but need not be simd - compatible . it is well understood that a model must take a stance on the distribution of outcomes @xmath14 _ a priori _ if it is to have any standing in formal bayesian model comparison , and that this requires a proper prior distribution . this requirement is fundamentally related to the generic structure of sequential posterior simulators , including those developed in section [ sec : algorithms ] , because they require a distribution of @xmath24 before the first observation is introduced . given a proper prior distribution the evaluation and simulation conditions are weak . simulation from the prior typically involves minimal computational cost and thus we do not require it to be simd - compatible . however , in practice it will often be so . [ cond : lf_evaluate](likelihood function evaluation ) the sequence of conditional densities @xmath25 can be evaluated with simd - compatible code for all @xmath12 . evaluation with simd - compatible code is important to computational efficiency because evaluation of ( [ model_conditional_pdf ] ) constitutes almost all of the floating point operations in typical applications for the algorithms developed in this paper . condition [ cond : lf_evaluate ] excludes situations in which unbiased simulation of ( [ model_conditional_pdf ] ) is possible but exact evaluation is not , which is often the case in nonlinear state space models , samples with missing data , and in general any case in which a closed form analytical expression for ( [ model_conditional_pdf ] ) is not available . a subsequent paper will take up this extension . [ cond : bounded_like ] ( bounded likelihood ) the data density @xmath26 is bounded above by @xmath27 for all @xmath28 . this is one of two sufficient conditions for the central limit theorem invoked in section [ subsec : nonadaptive ] . it is commonly but not universally satisfied . when it is not , it can often be attained by minor and innocuous modification of the likelihood function ; section [ sec : application ] provides an example . [ cond : prior_var](existence of prior moments ) if the algorithm is used to approximate @xmath29 $ ] , then @xmath30<\infty$ ] for some @xmath31 . in any careful implementation of posterior simulation the existence of relevant posterior moments must be verified analytically . this condition , together with condition [ cond : bounded_like ] , is sufficient for the existence of @xmath32 $ ] and @xmath33 $ ] . condition [ cond : prior_var ] also comes into play in establishing a central limit theorem . consider the implementation of any posterior simulator in a parallel computing environment like the one described in section [ subsec : compenv ] . following the conventional approach , we focus on the posterior simulation approximation of @xmath34 . \label{post_moment}\ ] ] the posterior simulator operates on parameter vectors , or particles , @xmath35 organized in @xmath36 groups of @xmath37 particles each ; define @xmath38 and @xmath39 . this organization is fundamental to the rest of the paper . each particle is associated with a core of the device . the cuda software described in section [ subsec : compenv ] copes with the case in which @xmath40 exceeds the number of cores in an efficient and transparent fashion . ideally the posterior simulator is simd - compatible , with identical instructions executed on all particles . some algorithms , like the sequential posterior simulators developed in section [ sec : algorithms ] , come quite close to this ideal . others , for example gibbs samplers with metropolis steps , may not . the theory in this section applies regardless . however , the advantages of implementing a posterior simulator in a parallel computing environment are driven by the extent to which the algorithm is simd - compatible . denote the evaluation of the function of interest at particle @xmath35 by @xmath41 and within - group means by @xmath42 @xmath43 . the posterior simulation approximation of @xmath44 is the grand mean @xmath45 in general we seek posterior simulators with the following properties . [ cond : normal_approx](asymptotic normality of posterior moment approximation ) the random variables @xmath46 are independently and identically distributed . there exists @xmath47 for which @xmath48 as @xmath49 . going forward , @xmath50 will denote a generic variance in a central limit theorem . convergence is part of the rigorous foundation of posterior simulators , e.g. geweke ( 1989 ) for importance sampling , tierney ( 1994 ) for mcmc , and chopin ( 2004 ) for a sequential posterior simulator . for importance sampling @xmath51 @xmath52 , where @xmath53 is the ratio of the posterior density kernel to the source density kernel . for to be of any practical use in assessing numerical accuracy there must also be a simulation - consistent approximation of @xmath50 . such approximations are immediate for importance sampling . they have proven more elusive for mcmc ; e.g. , see flegal and jones ( 2010 ) for discussion and an important contribution . to our knowledge there is no demonstrated simulation - consistent approximation of @xmath50 for sequential posterior simulators in the existing literature . however , for any simulator satisfying condition [ cond : normal_approx ] this is straightforward . given condition [ cond : normal_approx ] , it is immediate that @xmath54 define the estimated posterior simulation variance @xmath55 \sum_{j=1}^{j}\left ( \overline{g}_{j}^{n}-\overline{g}^{\left ( j , n\right ) } \right ) ^{2}\label{vhat}\ ] ] and the numerical standard error ( _ nse _ ) @xmath56 ^{1/2}=\left\ { \left [ j\left ( j-1\right ) \right ] ^{-1}\sum_{j=1}^{j}\left ( \overline{g}_{j}^{n}-\overline{g}^{\left ( j , n\right ) } \right)^2 \right\ } ^{1/2}. \label{nse_def}\ ] ] note the different scaling conventions in and : in the scaling is selected so that @xmath57 approximates @xmath50 in because we will use this expression mainly for methodological developments ; in the scaling is selected to make it easy to appraise the reliability of numerical approximations of posterior moments like those reported in section [ sec : application ] . one conventional assessment of the efficiency of a posterior simulator is its relative numerical efficiency ( _ rne _ ) ( geweke , 1989 ) , @xmath58 where @xmath59 , the simulation approximation of @xmath60 $ ] . [ prop : clt_var_approx]condition [ cond : normal_approx ] implies @xmath61 and @xmath62 ^{1/2}\overset{d}{\rightarrow } t\left ( j-1\right ) , \label{p1b}\ ] ] both as @xmath63 . conditions [ cond : prior_evaluate ] through [ cond : prior_var ] imply @xmath64 = var\left [ \overline{g}^{\left ( j , n\right ) } \right ] . \label{p1c}\ ] ] from condition [ cond : normal_approx ] , @xmath65 as @xmath63 . substituting in yields . condition [ cond : normal_approx ] also implies @xmath66 as @xmath63 . since and are independent in the limiting distribution , we have . conditions [ cond : prior_evaluate ] through [ cond : prior_var ] imply the existence of the first two moments of @xmath67 . then follows from the fact that the approximations @xmath68 are independent and identically distributed across @xmath69 . we seek a posterior simulator that is generic , requiring little or no intervention by the investigator in adapting it to new models beyond the provision of the software implicit in conditions [ cond : prior_evaluate ] and [ cond : lf_evaluate ] . it should reliably assess the numerical accuracy of posterior moment approximations , and this should be achieved in substantially less execution time than would be required using the same or an alternative posterior simulator for the same model in a serial computing environment . such simulators are necessarily adaptive : they must use the features of the evolving posterior distribution , revealed in the particles @xmath70 , to design the next steps in the algorithm . this practical requirement has presented a methodological conundrum in the sequential monte carlo literature , because the mathematical complications introduced by even mild adaptation lead to analytically intractable situations in which demonstration of the central limit theorem in condition [ cond : normal_approx ] is precluded . this section solves this problem and sets forth a particular adaptive sequential posterior simulator that has been successful in applications we have undertaken with a wide variety of models . section [ sec : application ] details one such application . section [ subsec : nonadaptive ] places a nonadaptive sequential posterior simulator ( chopin , 2004 ) that satisfies condition [ cond : normal_approx ] into the context developed in the previous section . section [ subsec : adaptive ] introduces a technique that overcomes the analytical difficulties associated with adaptive simulators . it is generic , simple and imposes little additional overhead in a parallel computing environment . section [ ss : details ] provides details on a particular variant of the algorithm that we have found to be successful for a variety of models representative of empirical research frontiers in economics and finance . section [ ss : software ] discusses a software package implementing the algorithm that we are making available . we rely on the following mild generalization of the sequential monte carlo algorithm of chopin ( 2004 ) , cast in the parallel computing environment detailed in the previous section . in this algorithm , the observation dates at which each cycle terminates ( @xmath71 ) and the parameters involved in specifying the metropolis updates ( @xmath72 ) are assumed to be fixed and known in advance , in conformance with the conditions specified by chopin ( 2004 ) . [ alg : general_nonadaptive](nonadaptive ) let @xmath73 be fixed integers with @xmath74 and let @xmath75 be fixed vectors . 1 . initialize @xmath76 and let @xmath77 . 2 . for @xmath78 1 . correction @xmath79 phase : 1 . 2 . for @xmath81@xmath82 3 . selection @xmath84 phase , applied independently to each group @xmath85 : using multinomial or residual sampling based on @xmath86 , select @xmath87 3 . mutation @xmath88 phase , applied independently across @xmath89 : @xmath90 where the drawings are independent and the p.d.f . satisfies the invariance condition @xmath91 3 . @xmath92 the algorithm is nonadaptive because @xmath93 and @xmath94 are predetermined before the algorithm starts . going forward it will be convenient to denote the cycle indices by @xmath95 . at the conclusion of the algorithm , the simulation approximation of a generic posterior moment is . [ prop_chopin ] if conditions [ cond : prior_evaluate ] through [ cond : prior_var ] are satisfied then algorithm [ alg : general_nonadaptive ] satisfies condition [ cond : normal_approx ] . the results follow from chopin ( 2004 ) , theorem 1 ( for multinomial resampling ) and theorem 2 ( for residual resampling ) . the assumptions made in theorems 1 and 2 are 1 . @xmath96 and @xmath97 @xmath98 , 2 . the functions @xmath99 are integrable on @xmath100 , and 3 . the moments @xmath101 exist . assumption 1 is merely a change in notation ; conditions [ cond : prior_evaluate ] through [ cond : bounded_like ] imply assumption 2 ; and conditions [ cond : prior_evaluate ] through [ cond : prior_var ] imply assumption 3 . theorems 1 and 2 in chopin ( 2004 ) are stated using the weighted sample at the end of the last @xmath102 phase , but as that paper states , they also apply to the unweighted sample at the end of the following @xmath103 phase . at the conclusion of each cycle @xmath104 , the @xmath36 groups of @xmath37 particles each , @xmath105 @xmath106 , are mutually independent because the @xmath107 phase is executed independently in each group . at the end of each cycle @xmath104 , all particles @xmath108 are identically distributed with common density @xmath109 . particles in different groups are independent , but particles within the same group are not . the amount of dependence within groups depends upon how well the @xmath103 phases succeed in rejuvenating particle diversity . in fact the sequences @xmath110 and @xmath111 for which the algorithm is sufficiently efficient for actual application are specific to each problem . as a practical matter these sequences must be tailored to the problem based on the characteristics of the particles @xmath112 produced by the algorithm itself . this leads to algorithms of the following type . [ alg : general_adaptive](adaptive ) algorithm [ alg : general_adaptive ] is the following generalization of algorithm [ alg : general_nonadaptive ] . in each cycle @xmath104 , 1 . [ alg : general_adaptive_cond1]@xmath113 may be random and depend on @xmath114 ; 2 . [ alg : general_adaptive_cond2]the value of @xmath115 may be random and depend on @xmath116 . while such algorithms can be effective and are what is needed for practical applied work , the theoretical foundation relevant to algorithm [ alg : general_nonadaptive ] does not apply . the groundwork for algorithm [ alg : general_nonadaptive ] was laid by chopin ( 2004 ) . with respect to the first condition of algorithm [ alg : general_adaptive ] , in chopin s development each cycle consists of a single observation , while in practice cycle lengths are often adaptive based on the effective sample size criterion ( liu and chen , 1995 ) . progress has been made toward demonstrating condition [ cond : normal_approx ] in this case only recently ( douc and moulines , 2008 ; del moral et al . , 2011 ) , and it appears that conditions [ cond : prior_evaluate ] through [ cond : prior_var ] are sufficient for the assumptions made in douc and moulines ( 2008 ) . with respect to the second condition of algorithm [ alg : general_adaptive ] , demonstration of condition [ cond : normal_approx ] for the adaptations that are necessary to render sequential posterior simulators even minimally efficient appears to be well beyond current capabilities . there are many specific adaptive algorithms , like the one described in section [ ss : details ] below , that will prove attractive to practitioners even in the absence of sound theory for appraising the accuracy of posterior moment approximations . in some cases condition [ cond : normal_approx ] will hold ; in others , it will not but the effects will be harmless ; and in still others condition [ cond : normal_approx ] will not hold and the relation of to will be an open question . investigators with compelling scientific questions are unlikely to forego attractive posterior simulators awaiting the completion of their proper mathematical foundations . the following algorithm resolves this dilemma , providing a basis for implementations that are both computationally effective and supported by established theory . [ alg : general_hybrid](hybrid ) 1 . [ alg : general_hybrid_step1]execute algorithm [ alg : general_adaptive ] . retain @xmath117 @xmath118 and discard @xmath119 @xmath120 . [ alg : general_hybrid_step3]execute algorithm [ alg : general_nonadaptive ] using the retained @xmath117 but with a new seed for the random number generator used in the @xmath107 and @xmath103 phases . [ prop : hybrid ] if conditions [ cond : prior_evaluate ] through [ cond : prior_var ] are satisfied then algorithm [ alg : general_hybrid ] satisfies condition [ cond : normal_approx ] . because the sequences @xmath121 and @xmath122 produced in step [ alg : general_hybrid_step1 ] are fixed ( predetermined ) with respect to the random particles @xmath123 generated in step [ alg : general_hybrid_step3 ] , the conditions of algorithm [ alg : general_nonadaptive ] are satisfied in step [ alg : general_hybrid_step3 ] . from proposition [ prop_chopin ] , condition [ cond : normal_approx ] is therefore satisfied . applications of bayesian inference are most often directed by investigators who are not specialists in simulation methodology . a generic version of algorithm [ alg : general_adaptive ] that works well for wide classes of existing and prospective models and data sets with minimal intervention by investigators is therefore attractive . while general descriptions of the algorithm such as provided in sections [ subsec : nonadaptive][subsec : adaptive ] give a useful basis for discussion , what is really needed for practical use is a detailed specification of the actual steps that need to be executed . this section provides the complete specification of a particular variant of the algorithm that has worked well across a spectrum of models and data sets characteristic of current research in finance and economics . the software package that we are making available with this paper provides a full implementation of the algorithm presented here ( see section [ ss : software ] ) . c phase termination : : the @xmath102 phase is terminated based on a rule assessing the effective sample size ( kong et al . , 1994 ; liu and chen , 1995 ) . following each update step @xmath124 as described in algorithm 1 , part 2(a)ii , compute @xmath125 ^{2}}{\sum_{j=1}^{j}\sum_{n=1}^{n}w_{jn}\left ( s\right ) ^{2}}.\ ] ] it is convenient to work with the relative sample size , @xmath126 , where @xmath40 is the total number of particles . terminate the @xmath102 phase and proceed to the @xmath107 phase if @xmath127 . + we have found that the specific threshold value used has little effect on the performance of the algorithm . higher thresholds imply that the @xmath102 phase is terminated earlier , but fewer metropolis steps are needed to obtain adequate diversification in the @xmath103 phase ( and inversely for lower thresholds ) . the default threshold in the software is 0.5 . this can be easily modified by users , but we have found little reason to do so . resampling method : : we make several resampling methods available . the results of chopin ( 2004 ) apply to multinomial and residual resampling ( baker , 1985 , 1987 ; liu and chen , 1998 ) . stratified ( kitagawa , 1996 ) and systematic ( carpenter et al . , 1999 ) samplers are also of potential interest . we use residual sampling as the default . it is substantially more efficient than multinomial at little additional computational cost . stratified and systematic resamplers are yet more efficient as well as being less costly and simpler to implement ; however , we are not aware of any available theory supporting their use and so recommend them only for experimental use at this point . metropolis steps : : the default is to use a sequence of gaussian random walk samplers operating on the entire parameter vector in a single block . at iteration @xmath128 of the @xmath103 phase in cycle @xmath104 , the variance of the sampler is obtained by @xmath129 where @xmath130 is the sample variance of the current population of particles and @xmath131 is a stepsize " scaling factor . the stepsize is initialized at 0.5 and incremented depending upon the metroplis acceptance rate at each iteration . using a target acceptance rate of 0.25 ( gelman et al . , 1996 ) , the stepsize is incremented by 0.1 if the acceptance rate is greater than the target and decremented by 0.1 otherwise , respecting the constraint @xmath132 . + an alternative that has appeared often in the literature is to use an independent gaussian sampler . this can give good performance for relatively simple models , and is thus useful for illustrative purposes using implementations that are less computationally efficient . but for more demanding applications , such as those that are likely to be encountered in practical work , the independent sampler lacks robustness . if the posterior is not well - represented by the gaussian sampler , the process of particle diversification can be very slow . the difficulties are amplified in the case of high - dimensional parameter vectors . and the sampler fails spectacularly in the presence of irregular posteriors ( e.g. , multimodality ) . + the random walk sampler gives up some performance in the case of near gaussian posteriors ( though it still performs well in this situation ) , but is robust to irregular priors , a feature that we consider to be important . the independent sampler is available in the software for users wishing to experiment . other alternatives for the metropolis updates are also possible , and we intend to investigate some of these in future work . m phase termination : : the simplest rule would be to terminate the @xmath103 phase after some fixed number of metropolis iterations , @xmath133 . but in practice , it sometimes occurs that the @xmath102 phase terminates with a very low @xmath134 . in this case , the @xmath103 phase begins with relatively few distinct particles and we would like to use more iterations in order to better diversify the population of particles . a better rule is to end the @xmath103 phase after @xmath135 iterations if @xmath136 and after @xmath137 iterations if @xmath138 ( the default settings in the software are @xmath139 , @xmath140 , @xmath141 and @xmath142 ) . we refer to this as the deterministic stopping rule . " + but , the optimal settings , especially for @xmath143 , are highly application dependent . setting @xmath143 too low results in poor rejuvenation of particle diversity and the _ rne _ associated with moments of functions of interest will typically be low . this problem can be addressed by either using more particles or a higher setting for @xmath143 . typically , the latter course of action is more efficient if _ rne _ drops below about 0.25 ; that is , finesse ( better rejuvenation ) is generally preferred over brute force ( simply adding more particles ) . on the other hand , there is little point in continuing @xmath103 phase iterations once _ rne _ gets close to one ( indicating that particle diversification is already good , with little dependence amongst the particles in each group ) . continuing past this point is simply a waste of computing resources . + _ rne _ thus provides a useful diagnostic of particle diversity . but it also provides the basis for a useful alternative termination criterion . the idea is to terminate @xmath103 phase iterations when the average _ rne _ of numerical approximations of selected test moments exceeds some threshold . we use a default threshold of @xmath144 . it is often useful to compute moments of functions of interest at particular observation dates . while it is possible to do this directly using weighted samples obtained within the @xmath102 phase , it is more efficient to terminate the @xmath102 phase and execute @xmath107 and @xmath103 phases at these dates prior to evaluating moments . in such cases , it is useful to target a higher _ rne _ threshold to assure more accurate approximations . we use a default of @xmath145 here . we also set an upper bound for the number of metropolis iterations , at which point the @xmath103 phase terminates regardless of _ the default is @xmath146 . + while the deterministic rule is simpler , the _ rne_-based rule has important advantages : it is likely to be more efficient than the deterministic rule in practice ; and it is fully automated , thus eliminating the need for user intervention . some experimental results illustrating these issues are provided in section [ sec : application ] . + an important point to note here is that after the @xmath107 phase has executed , _ no longer provides a useful as a measure of the diversity of the particle population ( sincle the particles are equally - weighted ) . _ rne _ is the only useful measure of which we are aware . parameterization : : bounded support of some parameters can impede the efficiency of the gaussian random walk in the @xmath103 phase , which has unbounded support . we have found that simple transformations , for example a positive parameter @xmath147 to @xmath148 or a parameter @xmath149 defined on @xmath150 to @xmath151 , can greatly improve performance . this requires attention to the jacobian of the transformation in the prior distribution , but it is often just as appealing simply to place the prior distribution directly on the transformed parameters . section 5.1 provides an example . the posterior simulator described in this section is adaptive ( and thus a special case of algorithm [ alg : general_adaptive ] ) . that is , key elements of the simulator are not known in advance , but are determined on the basis of particles generated in the course of executing it . in order to use the simulator in the context of the hybrid method ( algorithm [ alg : general_hybrid ] ) , these design elements must be saved for use in a second run . for the simulator described here , we need the observations @xmath152 at which each @xmath102 phase termination occurs , the variance matrices @xmath153 used in the metropolis updates , and the number of metropolis iterations executed in each @xmath103 phase @xmath154 . the hybrid method then involves running the simulator a second time , generating new particles but using the design elements saved from the first run . since the design elements are now fixed and known in advance , this is a special case of algorithm [ alg : general_nonadaptive ] and thus step 2 of algorithm [ alg : general_hybrid ] . and therefore , in particular , condition [ cond : normal_approx ] is satisfied and the results of section [ sec : parpostsims ] apply . the software package that we are making available with this paper ( http://www.quantosanalytics.org/garland/mp-sps_1.1.zip ) implements the algorithm developed in section [ ss : details ] . the various settings described there are provided as defaults but can be easily changed by users . switches are available allowing the user to choose whether the algorithm runs entirely on the cpu or using gpu acceleration , and whether to run the algorithm in adaptive or hybrid mode . the software provides log marginal likelihood ( and log score if a burn - in period is specified ) together with estimates of nse , as described in section [ subsec : ml ] . posterior mean and standard deviation of functions of interest at specified dates are provided along with corresponding estimates of rne and nse , as described in section [ sec : parpostsims ] . rss at each @xmath102 phase update and rne of specified test moments at each @xmath103 phase iteration are available for diagnostic purposes , and plots of these are generated as part of the standard output . for transparency and accessibility to economists doing practical applied work , the software uses a matlab shell ( with extensive use of the parallel / gpu toolbox ) , with calls to functions coded in c / cuda for the computationally intensive parts . for simple problems where computational cost is low to begin with , the overhead associated with using matlab is noticeable . but , for problems where computational cost is actually important , the cost is nearly entirely concentrated in c / cuda code . very little efficiency is lost by using a matlab shell , while the gain in transparency is substantial . the software is fully object - oriented , highly modular , and easily extensible . it is straightforward for users to write plug - ins implementing new models , providing access to the full benefits of gpu acceleration with little programming effort . all that is needed is a matlab class with methods that evaluate the data density and moments of any functions of interest . the requisite interfaces are all clearly specified by abstract classes . to use a model for a particular application , data and a description of the prior must be supplied as inputs to the class constructor . classes implementing simulation from and evaluation of some common priors are included with the software . extending the package to implement additional prior specifications is simply a matter of providing classes with this functionality . the software includes several models ( including the application described in section [ sec : application ] of this paper ) , and we are in the process of extending the library of available models and illustrative applications using this framework in ongoing work . geweke et al . ( 2013 ) provides an example of one such implementation for the logit model . marginal likelihoods are fundamental to the bayesian comparison , averaging and testing of models . sequential posterior simulators provide simulation - consistent approximations of marginal likelihoods as a by - product almost no additional computation is required . this happens because the marginal likelihood is the product of predictive likelihoods over the sample , and the @xmath102 phase approximates these terms in the same way that classical importance sampling can be used to approximate marginal likelihood ( kloek and van dijk , 1978 ; geweke , 2005 , section 8.2.2 ) . by contrast there is no such generic approach with mcmc posterior simulators . the results for posterior moment approximation apply only indirectly to marginal likelihood approximation . this section describes a practical adaptation of the results to this case , and a theory of approximation that falls somewhat short of condition [ cond : normal_approx ] for posterior moments . any predictive likelihood can be cast as a posterior moment by defining @xmath155 then @xmath156 : = \overline{g}\left(t , s\right ) . \label{pl__formula}\ ] ] [ prop_pl]if conditions [ cond : prior_evaluate ] , [ cond : lf_evaluate ] and [ cond : bounded_like ] are satisfied , then in algorithms [ alg : general_nonadaptive ] and [ alg : general_hybrid ] condition [ cond : normal_approx ] is satisfied for the function of interest ( [ pl_func ] ) and the sample @xmath157 . it suffices to note that conditions [ cond : prior_evaluate ] through [ cond : bounded_like ] imply that condition [ cond : prior_var ] is satisfied for . the utility of proposition [ prop_pl ] stems from the fact that simulation approximations of many moments ( [ pl__formula ] ) are computed as by - products in algorithm [ alg : general_nonadaptive ] . specifically , in the @xmath102 phase at for @xmath158 and @xmath159 @xmath160 from proposition [ prop_pl ] applied over the sample @xmath161 implies for @xmath158 and @xmath162 @xmath163 algorithms [ alg : general_nonadaptive ] and [ alg : general_hybrid ] therefore provide weakly consistent approximations to expressed in terms of the form . the logarithm of this approximation is a weakly consistent approximation of the more commonly reported log predictive likelihood . the numerical standard error of @xmath164 is given by , and the delta method provides the numerical standard error for the log predictive likelihood . the values of @xmath124 for which this result is useful are limited because as @xmath124 increases , @xmath165 becomes increasingly concentrated and the computational efficiency of the approximations @xmath166 of @xmath167 declines . indeed , this is why particle renewal ( in the form of @xmath107 and @xmath103 phases ) is undertaken when the effective sample size drops below a specified threshold . the sequential posterior simulator provides access to the posterior distribution at each observation in the @xmath102 phase using the particles with the weights computed at . if at this point one executes the auxiliary simulations @xmath168 @xmath169 then a simulation - consistent approximation of the cumulative distribution of a function @xmath170 , evaluated at the observed value @xmath171 , is @xmath172 } \left [ f\left ( y_{sjn}^{\left ( \ell -1\right ) } \right ) \right ] } { \sum_{j=1}^{j}\sum_{n=1}^{n}w_{jn}\left ( s-1\right)}.\ ] ] these evaluations are the essential element of a probability integral transform test of model specification ( rosenblatt , 1952 ; smith , 1985 ; diebold et al . , 1998 ; berkowitz , 2001 ; geweke and amisano , 2010 ) . thus by accessing the weights @xmath173 from the @xmath102 phase of the algorithm , the investigator can compute simulation consistent approximations of any set of predictive likelihoods ( 18 ) and can execute probability integral transform tests for any function of @xmath174 to develop a theory of sequential posterior simulation approximation to the marginal likelihood , some extension of the notation is useful . denote @xmath175 and then @xmath176 it is also useful to introduce the following condition , which describes an ideal situation that is not likely to be attained in practical applications but is useful for expositional purposes . [ cond : mphase]in the mutation phase of algorithm [ alg : general_nonadaptive ] @xmath177 with the addition of condition [ cond : mphase ] it would follow that @xmath178 $ ] in condition [ cond : normal_approx ] , and computed values of relative numerical efficiencies would be about 1 for all functions of interest @xmath179 . in general condition [ cond : mphase ] is unattainable in any interesting application of an sequential posterior simulator , for if it were the posterior distribution could be sampled by direct monte carlo . [ prop_ml]if conditions [ cond : prior_evaluate ] , [ cond : lf_evaluate ] and [ cond : bounded_like ] are satisfied then in algorithms [ alg : general_nonadaptive ] and [ alg : general_hybrid ] @xmath180 as @xmath63 , and @xmath181 ^{-1}\sum_{j=1}^{j } \left ( \overline{w}_{j}^{n}-\overline{w}^{\left ( j , n\right ) } \right ) ^{2}\right\ } = \mathrm{var}\left ( \overline{w}^{\left ( j , n\right ) } \right ) . \label{prop_4a}\ ] ] if , in addition , condition [ cond : mphase ] is satisfied then @xmath182 \overset{d}{\rightarrow } n\left ( 0,v\right ) . \label{prop_4b}\ ] ] from proposition [ prop_pl ] , condition [ cond : normal_approx ] is satisfied for @xmath183 . therefore @xmath184 as @xmath63 @xmath185 . the result follows from the decomposition @xmath186 condition [ cond : bounded_like ] implies that moments of @xmath187 of all orders exist . then follows from the mutual independence of @xmath188 . result follows from the mutual independence and asymptotic normality of the @xmath189 terms @xmath190 , and @xmath50 follows from the usual asymptotic expansion of the product in @xmath187 . note that without condition [ cond : mphase ] , @xmath191and @xmath192 are not independent and so condition [ cond : normal_approx ] does not provide an applicable central limit theorem for the product @xmath193 ( although condition [ cond : normal_approx ] does hold for each @xmath194 @xmath195 individually , as shown in section [ subsec : pl ] ) . let @xmath196 denote the term in braces in . proposition [ prop_ml ] motivates the working approximations @xmath197,\notag \\ & & \log \left ( \overline{w}^{\left ( j , n\right ) } \right ) \overset{\cdot } { \thicksim } n\left [ \log p\left ( y_{1:t}\right ) , \frac{{\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}}\left ( \overline{w}^{\left ( j , n\right ) } \right ) } { \left ( \overline{w}^{(j , n)}\right ) ^{2}}\right ] . \label{ml_se}\end{aligned}\ ] ] standard expansions of @xmath198 and @xmath199 suggest the same asymptotic distribution , and therefore comparison of these values in relation to the working standard error from ( [ ml_se ] ) provides one indication of the adequacy of the asymptotic approximation . condition [ cond : mphase ] suggests that these normal approximations will be more reliable as more metropolis iterations are undertaken in the algorithm detailed in section [ ss : details ] , and more generally , the more effective is the particle diversity generated by the @xmath103 phase . the predictive distributions of returns to financial assets are central to the pricing of their derivatives like futures contracts and options . the literature modeling asset return sequences as stochastic processes is enormous and has been a focus and motivation for bayesian modelling in general and application of sequential monte carlo ( smc ) methods in particular . one of these models is the exponential generalized autoregressive conditional heteroskedasticity ( egarch ) model introduced by nelson ( 1991 ) . the example in this section works with a family of extensions developed in durham and geweke ( 2013 ) that is highly competitive with many stochastic volatility models . in the context of this paper the egarch model is also of interest because its likelihood function is relatively intractable . the volatility in the model is the sum of several factors that are exchangeable in the posterior distribution . the return innovation is a mixture of normal distributions that are also exchangeable in the posterior distribution . both features are essential to the superior performance of the model ( durham and geweke , 2013 ) . permutations in the ordering of factors and mixture components induce multimodal distributions in larger samples . models with these characteristics have been widely used as a drilling ground to assess the performance of simulation approaches to bayesian inference with ill - conditioned posterior distributions ( e.g. , jasra et al . , 2007 ) . the models studied in this section have up to @xmath200 permutations , and potentially as many local modes . although it would be possible to optimize the algorithm for these characteristics , we intentionally make no efforts to do so . nonetheless , the irregularity of the posteriors turns out to pose no difficulties for the algorithm . most important , in our view , this example illustrates the potential large savings in development time and intellectual energy afforded by the algorithm presented in this paper compared with other approaches that might be taken . we believe that other existing approaches , including importance sampling and conventional variants on markov chain monte carlo ( mcmc ) , would be substantially more difficult . at the very least they would require experimentation with tuning parameters by bayesian statisticians with particular skills in these numerical methods , even after using the approach of geweke ( 2007 ) to deal with dimensions of posterior intractability driven by exchangeable parameters in the posterior distribution . the algorithm replaces this effort with a systematic updating of the posterior density , thereby releasing the time of investigators for more productive and substantive efforts . an egarch model for a sequence of asset returns @xmath201 has the form @xmath202 the return disturbance term @xmath203 is distributed as a mixture of @xmath204 normal distributions , @xmath205where @xmath206 is the gaussian density with mean @xmath207 and variance @xmath208 , @xmath209 @xmath210 and @xmath211 . the parameters of the model are identified by the conditions @xmath212 and @xmath213 ; equivalently , @xmath214 the models are indexed by @xmath215 , the number of volatility factors , and @xmath204 , the number of components in the return disturbance normal mixture , and we refer to the specification ( [ egarch_mod1])([egarch_mod2 ] ) as ` egarch_ki ` . the original form of the egarch model ( nelson , 1991 ) is with @xmath216 . all parameters have gaussian priors with means and standard deviations indicated below ( the prior distribution of @xmath217 is truncated below at @xmath218 ) . indices @xmath219 and @xmath220 take on the values @xmath221 and @xmath222 . .parameters and prior distributions for the egarch models [ cols="^,^,^,^",options="header " , ] [ t : rbar ] table [ t : rbar ] shows the outcome of an exercise exploring the relationship between the number of metropolis iterations undertaken in the @xmath103 phases and the performance of the algorithm . the first 8 rows of the table use the deterministic @xmath103 phase stopping rule with @xmath143 = 5 , 8 , 13 , 21 , 34 , 55 , 89 and 144 . the last row of the table , indicated by ` * ' in the @xmath143 field , uses the _ rne_-based stopping rule with @xmath144 , @xmath145 and @xmath223 . the last column of the table reports a measure of computational performance : precision ( i.e. , @xmath224 ) normalized by computational cost ( time in seconds ) . the ratio of precision to time is the relevant criterion for comparing two means of increasing numerical accuracy : more steps in the @xmath103 phase versus more particles . the ratio of precision to time is constant in the latter strategy . ( in fact , on a gpu , it increases up to a point because the gpu works more efficiently with more threads , but the application here with @xmath225 particles achieves full efficiency . ) therefore adding more steps in the @xmath103 phase is the more efficient strategy so long as the ratio of precision to time continues to increase . table [ t : rbar ] shows that in this example , adding iterations is dominant at @xmath226 and is likely dominant at @xmath227 . ideally , one would select @xmath143 to maximize the performance measure reported in the last column of the table . rne_-based stopping rule , shown in the last line of the table , does a good job of automatically picking an appropriate stopping point without requiring any user input or experimentation . the number of metropolis iterations varies from one @xmath103 phase to the next with this stopping rule , averaging just over 100 in this application . although the rne - based rule uses fewer metropolis steps overall relative to @xmath227 , total execution time is greater . this is because the metropolis iterations are more concentrated toward the end of the sample period with the _ rne_-based rule , where they are more computationally costly . the fact that so many metroplis iterations are needed to get good particle diversity is symptomatic of the extremely irregular posterior densities implied by this model ( we return to this issue in section [ ss : irregular ] ) . for models with posterior densities that are closer to gaussian , many fewer metropolis iterations will typically be needed . models for asset returns , like the egarch models considered here , are primarily of interest for their predictive distributions . we illustrate this application using the three functions of interest @xmath228 introduced in section [ subsec : egarch_performance ] . moments are evaluated by monte carlo approximation over the posterior distribution of @xmath229 using the particles obtained at time @xmath230 , as described in section [ sec : parpostsims ] . the last observation of the sample , march 31 , 2010 in this application ( @xmath231 ) , is typically of interest . for illustration , we specify the same date one year earlier , march 31 , 2009 ( @xmath232 ) , as an additional observation of interest . volatility is much higher on the earlier date than it is on the later date . at each date of interest , the @xmath102 phase is terminated ( regardless of the _ rss _ ) and @xmath107 and @xmath103 phases are executed . ; ( c ) @xmath226 ; ( d ) @xmath227 . ] figure [ f : rne ] shows _ rne _ for all three test functions @xmath233 at each metropolis iteration of each cycle @xmath104 . the figure reports results for four different @xmath103 phase stopping rules : the _ rne_-based rule ( with @xmath144 , @xmath145 and @xmath223 ) and the deterministic rule with @xmath143 = 34 , 89 and 144 . the beginning of each @xmath103 phase is indicated by a vertical line . the lower axis of each panel indicates the metropolis iteration ( cumulating across cycles ) ; the top axis indicates the observation date @xmath230 at which each @xmath103 phase takes place ; and the left axis indicates _ rne_. for reference , figure [ f : rne ] includes a horizontal line indicating @xmath234 . this is the default target for the _ rne_-based stopping rule and serves as a convenient benchmark for the other rules as well . in the early part of the sample , the deterministic rules execute many more metropolis steps than needed to achieve the nominal target of @xmath234 . however , these require relatively little time because sample size @xmath230 is small . as noted above , there is little point in undertaking additional metropolis iterations once _ rne _ approaches one , as happens toward the beginning of the sample for all three deterministic rules shown in the figure . toward the end of the sample , achieving any fixed _ target in the @xmath103 phase requires more iterations due to the extreme non - gaussianity of the posterior ( see section [ ss : irregular ] for a more detailed discussion of this issue ) . rne_-based rule adapts automatically , performing iterations only as needed to meet the _ rne _ target , implying more iterations as sample size increases in this application . at observation @xmath235 , november 15 , 1991 , the @xmath102 phase terminates with very low @xmath134 ( regardless of @xmath103 phase stopping rule used ) , the result of a return that is highly unlikely conditional on the model and past history of returns . the deterministic rules undertake additional metropolis iterations to compensate , as detailed in section 3.3 . rne_-based rule also requires more iterations than usual to meet the relevant threshold ; but in this case the number of additional iterations undertaken is determined algorithmically . cccccccccccccccccccc & & & & & & + @xmath143 & compute & e & sd & nse & rne & & e & sd & nse & rne & & e & sd & nse & rne + & time & & & & & & & & & & & & & & + + 5 & 91 & -37.405 & 0.373 & 0.027 & 0.003 & & 95.877 & 7.254 & 0.513 & 0.003 & & -1.999 & 0.823 & 0.063 & 0.003 + 8 & 127 & -37.388 & 0.390 & 0.029 & 0.003 & & 96.457 & 7.543 & 0.531 & 0.003 & & -2.204 & 0.837 & 0.060 & 0.003 + 13 & 205 & -37.347 & 0.366 & 0.020 & 0.005 & & 97.311 & 7.167 & 0.361 & 0.006 & & -2.348 & 0.774 & 0.034 & 0.008 + 21 & 304 & -37.382 & 0.352 & 0.014 & 0.009 & & 96.767 & 6.938 & 0.255 & 0.011 & & -2.530 & 0.788 & 0.030 & 0.011 + 34 & 482 & -37.339 & 0.349 & 0.011 & 0.015 & & 97.582 & 6.913 & 0.211 & 0.016 & & -2.569 & 0.812 & 0.021 & 0.023 + 55 & 776 & -37.340 & 0.353 & 0.007 & 0.034 & & 97.591 & 6.970 & 0.142 & 0.037 & & -2.563 & 0.811 & 0.011 & 0.079 + 89 & 1245 & -37.330 & 0.364 & 0.006 & 0.055 & & 97.735 & 7.133 & 0.109 & 0.065 & & -2.574 & 0.812 & 0.010 & 0.105 + 144 & 2120 & -37.332 & 0.355 & 0.004 & 0.141 & & 97.743 & 6.996 & 0.066 & 0.172 & & -2.580 & 0.816 & 0.007 & 0.230 + * & 2329 & -37.334 & 0.359 & 0.003 & 0.170 & & 97.687 & 7.037 & 0.063 & 0.192 & & -2.587 & 0.818 & 0.005 & 0.406 + + 5 & 91 & -50.563 & 0.309 & 0.014 & 0.007 & & 0.596 & 0.253 & 0.013 & 0.006 & & -2.078 & 0.816 & 0.063 & 0.003 + 8 & 127 & -50.514 & 0.309 & 0.011 & 0.012 & & 0.662 & 0.276 & 0.013 & 0.007 & & -2.274 & 0.838 & 0.059 & 0.003 + 13 & 205 & -50.517 & 0.309 & 0.009 & 0.016 & & 0.696 & 0.284 & 0.010 & 0.013 & & -2.425 & 0.771 & 0.033 & 0.008 + 21 & 304 & -50.512 & 0.310 & 0.007 & 0.031 & & 0.736 & 0.295 & 0.008 & 0.021 & & -2.599 & 0.780 & 0.028 & 0.012 + 34 & 482 & -50.492 & 0.309 & 0.006 & 0.044 & & 0.754 & 0.301 & 0.006 & 0.045 & & -2.616 & 0.795 & 0.018 & 0.030 + 55 & 776 & -50.491 & 0.310 & 0.004 & 0.097 & & 0.755 & 0.302 & 0.004 & 0.103 & & -2.610 & 0.797 & 0.010 & 0.093 + 89 & 1245 & -50.482 & 0.315 & 0.003 & 0.164 & & 0.765 & 0.308 & 0.003 & 0.162 & & -2.640 & 0.794 & 0.008 & 0.145 + 144 & 2120 & -50.483 & 0.314 & 0.002 & 0.381 & & 0.765 & 0.308 & 0.002 & 0.240 & & -2.632 & 0.800 & 0.006 & 0.276 + * & 2329 & -50.482 & 0.315 & 0.002 & 0.460 & & 0.769 & 0.307 & 0.002 & 0.588 & & -2.643 & 0.796 & 0.005 & 0.452 + table [ t : moments ] reports details on the posterior mean approximations for the two dates of interest . total computation time for running the simulator across the full sample is provided for reference . the last line in each panel , indicated by ` * ' in the @xmath143 field , uses the _ rne_-based stopping rule . for both dates , the _ nse _ of the approximation declines substantially as @xmath143 increases . comparison of the compute times reported in table [ t : moments ] again suggests that increasing @xmath143 is more efficient for reducing _ nse_than would be increasing @xmath37 , up to at least @xmath226 . some bias in the moment approximations is evident with low values of @xmath143 . the issue is most apparent for the 3% loss probability on march 31 , 2010 . since volatility is low on that date , the probability of realizing a 3% loss is tiny and arises from tails of the posterior distribution of @xmath229 , which is poorly represented in small samples . for example , with @xmath236 , _ rne _ is 0.013 , implying that each group of size @xmath237 has an effective sample size of only about 13 particles . there is no evidence of bias for @xmath238 or with the _ rne_-based rule . plots such as those shown in figure [ f : rne ] provide useful diagnostics and are provided as a standard output of the software . for example , it is easy to see that with @xmath239 ( panel ( b ) of the figure ) not enough iterations are performed in the @xmath103 phases , resulting in low _ rne _ toward the end of the sample . with lower values of @xmath143 the degradation in performance is yet more dramatic . as @xmath143 is increased to 89 and 144 in panels ( c ) and ( d ) of the figure , respectively , the algorithm is better able to maintain particle diversity through the entire sample . rne_-based rule does a good job at choosing an appropriate number of iterations in each @xmath103 phase and does so without the need for user input or experimentation . in the ` egarch_23 ` model there are 2 permutations of the factors @xmath240 and 6 permutations of the components of the normal mixture probability distribution function of @xmath241 . this presents a severe challenge for single - chain mcmc as discussed by celeux et al . ( 2000 ) and jasra et al . ( 2007 ) , and for similar reasons importance sampling is also problematic . the problem can be mitigated ( frhwirth - schnatter , 2001 ) or avoided entirely ( geweke , 2007 ) by exploiting the special mirror image structure of the posterior distribution . but these models are still interesting as representatives of multimodal and ill - behaved posterior distributions in the context of generic posterior simulators . we focus here on the 6 permutations of the normal mixture in ` egarch_23 ` . consider a @xmath242 parameter subvector @xmath243 with three distinct values of the triplets @xmath244 @xmath245 . there are six distinct ways in which these values could be assigned to components @xmath246 of the normal mixture in the ` egarch_23 ` model . these permutations define six points @xmath247 @xmath248 . for all sample sizes @xmath230 , the posterior densities @xmath249 at these six points are identical . let @xmath250 be a different parameter vector with analogous permutations @xmath251 @xmath252 . as the sample adds evidence @xmath253 or @xmath254 . thus , a specific triplet set of triplets @xmath255 @xmath256 and its permutations will emerge as pseudo - true values of the parameters ( geweke , 2005 , section 3.4 ) . from selected posterior distributions conditional on @xmath157 ( @xmath257 , @xmath258 , @xmath259 , @xmath260 , @xmath239 ) . ] the marginal distributions will exhibit these properties as well . consider the pair @xmath261 , which is the case portrayed in figure [ egarch_multimodal_sigma ] . the scatterplot is symmetric about the axis @xmath262 in all cases . as sample size @xmath230 increases six distinct and symmetric modes in the distribution gradually emerge . these reflect the full marginal posterior distribution for the normal mixture components of the ` egarch_23 ` model ( i.e. , marginalizing on all other parameters ) that is roughly centered on the components @xmath263 , @xmath264 and @xmath265 . the progressive decrease in entropy with increasing @xmath230 illustrates how the algorithm copes with ill - behaved posterior distributions . particles gradually migrate toward concentrations governed by the evidence in the sample . unlike mcmc there is no need for particles to migrate between modes , and unlike importance sampling there is no need to sample over regions eliminated by the data ( on the one hand ) or to attempt to construct multimodal source distributions ( on the other ) . similar phenomena are also evident for the other mixture parameters as well as for the parameters of the garch factors . the algorithm proposed in this paper adapts to these situations without specific intervention on a case - by - case basis . to benchmark the performance of the algorithm against a more conventional approach , we constructed a straightforward metropolis random walk mcmc algorithm , implemented in c code on a recent vintage cpu using the ` egarch_23 ` model . the variance matrix was tuned manually based on preliminary simulations , which required several hours of investigator time and computing time . the algorithm required 12,947 seconds for 500,000 iterations . the numerical approximation of the moment of interest @xmath266 $ ] for march 31 , 2010 , the same one addressed in table [ t : moments ] , produced the result @xmath267 and a _ nse_of @xmath268 . taking the square of _ nse _ to be inversely proportional to the number of mcmc iterations , an _ nse _ of @xmath269 ( the result for the _ rne_-based stopping rule in table [ t : moments ] ) would require about 1,840,000 iterations and 47,600 seconds computing time . thus posterior simulation for the full sample would require about 20 times as long using random walk metropolis in a conventional serial computing environment . although straightforward , this analysis severely understates the advantage of the sequential posterior simulator developed in this paper relative to conventional approaches . the mcmc simulator does not traverse all six mirror images of the posterior density . this fact greatly complicates attempts to recover marginal likelihood from the mcmc simulator output ; see celeux et al . ( 2000 ) . to our knowledge the only reliable way to attack this problem using mcmc is to compute predictive likelihoods from the prior distribution and the posterior densities @xmath270 @xmath271 . recalling that @xmath272 , and taking computation time to be proportional to sample size ( actually , it is somewhat greater ) yields a time requirement of @xmath273 seconds ( 2.28 cpu years ) , which is almost 100,000 times as long as was required in the algorithm and implementation used here . unless the function of interest @xmath274 is invariant to label switching ( geweke , 2007 ) , the mcmc simulator must traverse all the mirror images with nearly equal frequency . as argued in celeux et al . ( 2000 ) , for all practical purposes this condition can not be met with any reliability even with a simulator executed for several cpu centuries . this example shows that simple speedup factors may not be useful in quantifying the reduction in computing time afforded by massively parallel computing environments . for the scope of models set in this work that is , those satisfying conditions [ cond : prior_evaluate ] through [ cond : prior_var]sequential posterior simulation is much faster than posterior simulation in conventional serial computing environments this is a lower bound . there are a number of routine and reasonable objectives of posterior simulation , like those described in this section , that simply can not be achieved at all with serial computing but are fast and effortless with smc . most important , sequential posterior simulation in a massively parallel computing environment conserves the time , energy and talents of the investigator for more substantive tasks . recent innovations in parallel computing hardware and associated software provide the opportunity for dramatic increases in the speed and accuracy of posterior simulation . widely used mcmc simulators are not generically well - suited to this environment , whereas alternative approaches like importance sampling are . the sequential posterior simulator developed here has attractive properties in this context : inherent adaptability to new models ; computational efficiency relative to alternatives ; accurate approximation of marginal and predictive likelihoods ; reliable and well - grounded measures of numerical accuracy ; robustness to irregular posteriors ; and a well - developed theoretical foundation . establishing these properties required a number of contributions to the literature , summarized in section 1 and then developed in sections 2 , 3 and 4 . section 5 provided an application to a state - of - the - art model illustrating the properties of the simulator . the methods set forth in the paper reduce computing time dramatically in a parallel computing environment that is well within the means of academic investigators . relevant comparisons with conventional serial computing environments entail different algorithms , one for each environment . moreover , the same approximation may be relatively more advantageous in one environment than the other . this precludes generic conclusions about speed - up factors . in the example in section [ sec : application ] , for a simple posterior moment , the algorithm detailed in section [ ss : details ] was nearly 10 times faster than a competing random - walk metropolis simulator in a conventional serial computing environment . for predictive likelihoods and marginal likelihood it was 100,000 times faster . the parallel computations used a single graphics processing unit ( gpu ) . up to eight gpu s can be added to a standard desktop computer at a cost ranging from about $ 350 ( us ) for a mid - range card to about $ 2000 ( us ) for a high - performance tesla card . computation time is inversely proportional to the number of gpu s . these contributions address initial items on the research agenda opened up by the prospect of massively parallel computing environments for posterior simulation . in the near term it should be possible to improve on the specific contribution made here in the form of the algorithm detailed in section [ ss : details ] . looking forward on the research agenda , a major component is extending the class of models for which generic sequential posterior simulation will prove practical and reliable . large - scale hierarchical models , longitudinal data , and conditional data density evaluations that must be simulated all pose fertile ground for future work . the research reported here has been guided by two paramount objectives . one is to provide methods that are generic and minimize the demand for knowledge and effort on the part of applied statisticians who , in turn , seek primarily to develop and apply new models . the other is to provide methods that have a firm methodological foundation whose relevance is borne out in subsequent applications . it seems to us important that these objectives should be central in the research agenda . baker je ( 1985 ) . adaptive selection methods for genetic algorithms . in grefenstette j ( ed . ) , proceedings of the international conference on genetic algorithms and their applications , 101111 . malwah nj : erlbaum . chopin n , jacob p ( 2010 ) . free energy sequential monte carlo , application to mixture modelling . in : bernardo jm , bayarri mj , berger jo , dawid ap , heckerman d , smith afm , west m ( eds . ) , bayesian statistics 9 . oxford : oxford university press . chopin n , jacob pi , papaspiliopoulis o ( 2011 ) . smc@xmath275 : a sequential monte carlo algorithm with particle markov chain monte carlo updates . working paper . geweke j , durham g , xu h ( 2013 ) . bayesian inference for logistic regression models using sequential posterior simulation . lee l , yau c , giles mb , doucet a , homes cc ( 2010 ) . on the utility of graphics cards to perform massively parallel simulation of advanced monte carlo methods . journal of computational and graphical statistics 19 : 769789 . mcgrayne shb ( 2011 ) . the theory that would not die : how bayes rule cracked the enigma code , hunted down russian submarines , and emerged triumphant from two centuries of controversy . new haven : yale university press . souchard ma , wang q , chan c , frelinger j , cron a , west m ( 2010 ) . understanding gpu programming for statistical computation : studies in massively parallel massive mixtures . journal of computational graphics and statistics 19 : 419438 .

massively parallel desktop computing capabilities now well within the reach of individual academics modify the environment for posterior simulation in fundamental and potentially quite advantageous ways . but to fully exploit these benefits algorithms that conform to parallel computing environments are needed . sequential monte carlo comes very close to this ideal whereas other approaches like markov chain monte carlo do not . this paper presents a sequential posterior simulator well suited to this computing environment . the simulator makes fewer analytical and programming demands on investigators , and is faster , more reliable and more complete than conventional posterior simulators . the paper extends existing sequential monte carlo methods and theory to provide a thorough and practical foundation for sequential posterior simulation that is well suited to massively parallel computing environments . it provides detailed recommendations on implementation , yielding an algorithm that requires only code for simulation from the prior and evaluation of prior and data densities and works well in a variety of applications representative of serious empirical work in economics and finance . the algorithm is robust to pathological posterior distributions , generates accurate marginal likelihood approximations , and provides estimates of numerical standard error and relative numerical efficiency intrinsically . the paper concludes with an application that illustrates the potential of these simulators for applied bayesian inference . keywords : graphics processing unit ; particle filter ; posterior simulation ; sequential monte carlo ; single instruction multiple data jel classification : primary , c11 ; secondary , c630 .

You are an expert at summarizing long articles. Proceed to summarize the following text: deuteron - gold ( ) collisions have been extensively studied at the relativistic heavy ion collider ( rhic ) , both for their intrinsic interest and as a control experiment @xcite to judge the suppression seen in central gold - gold ( ) collisions at sufficiently high transverse momenta ( @xmath7 ) @xcite . unexpectedly , not only data , but recent extended @xmath7 coverage data also display a suppressed nuclear modification factor in central collisions@xcite . this motivates a study of possible mechanisms that may result in a nuclear modification factor smaller than unity at sufficiently high transverse momenta ( @xmath8 gev / c ) in central collisions . the nuclear modification factor @xmath9 compares the spectra of produced particles in collisions to a hypothetical scenario in which the nuclear collisions are assumed to be a superposition of the appropriate number of nucleon - nucleon collisions . in the transverse momentum window @xmath10 gev / c @xmath11 gev / c , the nuclear modification factors are dominated by the cronin peak @xcite . several physical pictures of this enhancement have been proposed @xcite . one family of models @xcite advances an explanation of the cronin effect in terms of the interplay between nuclear shadowing @xcite and the multiple scattering of particles propagating in the strongly - interacting medium ( multiscattering ) @xcite . using the hijing shadowing prescription @xcite , our model gave a reasonable description of the cronin effect in central collisions at midrapidity @xcite . at the same time , we obtained nuclear modification factors close to unity at high transverse momenta ( @xmath12 gev / c @xmath13 gev / c ) . in the high-@xmath7 region multiscattering no longer affects @xmath9 . it is then natural to ask if nuclear shadowing can explain suppression effects at high @xmath7 . in particular , since at @xmath14 @xmath15gev we are in the emc region of the shadowing function for @xmath12 gev / c @xmath13 gev / c , we ask if the emc effect @xcite plays a role in understanding these experiments . in this paper we first investigate collisions at the highest rhic energies and the role of the emc effect at transverse momenta up to @xmath16 gev / c . we then examine whether recent shadowing parameterizations incorporating theoretical uncertainties for the first time@xcite can account for the experimental information . we also display calculational results with a modest energy loss using energy - loss parameters applied earlier to data . finally we extend our considerations to the energy range of the large hadron collider ( lhc ) . figure [ fig1 ] displays recent phenix data ( triangles with error bars)@xcite for the most central collisions , where a high-@xmath7 suppression is clearly seen . while incoherent multiscattering can only lead to enhancement in @xmath9 , nuclear shadowing displays two regions where an @xmath17 can be expected : ( i ) at small @xmath18 ( @xmath19 ) , and ( ii ) in the emc region ( @xmath20 ) @xcite . at rhic energies the small-@xmath18 region is inconsequential at @xmath8 gev / c . thus we focus attention on the emc effect as a possible mechanism for the measured suppression . various shadowing parameterizations developed in the last 15 years @xcite show different behaviors at small-@xmath18 , but the emc region appears rather robust in most models . to see the effect of the emc region , we calculate pion production in a wide momentum range . for this purpose we use a perturbative qcd improved parton model @xcite . the model is based on the factorization theorem and generates the invariant cross section as a convolution of ( nuclear ) parton distribution functions @xmath21 , perturbative qcd cross sections @xmath22 , and fragmentation functions @xmath23 . we perform the calculation in leading order , following refs . @xcite : @xmath24 where @xmath25 and @xmath26 represent the factorization and fragmentation scales , respectively , @xmath27 , @xmath28 , and @xmath29 are momentum fractions , and @xmath30-s stand for two - dimensional transverse momentum vectors . the initial state effects of shadowing and multiscattering are included following the treatment in refs . @xcite . since the effects we investigate are on the @xmath31 level , it is customary to present the obtained results on a linear scale in terms of the nuclear modification factor @xmath32 here @xmath33 is the average number of binary collisions in the various impact - parameter bins . together with the data in central collisions , we display our results with several shadowing parameterizations in fig.[fig1 ] . we use the hijing shadowing including nuclear multiscattering ( solid lines ) , the eks shadowing ( where multiscattering is represented by strong anti - shadowing ) ( dashed line ) , and the hkn parameterization ( with and without nuclear multiscattering , dotted and dash - dotted lines , respectively ) . it can be seen in fig . [ fig1 ] that the suppression associated with the emc effect shows up at transverse momenta @xmath34 gev / c in all models considered , and does not explain the suppression in the data at around @xmath2 gev / c . the cronin peak at @xmath35 gev / c is best reproduced by the hijing parameterization . the `` hkn+multiscattering '' model appears to overshoot the data at low @xmath7 , while it gives very similar results to hijing at @xmath36 gev / c . since the uncertainty of the hkn nuclear parton distribution functions is also available , we next examine the theoretical uncertainty of the description . in fig . [ fig2 ] the effect of the uncertainty given by the hkn shadowing parameterization is illustrated on the nuclear modification factor , @xmath9 . the errors are calculated by the hessian method , using the original code of the hkn group . the mean value as a function of @xmath7 is represented by a solid line , surrounded by an error band of approximately @xmath37 . although the data fall within this band at low @xmath7 , for @xmath38 gev / c the observed suppression is stronger than allowed by this calculation . since we are not aware of any other initial - state suppression , we consider physics operating in the final state . in particular , jet energy loss suggests itself @xcite . this idea is supported by the presence of a minor energy loss effect in peripheral collisions @xcite . assuming @xmath39 fm for the average static transverse size of the traversed medium , and the usual glv parameters of @xmath40 fm mean free path and @xmath41 gev screening mass , we calculate the effect of jet energy loss on the nuclear modification factor for central collisions . this is displayed in fig . it can be seen that inclusion of this energy loss results in a parallel down - shift of @xmath9 relative to curves in fig [ fig2 ] . the slope of the data as a function of @xmath7 is still very different from that of the calculated results . nevertheless , taking into account all experimental and theoretical uncertainties , one could consider the displayed result with @xmath42 to be an acceptable compromise . jet energy loss depends on the transverse parton density , which relates to the measurable hadronic quantity @xmath43 , where @xmath44 is the transverse area of the deconfined region . assuming a realistic geometry for central and collisions and considering the experimental data on @xmath45 , we obtain only a factor of @xmath10 difference for the transverse parton densities between the and cases . on this basis , one could expect the jet - quenching effect in to be even stronger than shown by the calculated band in fig . however , jet energy loss in non - thermal matter is an open question , which we plan to investigate in a forthcoming paper @xcite . we expect that the emc region will shift to higher and higher transverse momenta with increasing collision energy . at the same time , due to the increasing parton density , the energy loss should also become larger . we repeat our calculation for @xmath46 collisions at @xmath47 gev and @xmath48 tev to display these tendencies . the results are shown in fig . 4 . following the ordering of the lines on the left of the figure , the top curve represents the results with hijing shadowing from fig . the second line corresponds to a similar calculation ( i.e. hijing shadowing , no energy loss ) at @xmath49 gev c.m . the next line shows the result at @xmath48 tev without jet quenching , while the dotted and dashed lines illustrate the effect of energy loss . first we took @xmath39 fm for the transverse size of the medium , and used the value of @xmath40 fm as earlier . the introduction of a smaller @xmath50 value to represent the increase in the transverse density of colored scattering centers with increasing c.m . energy gives @xmath51 . it can be seen that the dip corresponding to the emc effect shifts to higher transverse momenta with increasing energy as expected . while @xmath9 is above unity at @xmath35 gev / c for 0.2 tev ( cronin effect ) , at @xmath52 and @xmath48 tev , where we are deep in the shadowing region at these transverse momenta , at most a small ripple appears on @xmath9 , which is rising towards one with increasing transverse momentum in this region . the effect of varying @xmath50 at @xmath48 tev is minimal at transverse momenta @xmath53 gev / c , but quite significant at @xmath54 gev / c @xmath55 gev / c . in conclusion , while this solution appears initially tempting , the emc effect does not explain the unexpected suppression seen at @xmath56 gev / c in @xmath9 at @xmath14 @xmath15gev . taking into account experimental uncertainties as well as the uncertainties of the hkn nuclear parton distribution functions , the experimental and theoretical results can be brought into agreement using a non - negligible amount of final state parton energy loss ( with standard opacity parameters ) . the non - thermal nature of the system casts some doubt on the parameter values . the presence of jet quenching in the system is somewhat surprising at first sight , but is justified by the similarity ( within a factor of @xmath10 ) of the real transverse densities in the and systems . for lhc energies we do not have a fully reliable baseline calculation at present . it is clearly seen , however , that shadowing ( i.e suppression ) will dominate the momentum region up to @xmath57 gev / c . the effect of final state interactions ( jet energy loss ) may also reduce the nuclear modification factor , especially in this momentum region . the emc dip in the nuclear modification factor moves toward higher and higher transverse momenta with increasing energy . we thank shunzo kumano for providing the code and the data for calculating the error bars of the hkn parameterization of parton distribution functions . our work was supported in part by hungarian otka t043455 , t047050 , and nk62044 , by the u.s . department of energy under grant u.s . de - fg02 - 86er40251 , and jointly by the u.s . and hungary under mta - nsf - otka oise-0435701 . et al . _ ( phenix ) , phys . lett . * 91 * , 072303 ( 2003 ) ; phys . rev . * c74 * , 024904 ( 2006 ) . b.a . cole , nucl . phys . * a774 * 225 ( 2006 ) . j. adams _ et al . _ ( star ) , phys . lett . * 91 * , 072304 ( 2003 ) ; phys . rev . * c70 * , 064907 ( 2004 ) . et al . _ ( phenix ) , nucl . phys . * a757 * , 184 ( 2005 ) . j. adams _ et al . _ ( star ) , nucl . phys . * a757 * , 102 ( 2005 ) . et al . _ ( phenix ) , arxiv : nucl - ex/0610036 . 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we investigate the influence of modified nuclear parton distribution functions ( pdfs ) on high- hadron production at rhic and lhc energies using a pqcd - improved parton model . for application at rhic , we focus on the possible contribution of the emc modification of the nuclear pdfs in the @xmath0 region to the observed suppression of production at @xmath1 gev / c in collisions . we study three different parameterizations of the nuclear pdf modifications and find that they give consistent results for for neutral pions in the region @xmath2 gev / c @xmath3 gev / c . we find that the emc suppression of the parton distributions in the @xmath4 nucleus does not strongly influence the for in the region where the suppression is observed . using the hkn parameterization , we evaluate systematic errors in the theoretical resulting from uncertainties in the nuclear pdfs . the measured nuclear modification factor is inconsistent with the pqcd model result for @xmath1 gev / c even when the systematic uncertainties in the nuclear pdfs are accounted for . the inclusion of a small final - state energy loss can reduce the discrepancy with the data , but we can not perfectly reproduce the dependence of the measured . for the lhc , we find that shadowing of the nuclear pdfs produces a large suppression in the yield of hadrons with @xmath5 gev / c in @xmath6 collisions .

You are an expert at summarizing long articles. Proceed to summarize the following text: in this section we motivate our study of the ( two - dimensional ) voter model and its dual coalescing walks through their connection with a number of percolation models . in section 2 , we report on numerical results for the dimension of a natural `` chordal interface '' of the voter model . in section 3 we give rigorous ( and a few numerical ) results on the large coalescing classes for coalescing walks ( where vertices @xmath2 and @xmath3 in a box are in the same class if their walks coalesce before hitting the boundary ) . in the appendix , more details about our numerical results are provided . among the most important breakthroughs in statistical physics and probability in the last two decades is the work by schramm and coauthors @xcite and smirnov @xcite identifying ( or conjecturing ) members of the schramm - loewner evolution family of random curves as the scaling limits of various random walks and interfaces in two - dimensional spin systems . in particular smirnov @xcite ( see also camia and newman s paper @xcite ) has shown that the scaling limit of critical site - percolation on the triangular lattice @xmath4 is sle@xmath5 . to give a rough description of one version of this statement , take a rhombic box @xmath6 ( containing @xmath7 vertices ) in the triangular lattice in two dimensions . label the sides clockwise starting from the southwest corner as @xmath8 . a percolation configuration on @xmath6 is an element @xmath9 of @xmath10 defined as follows . fix the vertices in @xmath11 and @xmath12 to have value 0 ( or black or closed ) and those in @xmath13 and @xmath14 to be 1 ( or red or open ) . in the interior @xmath15 , set each vertex to be ( independently ) 0 or 1 , with probability @xmath16 each see figure [ fig : small_perc ] . with @xmath17 , with the exploration path shown in green . ] there exists a unique simple path @xmath18 of length @xmath19 from the southwest corner following edges in the dual hexagonal lattice to the opposite corner that keeps black / closed vertices on the left and red / open vertices on the right . @xmath20 is often referred to as the exploration path ; we will also call it the chordal interface . as @xmath21 the law of @xmath20 , after rescaling , converges weakly to a probability measure on continuous paths that is the law of _ chordal sle@xmath5 _ @xcite in a rhombic domain . one can use this to prove ( see ( * ? ? ? * prop . 2 ) ) that @xmath22\approx l^{7/4}$ ] , where @xmath23 in general for @xmath24 we have that @xmath25 if and only if @xmath26 note that @xmath27 uniquely determines the path @xmath18 . one can therefore ask about the limiting behavior of @xmath20 when the configurations @xmath28 are generated by some other process ( i.e. , not i.i.d . critical site percolation ) in the interior @xmath15 . in the case of the ising model ( where the states at two sites are not independent ) at the critical temperature , smirnov @xcite has identified that the limiting probability measure is instead chordal _ sle@xmath29_. we are interested in the limiting behavior of @xmath20 when the law of the configuration @xmath28 is the stationary distribution of the voter model ( or related models ) on @xmath15 . in this section we define our primary model of interest , on @xmath6 as described above , with boundary states set as 0 on one pair of adjacent sides and 1 on the other pair , while the law of the interior states @xmath30 is the stationary measure for the voter model @xmath31 on @xmath15 , as follows . each @xmath32 has its own independent poisson clock ( a poisson process @xmath33 ) of rate 1 . when the clock of a vertex @xmath34 rings we update the state @xmath35 of @xmath34 by choosing one of its six neighbors uniformly at random and adopting the state of the chosen neighbor . note that the neighbor may be one of the vertices in the boundary @xmath36 whose state is fixed . defined this way , @xmath37 is an irreducible markov process with finite state space @xmath38 , and therefore it has a unique invariant distribution . we will write @xmath39 for a random configuration sampled from this invariant distribution . the process admits a well - known graphical representation ( due to t.e . harris @xcite ) which we now review . for each @xmath40 , we draw a positive half line ( representing time ) in the third dimension , and on it we mark the times of poisson clock rings of that vertex . each mark on a time line represents a state update event which also has an arrow from @xmath34 to the uniformly chosen neighbor whose state is adopted . the lines of the boundary vertices have arrow marks to them , but not from them , as those states are fixed . fix an initial configuration @xmath41 . to determine the state of a vertex @xmath42 at time @xmath43 we start at height / time @xmath43 on the time line corresponding to @xmath34 and follow it down until we reach height / time 0 or we encounter an outgoing arrow ( whichever comes first ) at height @xmath44 . if we meet an outgoing arrow we follow it to the time line of a neighboring vertex @xmath45 and proceed as before , following this time line down from height @xmath46 until reaching height / time 0 or an outgoing arrow . we stop this procedure when we reach a boundary vertex or height 0 on some time line . thus from any @xmath42 and @xmath47 the path followed corresponds to a continuous time nearest neighbor simple random walk on @xmath48 stopped upon reaching a boundary vertex or height 0 . in either case the state @xmath49 at the terminal vertex is known and we set @xmath50 . such a system of `` state genealogy walks '' from all the vertices at time @xmath43 following backward in time is a dual model and is distributed as a system of _ coalescing _ simple symmetric continuous time random walks on the triangular lattice see for example @xcite . since @xmath48 is finite , if @xmath43 is large enough all the walks starting then will with high probability hit the boundary before reaching height 0 . indeed , if we continue the time lines and poisson clocks below height 0 ( and do not terminate the walks at height 0 ) then almost surely from any height @xmath43 there will be a random height @xmath51 at which the walks started from all vertices @xmath42 at height @xmath43 will have reached boundary vertices . what happens on the time lines below height @xmath52 does not affect @xmath53 since the states of the boundary vertices are fixed for all time . this is equivalent to saying that the voter model itself reaches stationarity by a random finite time ( distributed as @xmath54 ) . therefore to sample from @xmath39 it is enough to follow a system of coalescing continuous time simple random walks from each vertex @xmath34 of @xmath55 until they hit a boundary vertex @xmath56 , and set @xmath57 , i.e. @xmath58 if @xmath59 , and @xmath60 otherwise . one could instead sample from @xmath39 by setting @xmath61 for every @xmath62 ( so the state space would become @xmath63 ) and simulating the voter model dynamics until there is no vertex with state 2 . figure [ fig:1sample ] shows a simulation of @xmath39 with @xmath64 , obtained by simulating coalescing random walks from each vertex in the interior , until each one has reached the boundary . with @xmath64 , simulated using c++ the white curve is the exploration path or chordal interface . ] the duality discussed in the previous section tells us that @xmath65 ( we drop the superscript @xmath0 when there is no ambiguity ) is the probability that a simple random walk started at @xmath2 first hits the boundary at a 1-site . in other words , the one - dimensional distributions of our voter model on @xmath48 are equal to those of a model we would like to call _ harmonic percolation_. this is a model under which the states @xmath66 are independent of each other , and as we have already suggested , @xmath67 is equal to the probability that a simple random walk started from @xmath34 first hits the boundary @xmath68 at a 1-site ( i.e. , in @xmath69 ) . harmonic percolation on an infinite strip of thickness @xmath0 coincides with an independent percolation model called gradient percolation @xcite . in the case of gradient percolation the probability @xmath70 of a site @xmath2 being open changes linearly from one boundary where it is 0 to the other boundary where it is 1 . thus the function @xmath70 is harmonic inside the strip ( with specified boundary conditions ) . the difference between the voter and harmonic percolation models arises from the fact that the walks in the former are coalescing , whereas in the latter they are independent . to be more explicit , coalescence in the voter model leads to non - zero correlations as in the following simple lemma . [ lem : correlation ] for any @xmath71 , and any @xmath72 in the interior of @xmath48 , @xmath73 fix @xmath0 , @xmath74 and @xmath75 , and let @xmath76 and @xmath77 denote the elements of @xmath68 with fixed states 0 and 1 respectively . let @xmath78 and @xmath79 be two independent random walks starting from @xmath74 and @xmath75 respectively . let @xmath80 be the first time that @xmath78 and @xmath79 meet each other . let @xmath81 for all times and define @xmath82 so that @xmath83 and @xmath84 are coalescing walks started from @xmath74 and @xmath75 respectively . let @xmath85 and @xmath86 denote the respective hitting times of the boundary , and note that @xmath87 . then @xmath88 this proves the result for @xmath89 . a similar coupling argument can be made for any number @xmath90 of walkers starting from vertices @xmath91 ( choosing the lower indexed random walker to continue when any two meet ) , establishing the claim . for any @xmath92 , if @xmath93 and @xmath94 are distance at least @xmath95 from each other and the boundary @xmath96 then there exist @xmath97 and @xmath98 such that @xmath99 for @xmath100 and all @xmath0 , while @xmath101 where @xmath102 and @xmath103 are times when the difference random walk @xmath104 started at @xmath105 first hits the origin and the boundary of the box @xmath106 respectively , and the last equality follows from proposition 6.4.3 of @xcite . then implies that the correlation @xmath107 between the votes at @xmath74 and @xmath75 goes to zero as per the following . [ lem : corgoeszero ] let @xmath92 , and @xmath93 and @xmath94 be distance at least @xmath95 from each other and the boundary @xmath96 . then @xmath108 as @xmath1 . one can consider i.i.d . percolation , harmonic percolation , and the stationary voter model on @xmath48 as special cases of a general 2-parameter family of models as follows . start a continuous - time walker from each site . each walker initially wears a hat . two walkers wearing hats coalesce when they meet , and instantly become a single walker wearing a hat . walkers not wearing hats do not coalesce with any other walkers . in addition a poisson clock is assigned to each walker . when such a clock rings , the walker takes a random walk step , but before doing so removes her coalescence hat with probability @xmath109 . if a walker wearing a hat steps into a site with another walker with a hat on , the walker that just made its step becomes part of the coalescence set of the walker that was already at the site . upon hitting a boundary site , with probability @xmath110 a walker ( and its entire coalescence set ) is assigned the vote of the boundary vertex it hit , and otherwise ( i.e. , with probability @xmath111 ) its entire coalescence set attains an independently and uniformly chosen vote . varying the boundary and coalescence noise parameters @xmath111 and @xmath109 between 0 and 1 allows us to interpolate between the four corner models : the voter model @xmath112 ; harmonic percolation @xmath113 ; i.i.d . percolation @xmath114 ; and the case @xmath115 corresponds to a model we would like to call cow ( coalescing walk ) percolation . recall that @xmath19 denotes the length of the interface . since this path is a nearest neighbor simple path , there exist @xmath116 such that @xmath117 almost surely . we conjecture that @xmath118\approx l^{d}\end{aligned}\ ] ] for some @xmath119 $ ] . in the case of critical i.i.d . percolation , holds with @xmath120 which is also the hausdorff dimension of the limiting law ( i.e. , of sle@xmath5 , see @xcite ) . for gradient percolation on an infinite strip , the interface curve between the occupied cluster and empty cluster is a.s . unique and has expected length approximately @xmath121 , where @xmath122 is the horizontal length of the piece of strip in which we measure boundary length ( * ? ? ? * proposition 11 ) . so , for any @xmath92 , for all sufficiently large @xmath0 , if we take a piece of strip which is @xmath0 long ( and @xmath0 thick ) , the expected length of the interface curve @xmath123 satisfies @xmath124 . for any @xmath125 , with probability going to 1 with @xmath0 , the curve stays in the central band ( around the central @xmath126 line where @xmath127 ) of width @xmath128 ( * ? ? ? * theorem 6 ) . thus , as @xmath1 , unless we appropriately zoom in around the central line , we expect to see the rescaled interface curve converge to a straight line in the center . since the harmonic function inside a rhombic area with our boundary condition looks almost linear along the diagonal that connects the middle corner of the 1 valued boundary to the middle corner of the 0 valued boundary ( or indeed along any parallel line ) , we expect that the interface curve for harmonic percolation inside our rhombus should scale to a straight line as well . writing @xmath129 for some function @xmath130 which makes the equality true we have that @xmath131 computing the average interface curve length @xmath132 from @xmath133 independent realizations of @xmath39 we obtain the following estimators for @xmath134 based on @xmath135 we say that an estimator @xmath136 ( more precisely a family of estimators @xmath137 ) is a _ consistent _ estimator of some quantity @xmath138 if @xmath139 it is easy to show that @xmath140 is a consistent estimator of @xmath134 if and only if @xmath141 as @xmath142 ( i.e. , if and only if @xmath143 ) , while @xmath144 is a consistent estimator for @xmath134 if and only if @xmath145 as @xmath1 . thus both estimators are consistent if @xmath146 is slowly varying at @xmath147 . if we are willing to assume that the random interface length @xmath19 in a box of size @xmath0 satisfies @xmath148 , where @xmath149=0 $ ] , then it is natural to consider the ordinary least squares estimator @xmath150 for the slope coefficient @xmath134 of the simple linear regression model @xmath151 where @xmath152 are interface lengths on boxes of side lengths @xmath153 , and the @xmath154 are random variables with mean 0 . note that @xmath130 is constant under this assumption . the results from independent simulations of computing the average lengths of the interface curve and estimates @xmath155 ( with @xmath156 ) appear in table [ tab : d_estimates ] . for ordinary percolation and harmonic percolation the values are known ( or expected ) to be @xmath157 and @xmath158 , so for these models @xmath144 ( with @xmath159 ) appears to do best . for the voter model the value of @xmath144 is about 1.46 . .estimates of @xmath134 with @xmath156 and with @xmath160 ( and @xmath161 ) , rounded to 4 decimal places . for harmonic percolation and percolation , @xmath144 is the closest estimator to the true values of 1.42857 and 1.75 respectively . [ cols= " < , < , < , < , < , < , < , < , > " , ]

consider the voter model on a box of side length @xmath0 ( in the triangular lattice ) with boundary votes fixed forever as type 0 or type 1 on two different halves of the boundary . motivated by analogous questions in percolation , we study several geometric objects at stationarity , as @xmath1 . one is the interface between the ( large i.e. , boundary connected ) 0-cluster and 1-cluster . another is the set of large `` coalescing classes '' determined by the coalescing walk process dual to the voter model .

You are an expert at summarizing long articles. Proceed to summarize the following text: factor analysis is one of the most useful tools for modeling common dependence among multivariate outputs . suppose that we observe data @xmath0 that can be decomposed as @xmath1 where @xmath2 are unobservable common factors ; @xmath3 are corresponding factor loadings for variable @xmath4 , and @xmath5 denotes the idiosyncratic component that can not be explained by the static common component . here , @xmath6 and @xmath7 , respectively , denote the dimension and sample size of the data . model ( [ eq1.1 ] ) has broad applications in the statistics literature . for instance , @xmath8 can be expression profiles or blood oxygenation level dependent ( bold ) measurements for the @xmath9th microarray , proteomic or fmri - image , whereas @xmath4 represents a gene or protein or a voxel . see , for example , @xcite . the separations between the common factors and idiosyncratic components are carried out by the low - rank plus sparsity decomposition . see , for example , @xcite . the factor model ( [ eq1.1 ] ) has also been extensively studied in the econometric literature , in which @xmath10 is the vector of economic outputs at time @xmath9 or excessive returns for individual assets on day @xmath9 . the unknown factors and loadings are typically estimated by the principal component analysis ( pca ) and the separations between the common factors and idiosyncratic components are characterized via static pervasiveness assumptions . see , for instance , @xcite among others . in this paper , we consider static factor model , which differs from the dynamic factor model [ @xcite , @xcite ( @xcite ) ] . the dynamic model allows more general infinite dimensional representations . for this type of model , the frequency domain pca [ @xcite ] was applied on the spectral density . the so - called _ dynamic pervasiveness _ condition also plays a crucial role in achieving consistent estimation of the spectral density . accurately estimating the loadings and unobserved factors are very important in statistical applications . in calculating the false - discovery proportion for large - scale hypothesis testing , one needs to adjust accurately the common dependence via subtracting it from the data in ( [ eq1.1 ] ) [ @xcite ] . in financial applications , we would like to understand accurately how each individual stock depends on unobserved common factors in order to appreciate its relative performance and risks . in the aforementioned applications , dimensionality is much higher than sample - size . however , the existing asymptotic analysis shows that the consistent estimation of the parameters in model ( [ eq1.1 ] ) requires a relatively large @xmath7 . in particular , the individual loadings can be estimated no faster than @xmath11 . but large sample sizes are not always available . even with the availability of `` big data , '' heterogeneity and other issues make direct applications of ( [ eq1.1 ] ) with large @xmath7 infeasible . for instance , in financial applications , to pertain the stationarity in model ( [ eq1.1 ] ) with time - invariant loading coefficients , a relatively short time series is often used . to make observed data less serially correlated , monthly returns are frequently used to reduce the serial correlations , yet a monthly data over three consecutive years contain merely 36 observations . to overcome the aforementioned problems , and when relevant covariates are available , it may be helpful to incorporate them into the model . let @xmath12 be a vector of @xmath13-dimensional covariates associated with the @xmath4th variables . in the seminal papers by @xcite and @xcite , the authors studied the following semi - parametric factor model : @xmath14 where loading coefficients in ( [ eq1.1 ] ) are modeled as @xmath15 for some functions @xmath16 . for instance , in health studies , @xmath17 can be individual characteristics ( e.g. , age , weight , clinical and genetic information ) ; in financial applications @xmath17 can be a vector of firm - specific characteristics ( market capitalization , price - earning ratio , etc . ) . the semiparametric model ( [ eq1.2 ] ) , however , can be restrictive in many cases , as it requires that the loading matrix be fully explained by the covariates . a natural relaxation is the following semiparametric model : @xmath18 where @xmath19 is the component of loading coefficient that can not be explained by the covariates @xmath17 . let @xmath20 . we assume that @xmath21 have mean zero , and are independent of @xmath22 and @xmath23 . in other words , we impose the following factor structure : @xmath24 which reduces to model ( [ eq1.2 ] ) when @xmath25 and model ( [ eq1.1 ] ) when @xmath26 . when @xmath17 genuinely explains a part of loading coefficients @xmath27 , the variability of @xmath28 is smaller than that of @xmath27 . hence , the coefficient @xmath19 can be more accurately estimated by using regression model ( [ eq1.3 ] ) , as long as the functions @xmath29 can be accurately estimated . let @xmath30 be the @xmath31 matrix of @xmath32 , @xmath33 be the @xmath34 matrix of @xmath35 , @xmath36 be the @xmath37 matrix of @xmath38 , @xmath39 be the @xmath37 matrix of @xmath19 and @xmath40 be @xmath31 matrix of @xmath5 . then model ( [ eq1.4 ] ) can be written in a more compact matrix form : @xmath41 we treat the loadings @xmath36 and @xmath39 as realizations of random matrices throughout the paper . this model is also closely related to the _ supervised singular value decomposition _ model , recently studied by @xcite . the authors showed that the model is useful in studying the gene expression and single - nucleotide polymorphism ( snp ) data , and proposed an em algorithm for parameter estimation . we propose a projected - pca estimator for both the loading functions and factors . our estimator is constructed by first projecting @xmath30 onto the sieve space spanned by @xmath22 , then applying pca to the projected data or fitted values . due to the approximate orthogonality condition of @xmath42 , @xmath40 and @xmath39 , the projection of @xmath30 is approximately @xmath43 , as the smoothing projection suppresses the noise terms @xmath39 and @xmath40 substantially . therefore , applying pca to the projected data allows us to work directly on the sample covariance of @xmath43 , which is @xmath44 under normalization conditions . this substantially improves the estimation accuracy , and also facilitates the theoretical analysis . in contrast , the traditional pca method for factor analysis [ e.g. , @xcite , @xcite ] is no longer suitable in the current context . moreover , the idea of projected - pca is also potentially applicable to dynamic factor models of @xcite , by first projecting the data onto the covariate space . the asymptotic properties of the proposed estimators are carefully studied . we demonstrate that as long as the projection is genuine , the consistency of the proposed estimator for latent factors and loading matrices requires only @xmath45 , and @xmath7 does not need to grow , which is attractive in the typical high - dimension - low - sample - size ( hdlss ) situations [ e.g. , @xcite ] . in addition , if both @xmath6 and @xmath7 grow simultaneously , then with sufficiently smooth @xmath29 , using the sieve approximation , the rate of convergence for the estimators is much faster than those of the existing results for model ( [ eq1.1 ] ) . typically , the loading functions can be estimated at a convergence rate @xmath46 , and the factor can be estimated at @xmath47 . throughout the paper , @xmath48 and @xmath49 are assumed to be constant and do not grow . let @xmath50 be a @xmath37 matrix of @xmath51 . model ( [ eq1.3 ] ) implies a decomposition of the loading matrix : @xmath52 where @xmath36 and @xmath39 are orthogonal loading components in the sense that @xmath53 . we conduct two specification tests for the hypotheses : @xmath54 the first problem is about testing whether the observed covariates have explaining power on the loadings . if the null hypothesis is rejected , it gives us the theoretical basis to employ the projected - pca , as the projection is now genuine . our empirical study on the asset returns shows that firm market characteristics do have explanatory power on the factor loadings , which lends further support to our projected - pca method . the second tests whether covariates fully explain the loadings . our aforementioned empirical study also shows that model ( [ eq1.2 ] ) used in the financial econometrics literature is inadequate and more generalized model ( [ eq1.5 ] ) is necessary . as claimed earlier , even if @xmath55 does not hold , as long as @xmath56 , the projected - pca can still consistently estimate the factors as @xmath45 , and @xmath7 may or may not grow . our simulated experiments confirm that the estimation accuracy is gained more significantly for small @xmath7 s . this shows one of the benefits of using our projected - pca method over the traditional methods in the literature . in addition , as a further illustration of the benefits of using projected data , we apply the projected - pca to consistently estimate the number of factors , which is similar to those in @xcite and @xcite . different from these authors , our method applies to the projected data , and we demonstrate numerically that this can significantly improve the estimation accuracy . we focus on the case when the observed covariates are time - invariant . when @xmath7 is small , these covariates are approximately locally constant , so this assumption is reasonable in practice . on the other hand , there may exist individual characteristics that are time - variant [ e.g. , see @xcite ] . we expect the conclusions in the current paper to still hold if some smoothness assumptions are added for the time varying components of the covariates . due to the space limit , we provide heuristic discussions on this case in the supplementary material of this paper [ @xcite ] . in addition , note that in the usual factor model , @xmath50 was assumed to be deterministic . in this paper , however , @xmath50 is mainly treated to be stochastic , and potentially depend on a set of covariates . but we would like to emphasize that the results presented in section [ 1541512515 ] under the framework of more general factor models hold regardless of whether @xmath50 is stochastic or deterministic . finally , while some financial applications are presented in this paper , the projected - pca is expected to be useful in broad areas of statistical applications [ e.g. , see @xcite for applications in gene expression data analysis ] . throughout this paper , for a matrix @xmath57 , let @xmath58 and @xmath59 , @xmath60 denote its frobenius , spectral and max- norms . let @xmath61 and @xmath62 denote the minimum and maximum eigenvalues of a square matrix . for a vector @xmath63 , let @xmath64 denote its euclidean norm . the rest of the paper is organized as follows . section [ sec2 ] introduces the new projected - pca method and defines the corresponding estimators for the loadings and factors . sections [ 1541512515 ] and [ s4 ] provide asymptotic analysis of the introduced estimators . section [ sec5 ] introduces new specification tests for the orthogonal decomposition of the semiparametric loadings . section [ sec6 ] concerns about estimating the number of factors . section [ sec7 ] presents numerical results . finally , section [ sec8 ] concludes . all the proofs are given in the and the supplementary material [ @xcite ] . in the high - dimensional factor model , let @xmath50 be the @xmath37 matrix of loadings . then the general model ( [ eq1.1 ] ) can be written as @xmath65 suppose we additionally observe a set of covariates @xmath66 . the basic idea of the projected - pca is to smooth the observations @xmath67 for each given day @xmath9 against its associated covariates . more specifically , let @xmath68 be the fitted value after regressing @xmath67 on @xmath69 for each given @xmath9 . this results in a smooth or projected observation matrix @xmath70 , which will also be denoted by @xmath71 . the projected - pca then estimates the factors and loadings by running the pca based on the projected data @xmath70 . here , we heuristically describe the idea of projected - pca ; rigorous analysis will be carried out afterward . let @xmath72 be a space spanned by @xmath73 , which is orthogonal to the error matrix @xmath40 . let @xmath74 denote the projection matrix onto @xmath72 [ whose formal definition will be given in ( [ eq2.5 ] ) below . at the population level , @xmath74 approximates the conditional expectation operator @xmath75 , which satisfies @xmath76 , then @xmath77 and @xmath78 . hence , analyzing the projected data @xmath79 is an approximately noiseless problem , and the sample covariance has the following approximation : @xmath80 we now argue that @xmath33 and @xmath81 can be recovered from the projected data @xmath70 under some suitable normalization condition . the normalization conditions we impose are @xmath82 under this normalization , using ( [ eq2.1a ] ) , @xmath83 . we conclude that the columns of @xmath84 are approximately @xmath85 times the first @xmath86 eigenvectors of the @xmath87 matrix @xmath88 . therefore , the projected - pca naturally defines a factor estimator @xmath89 using the first @xmath86 principal components of @xmath90 . the projected loading matrix @xmath81 can also be recovered from the projected data @xmath71 in two ( equivalent ) ways . given @xmath33 , from @xmath91 , we see @xmath92 . alternatively , consider the @xmath93 projected sample covariance : @xmath94 where @xmath95 is a remaining term depending on @xmath96 . right multiplying @xmath81 and ignoring @xmath95 , we obtain @xmath97 . hence , the ( normalized ) columns of @xmath81 approximate the first @xmath86 eigenvectors of @xmath98 , the @xmath93 sample covariance matrix based on the projected data . therefore , we can either estimate @xmath81 by @xmath99 given @xmath89 , or by the leading eigenvectors of @xmath98 . in fact , we shall see later that these two estimators are equivalent . if in addition , @xmath100 , that is , the loading matrix belongs to the space @xmath72 , then @xmath50 can also be recovered from the projected data . the above arguments are the fundament of the projected - pca , and provide the rationale of our estimators to be defined in section [ sec2.3 ] . we shall make the above arguments rigorous by showing that the projected error @xmath101 is asymptotically negligible and , therefore , the idiosyncratic error term @xmath40 can be completely removed by the projection step . as one of the useful examples of forming the space @xmath102 and the projection operator , this paper considers model ( [ eq1.4 ] ) , where @xmath17 s and @xmath32 s are the only observable data , and @xmath103 are unknown nonparametric functions . the specific case ( [ eq1.2 ] ) ( with @xmath104 ) was used extensively in the financial studies by @xcite , @xcite and @xcite , with @xmath17 s being the observed `` market characteristic variables . '' we assume @xmath86 to be known for now . in section [ sec6 ] , we will propose a projected - eigenvalue - ratio method to consistently estimate @xmath86 when it is unknown . we assume that @xmath105 does not depend on @xmath9 , which means the loadings represent the cross - sectional heterogeneity only . such a model specification is reasonable since in many applications using factor models , to pertain the stationarity of the time series , the analysis can be conducted within each fixed time window with either a fixed or slowly - growing @xmath7 . through localization in time , it is not stringent to require the loadings be time - invariant . this also shows one of the attractive features of our asymptotic results : under mild conditions , our factor estimates are consistent even if @xmath7 is finite . to nonparametrically estimate @xmath105 without the curse of dimensionality when @xmath17 is multivariate , we assume @xmath29 to be additive : for each @xmath106 , there are @xmath107 nonparametric functions such that @xmath108 each additive component of @xmath109 is estimated by the sieve method . define @xmath110 to be a set of basis functions ( e.g. , b - spline , fourier series , wavelets , polynomial series ) , which spans a dense linear space of the functional space for @xmath111 . then for each @xmath112 , @xmath113 here , @xmath114 are the sieve coefficients of the @xmath115th additive component of @xmath105 , corresponding to the @xmath116th factor loading ; @xmath117 is a `` remaining function '' representing the approximation error ; @xmath118 denotes the number of sieve terms which grows slowly as @xmath45 . the basic assumption for sieve approximation is that @xmath119 as @xmath120 . we take the same basis functions in ( [ eq2.4 ] ) purely for simplicity of notation . define , for each @xmath121 and for each @xmath122 , @xmath123 then we can write @xmath124 let @xmath125 be a @xmath126 matrix of sieve coefficients , @xmath127 be a @xmath128 matrix of basis functions , and @xmath129 be @xmath37 matrix with the @xmath130th element @xmath131 . then the matrix form of ( [ eq2.3 ] ) and ( [ eq2.4 ] ) is @xmath132 substituting this into ( [ eq1.5 ] ) , we write @xmath133 we see that the residual term consists of two parts : the sieve approximation error @xmath134 and the idiosyncratic @xmath40 . furthermore , the random effect assumption on the coefficients @xmath39 makes it also behave like noise , and hence negligible when the projection operator @xmath74 is applied . based on the idea described in section [ sec2.1 ] , we propose a projected - pca method , where @xmath72 is the sieve space spanned by the basis functions of @xmath42 , and @xmath74 is chosen as the projection matrix onto @xmath72 , defined by the @xmath93 projection matrix @xmath135 the estimators of the model parameters in ( [ eq1.5 ] ) are defined as follows . the columns of @xmath136 are defined as the eigenvectors corresponding to the first @xmath86 largest eigenvalues of the @xmath87 matrix @xmath137 , and @xmath138 is the estimator of @xmath36 . the intuition can be readily seen from the discussions in section [ sec2.1 ] , which also provides an alternative formulation of @xmath139 as follows : let @xmath140 be a @xmath141 diagonal matrix consisting of the largest @xmath86 eigenvalues of the @xmath93 matrix @xmath142 . let @xmath143 be a @xmath37 matrix whose columns are the corresponding eigenvectors . according to the relation @xmath144 described in section [ sec2.1 ] , we can also estimate @xmath36 or @xmath81 by @xmath145 we shall show in lemma [ la.1add ] that this is equivalent to ( [ eq2.6 ] ) . therefore , unlike the traditional pca method for usual factor models [ e.g. , @xcite , @xcite ] , the projected - pca takes the principal components of the projected data @xmath71 . the estimator is thus invariant to the rotation - transformations of the sieve bases . the estimation of the loading component @xmath39 that can not be explained by the covariates can be estimated as follows . with the estimated factors @xmath89 , the least - squares estimator of loading matrix is @xmath146 , by using ( [ eq2.1 ] ) and ( [ eq2.2 ] ) . therefore , by ( [ eq1.5 ] ) , a natural estimator of @xmath147 is @xmath148 consider a panel data model with time - varying coefficients as follows : @xmath149 where @xmath17 is a @xmath13-dimensional vector of time - invariant regressors for individual @xmath4 ; @xmath150 denotes the unobservable random time effect ; @xmath5 is the regression error term . the regression coefficient @xmath151 is also assumed to be random and time - varying , but is common across the cross - sectional individuals . the semiparametric factor model admits ( [ eq2.8 ] ) as a special case . note that ( [ eq2.8 ] ) can be rewritten as @xmath152 with @xmath153 unobservable `` factors '' @xmath154 and `` loading '' @xmath155 . the model ( [ eq1.4 ] ) being considered , on the other hand , allows more general nonparametric loading functions . let us first consider the asymptotic performance of the projected - pca in the conventional factor model : @xmath156 in the usual statistical applications for factor analysis , the latent factors are assumed to be serially independent , while in financial applications , the factors are often treated to be weakly dependent time series satisfying strong mixing conditions . we now demonstrate by a simple example that latent factors @xmath33 can be estimated at a faster rate of convergence by projected - pca than the conventional pca and that they can be consistently estimated even when sample size @xmath7 is finite . [ ex3.1 ] to appreciate the intuition , let us consider a specific case in which @xmath157 so that model ( [ eq1.4 ] ) reduces to @xmath158 assume that @xmath159 is so smooth that it is in fact a constant @xmath160 ( otherwise , we can use a local constant approximation ) , where @xmath161 . then the model reduces to @xmath162 the projection in this case is averaging over @xmath4 , which yields @xmath163 where @xmath164 , @xmath165 and @xmath166 denote the averages of their corresponding quantities over @xmath4 . for the identification purpose , suppose @xmath167 , and @xmath168 . ignoring the last two terms , we obtain estimators @xmath169 these estimators are special cases of the projected - pca estimators . to see this , define @xmath170 , and let @xmath171 be a @xmath6-dimensional column vector of ones . take a naive basis @xmath172 ; then the projected data matrix is in fact @xmath173 . consider the @xmath87 matrix @xmath174 , whose largest eigenvalue is @xmath175 . from @xmath176 we have the first eigenvector of @xmath137 equals @xmath177 . hence , the projected - pca estimator of factors is @xmath178 . in addition , the projected - pca estimator of the loading vector @xmath179 is @xmath180 hence , the projected - pca - estimator of @xmath181 equals @xmath182 . these estimators match with ( [ e3.2 ] ) . moreover , since the ignored two terms @xmath183 and @xmath184 are of order @xmath185 , @xmath186 and @xmath187 converge whether or not @xmath7 is large . note that this simple example satisfies all the assumptions to be stated below , and @xmath188 and @xmath189 achieve the same rate of convergence as that of theorem [ th4.1 ] . we shall present more details about this example in appendix g in the supplementary material [ @xcite ] . we now state the conditions and results formally in the more general factor model ( [ eq3.1 ] ) . recall that the projection matrix is defined as @xmath190 the following assumption is the key condition of the projected - pca . [ ass3.1 ] there are positive constants @xmath191 and @xmath192 such that , with probability approaching one ( as @xmath193 ) , @xmath194 since the dimensions of @xmath195 and @xmath50 are , respectively , @xmath196 and @xmath37 , assumption [ ass3.1 ] requires @xmath197 , which is reasonable since we assume @xmath86 , the number of factors , to be fixed throughout the paper . assumption [ ass3.1 ] is similar to the _ pervasive _ condition on the factor loadings [ @xcite ] . in our context , this condition requires the covariates @xmath42 have nonvanishing explaining power on the loading matrix , so that the projection matrix @xmath198 has spiked eigenvalues . note that it rules out the case when @xmath42 is completely unassociated with the loading matrix @xmath50 ( e.g. , when @xmath42 is pure noise ) . one of the typical examples that satisfies this assumption is the semiparametric factor model [ model ( [ eq1.4 ] ) ] . we shall study this specific type of factor model in section [ s4 ] , and prove assumption [ ass3.1 ] in the supplementary material [ @xcite ] . note that @xmath33 and @xmath50 are not separately identified , because for any nonsingular @xmath199 , @xmath200 . therefore , we assume the following . [ ass3.2 ] almost surely , @xmath201 and @xmath198 is a @xmath141 diagonal matrix with distinct entries . this condition corresponds to the pc1 condition of @xcite , which separately identifies the factors and loadings from their product @xmath202 . it is often used in factor analysis for identification , and means that the columns of factors and loadings can be orthogonalized [ also see @xcite ] . [ ass3.3 ] ( i ) there are @xmath203 and @xmath204 so that with probability approaching one ( as @xmath193 ) , @xmath205 \(ii ) @xmath206 . note that @xmath207 and @xmath208 is a vector of dimensionality @xmath209 . thus , condition ( i ) can follow from the strong law of large numbers . for instance , @xmath22 are weakly correlated and in the population level @xmath210 is well - conditioned . in addition , this condition can be satisfied through proper normalizations of commonly used basis functions such as b - splines , wavelets , fourier basis , etc . in the general setup of this paper , we allow @xmath211 s to be cross - sectionally dependent and nonstationary . regularity conditions about weak dependence and stationarity are imposed only on @xmath212 as follows . we impose the strong mixing condition . let @xmath213 and @xmath214 denote the @xmath215-algebras generated by @xmath216 and @xmath217 , respectively . define the mixing coefficient @xmath218 [ ass3.4 ] ( i ) @xmath219 is strictly stationary . in addition , @xmath220 for all @xmath221 ; @xmath222 is independent of @xmath223 . strong mixing : there exist @xmath224 such that for all @xmath225 , @xmath226 weak dependence : there is @xmath227 so that @xmath228 exponential tail : there exist @xmath229 satisfying @xmath230 and @xmath231 , such that for any @xmath232 , @xmath122 and @xmath233 , @xmath234 assumption [ ass3.4 ] is standard , especially condition ( iii ) is commonly imposed for high - dimensional factor analysis [ e.g. , @xcite ] , which requires @xmath235 be weakly dependent both serially and cross - sectionally . it is often satisfied when the covariance matrix @xmath236 is sufficiently sparse under the strong mixing condition . we provide primitive conditions of condition ( iii ) in the supplementary material [ @xcite ] . formally , we have the following theorem : [ th3.1 ] consider the conventional factor model ( [ eq3.1 ] ) with assumptions [ ass3.1][ass3.4 ] . the projected - pca estimators @xmath237 and @xmath238 defined in section [ sec2.3 ] satisfy , as @xmath239 [ @xmath240 may either grow simultaneously with @xmath6 satisfying @xmath241 or stay constant with @xmath242 , @xmath243 to compare with the traditional pca method , the convergence rate for the estimated factors is improved for small @xmath7 . in particular , the projected - pca does not require @xmath244 , and also has a good rate of convergence for the loading matrix up to a projection transformation . hence , we have achieved a finite-@xmath7 consistency , which is particularly interesting in the `` high - dimensional - low - sample - size '' ( hdlss ) context , considered by @xcite . in contrast , the traditional pca method achieves a rate of convergence of @xmath245 for estimating factors , and @xmath246 for estimating loadings . see remarks [ re4.1 ] , [ re4.2 ] below for additional details . let @xmath247 be the @xmath93 covariance matrix of @xmath248 . convergence ( [ eq3.4add ] ) in theorem [ th3.1 ] also describes the relationship between the leading eigenvectors of @xmath98 and those of @xmath249 . to see this , let @xmath250 be the eigenvectors of @xmath249 corresponding to the first @xmath86 eigenvalues . under the _ pervasiveness condition _ , @xmath251 can be approximated by @xmath50 multiplied by a positive definite matrix of transformation [ @xcite ] . in the context of projected - pca , by definition , @xmath252 ; here we recall that @xmath253 is a diagonal matrix consisting of the largest @xmath86 eigenvalues of @xmath98 , and @xmath254 is a @xmath37 matrix whose columns are the corresponding eigenvectors . then ( [ eq3.4add ] ) immediately implies the following corollary , which complements the pca consistency in _ spiked covariance models _ [ e.g. , @xcite and @xcite ] . [ th3.2 ] under the conditions of theorem [ th3.1 ] , there is a @xmath141 positive definite matrix @xmath255 , whose eigenvalues are bounded away from both zero and infinity , so that as @xmath193 [ @xmath240 may either grow simultaneously with @xmath6 satisfying @xmath241 or stay constant with @xmath242 , @xmath256 in the semiparametric factor model , it is assumed that @xmath257 , where @xmath105 is a nonparametric smooth function for the observed covariates , and @xmath19 is the unobserved random loading component that is independent of @xmath17 . hence , the model is written as @xmath258 in the matrix form , @xmath259 and @xmath36 does not vanish ( pervasive condition ; see assumption [ ass4.2 ] below ) . the estimators @xmath237 and @xmath238 are the projected - pca estimators as defined in section [ sec2.3 ] . we now define the estimator of the nonparametric function @xmath29 , @xmath260 . in the matrix form , the projected data has the following sieve approximated representation : @xmath261 where @xmath262 is `` small '' because @xmath39 and @xmath40 are orthogonal to the function space spanned by @xmath42 , and @xmath129 is the sieve approximation error . the sieve coefficient matrix @xmath263 can be estimated by least squares from the projected model ( [ eq4.1 ] ) : ignore @xmath264 , replace @xmath33 with @xmath237 , and solve ( [ eq4.1 ] ) to obtain @xmath265^{-1}\phi ( \bx ) ' \by{\widehat\bf}.\ ] ] we then estimate @xmath29 by @xmath266 where @xmath267 denotes the support of @xmath17 . when @xmath268 , @xmath36 can be understood as the projection of @xmath50 onto the sieve space spanned by @xmath42 . hence , the following assumption is a specific version of assumptions [ ass3.1 ] and [ ass3.2 ] in the current context . [ ass4.1 ] ( i ) almost surely , @xmath201 and @xmath269 is a @xmath141 diagonal matrix with distinct entries . \(ii ) there are two positive constants @xmath191 and @xmath192 so that with probability approaching one ( as @xmath193 ) , @xmath270 in this section , we do not need to assume @xmath271 to be i.i.d . for the estimation purpose . cross - sectional weak dependence as in assumption [ ass4.2](ii ) below would be sufficient . the i.i.d . assumption will be only needed when we consider specification tests in section [ sec5 ] . write @xmath272 , and @xmath273 [ ass4.2 ] ( i ) @xmath274 and @xmath22 is independent of @xmath275 . \(ii ) @xmath276 , @xmath277 and @xmath278 the following set of conditions is concerned about the accuracy of the sieve approximation . [ ass4.3 ] @xmath279 , \(i ) the loading component @xmath280 belongs to a hlder class @xmath281 defined by @xmath282 for some @xmath283 ; \(ii ) the sieve coefficients @xmath284 satisfy for @xmath285 , as @xmath286 , @xmath287 where @xmath288 is the support of the @xmath115th element of @xmath17 , and @xmath118 is the sieve dimension . \(iii ) @xmath289 . condition ( ii ) is satisfied by common basis . for example , when @xmath290 is polynomial basis or b - splines , condition ( ii ) is implied by condition ( i ) [ see , e.g. , @xcite and @xcite ] . [ th4.1 ] suppose @xmath241 . under assumptions [ ass3.3 ] , [ ass3.4 ] , [ ass4.1][ass4.3 ] , as @xmath291 , @xmath7 can be either divergent or bounded , we have that @xmath292 in addition , if @xmath244 simultaneously with @xmath6 and @xmath118 , then @xmath293 the optimal @xmath294 simultaneously minimizes the convergence rates of the factors and nonparametric loading function @xmath29 . it also satisfies the constraint @xmath295 as @xmath296 . with @xmath297 , we have @xmath298 and @xmath299 satisfies @xmath300 some remarks about these rates of convergence compared with those of the conventional factor analysis are in order . [ re4.1]the rates of convergence for factors and nonparametric functions do not require @xmath244 . when @xmath301 , @xmath302 the rates still converge fast when @xmath6 is large , demonstrating the blessing of dimensionality . this is an attractive feature of the projected - pca in the hdlss context , as in many applications , the stationarity of a time series and the time - invariance assumption on the loadings hold only for a short period of time . in contrast , in the usual factor analysis , consistency is granted only when @xmath303 . for example , according to @xcite ( lemma c.1 ) , the regular pca method has the following convergence rate : @xmath304 which is inconsistent when @xmath7 is bounded . [ re4.2]when both @xmath6 and @xmath7 are large , the projected - pca estimates factors as well as the regular pca does , and achieves a faster rate of convergence for the estimated loadings when @xmath19 vanishes . in this case , @xmath305 , the loading matrix is estimated by @xmath306 , and @xmath307 in contrast , the regular pca method as in @xcite yields @xmath308 comparing these rates , we see that when @xmath29 s are sufficiently smooth ( larger @xmath309 ) , the rate of convergence for the estimated loadings is also improved . the loading matrix always has the following orthogonal decomposition : @xmath310 where @xmath39 is interpreted as the loading component that can not be explained by @xmath42 . we consider two types of specification tests : testing @xmath311 , and @xmath312 . the former tests whether the observed covariates have explaining powers on the loadings , while the latter tests whether the covariates fully explain the loadings . the former provides a diagnostic tool as to whether or not to employ the projected - pca ; the latter tests the adequacy of the semiparametric factor models in the literature . testing whether the observed covariates have explaining powers on the factor loadings can be formulated as the following null hypothesis : @xmath314 due to the approximate orthogonality of @xmath42 and @xmath39 , we have @xmath315 . hence , the null hypothesis is approximately equivalent to @xmath316 this motivates a statistic @xmath317 for a consistent loading estimator @xmath318 . normalizing the test statistic by its asymptotic variance leads to the test statistic @xmath319 where the @xmath141 matrix @xmath320 is the weight matrix . the null hypothesis is rejected when @xmath321 is large . the projected - pca estimator is inappropriate under the null hypothesis as the projection is not genuine . we therefore use the least squares estimator @xmath322 , leading to the test statistic @xmath323 here , we take @xmath324 as the traditional pca estimator : the columns of @xmath325 are the first @xmath86 eigenvectors of the @xmath87 data matrix @xmath326 . connor , hagmann and linton ( @xcite ) applied the semiparametric factor model to analyzing financial returns , who assumed that @xmath328 , that is , the loading matrix can be fully explained by the observed covariates . it is therefore natural to test the following null hypothesis of specification : @xmath329 recall that @xmath330 so that @xmath331 . therefore , essentially the specification testing problem is equivalent to testing @xmath332 that is , we are testing whether the loading matrix in the factor model belongs to the space spanned by the observed covariates . a natural test statistic is thus based on the weighted quadratic form @xmath333 for some @xmath334 positive definite weight matrix @xmath335 , where @xmath237 is the projected - pca estimator for factors and @xmath336 . to control the size of the test , we take @xmath337 , where @xmath338 is a diagonal covariance matrix of @xmath339 under @xmath340 , assuming that @xmath341 are uncorrelated . we replace @xmath342 with its consistent estimator : let @xmath343 . define @xmath344 then the operational test statistic is defined to be @xmath345 the null hypothesis is rejected for large values of @xmath346 . for the testing purpose , we assume @xmath347 to be i.i.d . , and let @xmath348 simultaneously . the following assumption regulates the relation between @xmath7 and @xmath6 . [ ass5.1 ] suppose ( i ) @xmath349 are independent and identically distributed ; @xmath350 , and @xmath351 ; @xmath118 and @xmath309 satisfy : @xmath352 , and @xmath353 . condition ( ii ) requires a balance of the dimensionality and the sample size . on one hand , a relatively large sample size is desired [ @xmath354 so that the effect of estimating @xmath342 is negligible asymptotically . on the other hand , as is common in high - dimensional factor analysis , a lower bound of the dimensionality is also required [ condition @xmath350 ] to ensure that the factors are estimated accurately enough . such a required balance is common for high - dimensional factor analysis [ e.g. , @xcite , @xcite ] and in the recent literature for pca [ e.g. , @xcite , @xcite ] . the i.i.d . assumption of covariates @xmath17 in condition ( i ) can be relaxed with further distributional assumptions on @xmath356 ( e.g. , assuming @xmath356 to be gaussian ) . the conditions on @xmath118 in condition ( iii ) is consistent with those of the previous sections . we focus on the case when @xmath357 is gaussian , and show that under @xmath358 , @xmath359 and under @xmath55 @xmath360 whose conditional distributions ( given @xmath33 ) under the null are @xmath361 with degree of freedom , respectively , @xmath362 and @xmath363 . we can derive their standardized limiting distribution as @xmath364 . this is given in the following result . [ th5.1 ] suppose assumptions [ ass3.3 ] , [ ass3.4 ] , [ ass4.2 ] , [ ass5.1 ] hold . then under @xmath358 , @xmath365 where @xmath48 and @xmath49 . in addition , suppose assumptions [ ass4.1 ] and [ ass4.3 ] further hold , @xmath366 is i.i.d . @xmath367 with a diagonal covariance matrix @xmath338 whose elements are bounded away from zero and infinity . then under @xmath55 , @xmath368 in practice , when a relatively small sieve dimension @xmath118 is used , one can instead use the upper @xmath369-quantile of the @xmath370 distribution for @xmath371 . we require @xmath5 be independent across @xmath9 , which ensures that the covariance matrix of the leading term @xmath372 to have a simple form @xmath373 . this assumption can be relaxed to allow for weakly dependent @xmath374 , but many autocovariance terms will be involved in the covariance matrix . one may regularize standard autocovariance matrix estimators such as @xcite and @xcite to account for the high dimensionality . moreover , we assume @xmath338 be diagonal to facilitate estimating @xmath342 , which can also be weakened to allow for a nondiagonal but sparse @xmath338 . regularization methods such as thresholding [ @xcite ] can then be employed , though they are expected to be more technically involved . we now address the problem of estimating @xmath48 when it is unknown . once a consistent estimator of @xmath86 is obtained , all the results achieved carry over to the unknown @xmath86 case using a conditioning argument . , then argue that the results still hold unconditionally as @xmath375 . ] in principle , many consistent estimators of @xmath86 can be employed , for example , @xcite , @xcite , @xcite , @xcite . more recently , @xcite and @xcite proposed to select the largest ratio of the adjacent eigenvalues of @xmath326 , based on the fact that the @xmath86 largest eigenvalues of the sample covariance matrix grow as fast as @xmath6 increases , while the remaining eigenvalues either remain bounded or grow slowly . we extend ahn and horenstein s ( @xcite ) theory in two ways . first , when the loadings depend on the observable characteristics , it is more desirable to work on the projected data @xmath71 . due to the orthogonality condition of @xmath40 and @xmath42 , the projected data matrix is approximately equal to @xmath43 . the projected matrix @xmath376 thus allows us to study the eigenvalues of the principal matrix component @xmath377 , which directly connects with the strengths of those factors . since the nonvanishing eigenvalues of @xmath376 and @xmath378 are the same , we can work directly with the eigenvalues of the matrix @xmath379 . second , we allow @xmath380 . let @xmath381 denote the @xmath116th largest eigenvalue of the projected data matrix @xmath137 . we assume @xmath382 , which naturally holds if the sieve dimension @xmath118 slowly grows . the estimator is defined as @xmath383 the following assumption is similar to that of @xcite . recall that @xmath384 is a @xmath31 matrix of the idiosyncratic components , and @xmath385 denotes the @xmath386 covariance matrix of @xmath339 . [ ass6.1 ] the error matrix @xmath40 can be decomposed as @xmath387 where : the eigenvalues of @xmath338 are bounded away from zero and infinity , @xmath388 is a @xmath7 by @xmath7 positive semidefinite nonstochastic matrix , whose eigenvalues are bounded away from zero and infinity , @xmath389 is a @xmath31 stochastic matrix , where @xmath390 is independent in both @xmath4 and @xmath9 , and @xmath391 are i.i.d . isotropic sub - gaussian vectors , that is , there is @xmath392 , for all @xmath232 , @xmath393 there are @xmath394 , almost surely , @xmath395 this assumption allows the matrix @xmath40 to be both cross - sectionally and serially dependent . the @xmath87 matrix @xmath388 captures the serial dependence across @xmath9 . in the special case of no - serial - dependence , the decomposition ( [ eq5.1 ] ) is satisfied by taking @xmath396 . in addition , we require @xmath339 to be sub - gaussian to apply random matrix theories of @xcite . for instance , when @xmath339 is @xmath397 , for any @xmath398 , @xmath399 , and thus condition ( iii ) is satisfied . finally , the _ almost surely _ condition of ( iv ) seems somewhat strong , but is still satisfied by bounded basis functions ( e.g. , fourier basis ) . we show in the supplementary material [ @xcite ] that when @xmath338 is diagonal ( @xmath5 is cross - sectionally independent ) , both the sub - gaussian assumption and condition ( iv ) can be relaxed . the following theorem is the main result of this section . [ th6.1 ] under assumptions of theorem [ th4.1 ] and assumption [ ass6.1 ] , as @xmath400 , if @xmath118 satisfies @xmath401 and @xmath402 ( @xmath118 may either grow or stay constant ) , we have @xmath403 this section presents numerical results to demonstrate the performance of projected - pca method for estimating loadings and factors using both real data and simulated data . we collected stocks in s&p 500 index constituents from crsp which have complete daily closing prices from year 2005 through 2013 , and their corresponding market capitalization and book value from compustat . there are @xmath404 stocks in our data set , whose daily excess returns were calculated . we considered four characteristics @xmath42 as in @xcite for each stock : size , value , momentum and volatility , which were calculated using the data before a certain data analyzing window so that characteristics are treated known . see @xcite for detailed descriptions of these characteristics . all four characteristics are standardized to have mean zero and unit variance . note that the construction makes their values independent of the current data . we fix the time window to be the first quarter of the year 2006 , which contains @xmath405 observations . given the excess returns @xmath406 and characteristics @xmath17 as the input data and setting @xmath407 , we fit loading functions @xmath408 for @xmath409 using the projected - pca method . the four additive components @xmath280 are fitted using the cubic spline in the r package `` gam '' with sieve dimension @xmath410 . all the four loading functions for each factor are plotted in figure [ fig : gcurves ] . the contribution of each characteristic to each factor is quite nonlinear . , @xmath411 from financial returns of 337 stocks in s&p 500 index . they are taken as the true functions in the simulation studies . in each panel ( fixed @xmath115 ) , the true and estimated curves for @xmath412 are plotted and compared . the solid , dashed and dotted red curves are the true curves corresponding to the first , second and third factors , respectively . the blue curves are their estimates from one simulation of the calibrated model with @xmath413 , @xmath414 . ] we now treat the estimated functions @xmath280 as the true loading functions , and calibrate a model for simulations . the `` true model '' is calibrated as follows : take the estimated @xmath280 from the real data as the true loading functions . for each @xmath6 , generate @xmath366 from @xmath415 where @xmath416 is diagonal and @xmath417 sparse . generate the diagonal elements of @xmath416 from gamma(@xmath418 ) with @xmath419 , @xmath420 ( calibrated from the real data ) , and generate the off - diagonal elements of @xmath417 from @xmath421 with @xmath422 , @xmath423 . then truncate @xmath417 by a threshold of correlation @xmath424 to produce a sparse matrix and make it positive definite by r package `` nearpd . '' generate @xmath425 from the i.i.d . gaussian distribution with mean @xmath426 and standard deviation @xmath427 , calibrated with real data . generate @xmath428 from a stationary var model @xmath429 where @xmath430 . the model parameters are calibrated with the market data and listed in table [ table : calibfactor ] . finally , generate @xmath431 . here @xmath432 is a @xmath433 correlation matrix estimated from the real data . @lccd2.4d2.4c@ & + 0.9076 & 0.0049 & 0.0230 & -0.0371 & -0.1226 & @xmath434 + 0.0049 & 0.8737 & 0.0403 & -0.2339 & 0.1060 & @xmath435 + 0.0230 & 0.0403 & 0.9266 & 0.2803 & 0.0755 & @xmath436 + by projected - pca ( p - pca , red solid ) and traditional pca ( dashed blue ) and @xmath437 , @xmath438 by p - pca over 500 repetitions . left panel : @xmath439 , right panel : @xmath440 . ] and @xmath441 over 500 repetitions , by projected - pca ( p - pca , solid red ) and traditional pca ( dashed blue ) . ] we simulate the data from the calibrated model , and estimate the loadings and factors for @xmath442 and @xmath443 with @xmath6 varying from @xmath444 through @xmath445 . the `` true '' and estimated loading curves are plotted in figure [ fig : gcurves ] to demonstrate the performance of projected - pca . note that the `` true '' loading curves in the simulation are taken from the estimates calibrated using the real data . the estimates based on simulated data capture the shape of the true curve , though we also notice slight biases at boundaries . but in general , projected - pca fits the model well . we also compare our method with the traditional pca method [ e.g. , @xcite ] . the mean values of @xmath446 , @xmath447 , @xmath448 and @xmath449 are plotted in figures [ fig : calibg ] and [ fig : calibf ] where @xmath450 [ see section [ design2 ] for definitions of @xmath451 and @xmath452 . the breakdown error for @xmath453 and @xmath39 are also depicted in figure [ fig : calibg ] . in comparison , projected - pca outperforms pca in estimating both factors and loadings including the nonparametric curves @xmath36 and random noise @xmath39 . the estimation errors for @xmath36 of projected - pca decrease as the dimension increases , which is consistent with our asymptotic theory . and @xmath454 over 500 repetitions . p - pca , pca and sls , respectively , represent projected - pca , regular pca and sieve least squares with known factors : design 2 . here , @xmath328 , so @xmath455 . upper two panels : @xmath6 grows with fixed @xmath7 ; bottom panels : @xmath7 grows with fixed @xmath6 . ] and @xmath456 by projected - pca ( solid red ) and pca ( dashed blue ) : design 2 . upper two panels : @xmath6 grows with fixed @xmath7 ; bottom panels : @xmath7 grows with fixed @xmath6 . ] consider a different design with only one observed covariate and three factors . the three characteristic functions are @xmath457 with the characteristic @xmath458 being standard normal . generate @xmath459 from the stationary var(1 ) model , that is , @xmath460 where @xmath461 . we consider @xmath462 . we simulate the data for @xmath442 or @xmath443 and various @xmath6 ranging from @xmath444 to @xmath445 . to ensure that the true factor and loading satisfy the identifiability conditions , we calculate a transformation matrix @xmath199 such that @xmath463 , @xmath464 is diagonal . let the final true factors and loadings be @xmath465 , @xmath466 . for each @xmath6 , we run the simulation for @xmath445 times . we estimate the loadings and factors using both projected - pca and pc . for projected - pca , as in our theorem , we choose @xmath467 , with @xmath468 and @xmath469 . to estimate the loading matrix , we also compare with a third method : sieve - least - squares ( sls ) , assuming the factors are observable . in this case , the loading matrix is estimated by @xmath470 , where @xmath471 is the true factor matrix of simulated data . the estimation error measured in max and standardized frobenius norms for both loadings and factors are reported in figures [ fig : simpleg ] and [ fig : simplef ] . the plots demonstrate the good performance of projected - pca in estimating both loadings and factors . in particular , it works well when we encounter small @xmath7 but a large @xmath6 . in this design , @xmath328 , so the accuracy of estimating @xmath472 is significantly improved by using the projected - pca . figure [ fig : simplef ] shows that the factors are also better estimated by projected - pca than the traditional one , particularly when @xmath7 is small . it is also clearly seen that when @xmath6 is fixed , the improvement on estimating factors is not significant as @xmath7 grows . this matches with our convergence results for the factor estimator . it is also interesting to compare projected - pca with sls ( sieve least - squares with observed factors ) in estimating the loadings , which corresponds to the cases of unobserved and observed factors . as we see from figure [ fig : simpleg ] , when @xmath6 is small , the projected - pca is not as good as sls . but the two methods behave similarly as @xmath6 increases . this further confirms the theory and intuition that as the dimension becomes larger , the effects of estimating the unknown factors are negligible . we now demonstrate the effectiveness of estimating @xmath86 by the projected - pc s eigenvalue - ratio method . the data are simulated in the same way as in design 2 . @xmath442 or @xmath443 and we took the values of @xmath6 ranging from @xmath444 to @xmath445 . we compare our projected - pca based on the projected data matrix @xmath137 to the eigenvalue - ratio test ( ah ) of @xcite and @xcite , which works on the original data matrix @xmath326 . . p - pca and ah , respectively , represent the methods of projected - pca and @xcite . left panel : mean ; right panel : standard deviation . ] for each pair of @xmath473 , we repeat the simulation for @xmath443 times and report the mean and standard deviation of the estimated number of factors in figure [ fig : estimatek ] . the projected - pca outperforms ah after projection , which significantly reduces the impact of idiosyncratic errors . when @xmath413 , we can recover the number of factors almost all the time , especially for large dimensions ( @xmath474 ) . on the other hand , even when @xmath475 , projected - pca still obtains a closer estimated number of factors . we test the loading specifications on the real data . we used the same data set as in section [ sec7.1 ] , consisting of excess returns from 2005 through 2013 . the tests were conducted based on rolling windows , with the length of windows spanning from 10 days , a month , a quarter and half a year . for each fixed window - length ( @xmath7 ) , we computed the standardized test statistic of @xmath321 and @xmath476 , and plotted them along the rolling windows respectively in figure [ fig : testing ] . in almost all cases , the number of factors is estimated to be one in various combinations of @xmath477 . figure [ fig : testing ] suggests that the semiparametric factor model is strongly supported by the data . judging from the upper panel [ testing @xmath478 , we have very strong evidence of the existence of nonvanishing covariate effect , which demonstrates the dependence of the market beta s on the covariates @xmath42 . in other words , the market beta s can be explained at least partially by the characteristics of assets . the results also provide the theoretical basis for using projected - pca to get more accurate estimation . from 2006/01/03 to 2012/11/30 . the dotted lines are @xmath479 . ] in the bottom panel of figure [ fig : testing ] ( testing @xmath480 ) , we see for a majority of periods , the null hypothesis is rejected . in other words , the characteristics of assets can not fully explain the market beta as intuitively expected , and model ( [ eq1.2 ] ) in the literature is inadequate . however , fully nonparametric loadings could be possible in certain time range mostly before financial crisis . during 20082010 , the market s behavior had much more complexities , which causes more rejections of the null hypothesis . the null hypothesis @xmath328 is accepted more often since 2012 . we also notice that larger @xmath7 tends to yield larger statistics in both tests , as the evidence against the null hypothesis is stronger with larger @xmath7 . after all , the semiparametric model being considered provides flexible ways of modeling equity markets and understanding the nonparametric loading curves . this paper proposes and studies a high - dimensional factor model with nonparametric loading functions that depend on a few observed covariate variables . this model is motivated by the fact that observed variables can explain partially the factor loadings . we propose a projected - pca to estimate the unknown factors , loadings , and number of factors . after projecting the response variable onto the sieve space spanned by the covariates , the projected - pca yields a significant improvement on the rates of convergence than the regular methods . in particular , consistency can be achieved without a diverging sample size , as long as the dimensionality grows . this demonstrates that the proposed method is useful in the typical hdlss situations . in addition , we propose new specification tests for the orthogonal decomposition of the loadings , which fill the gap of the testing literature for semiparametric factor models . our empirical findings show that firm characteristics can explain partially the factor loadings , which provide theoretical basis for employing projected - pca method . on the other hand , our empirical study also shows that the firm characteristics can not fully explain the factor loadings so that the proposed generalized factor model is more appropriate . throughout the proofs , @xmath45 and @xmath7 may either grow simultaneously with @xmath6 or stay constant . for two matrices @xmath481 with fixed dimensions , and a sequence @xmath482 , by writing @xmath483 , we mean @xmath484 . in the regular factor model @xmath485 , let @xmath486 denote a @xmath141 diagonal matrix of the first @xmath86 eigenvalues of @xmath487 . then by definition , @xmath488 . let @xmath489 . then @xmath490 where @xmath491 still by the equality ( [ ea.1add ] ) , @xmath501 . hence , this step is achieved by bounding @xmath502 for @xmath503 . note that in this step , we shall not apply a simple inequality @xmath504 , which is too crude . instead , with the help of the result @xmath505 achieved in step 1 , sharper upper bounds for @xmath502 can be achieved . we do so in lemma b.2 in the supplementary material [ @xcite ] . consider the singular value decomposition : @xmath508 , where @xmath509 is a @xmath93 orthogonal matrix , whose columns are the eigenvectors of @xmath98 ; @xmath510 is a @xmath511 matrix whose columns are the eigenvectors of @xmath512 ; @xmath513 is a @xmath31 rectangular diagonal matrix , with diagonal entries as the square roots of the nonzero eigenvalues of @xmath98 . in addition , by definition , @xmath253 is a @xmath141 diagonal matrix consisting of the largest @xmath86 eigenvalues of @xmath98 ; @xmath254 is a @xmath37 matrix whose columns are the corresponding eigenvectors . the columns of @xmath514 are the eigenvectors of @xmath88 , corresponding to the first @xmath86 eigenvalues . by assumption [ ass3.3 ] , @xmath528 , @xmath529 hence , @xmath530 by lemma b.1 in the supplementary material [ @xcite ] , @xmath531 . similarly , @xmath532 using the inequality that for the @xmath116th eigenvalue , @xmath533 , we have @xmath534 , for @xmath260 . hence , it suffices to prove that the first @xmath86 eigenvalues of @xmath320 are bounded away from both zero and infinity , which are also the first @xmath535 eigenvalues of @xmath536 . this holds under the theorem s assumption ( assumption [ ass3.1 ] ) . thus , @xmath537 , which also implies @xmath523 . fan , j. , liao , y. and mincheva , m. ( 2013 ) . large covariance estimation by thresholding principal orthogonal complements ( with discussion ) . _ journal of the royal statistical society , series b _ * 75 * 603680 .

this paper introduces a projected principal component analysis ( projected - pca ) , which employs principal component analysis to the projected ( smoothed ) data matrix onto a given linear space spanned by covariates . when it applies to high - dimensional factor analysis , the projection removes noise components . we show that the unobserved latent factors can be more accurately estimated than the conventional pca if the projection is genuine , or more precisely , when the factor loading matrices are related to the projected linear space . when the dimensionality is large , the factors can be estimated accurately even when the sample size is finite . we propose a flexible semiparametric factor model , which decomposes the factor loading matrix into the component that can be explained by subject - specific covariates and the orthogonal residual component . the covariates effects on the factor loadings are further modeled by the additive model via sieve approximations . by using the newly proposed projected - pca , the rates of convergence of the smooth factor loading matrices are obtained , which are much faster than those of the conventional factor analysis . the convergence is achieved even when the sample size is finite and is particularly appealing in the high - dimension - low - sample - size situation . this leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings . the proposed method is illustrated by both simulated data and the returns of the components of the s&p 500 index . ./style / arxiv - general.cfg ,

You are an expert at summarizing long articles. Proceed to summarize the following text: in contrast to most astronomical objects ( such as planets , stars and galaxies ) , the universe as a whole is usually considered to be non - rotational . however , the possibility that the universe rotates should not be ignored , since solutions of gr corresponding to a rotating universe have been found @xcite indicating that a global rotation is physically allowed . although it is widely assumed that galaxies align randomly and have zero net angular momentum , there have been many investigations on the general alignment of galaxies . some even claim that a weak alignment of galaxies does exist@xcite . such an alignment may be used to explain@xcite the recently discovered non - gaussian properties@xcite of the cmba . furthermore , jaffe et al . @xcite suggest that the bianchi type vii@xmath3 model with a global rotation may be used to explain some anomalies of the cmba . in addition , the existence of a global rotation may contradict the inflationary model of the early universe @xcite and mach s principle @xcite . rotational perturbations may also be used to determine whether or not the universe is open or closed @xcite . therefore , the study of global rotation is of interest in many different aspects of cosmology , and constraint of the rotation speed of the universe is important . the most popular approach to constrain the magnitude of the global rotation speed is to make use of data from the cmba because of their precision . most discussions @xcite have focused on homogeneous cosmological models , i.e. bianchi models @xcite . to be consistent with obeservations , only bianchi type v , vii@xmath4 , vii@xmath3 and ix models , which include the robertson - walker model as a special case , are considered . the constraints of the global rotation speed obtained depend on the parameters of the models . besides , shear and vorticity are inseparable in these works @xcite , i.e. zero shear automatically implies zero vorticity . there are many other approaches to constrain the global rotation . based on the idea that a global rotation induces a total net spin of galaxies , the global rotation can be limited @xcite . moreover , empirical relations between angular momenta and mass of galaxies / clusters , such as @xmath5 for spiral galaxies and @xmath6 for clusters can be explained by the global rotation @xcite . the acceleration caused by the global rotation may be used to explain parts of the accelerating expansion of our universe , and thus the global rotation can be constrained by supernova type ia data @xcite . recently , some studies of the cmb polarization induced by the global rotation are published @xcite providing potential constraints in the future . to develop a model that preserves the homogeneity and isotropy of the mean cmb , we study the rotation of the universe as a perturbation in the robertson - walker framework with a cosmological constant in this paper . unlike the bianchi models , such an approach allows to have non - zero rotation but trivial shear . since the global rotation does not have any influences on the 1st - order sachs - wolfe effect ( sw effect ) , we need to calculate the metric up to 2nd - order perturbations and the 2nd - order sw effect . then , we will constrain the angular speed of the rotation using recent data on cmba @xcite . our model is inhomogeneous with an axial symmetry in general . the global rotation in our model is not only time - dependent but also radial - dependent . the line element of a flat rotational universe possesses an axial symmetry and can be written in the form of @xcite @xmath7d\eta^2-[1-h(r,\eta)]dr^2-[1-h(r,\eta)]r^2d\theta^2- [ 1-k(r,\eta)]dz^2\nonumber\\ & ~&+2r^2a(\eta)\omega(r,\eta)d\theta d\eta\},\end{aligned}\ ] ] where @xmath8 and @xmath9 , @xmath10 , @xmath11 is the conformal time defined by @xmath12 with @xmath13 the cosmological time , @xmath14 , @xmath15 and @xmath16 are the cylindrical coordinates in the comoving frame of the universe , @xmath17 is the axis of rotation , @xmath18 is the scale factor of the universe with @xmath19 at the present time , @xmath20 is the angular velocity of the metric observed from an inertial frame whose origin is on the rotational axis , and @xmath21 , @xmath22 and @xmath23 are the perturbations on the ( 0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 ) , ( 3 , 3)-components of the metric due to the rotation . because of the cylindrical symmetry , the perturbation functions due to the rotation are also independent of @xmath15 and @xmath16 . here , we assume that the norm of @xmath24@xmath25@xmath26 , where @xmath26 is the unperturbed metric , is much smaller than that of @xmath24 . explicitly , we assume that the rotation is slow , so that @xmath27 , and we can think of @xmath28 for @xmath14 , @xmath11 within the last scattering surface as the perturbation parameter . by parity consideration , we can see that @xmath20 is composed of only odd powers of @xmath29 , whereas @xmath21 , @xmath22 and @xmath23 , being density and pressure perturbations , only even powers . since we are interested only up to second - order perturbations , we will consider @xmath20 to be first - order and @xmath21 , @xmath22 and @xmath23 to be second - order . the metric eq . ( 1.1 ) in ref . @xcite will be recovered if we truncate ours up to the first - order . since the effect of the rotation on the cmba is independent of the parity , we expect that the sw effect due to rotation occurs in even orders of @xmath29 only . the einstein field equations ( efes ) for a universe with cosmological constant @xmath0 are @xmath30 where @xmath31 is the stress - energy tensor for a perfect fluid , @xmath32 is the ricci curvature tensor , @xmath33 is the scalar curvature , @xmath34 is the mass - energy density , @xmath35 is the pressure and @xmath36 is the four - velocity of the fluid in the comoving frame . here , we set @xmath37 . if @xmath38 , the universe is homogeneous and @xmath39 expresses the angular velocity of the universe observed anywhere in the comoving frame . otherwise , the universe is inhomogeneous and the observer at the rotating axis passing through the origin is distinct . to solve eq . ( 3 ) up to 2nd - order in @xmath29 , we expand all quantities : @xmath40 where the subscripts indicate the corresponding orders of perturbations . the zeroth - order efes give rise to the standard friedmann equations : @xmath41 once the equation of state ( eos ) of the fluid is given , we can determine the scale factor @xmath18 , the density @xmath42 and the pressure @xmath43 with the equations above . in this paper , we consider a universe with @xmath44 . however , the following formalism can be applied to any fluid with a specified eos . from the temporal - spatial efes and the condition @xmath45 , we have @xmath46 and @xmath47 . the first - order efes then give : @xmath48=\frac{a^4(\eta)}{8\dot{a}^2(\eta)- 4a(\eta)\ddot{a}(\eta)}[3r\omega ' ( r,\eta)+r^2\omega ' ' ( r,\eta)],\\ 0&=&3\dot{a}(\eta)\omega ' ( r,\eta)+a(\eta)\dot{\omega}'(r,\eta),\end{aligned}\ ] ] where the dots refer to derivatives with respect to the conformal time @xmath11 , and primes mean derivatives with respect to @xmath14 . as seen from the equations above , a 1st - order rotational perturbation can not generate 1st - order perturbations of the mass - energy density and pressure . this is expected because @xmath34 and @xmath35 should be unchanged under the inversion of the rotation . for the same reason , @xmath49 . from eq . ( 11 ) , we see that if @xmath20 is independent of @xmath14 , then @xmath50 . that is , the fluid in the universe rotates with the metric at the same pace . nevertheless , an @xmath14-dependent @xmath20 allows us to discuss the centrifugal force for the universe as that discussed in ref . @xcite for relativistic stars . ( 12 ) implies that @xmath20 must be in the form of @xmath51 , where @xmath52 and @xmath53 are arbitrary functions . moreover , if the fluid is viscous , the r.h.s . of eq . ( 12 ) will be equal to the 1st - order shear term of @xmath54 and this will free @xmath20 from the form above . without loss of generality , we perform the following transformations : @xmath55 where @xmath56 and @xmath57 are arbitrary functions depending on @xmath21 and @xmath22 . the first term of @xmath21 comes from the transformation @xmath58 . using these transformations to formulate the second - order efes , we find that @xmath59 ^ 2}{r^2a^2(\eta ) } + k(r,\eta)+t(r,\eta)\right\},\\ _ 2u_1(r,\eta)&=&\frac{-a^2(\eta)}{8\dot{a}^2(\eta)-4a(\eta)\ddot{a}(\eta ) } \{-\dot{a}(\eta)[2k'(r,\eta)+2t'(r,\eta)]\nonumber\\ & ~&+a(\eta)[-2\dot{k}'(r,\eta)+\dot{l}'(r,\eta)]\},\\ _ 2\rho(r,\eta)&=&-\frac{1}{32\pi a^2(\eta)}\{-4\lambda a^2(\eta)[k(r,\eta)+t(r,\eta ) ] + \frac{4\dot{a}(\eta)[3\dot{k}(r,\eta)-2\dot{l}(r,\eta)]}{a(\eta)}\nonumber\\ & ~&-\frac{2[2k'(r,\eta)-l'(r,\eta)]}{r}+12ra^2(\eta)\omega(r,\eta)\omega ' ( r,\eta)+r^2a^2(\eta)\omega ' ^2(r,\eta)-4k''(r,\eta)\nonumber\\ & ~&+2l''(r,\eta)+4r^2a^2(\eta)\omega(r,\eta)\omega ' ' ( r,\eta ) + 4\left[\lambda a^2(\eta)+\frac{3\dot{a}^2(\eta)}{a^2(\eta)}\right]\{k(r,\eta ) + t(r,\eta)\nonumber\\ & ~&-r^2a^2(\eta)\omega^2(r,\eta)+\left[ra(\eta)\omega(r,\eta)- \frac{_1u_2(r,\eta)}{ra(\eta)}\right]^2\}\},\\ _ 2p(r,\eta)&=&\frac{1}{32\pi ra^4(\eta)}\ { 4ra(\eta)[3\dot{a}(\eta)\dot{k}(r,\eta ) -\dot{a}(\eta)\dot{l}(r,\eta)+\dot{a}(\eta)\dot{t}(r,\eta)]+ 2a^2(\eta)[2r\ddot{k}(r,\eta)\nonumber\\ & ~&-r\ddot{l}(r,\eta)+t'(r,\eta ) ] + ra^4(\eta)[-4\lambda k(r,\eta)-4\lambda t(r,\eta)+r^2\omega ' ^2(r,\eta)]\},\\ _ 2p(r,\eta)&=&\frac{1}{32\pi r^2a^6(\eta)}\{-12\dot{a}^2(\eta)[_1u_2(r,\eta)]^2 + 4r^2a^3(\eta)[3\dot{a}(\eta)\dot{k}(r,\eta ) -\dot{a}(\eta)\dot{l}(r,\eta)\nonumber\\ & ~&+\dot{a}(\eta)\dot{t}(r,\eta)]-r^2a^6(\eta)[4\lambda k(r,\eta)+4\lambda t(r,\eta)+3r^2\omega ' ^2(r,\eta)]\nonumber\\ & ~&+a^4(\eta)\{-4\lambda[_1u_2(r,\eta)]^2 + 2r^2[2\ddot{k}(r,\eta)-\ddot{l}(r,\eta)+t''(r,\eta)]\}\},\\ _ 2p(r,\eta)&=&\frac{1}{32\pi ra^4(\eta)}\ { 4ra(\eta)\dot{a}(\eta)[3\dot{k}(r,\eta)-2\dot{l}(r,\eta)+\dot{t}(r,\eta ) ] -ra^4(\eta)[4\lambda k(r,\eta)\nonumber\\ & ~&+4\lambda t(r,\eta)+r^2\omega ' ^2(r,\eta ) ] + 2a^2(\eta)[2r\ddot{k}(r,\eta)-2r\ddot{l}(r,\eta)+l'(r,\eta)+t'(r,\eta)\nonumber\\ & ~&+rl''(r,\eta)+rt''(r,\eta)]\}.\end{aligned}\ ] ] eqs . ( 19)-(21 ) are three different expressions for @xmath60 derived by the three 2nd - order spatial - spatial efes . ( 15 ) is expected by considering the symmetries of @xmath15(odd)- , @xmath16(even)-components of the four - velocity @xmath61 under the inversion of the rotation . @xmath62 , which is non - zero in general and corresponds to the dynamical changes of @xmath63 and @xmath60 for an @xmath14-dependent rotational speed . in order to calculate these 2nd - order perturbations , we have to find the solutions of @xmath23 , @xmath57 and @xmath56 . since the pressure is the same along different directions at one point , eqs . ( 19)-(21 ) are equivalent . substracting eqs . ( 19)-(21 ) from each other leads to two equations for solving @xmath57 and @xmath56 when @xmath20 is specified while @xmath23 is regarded as an arbitrary function independent of the rotation . the detailed derivations of these solutions are shown in appendix i. as the sw effect is invariant under the inversion of the rotation , the first non - zero sw effect due to rotation occurs in 2nd - order perturbations . the general formalism of the 2nd - oder sw effect has been comprehensively discussed . in the following , we will make use of the ideas in @xcite and derive the 2nd - order sw effect of a rotating universe . the cosmic microwave background ( cmb ) temperature observed at the origin towards a direction @xmath64 can be written as @xmath65 where @xmath66 , the subscripts ( @xmath67 and @xmath68 ) denoting the origin and the last scattering hypersurface ( lsh ) respectively , @xmath36 is the four - velocity of the fluid in the comoving frame , @xmath69 is the wave vector of a light ray in the conformal metric with an affine parameter @xmath70 , @xmath71 is the temperature measured at the point @xmath72 on the lsh , and @xmath73 is the direction of the light ( passing through the point @xmath72 ) observed at the origin . we show in appendix ii that the 1st - order sw effect due to rotation is zero and the 2nd - order sw effect is @xmath74|_{\eta_{\epsilon}}^{\eta_0}\nonumber\\ & = & \frac{\omega_{\lambda}\sin\phi}{2\lambda(1-\omega_{\lambda } ) } [ 2\dot{a}(\eta_{\epsilon})t'(r_{\epsilon},\eta_{\epsilon})- a(\eta_{\epsilon})\dot{l}'(r_{\epsilon},\eta_{\epsilon})]\nonumber\\ & ~&-\frac{1}{2}\left\{\frac{\omega_{\lambda}^2a^4(\eta_{\epsilon})}{4\lambda^2(1-\omega_{\lambda})^2 } [ 3\omega ' ( r_{\epsilon},\eta_{\epsilon})+r_{\epsilon}\omega ' ' ( r_{\epsilon},\eta_{\epsilon})]^2+t(r_{\epsilon},\eta_{\epsilon})\right\}\nonumber\\ & ~&+\int_{\eta_{\epsilon}}^0\left[-\frac{\dot{t}(-\lambda\sin\phi,\lambda)}{2}+ t'(-\lambda\sin\phi,\lambda)\sin\phi-\frac{\dot{l}(-\lambda\sin\phi,\lambda)}{2}\sin^2\phi\right]d\lambda,\end{aligned}\ ] ] where @xmath75 and @xmath76 denotes the conformal time of the last scattering . ( 23 ) determines the cmba produced by the rotation of the universe once @xmath20 is specified . as an example , we consider the simplest case stationary homogeneous rotation ( i.e. @xmath77 , @xmath53 is an arbitrary function ) . then , we have @xmath78 and @xmath79 where @xmath80 is a constant . it is straight - forward to find that @xmath81 to explain this , we recall that @xmath82 if @xmath20 is independent of @xmath14 , which means that the fluid is rotating with the same phase as the metric . therefore , the effect of the rotating metric cancels the relativistic doppler effect caused by the sources rotating in a stationary metric . we make use of the previous example , i.e. @xmath83 with @xmath84 in eqs . ( a.10)-(a.18 ) , to constrain the rotation of the universe . using eqs . ( 23 ) , ( a.10 ) , ( a.16)-(a.18 ) , we expand the cmba as @xmath85 the values of @xmath86 s are listed in table 1 . + * table 1 * + [ cols="<,<,<,<,<",options="header " , ] + we notice that the spherical harmonic expansion has non - zero coefficients only when @xmath87 and even @xmath88 for which @xmath89 has cylindrical and parity symmetries . the result is unlikely to be related to the ` axis of evil ' @xcite , a preferred direction of several low multipoles ( especially quadrupole and octopole ) . however , in the general case , when we are located off the rotational axis , the cylindrical symmetry is broken and non - zero coefficients for other multipoles are allowed . for example , the cmba on the two sides of the rotation axis will be affected differently by the rotation in general . such an asymmetric effect enhances the dipole moment of the cmba . thus , its potential to explain the ` axis of evil ' can not be eliminated without further study . with @xmath90 , @xmath91km / s / mpc and @xmath92 , we constrain @xmath29 to be less than @xmath93 m in si unit . that is , @xmath94 is less than @xmath95 rad yr@xmath2 in usual unit at the last scattering surface . some cmba maps generated with the rotation of the universe are shown in fig . 1 as examples . nevertheless , our result can be regarded as the first constraint of the rotation of a @xmath0cdm universe . in fig . 2 , some normalized 2nd - order perturbed quantities along the light path of the last scattered photons are plotted as a function of @xmath96 with @xmath83 and @xmath97 m. as expected , the perturbed quantities increase with the rotating speed . in fig . 3 , the angular velocity of matter @xmath98 and its difference from that of the metric along the light path of the last scattered photons are plotted against time . we can see that the angular velocity of matter can be negative while the rotation speed of the universe is always positive . because of the @xmath14-dependence of @xmath20 , the angular velocity of matter can be different from that of the metric in general as indicated in eq . these quantities are useful for studying the frame - dragging of the universe in the future . the distributions of the 2nd - order perturbed densities of matter are shown at two different times in fig . 4 . as shown in fig . 2 , @xmath99 is always positive , which means that matter is moving away from the rotating axis and hence the density is expected to be decreasing with time ( shown in fig . 4 ) . under the mollweide projection with the z - direction pointing to @xmath100 with @xmath97 m , which is the maximum allowed by current cmb data . the middle map shows the original 5-year wmap map @xcite , and the bottom map is a combined map of the two above.,title="fig:",width=340,height=188 ] under the mollweide projection with the z - direction pointing to @xmath100 with @xmath97 m , which is the maximum allowed by current cmb data . the middle map shows the original 5-year wmap map @xcite , and the bottom map is a combined map of the two above.,title="fig:",width=340,height=188 ] under the mollweide projection with the z - direction pointing to @xmath100 with @xmath97 m , which is the maximum allowed by current cmb data . the middle map shows the original 5-year wmap map @xcite , and the bottom map is a combined map of the two above.,title="fig:",width=340,height=188 ] along the light paths of the last - scattered photons , where @xmath101 and @xmath102 . we set @xmath103.,width=604,height=453 ] and its difference from the rotation speed of the universe are plotted against time . similar to fig . 2 , @xmath101 , @xmath102 and @xmath103 . we note that the rotation of matter in the universe can be different and even opposite to the rotation of the universe.,width=604,height=453 ] .,width=604,height=453 ] in this paper , we have developed a cosmological model that has a non - zero rotation but trivial shear in the robertson - walker framework with a cosmological constant . we have solved the efe s up to 2nd - order perturbations of a flat @xmath0cdm universe with rotation as a 1st - order perturbation . we also set up the formulation for the 2nd - order sw effect due to the rotational perturbation and find that the effect only influences the spherical harmonics with even @xmath88 s . by making use of recent cmba data , the angular speed of the rotation is constrained to be less than @xmath1 rad yr@xmath2 at the last scattering surface . the model of the universe here is different from the bianchi models used in the literatures . first of all , our model is inhomogeneous with an axial symmetry in general while bianchi models are homogeneous . moreover , our model is shear - free and thus has the advantage that the sw effect and the constraint obtained are purely due to the global rotation . compared with previous works , the constraint is much weaker . for example , barrow et al . @xcite put a constraint of @xmath104 rad yr@xmath2 on the rotaion of flat bianchi models . this can be understood mainly because the effects of rotation in our model here show up as 2nd - order sw effects while in previous works they are 1st - order sw effects . for further study , we notice that our model here produces a 2nd - order outward radial velocity . it may be used to explain parts of the accelerating expansion of the universe , and therefore , constraints on the global rotation can be obtained with type ia supernova data as proposed in @xcite . although we only study a flat universe here , it is interesting to study the closed and open cases of our model in view of the significantly different constraints on closed and open bianchi models@xcite . @xmath105 ^ 2 } \{64r^3\dot{a}^4(\eta)\omega ' ^2(r,\eta ) + 16r^3a^2(\eta)\ddot{a}^2(\eta)\omega ' ^2(r,\eta)\nonumber\\ & ~&+r\lambda a^6(\eta)[3\omega ' ( r,\eta)+r\omega ' ' ( r,\eta)]^2 + ra(\eta)\dot{a}^2(\eta)\{-64r^2\ddot{a}(\eta)\omega ' ^2(r,\eta)\nonumber\\ & ~&+3a(\eta)[3\omega ' ( r,\eta)+r\omega ' ' ( r,\eta)]^2\}\},\\ 0&=&2r\dot{a}^2(\eta)l(r,\eta)+ra(\eta)[-4\ddot{a}(\eta)l(r,\eta)+ 2\dot{a}(\eta)\dot{l}(r,\eta)]\nonumber\\ & ~&+a^4(\eta)[-2r\lambda l(r,\eta ) + r^3\omega ' ^2(r,\eta)]+a^2(\eta)[r\ddot{l}(r,\eta)-l'(r,\eta)\nonumber\\ & ~&-rl''(r,\eta)-rt''(r,\eta)].\end{aligned}\ ] ] these two equations are independent of @xmath23 . that is , @xmath23 is only an arbitrary function unrelated to @xmath20 in general . physically , @xmath23 comes from the 2nd - order perturbation of mass density ( analogous to the diagonal perturbations of schwarzschild metric ) . as we are interested in the effects of the rotation only , we set @xmath106 for simplicity . in this paper , we focus on a non - viscous fluid in the universe . using the fact that @xmath107 $ ] for a flat @xmath0cdm universe and eq . ( 8) , we can simplify these two equations further @xmath108 ^ 2}{2\lambda(1-\omega_{\lambda})a^3(\eta)},\\ \frac{r}{a(\eta)}\frac{d}{d\eta}[a^2(\eta)\dot{l}(r,\eta)]-a(\eta)[l'(r,\eta ) + rl''(r,\eta)]=ra(\eta)t''(r,\eta)-r^3a^{-3}(\eta)a'^2(r).\end{aligned}\ ] ] here , we remark that @xmath53 , which disappears from these two equations , is arbitrary and does not affect the sachs - wolfe effect . such an arbitrariness is unrelated to the effects of the rotational universe on the cmba . without loss of generality , we expand @xmath52 into taylor series : @xmath109 ( we will explain below why it does not start from @xmath110 ) . substituting the series into eq . ( a.3 ) , @xmath111 where @xmath112 for @xmath113 . by separation of variables , we find that @xmath114,\end{aligned}\ ] ] where @xmath115 as @xmath116 and @xmath117 are arbitrary functions of the homogeneous solutions for eq . ( a.5 ) and thus are independent of @xmath20 , we are free to set them zero . the series of @xmath52 starts from @xmath118 because the @xmath110 term can be absorbed into @xmath53 while the @xmath119 term , which produces @xmath120 as the particular solution of @xmath56 , is rejected because of the singularity at @xmath121 . similarly , we expand @xmath122 and substitute it into eq . ( a.4 ) : @xmath123 -(n+1)^2a(\eta)e_{n+1}(\eta)\right\}r^n\nonumber\\ & = & -\sum_{n=5}^{\infty}\sum_{l=2}^{\infty}\frac{l(3n-1)(n - l-1)}{(n-1)a^3(\eta ) } c_lc_{n - l-1}r^{n}\nonumber\\ & ~&-\sum_{n=3}^{\infty}\sum_{l=2}^{\infty } \frac{nl(l+2)(n - l+1)(n - l+3)\omega_{\lambda}}{2\lambda(1-\omega_{\lambda})(n-1 ) a^2(\eta)}c_lc_{n - l+1}r^{n},\end{aligned}\ ] ] where @xmath124 for @xmath125 . in general , by comparing the @xmath126 terms on both sides , we can obtain the recurrence relations for solving @xmath127 . as an example , we work out the simplest case where @xmath128 ( @xmath129 is a constant ) : @xmath130 - 4a(\eta)e_2(\eta),\\ -\frac{48\omega_{\lambda}\alpha^2}{\lambda(1-\omega_{\lambda})a^2(\eta)}&=&\frac{1}{a(\eta ) } \frac{d}{d\eta}[a^2(\eta)\dot{e}_2(\eta)]-16a(\eta)e_4(\eta),\\ -\frac{14\alpha^2}{a^3(\eta)}&=&\frac{1}{a(\eta ) } \frac{d}{d\eta}[a^2(\eta)\dot{e}_4(\eta)]-36a(\eta)e_6(\eta),\\ & \vdots&\nonumber\\ 0&=&\frac{1}{a(\eta)}\frac{d}{d\eta}[a^2(\eta)\dot{e}_{2n-2}(\eta)]-4n^2a(\eta)e_{2n}(\eta).\end{aligned}\ ] ] we notice that there is a freedom to choose one of @xmath131 arbitrarily , independent of @xmath20 . to prevent infinite series , we set @xmath132 so that @xmath133 for @xmath134 and all non - zero @xmath131 s depend on @xmath135 ( due to the rotation ) . we have @xmath136d\eta ' ' d\eta ' , \\ e_0(\eta)&=&\int_{\eta}^0\frac{1}{a^2(\eta ' ) } \int_{\eta ' } ^0 4a^2(\eta ' ' ) e_2(\eta ' ' ) d\eta ' ' d\eta ' , \end{aligned}\ ] ] which can be solved numerically . ( b.1 ) , we need to solve the geodesic equations for the light rays of the cmb , which are @xmath139 as before , we expand the geodesic equations into different orders of @xmath20 . to be consistent , @xmath140 has to be expanded into @xmath141 for the first - order , @xmath144=0,\\ \frac{d}{d\lambda}\left[_0g_{ii}\frac{d_1x^i}{d\lambda}\right]=0,\end{aligned}\ ] ] for @xmath145 . therefore , @xmath146 where @xmath86 s are constants to be determined . for simplicity , we assume that we are located on the rotating axis . therefore , @xmath102 and @xmath101 due to the cylindrical symmetry . although the general case that we may be off the rotating axis is more realistic , the constraint here can be regarded as a good approximation provided that our distance to the rotating axis is small compared to that of the last scattering surface . the general case can be found by assigning a suitable dependence of @xmath15 on @xmath147 and @xmath148 . the term @xmath149 refers to the angular velocity of the comoving metric . for the second - order , @xmath150 by setting @xmath106 and the requirement of a null geodesic ( @xmath151 ) , we obtain @xmath152 where @xmath153\,d\lambda ' , \\ _ 2k^1(\lambda)&=&\int\left[\frac{t'(r_{\lambda ' } , \eta_{\lambda ' } ) } { 2}+ \frac{l'(r_{\lambda ' } , \eta_{\lambda ' } ) } { 2}\sin^2\phi -\dot{l}(r_{\lambda ' } , \eta_{\lambda ' } ) \sin\phi\right]\,d\lambda ' .\end{aligned}\ ] ] to have a null geodesic , @xmath154 for @xmath155 . from the results of @xmath156 and @xmath157 , we can easily verify the argument that the 1st - order perturbations of sw effect due to the rotation is zero and calculate the 2nd - order sw effect as @xmath74|_{\eta_{\epsilon}}^{\eta_0}\nonumber\\ & = & \frac{\omega_{\lambda}\sin\phi}{2\lambda(1-\omega_{\lambda } ) } [ 2\dot{a}(\eta_{\epsilon})t'(r_{\epsilon},\eta_{\epsilon})- a(\eta_{\epsilon})\dot{l}'(r_{\epsilon},\eta_{\epsilon})]\nonumber\\ & ~&-\frac{1}{2}\left\{\frac{\omega_{\lambda}^2a^4(\eta_{\epsilon})}{4\lambda^2(1-\omega_{\lambda})^2 } [ 3\omega ' ( r_{\epsilon},\eta_{\epsilon})+r_{\epsilon}\omega ' ' ( r_{\epsilon},\eta_{\epsilon})]^2+t(r_{\epsilon},\eta_{\epsilon})\right\}\nonumber\\ & ~&+\int_{\eta_{\epsilon}}^0\left[-\frac{\dot{t}(-\lambda\sin\phi,\lambda)}{2}+ t'(-\lambda\sin\phi,\lambda)\sin\phi-\frac{\dot{l}(-\lambda\sin\phi,\lambda)}{2}\sin^2\phi\right]d\lambda,\end{aligned}\ ] ] where @xmath75 and @xmath76 denotes the occuring conformal time of the last scattering . we made use of healpix @xcite to produce fig . this work is supported by grants from the research grant council of the hong kong special administrative region , china ( project nos . 400707 and 400803 ) . 20 k. gdel , rev . modern phys . * 21 * , 3 ( 1949 ) . s. hawking , mnras * 142 * , 129 ( 1969 ) . j. d. barrow @xmath158 @xmath159 , mnras * 213 * , 917 ( 1985 ) . a. de oliveira - 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models of a rotating universe have been studied widely since gdel @xcite , who showed an example that is consistent with general relativity ( gr ) . by now , the possibility of a rotating universe has been discussed comprehensively in the framework of some types of bianchi s models , such as type v , vii and ix @xcite , and different approaches have been proposed to constrain the rotation . recent discoveries of some non - gaussian properties of the cosmic microwave background anisotropies ( cmba ) @xcite , such as the suppression of the quadrupole and the alignment of some multipoles draw attention to some bianchi models with rotation @xcite . however , cosmological data , such as those of the cmba , strongly prefer a homogeneous and isotropic model . therefore , it is of interest to discuss the rotation of the universe as a perturbation of the robertson - walker metric , to constrain the rotating speed by cosmological data and to discuss whether it could be the origin of the non - gaussian properties of the cmba mentioned above . here , we derive the general form of the metric ( up to 2nd - order perturbations ) which is compatible with the rotation perturbation in a flat @xmath0-cdm universe . by comparing the 2nd - order sachs - wolfe effect @xcite due to rotation with the cmba data , we constrain the angular speed of the rotation to be less than @xmath1 rad yr@xmath2 at the last scattering surface . this provides the first constraint on the shear - free rotation of a @xmath0cdm universe .

You are an expert at summarizing long articles. Proceed to summarize the following text: the dark energy survey data management ( desdm ) is the part of the dark energy survey project @xcite that will transfer , process and distribute the data generated by survey s camera decam @xcite . desdm is a large , scalable system led by the national center for supercomputing applications at the university of illinois at urbana - champaign ( ncsa / uiuc ) consisting of : 1 . an archive system for different levels of data . 2 . scientific codes to process raw data . 3 . database to support calibration , provenance and data analyses . 4 . web portals providing process control and easy access to images and catalogs . 5 . hardware platforms for execution and storage . as of august 2011 , the desdm team consists of around 20 computing professionals and physicists from several institutions around the world , each providing their own expertise to its development . in this contribution we describe the basic design of the desdm and make particular emphasis on how the raw data in the form of ccd images is reduced to its final form as science - ready catalogs . we also present the testing campaigns that the system is undergoing and the outlook in the short term , in the context of the des project . for more information on other aspects of desdm , see also @xcite @xcite . the main functions of the system are schematically summarized in fig . [ fig : desdm_diagram ] . the design is driven by the science requirements document of the des project , flowed down to the technical level . additional requirements include the timely and reliable processing and the archiving of the data . the functions of the desdm are the following : 1 . * transfer . * raw images must be transferred from the telescope site at the rate of approximately 100 mbps . this rate takes into account that 360 science and associated calibration exposures are produced each night , totalling @xmath0 gb of data that have to be transported in less than 18 hours , to allow for nightly processing and feedback . some overhead is included to consider possible network outages . recent studies indicate normal data delivery will occur in near real time . this step is taken care of by the national optical astronomy observatory data transport system through a microwave downlink off the mountain and then by network to ncsa . 2 . * processing . * as the data arrive at ncsa , it is ingested into the system , and sent to high performance computing ( hpc ) resources on xsede @xcite and possibly the open science grid @xcite . data are staged using using gridftp and globusonline @xcite . the pipeline containing the parallelization , scientific codes ( see section [ sec : algorithms ] ) and quality assurance is executed for the night s images . archive and distribution . * results are then returned to the ncsa primary archive , and releases built . data files in releases are replicated to the secondary archive at fermilab and tertiary archives at collaborator s sites . data are also released generally to the collaboration . catalogs of objects from coadded images and single epoch images , as well as other meta - data are served to the collaboration using an ncsa - provided oracle rac system . after a proprietary period , data will be released to the community using the same methods . we expect the system will evolve to use vao protocols and tools in time for the public release . ] in this section we describe what algorithms are involved in the reduction of the raw data once the process that sends it to the hpc platform has been started . the basic event in an astronomical observation is the _ exposure_. the decam camera is exposed to the night sky for @xmath1 seconds , generating a file of slightly less than 1 gb in size , which is written in fits format . an exposure ( see fig.[fig : sample_image ] for an example ) consists of 62 ccd images of a part of the sky , covering a total solid angle of 3 square degrees , showing multiple sources and instrumental effects . around 300 scientific images are generated per night , for different pointings , together with about 60 calibration images . the former have to be corrected for instrumental effects ( detrending ) as well as calibrated for the absolute position ( astrometry ) and absolute flux ( photometry ) . ] this pipeline contains several astronomy modules that together remove the decam instrumental signatures . it includes the following steps : * exposure segmentation and crosstalk correction . the exposure is divided into 62 ccd images and crosstalk between them is accounted for , using the crosstalk coefficients measured from the calibration dataset @xmath2 where @xmath3 stands for the image of a particular ccd and @xmath4 is the crosstalk coefficient describing the fraction of image @xmath5 appearing in image @xmath6 . * image correction . this involves correcting for the pedestal or bias level of the ccds , eliminating non - imaging sections of the exposure , eliminating ghost images from multiple scattered light in the optics , removing illumination and fringing effects and correcting for pixel - to - pixel sensitivity variations ( flat - fielding ) . thermal noise from electrons is also treated but is usually negligible at the camera s working temperature . in this step additional calibration images ( taken during the same night , every season or during commissioning ) are required and must be part of the job submitted to the hpc resource . after this process , raw data has been turned into reduced images with associated maps containing information on bad pixels and the weights of the image at each particular pixel ( inverse variance ) . this step requires the identification of positional standards in the exposure , i.e. , stars with very well - known positions in celestial coordinates . the coordinates of the brightest sources in the exposure are extracted as well ( using the sextractor @xcite package ) in the exposures s own reference system in _ x_-_y _ coordinates . knowing an approximate initial solution ( provided by the telescope s control system ) a match of both catalogs can be performed , as well as the fitting of the best transformation parameters from the image _ x_-_y _ system to the celestial reference system ( absolute system in spherical coordinates ) . this part is performed by the scamp @xcite software . additionally , the effect of the distortion caused by the optics towards the edges of the field , has to be included when calculating the astrometric solution . the actual digital counts observed in each pixel of the detrended image , have to be translated into physical flux units . in order to make this conversion , on every night in which conditions are sufficiently good ( moonless , stable skies ) specific calibration star fields will be imaged . for these stars , the flux is known or can be obtained ( @xmath7 ) and this can be compared to the total light measured in the image ( @xmath8 ) . this relationship is called the photometric equation and contains several unknowns which are ccd and filter dependent : @xmath9 where @xmath10 is the photometric zeropoint , representing the normalization value of the image ; @xmath11 is an instrumental coefficient known as _ color term _ which takes into account the shape of the response of the ccd to light in different filters ( @xmath12 ) ; and @xmath13 is the atmospheric extinction coefficient which considers the increased absorption by the atmosphere at the airmass denoted by @xmath14 ( dependent on sky angle ) . measuring multiple reference stars in the ccds for different filters and angles in the sky provides us with enough equations to solve the system , thus allowing us to find the photometric solution ( values of these constants and their errors ) . in order to reach the scientific requirements of des in terms of depth , it is necessary to perform a process called image coaddition . as the name implies , this is the combination of several overlapping single - epoch images in a given filter to improve the signal to noise of real sources in the image . moreover , if the different images thus co - added are slightly offset from one another , this procedure has the added bonus of improving the photometric calibration with a more robust determination of the solution , as the photometric solution will incorporate information from different parts of the sky simultaneously . another advantage is the possibility of eliminating transient effects from the final coadded image such as satellite trails or cosmic rays , as they will show up in only one of the images being added and are easily identifiable . prior to the coaddition itself , it is necessary to transform the flux values in the individual overlapping ccd pixels into a uniform pixel grid in which the coaddition can be performed ( _ remapping _ ) . to do this , artificial _ tiles _ in the sky are created , one degree on a side , and one coadded image per band is produced for every tile . for each single - epoch image , it is determined to which tiles it contributes to ( using swarp @xcite ) . the photometric solution ( in particular , the zeropoints ) is re - evaluated for these new images . there are two main caveats to be pointed out : * the color terms of the ccds , which are simple to take into account in single - epoch images , are not easily combined if they vary across the field . a solution is currently under implementation ; * the point spread function ( psf ) changes within an image and from one image to another , due to varying conditions of the exposure and quality of the sky . therefore the psf of the coadded image is subject to discontinuous jumps . to deal with the second point above , a psf homogenization procedure has been developed . the model psf and its variation has been computed for each image during nightly processing using the psfex @xcite package . we define the target psf to be used as a circular moffat @xcite function with a full width at half - maximum which is the median of the seeing distribution one in the whole set of images contributing to the coadd . going back to the individual component images with psf @xmath15 , we find the kernel @xmath16 which minimizes the difference with respect to this median target psf @xmath17 . @xmath18 where @xmath19 are the elements of a polynomial basis in @xmath20 . the kernel elements are stored and during the homogenization process , they are recovered to convolve with the image . once every image has been treated this way , the coaddition will not introduce any discontinuities from the psf variations . image coaddition takes place off - season due to its increased cpu requirements with respect to the nightly processing . coadd construction is to be carried out multiple times depending on the specific needs of each science working group , which have different requirements in the balance between depth and source morphology . for instance , the weak lensing group would be interested in no coaddition at all given their stringent requirements on the shape measurement ( though they will benefit from multiple imaging of the same objects , improving the solution of the extracted shear , see section [ sec : wl ] ) . the final step in the processing consists in transforming the coadded images in different filters into single object entries in a catalog . desdm uses the sextractor software in this step too , running over a master image which uses coadd images in all filters to detect where the sources are , and then extracting the relevant information from single - filter coadded images . an object is identified as such when the convolution of the psf with the image is above a certain local background estimation ( see @xcite for details ) . in this step , it is important to approach the deblending of sources . this is currently being done by producing several isophotal layers for each object and at each layer where two light-islands join making the decision on whether to merge them into the same object using as a criterion the relative integrated intensity between the branch and full object . the type of information extracted in this step is positional , photometric and morphological , besides identification and other bookkeeping variables . concerning the astrometry , the barycenter for each object is derived using several estimators . the most reliable one makes use of an iterative calculation through a gaussian window . this is transformed to celestial coordinates using the astrometric solution found during nightly processing . photometry is measured using several methods : * simple flux counting inside a fixed circular aperture for several apertures ( in the arcseconds range ) ; * using an elliptical aperture adjusted to the morphological properties of the object . from the second order moments of the object , we would find the elongation and orientation of the ellipse representing it . the ellipse scaling factor is derived from the first order moment of the radial distribution @xcite ; * using a fit to the measured psf shape ( suitable for stars ) ; * using a one- or two- component model convolved with a local model of the psf ( exponential and/or spheroidal , suitable for galaxies ) . among the morphological measurements , currently there are two variables appraising the deviation of the object from a point - like shape , therefore providing a star - galaxy separation handle . one of them relies on a previously trained neural network and is the well - known stellarity parameter class_star from sextractor . the other arises from the calculation of a discriminant function measuring the deviation of the image from the psf shape . most of the variables in the catalogs have their corresponding errors and quality flags to guide the selection of sources for analysis . additional columns are included in the final catalog , and these are produced in pipelines running over images and catalogs , to produce the final coadded catalogs . two of these are briefly described in the following sections . an additional catalog is built to identify transient objects , as described in section [ sec : diff_imaging ] . redshifts in the des will be estimated from the photometric information of the objects in different filters . traditionally , the approach to this has been to either find the best fit of the spectral energy distribution to a collection of templates for different types of galaxies at different redshifts , or use a neural network trained with spectroscopic information ( see @xcite and @xcite for examples of both ) . currently the default estimation in desdm uses the latter method , using ten input magnitudes ( fixed circular and automatic elliptical apertures , described above , for five filters each ) . in fig . [ fig : photoz ] a comparison of photometric redshifts versus spectroscopic ( true ) ones is shown , as obtained in one of the latest data challenges ( section [ sec : dcs ] ) . there is a large code comparison project within a specific photo - z science working group . ] the weak lensing probe will require measuring the distortion in the shapes of galaxies to extract the shear produced by gravity from the intervening matter between the source and observer . it will require a very precise measurement of the local psf shape independently of the standard determination used in the pipeline . this is done using bright isolated stars and additional instrument data . the psf is interpolated with a polynomial throughout the image , and it is deconvolved from each galaxy . the shear will be extracted using all the images where the object shows up ( in multiple filters ) to provide a more robust determination . the des will contain two modes of operation : the survey mode in which a wide area of the sky will be scanned several times for each filter over the course of the five - year survey ; and a time - domain survey where particular regions will be observed repeatedly on short time - scales ( @xmath21 weekly ) in search of transient phenomena ( from which supernovae of type ia have to be identified ) . from every reduced image coming from the nightly processing of these regions , a template is extracted from the coadd corresponding to that position . this template is subtracted from the reduced image , and all objects remaining above a certain threshold are cataloged . the desdm also plans to automate the generation of a polygon - based survey mask which would encode information on which single epoch images contributed to the coadded image at a given point and track the coadded magnitude depth , coadded color terms , regions blocked by saturated stars and any other information which is relevant to be included as a survey map rather than in a source - by - source basis . the mask is generated offline using the mangle @xcite software , which is currently being incorporated to the pipeline . during the execution of the pipelines , in - built quality assurance modules make sure that the images and catalogs are good enough for scientific analysis and diagnose possible problems . some sample results are shown in fig.[fig : photometry ] and fig.[fig : astrometry ] . a broader approach to the testing and validation of the desdm is the data challenge ( dc ) process , in which a small sample of the survey is simulated in detail and fed to the pipelines to generate realistic images and catalogs . the scale for the sample ranges from a single 0.6 square degree tile to 200 square degrees corresponding to about 10 nights of observations . it starts with the creation of galaxy catalogs stemming from an n - body simulation@xcite and detailed models of the milky way galaxy for the star component @xcite . these are merged and fed to an image simulator which includes atmospheric and instrumental effects . the resulting images serve as inputs for the desdm , as if they had been really observed at the telescope . this whole process is a joint effort of the stanford , brazil and barcelona teams , for the catalog side , and fermilab , for the image simulation aspect . the resulting catalogs are then examined by members of the desdm and science working groups . several of these dcs have taken place during the project s lifetime . currently the testing approach consists on large data challenges every 6 months to verify the compliance with science requirements , while maintaining a 2-week cycle with the generation of small samples for quick feedback to the developers . in addition , the system is being used on real images of the blanco cosmology survey @xcite which are proving to be very valuable for the development of the algorithms . the desdm system has been fully developed and all elements are in place to process the large dataset that will be generated by the des camera . current testing results point to requirements being met in terms of astrometry , photometry and depth . others such as completeness and star - galaxy separation are compliant up to shallower magnitudes though improvements are in the works . in parallel , a community pipeline is being developed using essentially the same structure and algorithms but with the goal of addressing non - des user needs , given that two - thirds of the year the instrument will be available to the general astronomical community with access to noao facilities . funding for the des projects has been provided by the u.s . department of energy , the u.s . national science foundation , the ministerio of ciencia y educacin of spain , the science and technology facilities council of the united kingdom , the higher education funding council for england , the national center for supercomputing applications at the university of illinois at urbana - champaign , the kavli institute for cosmological physics at the university of chicago , financiadora de estudos e projetos , fundao carlos chagas filho de amparo pesquisa do estado do rio de janeiro , conselho nacional de desenvolvimento cientfico e tecnolgico and the ministrio da cincia e tecnologia , the deutsche forschungsgemeinschaft and the collaborating institutions in the dark energy survey . the collaborating institutions are argonne national laboratories , the university of california at santa cruz , the university of cambridge , centro de investigaciones energticas , medioambientales y tecnolgicas - madrid , the university of chicago , university college london , des - brazil , fermilab , the university of edinburgh , the university of illinois at urbana - champaign , the institut de cincies de lespai ( ieec / csic ) , the institut de fsica daltes energies , the lawrence berkeley national laboratory , ludwig - maximilians universitt and the associated excellence cluster universe , the university of michigan , the national optical astronomy observatory , the university of nottingham , the ohio state university , the university of pennsylvania , the university of portsmouth , universidade federal do rio grande do sul , slac , stanford university , the university of sussex and texas a&m university . 99 f.abdalla these proceedings . j.hao these proceedings . j.mohr et al . observatory operations : strategies , proceses , and systems ii . proceedings of the spie 7016(2008)70160l k.kotwani et al . 8th international workshop on middleware for grids , clouds and e - science mgc ( 2010 ) http://mgc2010.lncc.br/ n.wilkins-diehr et al . 41(11)(2008)32 r.pordes et al . journal of physics : conference series 78(2007)012057 i.foster ieee internet computing may - june(2011)70 s.arnouts & e.bertin a&as 117(1996)393 e.bertin astronomical data analysis software and systems xv . astronomical society of the pacific conference series 351(2006)112 e.bertin et al . astronomical data analysis software and systems xi . astronomical society of the pacific conference series 281(2002)228 e.bertin astronomical data analysis software and systems xx . astronomical society of the pacific conference series 442(2011)435 a.f.j.moffat a&a 3(1969)455 r.g.kron apjs 43(1980)305 n.bentez apj 536(2000)571 h.oyaizu et al . apj 674(2008)768 m.swanson et al . mnras 387(2008)1391 http://space.mit.edu/@xmath21molly/mangle/ m.busha et al . ( 2011 ) in preparation b.rosetto et al . aj 141(2011)185 j.mohr http://www.usm.uni-muenchen.de/@xmath21jmohr/bcs

the dark energy survey ( des ) is a project with the goal of building , installing and exploiting a new 74 ccd - camera at the blanco telescope , in order to study the nature of cosmic acceleration . it will cover 5000 square degrees of the southern hemisphere sky and will record the positions and shapes of 300 million galaxies up to redshift 1.4 . the survey will be completed using 525 nights during a 5-year period starting in 2012 . about o(1 tb ) of raw data will be produced every night , including science and calibration images . the des data management system has been designed for the processing , calibration and archiving of these data . it is being developed by collaborating des institutions , led by ncsa . in this contribution , we describe the basic functions of the system , what kind of scientific codes are involved and how the data challenge process works , to improve simultaneously the data management system algorithms and the science working group analysis codes .

You are an expert at summarizing long articles. Proceed to summarize the following text: let @xmath5 be an abelian variety defined over a finite field @xmath6 , and @xmath9 a prime number dividing the order of the group @xmath10 . then the _ embedding degree _ of @xmath5 with respect to @xmath7 is the degree of the field extension @xmath11 obtained by adjoining a primitive @xmath7-th root of unity @xmath12 to @xmath6 . the embedding degree is a natural notion in pairing - based cryptography , where @xmath5 is taken to be the jacobian of a curve defined over @xmath6 . in this case , @xmath5 is principally polarized and we have the non - degenerate _ weil pairing _ @xmath13\times a[r ] \longrightarrow \mu_r\ ] ] on the subgroup scheme @xmath14 $ ] of @xmath7-torsion points of @xmath5 with values in the @xmath7-th roots of unity . if @xmath6 contains @xmath12 , we also have the non - trivial _ tate pairing _ @xmath15({{\mathbf f } } ) \times a({{\mathbf f}})/ra({{\mathbf f } } ) \to \mathbf{f}^*/({{\mathbf f}}^*)^r.\ ] ] the weil and tate pairings can be used to ` embed ' @xmath7-torsion subgroups of @xmath10 into the multiplicative group @xmath16 , and thus the discrete logarithm problem in @xmath17 $ ] can be ` reduced ' to the same problem in @xmath16 @xcite . in pairing - based cryptographic protocols @xcite , one chooses the prime @xmath7 and the embedding degree @xmath8 such that the discrete logarithm problems in @xmath17 $ ] and @xmath16 are computationally infeasible , and of roughly equal difficulty . this means that @xmath7 is typically large , whereas @xmath8 is small . jacobians of curves meeting such requirements are often said to be _ pairing - friendly_. if @xmath6 has order @xmath3 , the embedding degree @xmath18 $ ] is simply the multiplicative order of @xmath3 in @xmath19 . as ` most ' elements in @xmath19 have large order , the embedding degree of @xmath5 with respect to a large prime divisor @xmath7 of @xmath20 will usually be of the same size as @xmath7 , and @xmath5 will not be pairing - friendly . one is therefore led to the question of how to efficiently construct @xmath5 and @xmath6 such that @xmath10 has a ( large ) prime factor @xmath7 and the embedding degree of @xmath5 with respect to @xmath7 has a prescribed ( small ) value @xmath8 . the current paper addresses this question on two levels : the _ existence _ and the actual _ construction _ of @xmath5 and @xmath6 . section [ s : weilnumbers ] focuses on the question whether , for given @xmath7 and @xmath8 , there exist abelian varieties @xmath5 that are defined over a finite field @xmath6 , have an @xmath6-rational point of order @xmath7 , and have embedding degree @xmath8 with respect to @xmath7 . we consider only abelian varieties @xmath5 that are _ simple _ , that is , not isogenous ( over @xmath6 ) to a product of lower - dimensional varieties , as we can always reduce to this case . by honda - tate theory @xcite , isogeny classes of simple abelian varieties @xmath5 over the field @xmath6 of @xmath3 elements are in one - to - one correspondence with @xmath21-conjugacy classes of _ @xmath3-weil numbers _ , which are algebraic integers @xmath22 with the property that all embeddings of @xmath22 into @xmath23 have absolute value @xmath24 . this correspondence is given by the map sending @xmath5 to its @xmath3-th power frobenius endomorphism @xmath22 inside the number field @xmath25 . the existence of abelian varieties with the properties we want is thus tantamount to the existence of suitable weil numbers . our main result , algorithm [ alg : construct - pi ] , constructs suitable @xmath3-weil numbers @xmath22 in a given _ cm - field _ it exhibits @xmath22 as a _ type norm _ of an element in a _ reflex field _ of @xmath0 satisfying certain congruences modulo @xmath7 . the abelian varieties @xmath5 in the isogeny classes over @xmath6 that correspond to these weil numbers have an @xmath6-rational point of order @xmath7 and embedding degree @xmath8 with respect to @xmath7 . moreover , they are _ ordinary _ , i.e. , @xmath26=p^g$ ] , where @xmath27 is the characteristic of @xmath6 . theorem [ thm : ispoltime ] shows that for fixed @xmath0 , the expected run time of our algorithm is heuristically polynomial in @xmath28 . for an abelian variety of dimension @xmath29 over the field @xmath6 of @xmath3 elements , the group @xmath10 has roughly @xmath30 elements , and one compares this size to @xmath7 by setting @xmath31 in cryptographic terms , @xmath32 measures the ratio of a pairing - based system s required bandwidth to its security level , so small @xmath32-values are desirable . _ supersingular _ abelian varieties can achieve @xmath32-values close to @xmath33 , but their embedding degrees are limited to a few values that are too small to be practical @xcite . theorem [ thm : heur ] discusses the distribution of the ( larger ) @xmath32-values we obtain . in section [ s : construct ] , we address the issue of the actual construction of abelian varieties corresponding to the weil numbers found by our algorithm . this is accomplished via the construction in characteristic zero of the abelian varieties having cm by the ring of integers @xmath34 of @xmath0 , a hard problem that is far from being algorithmically solved . we discuss the elliptic case @xmath35 , for which reasonable algorithms exist , and the case @xmath36 , for which such algorithms are still in their infancy . for genus @xmath37 , we restrict attention to a few families of curves that we can handle at this point . our final section [ s : examples ] provides numerical examples . let @xmath6 be a field of @xmath3 elements , @xmath5 a @xmath29-dimensional simple abelian variety over @xmath6 , and @xmath38 the number field generated by the frobenius endomorphism @xmath22 . then @xmath22 is a _ @xmath3-weil number _ in @xmath0 : an algebraic integer with the property that all of its embeddings in @xmath39 have complex absolute value @xmath24 . the @xmath3-weil number @xmath22 determines the group order of @xmath10 : the @xmath6-rational points of @xmath5 form the kernel of the endomorphism @xmath40 , and in the case where @xmath41 is the full endomorphism algebra @xmath42 we have @xmath43 in the case @xmath44 we will focus on , @xmath0 is a _ cm - field _ of degree @xmath45 as in ( * ? ? ? * section 1 ) , i.e. , a totally complex quadratic extension of a totally real subfield @xmath46 . [ p : exist - embed ] let @xmath5 , @xmath6 and @xmath22 be as above , and assume @xmath41 equals @xmath47 . let @xmath8 be a positive integer , @xmath48 the @xmath8-th cyclotomic polynomial , and @xmath49 a prime number . if we have @xmath50 then @xmath5 has embedding degree @xmath8 with respect to @xmath7 . the first condition tells us that @xmath7 divides @xmath20 , the second that the order of @xmath51 in @xmath19 , which is the embedding degree of @xmath5 with respect to @xmath7 , equals @xmath8 . by honda - tate theory @xcite , all @xmath3-weil numbers arise as frobenius elements of abelian varieties over @xmath6 . thus , we can prove the _ existence _ of an abelian variety @xmath5 as in proposition [ p : exist - embed ] by exhibiting a @xmath3-weil number @xmath52 as in that proposition . the following lemma states what we need . [ l : honda - tate ] let @xmath22 be a @xmath3-weil number and @xmath6 be the field of @xmath3 elements . then there exists a unique isogeny class of simple abelian varieties @xmath53 with frobenius @xmath22 . if @xmath41 is totally imaginary of degree @xmath45 and @xmath3 is prime , then such @xmath5 have dimension @xmath29 , and @xmath0 is the full endomorphism algebra @xmath54 . if furthermore @xmath3 is unramified in @xmath0 , then @xmath5 is ordinary . the main theorem of @xcite yields existence and uniqueness , and shows that @xmath55 is a central simple algebra over @xmath41 satisfying @xmath56^{\frac{1}{2}}[k:{\mathbf{q}}].\ ] ] for @xmath0 totally imaginary of degree @xmath45 and @xmath3 prime , waterhouse ( * ? ? ? * theorem 6.1 ) shows that we have @xmath57 and @xmath58 . by ( 7.1 ) , @xmath5 is ordinary if and only if @xmath59 is prime to @xmath60 in @xmath34 . thus if @xmath5 is not ordinary , the ideals @xmath61 and @xmath62 have a common divisor @xmath63 with @xmath64 , so @xmath3 ramifies in @xmath0 . [ cycliccase ] our general construction is motivated by the case where @xmath0 is a galois cm - field of degree @xmath45 , with cyclic galois group generated by @xmath65 . here @xmath66 is complex conjugation , so we can construct an element @xmath4 satisfying @xmath67 by choosing any @xmath68 and letting @xmath69 . for such @xmath22 , we have @xmath70 . if @xmath71 is a prime @xmath3 , then @xmath22 is a @xmath3-weil number in @xmath0 . now we wish to impose the conditions of proposition [ p : exist - embed ] on @xmath22 . let @xmath7 be a rational prime that splits completely in @xmath0 , and @xmath72 a prime of @xmath34 over @xmath7 . for @xmath73 , put @xmath74 ; then the factorization of @xmath7 in @xmath34 is @xmath75 . if @xmath76 is the residue class of @xmath77 modulo @xmath78 , then @xmath79 modulo @xmath72 is also @xmath80 , so the residue class of @xmath22 modulo @xmath72 is @xmath81 . furthermore , the residue class of @xmath82 modulo @xmath72 is @xmath83 . if we choose @xmath77 to satisfy @xmath84 we find @xmath85 and thus @xmath86 . by choosing @xmath77 such that in addition @xmath87 is a primitive @xmath8-th root of unity in @xmath88 , we guarantee that @xmath89 is a primitive @xmath8-th root of unity modulo @xmath7 . thus we can try to find a weil number as in proposition [ p : exist - embed ] by picking residue classes @xmath90 for @xmath73 meeting the two conditions above , computing some ` small ' lift @xmath91 with @xmath92 , and testing whether @xmath69 has prime norm . as numbers of moderate size have a high probability of being prime by the prime number theorem , a small number of choices @xmath93 should suffice . there are @xmath94 possible choices for @xmath95 , where @xmath96 is the euler totient function , so for @xmath97 and large @xmath7 we are very likely to succeed . for @xmath35 , there are only a few choices @xmath98 , but one can try various lifts and thus recover what is known as the cocks - pinch algorithm ( * ? ? ? * theorem 4.1 ) for finding pairing - friendly elliptic curves . for arbitrary cm - fields @xmath0 , the appropriate generalization of the map @xmath99 in example [ cycliccase ] is provided by the _ type norm_. a _ cm - type _ of a cm - field @xmath0 of degree @xmath45 is a set @xmath100 of embeddings of @xmath0 into its normal closure @xmath101 such that @xmath102 is the complete set of embeddings of @xmath0 into @xmath101 . the _ type norm _ @xmath103 with respect to @xmath104 is the map @xmath105 which clearly satisfies @xmath106 if @xmath0 is not galois , the type norm @xmath107 does not map @xmath0 to itself , but to its _ reflex field _ @xmath108 with respect to @xmath104 . to end up in @xmath0 , we can however take the type norm with respect to the _ reflex type _ @xmath109 , which we will define now ( cf . * section 8) ) . let @xmath110 be the galois group of @xmath111 , and @xmath112 the subgroup fixing @xmath0 . then the @xmath45 left cosets of @xmath112 in @xmath110 can be viewed as the embeddings of @xmath0 in @xmath101 , and this makes the cm - type @xmath104 into a set of @xmath29 left cosets of @xmath112 for which we have @xmath113 . let @xmath114 be the union of the left cosets in @xmath104 , and put @xmath115 . let @xmath116 be the stabilizer of @xmath114 in @xmath110 . then @xmath117 defines a subfield @xmath108 of @xmath101 , and as we have @xmath118 we can interpret @xmath119 as a union of left cosets of @xmath117 inside @xmath110 . these cosets define a set of embeddings @xmath109 of @xmath108 into @xmath101 . we call @xmath108 the _ reflex field _ of @xmath120 and we call @xmath109 the _ reflex type_. [ l : reflex ] the field @xmath108 is a cm - field . it is generated over @xmath121 by the sums @xmath122 for @xmath123 , and @xmath109 is a cm - type of @xmath108 . the type norm @xmath107 maps @xmath0 to @xmath108 . the first two statements are proved in ( * ? ? ? * chapter ii , proposition 28 ) ( though the definition of @xmath117 differs from ours , because shimura lets @xmath110 act from the right ) . for the last statement , notice that for @xmath124 , we have @xmath125 , so @xmath126 . a cm - type @xmath104 of @xmath0 is _ induced _ from a cm - subfield @xmath127 if it is of the form @xmath128 for some cm - type @xmath129 of @xmath130 . in other words , @xmath104 is induced from @xmath130 if and only if @xmath114 as above is a union of left cosets of @xmath131 . we call @xmath104 _ primitive _ if it is not induced from a strict subfield of @xmath0 ; primitive cm - types correspond to simple abelian varieties @xcite . notice that the reflex type @xmath109 is primitive by definition of @xmath108 , and that @xmath120 is induced from the reflex of its reflex . in particular , if @xmath104 is primitive , then the reflex of its reflex is @xmath120 itself . for @xmath0 galois and @xmath104 primitive we have @xmath132 , and the reflex type of @xmath104 is @xmath133 . for cm - fields @xmath0 of degree @xmath134 or @xmath135 with primitive cm - types , the reflex field @xmath108 has the same degree as @xmath0 . this fails to be so for @xmath136 . [ l : reflexdeg ] if @xmath0 has degree @xmath45 , then the degree of @xmath108 divides @xmath137 . we have @xmath138 , with @xmath139 totally real and @xmath140 totally negative . the normal closure @xmath101 of @xmath0 is obtained by adjoining to the normal closure @xmath141 of @xmath139 , which has degree dividing @xmath142 , the square roots of the @xmath29 conjugates of @xmath143 . thus @xmath101 is of degree dividing @xmath137 , and @xmath108 is a subfield of @xmath101 . for a ` generic ' cm field @xmath0 the degree of @xmath101 is exactly @xmath144 , and @xmath145 is a field of degree @xmath146 generated by @xmath147 , with @xmath65 ranging over @xmath148 . from and lemma [ l : reflex ] , we find that for every @xmath149 , the element @xmath150 is an element of @xmath151 that satisfies @xmath152 . to make @xmath22 satisfy the conditions of proposition [ p : exist - embed ] , we need to impose conditions modulo @xmath7 on @xmath77 in @xmath108 . suppose @xmath7 splits completely in @xmath0 , and therefore in its normal closure @xmath101 and in the reflex field @xmath108 with respect to @xmath104 . pick a prime @xmath153 over @xmath7 in @xmath101 , and write @xmath154 for @xmath155 . then the factorization of @xmath7 in @xmath156 is @xmath157 [ t : construct - pi ] let @xmath120 be a cm - type and @xmath158 its reflex . let @xmath159 be a prime that splits completely in @xmath0 , and write its factorization in @xmath156 as in . given @xmath160 , write @xmath161 and @xmath162 for @xmath163 . if we have @xmath164 for some primitive @xmath8-th root of unity @xmath165 , then @xmath166 satisfies @xmath152 and @xmath167 this is a straightforward generalization of the argument in example [ cycliccase ] . the conditions generalize and , and imply in the present context that @xmath168 and @xmath169 are in the prime @xmath170 over @xmath7 that underlies the factorization . if the element @xmath22 in theorem [ t : construct - pi ] generates @xmath0 and @xmath171 is a prime @xmath3 that is unramified in @xmath0 , then by lemma [ l : honda - tate ] @xmath22 is a @xmath3-weil number corresponding to an ordinary abelian variety @xmath5 over @xmath172 with endomorphism algebra @xmath0 and frobenius element @xmath22 . by proposition [ p : exist - embed ] , @xmath5 has embedding degree @xmath8 with respect to @xmath7 . this leads to the following algorithm . [ alg : construct - pi ] input : a cm - field @xmath0 of degree @xmath173 , a primitive cm - type @xmath104 of @xmath0 , a positive integer @xmath8 , and a prime @xmath159 that splits completely in @xmath0 . output : a prime @xmath3 and a @xmath3-weil number @xmath174 corresponding to an ordinary , simple abelian variety @xmath175 with embedding degree @xmath8 with respect to @xmath7 . 1 . compute a galois closure @xmath101 of @xmath0 and the reflex @xmath158 of @xmath120 . set @xmath176 and write @xmath177 . [ st : factor ] fix a prime @xmath178 of @xmath179 , and compute the factorization of @xmath7 in @xmath156 as in . compute a primitive @xmath8-th root of unity @xmath165 . [ st : random ] choose random @xmath180 . 5 . set @xmath181 and @xmath182 . 6 . [ st : crt ] compute @xmath183 such that @xmath184 and @xmath185 for @xmath186 . [ st : q ] set @xmath187 . if @xmath3 is not prime , go to step . [ st : pi ] set @xmath188 . if @xmath3 is not unramified in @xmath0 , or @xmath22 does not generate @xmath0 , go to step . [ st : output ] return @xmath3 and @xmath22 . we require @xmath189 in algorithm [ alg : construct - pi ] , as the case @xmath35 is already covered by example [ cycliccase ] , and requires a slight adaptation . the condition that @xmath7 be prime is for simplicity of presentation only ; the algorithm easily extends to square - free values of @xmath7 that are given as products of splitting primes . such @xmath7 are required , for example , by the cryptosystem of @xcite . [ thm : ispoltime ] if the field @xmath0 is fixed , then the heuristic expected run time of algorithm [ alg : construct - pi ] is polynomial in @xmath28 . the algorithm consists of a precomputation for the field @xmath0 in steps ( 1)(3 ) , followed by a loop in steps that is performed until an element @xmath77 is found that has prime norm @xmath190 , and we also find in step that @xmath3 is unramified in @xmath0 and the type norm @xmath191 generates @xmath0 . the primality condition in step is the ` true ' condition that becomes harder to achieve with increasing @xmath7 , whereas the conditions in step , which are necessary to guarantee correctness of the output , are so extremely likely to be fulfilled ( especially in cryptographic applications where @xmath0 is small and @xmath7 is large ) that they will hardly ever fail in practice and only influence the run time by a constant factor . as @xmath77 is computed in step as the lift to @xmath156 of an element @xmath192 , its norm can be bounded by a constant multiple of @xmath193 . heuristically , @xmath194 behaves as a random number , so by the prime number theorem it will be prime with probability at least @xmath195 , and we expect that we need to repeat the loop in steps about @xmath196 times before finding @xmath77 of prime norm @xmath3 . as each of the steps is polynomial in @xmath28 , so is the expected run time up to step , and we are done if we show that the conditions in step are met with some positive probability if @xmath0 is fixed and @xmath7 is sufficiently large . for @xmath3 being unramified in @xmath0 , one simply notes that only finitely many primes ramify in the field @xmath0 ( which is fixed ) and that @xmath3 tends to infinity with @xmath7 , since @xmath7 divides @xmath197 . finally , we show that @xmath22 generates @xmath0 with probability tending to @xmath33 as @xmath7 tends to infinity . suppose that for every vector @xmath198 that is not all 0 or 1 , we have @xmath199 this set of @xmath200 ( dependent ) conditions on the @xmath201 independent random variables @xmath202 for @xmath203 is satisfied with probability at least @xmath204 . for any automorphism @xmath205 of @xmath101 , the set @xmath206 is a cm - type of @xmath145 and there is a @xmath198 such that @xmath207 if @xmath208 contains @xmath209 and @xmath210 otherwise . then @xmath80 is @xmath211 , while @xmath212 is @xmath213 , so @xmath214 is @xmath215 . by , if this expression is @xmath33 then @xmath216 or @xmath217 , so @xmath218 or @xmath219 , which by definition of the reflex is equivalent to @xmath205 or @xmath220 being trivial on @xmath0 , i.e. , to @xmath205 being trivial on the maximal real subfield @xmath139 . thus if holds , then @xmath221 implies that @xmath205 is trivial on @xmath139 , hence @xmath222 . since @xmath174 is not real ( otherwise , @xmath223 ramifies in @xmath0 ) , this implies that @xmath224 . in order to maximize the likelihood of finding prime norms , one should minimize the norm of the lift @xmath77 computed in the chinese remainder step . this involves minimizing a norm function of degree @xmath225 in @xmath225 integral variables , which is already infeasible for @xmath226 . in practice , for given @xmath7 , one lifts a standard basis of @xmath227 to @xmath156 . multiplying those lifts by integer representatives for the elements @xmath80 and @xmath212 of @xmath228 , one quickly obtains lifts @xmath77 . we also choose , independently of @xmath7 , a @xmath229-basis of @xmath230 consisting of elements that are ` small ' with respect to all absolute values of @xmath145 . we translate @xmath77 by multiples of @xmath7 to lie in @xmath231 , where @xmath232 is the fundamental parallelotope in @xmath233 consisting of those elements that have coordinates in @xmath234 $ ] with respect to our chosen basis . if we denote the maximum on @xmath235 of all complex absolute values of @xmath108 by @xmath236 , we have @xmath237 for the @xmath32-value we find @xmath238 which is approximately @xmath239 if @xmath7 gets large with respect to @xmath236 . we would like @xmath32 to be small , but this is not what one obtains by lifting random admissible choices of @xmath240 . [ thm : heur ] if the field @xmath0 is fixed and @xmath7 is large , we expect that ( 1 ) the output @xmath3 of algorithm [ alg : construct - pi ] yields @xmath241 , and ( 2 ) an optimal choice of @xmath242 satisfying the conditions of theorem [ t : construct - pi ] yields @xmath243 . find an efficient algorithm to compute an element @xmath242 satisfying the conditions of theorem [ t : construct - pi ] for which @xmath243 . we will prove theorem [ thm : heur ] via a series of lemmas . let @xmath244 be the subset of the parallelotope @xmath245 consisting of those @xmath246 that satisfy the two congruence conditions for a given embedding degree @xmath8 . heuristically , we will treat the elements of @xmath244 as random elements of @xmath231 with respect to the distributions of complex absolute values and norm functions . we will also use the fact that , as @xmath108 is totally complex of degree @xmath247 , the @xmath248-algebra @xmath249 is naturally isomorphic to @xmath250 . we assume throughout that @xmath251 . [ l : lower - max ] fix the field @xmath0 . under our heuristic assumption , there exists a constant @xmath252 such that for all @xmath253 , the probability that a random @xmath254 satisfies @xmath255 is less than @xmath256 . the probability that a random @xmath77 lies in the set @xmath257 is the quotient of the volume of @xmath258 by the volume @xmath259 of @xmath231 , where @xmath260 is the discriminant of @xmath108 . now @xmath258 is contained inside @xmath261 , which has volume @xmath262^{{\widehat{g}}}\\ \prod { \lvert x_i \rvert}^2\leq r^{2({{\widehat{g}}}-\varepsilon ) } } $ } } \prod|x_i| dx \ \ < \ \ ( 2\pi)^{{\widehat{g}}}\mathop{\int}_{\hbox to0pt{\hss $ \scriptstyle x\in [ 0,rm_{\widehat{k}}]^{{\widehat{g}}}$ } } r^{{{\widehat{g}}}-\varepsilon}dx \ \ = \ \ ( 2\pi m_{\widehat{k}})^{{\widehat{g}}}r^{2{{\widehat{g}}}-\varepsilon},\ ] ] so a random @xmath77 lies in @xmath258 with probability less than @xmath263 . [ lem : aux ] there exists a number @xmath264 , depending only on @xmath145 , such that for any positive real number @xmath265 , the expected number of @xmath254 with all absolute values below @xmath266 is @xmath267 let @xmath268 be a lower bound on @xmath269 for the maximum of all complex absolute values , so the box @xmath270 consisting of those elements that have all absolute values below @xmath266 lies completely inside @xmath271 . the volume of @xmath272 in @xmath249 is @xmath273 , while @xmath231 has volume @xmath274 . the expected number of @xmath254 satisfying @xmath275 for all absolute values is @xmath276 times the quotient of these volumes . [ l : upper - min ] fix the field @xmath0 . under our heuristic assumption , there exists a constant @xmath277 such that for all positive @xmath278 , if @xmath7 is sufficiently large , then we expect the number of @xmath254 satisfying @xmath279 to be at least @xmath280 . any @xmath77 as in lemma [ lem : aux ] satisfies @xmath281 , so we apply the lemma to @xmath282 , which is less than @xmath283 for large enough @xmath7 and @xmath284 . [ l : lower - min ] fix the field @xmath0 . under our heuristic assumption , for all @xmath253 , if @xmath7 is large enough , we expect there to be no @xmath254 satisfying @xmath285 . let @xmath286 be the ring of integers of the maximal real subfield of @xmath145 . let @xmath287 be the subgroup of norm one elements of @xmath288 . we embed @xmath287 into @xmath289 by mapping @xmath290 to the vector @xmath291 of logarithms of absolute values of @xmath292 . the image is a complete lattice in the @xmath293-dimensional space of vectors with coordinate sum @xmath294 . fix a fundamental parallelotope @xmath295 for this lattice . let @xmath296 be the element of @xmath244 of smallest norm . since the conditions , as well as the norm of @xmath296 , are invariant under multiplication by elements of @xmath287 , we may assume without loss of generality that @xmath297 is inside @xmath298 . then every difference of two entries of @xmath297 is bounded , and hence every quotient of absolute values of @xmath296 is bounded from below by a positive constant @xmath299 depending only on @xmath0 . in particular , if @xmath300 is the maximum of all absolute values of @xmath296 , then @xmath301 . now suppose @xmath296 has norm below @xmath302 . then all absolute values of @xmath296 are below @xmath303 , and @xmath304 for @xmath7 sufficiently large . now lemma [ lem : aux ] implies that the expected number of @xmath305 with all absolute values below @xmath266 is a constant times @xmath306 , so for any sufficiently large @xmath7 we expect there to be no such @xmath77 , a contradiction . the upper bound @xmath307 follows from . lemma [ l : lower - max ] shows that for any @xmath253 , the probability that @xmath32 is smaller than @xmath308 tends to zero as @xmath7 tends to infinity , thus proving the lower bound @xmath309 . lemma [ l : upper - min ] shows that for any @xmath253 , if @xmath7 is sufficiently large then we expect there to exist a @xmath77 with @xmath32-value at most @xmath310 , thus proving the bound @xmath311 . lemma [ l : lower - min ] shows that we expect @xmath312 for the optimal @xmath77 , which proves the bound @xmath313 . for very small values of @xmath7 we are able to do a brute - force search for the smallest @xmath3 by testing all possible values of @xmath314 in step [ st : random ] of algorithm [ alg : construct - pi ] . we performed two such searches , one in dimension 2 and one in dimension 3 . the experimental results support our heuristic evidence that @xmath243 is possible with a smart choice in the algorithm , and that @xmath315 is achieved with a randomized algorithm . take @xmath316 , and let @xmath317 be the cm - type of @xmath0 defined by @xmath318 . we ran algorithm [ alg : construct - pi ] with @xmath319 and @xmath320 , and tested all possible values of @xmath321 . the total number of primes @xmath3 found was @xmath322 , and the corresponding @xmath32-values were distributed as follows : @xmath323{plot1}\quad\includegraphics[width=4.5cm]{plot2}\ ] ] the smallest @xmath3 found was @xmath324 , giving a @xmath32-value of @xmath325 . the curve over @xmath326 for which the jacobian has this @xmath32-value is @xmath327 , and the number of points on its jacobian is @xmath328 . take @xmath329 , and let @xmath330 be the cm - type of @xmath0 defined by @xmath331 . we ran algorithm [ alg : construct - pi ] with @xmath332 and @xmath333 , and tested all possible values of @xmath334 . the total number of primes @xmath3 found was @xmath335 , and the corresponding @xmath32-values were distributed as follows : @xmath323{plot3}\quad\includegraphics[width=4.5cm]{plot4}\ ] ] the smallest @xmath3 found was @xmath336 , giving a @xmath32-value of @xmath337 . the curve over @xmath326 for which the jacobian has this @xmath32-value is @xmath338 , and the number of points on its jacobian is @xmath339 . take @xmath316 , and let @xmath317 be the cm - type of @xmath0 defined by @xmath340 . we ran algorithm [ alg : construct - pi ] with @xmath341 and @xmath342 , and tested @xmath343 random values of @xmath321 . the total number of primes @xmath3 found was @xmath344 . of these primes , 6509 ( 91.6% ) produced @xmath32-values between 7.9 and 8.0 , while 592 ( 8.3% ) had @xmath32-values between 7.8 and 7.9 . the smallest @xmath3 found had @xmath345 binary digits , giving a @xmath32-value of @xmath346 . our algorithm [ alg : construct - pi ] yields @xmath3-weil numbers @xmath174 that correspond , in the sense of honda and tate @xcite , to isogeny classes of ordinary , simple abelian varieties over prime fields that have a point of order @xmath7 and embedding degree @xmath8 with respect to @xmath7 . it does not give a method to explicitly construct an abelian variety @xmath5 with frobenius @xmath174 . in this section we focus on the problem of explicitly constructing such varieties using complex multiplication techniques . the key point of the complex multiplication construction is the fact that every ordinary , simple abelian variety over @xmath172 with frobenius @xmath174 arises as the reduction at a prime over @xmath3 of some abelian variety @xmath347 in characteristic zero that has cm by the ring of integers of @xmath0 . thus if we have fixed our @xmath0 as in algorithm [ alg : construct - pi ] , we can solve the construction problem for all ordinary weil numbers coming out of the algorithm by compiling the finite list of @xmath348-isogeny classes of abelian varieties in characteristic zero having cm by @xmath34 . there will be one @xmath348-isogeny class for each equivalence class of primitive cm - types of @xmath0 , where @xmath104 and @xmath129 are said to be equivalent if we have @xmath349 for an automorphism @xmath65 of @xmath0 . as we can choose our favorite field @xmath0 of degree @xmath45 to produce abelian varieties of dimension @xmath29 , we can pick fields @xmath0 for which such lists already occur in the literature . from representatives of our list of isogeny classes of abelian varieties in characteristic zero having cm by @xmath34 , we obtain a list @xmath350 of abelian varieties over @xmath6 with cm by @xmath34 by reducing at some fixed prime @xmath351 over @xmath3 . changing the choice of the prime @xmath351 amounts to taking the reduction at @xmath351 of a conjugate abelian variety , which also has cm by @xmath34 and hence is @xmath352-isogenous to one already in the list . for every abelian variety @xmath353 , we compute the set of its twists , i.e. , all the varieties up to @xmath6-isomorphism that become isomorphic to @xmath5 over @xmath352 . there is at least one twist @xmath354 of an element @xmath355 satisfying @xmath356 , and this @xmath354 has a point of order @xmath7 and the desired embedding degree . note that while efficient point - counting algorithms do not exist for varieties of dimension @xmath357 , we can determine probabilistically whether an abelian variety has a given order by choosing a random point , multiplying by the expected order , and seeing if the result is the identity . the complexity of the construction problem rapidly increases with the genus @xmath358/2 $ ] , and it is fair to say that we only have satisfactory general methods at our disposal in very small genus . in genus one , we are dealing with elliptic curves . the @xmath359-invariants of elliptic curves over @xmath23 with cm by @xmath34 are the roots of the _ hilbert class polynomial _ of @xmath0 , which lies in @xmath360 $ ] . the degree of this polynomial is the class number @xmath361 of @xmath0 , and it can be computed in time @xmath362 . for genus 2 , we have to construct abelian surfaces . any principally polarized abelian surface is the jacobian of a genus 2 curve , and all genus 2 curves are hyperelliptic . there is a theory of class polynomials analogous to that for elliptic curves , as well as several algorithms to compute these polynomials , which lie in @xmath363 $ ] . the genus 2 algorithms are not as well - developed as those for elliptic curves ; at present they can handle only very small quartic cm - fields , and there exists no rigorous run time estimate . from the roots in @xmath6 of these polynomials , we can compute the genus 2 curves using mestre s algorithm . any three - dimensional principally polarized abelian variety is isogenous to the jacobian of a genus 3 curve . there are two known families of genus 3 curves over @xmath23 whose jacobians have cm by an order of dimension @xmath364 . the first family , due to weng @xcite , gives hyperelliptic curves whose jacobians have cm by a degree-6 field containing @xmath365 . the second family , due to koike and weng @xcite , gives picard curves ( curves of the form @xmath366 with @xmath367 ) whose jacobians have cm by a degree-6 field containing @xmath368 . explicit cm - theory is mostly undeveloped for dimension @xmath369 . moreover , most principally polarized abelian varieties of dimension @xmath370 are not jacobians , as the moduli space of jacobians has dimension @xmath371 , while the moduli space of abelian varieties has dimension @xmath372 . for implementation purposes we prefer jacobians or even hyperelliptic jacobians , as these are the only abelian varieties for which group operations can be computed efficiently . in cases where we can not compute every abelian variety in characteristic zero with cm by @xmath34 , we use a single such variety @xmath5 and run algorithm [ alg : construct - pi ] for each different cm - type of @xmath0 until it yields a prime @xmath3 for which the reduction of @xmath5 mod @xmath3 is in the correct isogeny class . an example for @xmath373 with @xmath27 prime is given by the jacobian of @xmath374 , which has dimension @xmath375 . we implemented algorithm [ alg : construct - pi ] in magma and used it to compute examples of hyperelliptic curves of genus 2 and 3 over fields of cryptographic size for which the jacobians are pairing - friendly . the subgroup size @xmath7 is chosen so that the discrete logarithm problem in @xmath14 $ ] is expected to take roughly @xmath376 steps . the embedding degree @xmath8 is chosen so that @xmath377 ; this would be the ideal embedding degree for the 80-bit security level if we could construct varieties over @xmath378 with @xmath379 . space constraints prevent us from giving the group orders for each jacobian , but we note that a set of all possible @xmath3-weil numbers in @xmath0 , and hence all possible group orders , can be computed from the factorization of @xmath3 in @xmath0 . let @xmath380 and let @xmath0 be the degree-4 galois cm field @xmath381 . let @xmath317 be the cm type of @xmath0 such that @xmath382 . we ran algorithm [ alg : construct - pi ] with cm type @xmath120 , @xmath383 , and @xmath384 . the algorithm output the following field size : @xmath385 there is a single @xmath386-isomorphism class of curves over @xmath387 whose jacobians have cm by @xmath34 and it has been computed in @xcite ; the desired twist turns out to be @xmath388 . the @xmath32-value of @xmath389 is @xmath390 . let @xmath391 and let @xmath0 be the degree-4 non - galois cm field @xmath381 . the reflex field @xmath108 is @xmath392 where @xmath393 . let @xmath109 be the cm type of @xmath0 such that @xmath394 . we ran algorithm [ alg : construct - pi ] with the cm type @xmath395 , subgroup size @xmath396 , and embedding degree @xmath384 . the algorithm output the following field size : @xmath397 the class polynomials for @xmath0 can be found in the preprint version of @xcite . we used the roots of the class polynomials mod @xmath3 to construct curves over @xmath387 with cm by @xmath34 . as @xmath0 is non - galois with class number @xmath135 , there are 8 isomorphism classes of curves in 2 isogeny classes . we found a curve @xmath398 in the correct isogeny class with equation @xmath399 , with @xmath400 the @xmath32-value of @xmath389 is @xmath401 . let @xmath0 be the degree-6 galois cm field @xmath402 , and let @xmath330 be the cm type of @xmath0 such that @xmath403 . we used the cm type @xmath120 to construct a curve @xmath398 whose jacobian has embedding degree @xmath404 with respect to @xmath405 . since @xmath0 has class number @xmath33 and one equivalence class of primitive cm types , there is a unique isomorphism class of curves in characteristic zero whose jacobians are simple and have cm by @xmath0 ; these curves are given by @xmath406 . algorithm [ alg : construct - pi ] output the following field size : @xmath407 the equation of the curve @xmath398 is @xmath408 . the @xmath32-value of @xmath409 is @xmath410 . we conclude with an example of an 8-dimensional abelian variety found using our algorithms . we started with a single cm abelian variety @xmath5 in characteristic zero and applied our algorithm to different cm - types until we found a prime @xmath3 for which the reduction has the given embedding degree . let @xmath411 . we set @xmath319 and @xmath412 and ran algorithm [ alg : construct - pi ] repeatedly with different cm types for @xmath0 . given the output , we tested the jacobians of twists of @xmath413 for the specified number of points . we found that the curve @xmath414 has embedding degree @xmath415 with respect to @xmath7 over the field @xmath6 of order @xmath416 the cm type was @xmath417 where @xmath418 . the @xmath32-value of @xmath389 is @xmath419 .

we present an algorithm that , on input of a cm - field @xmath0 , an integer @xmath1 , and a prime @xmath2 , constructs a @xmath3-weil number @xmath4 corresponding to an ordinary , simple abelian variety @xmath5 over the field @xmath6 of @xmath3 elements that has an @xmath6-rational point of order @xmath7 and embedding degree @xmath8 with respect to @xmath7 . we then discuss how cm - methods over @xmath0 can be used to explicitly construct @xmath5 .

You are an expert at summarizing long articles. Proceed to summarize the following text: water ice is believed to play many important roles in the planet formation theories . for example , ice enhances the surface density of solid material in the cold outer part of a protoplanetary disk , which promotes the formation of massive cores of gaseous planets ( e.g. , * ? ? ? * ) . thus the ice sublimation / condensation front called snowline , is considered to be the boundary of the forming regions of the terrestrial and jovian planets . snowline is also suggested as a possible forming site of the planetesimals @xcite . furthermore , icy planetesimals or comets may bring water to the earth ( e.g. , * ? ? ? recently , numerous water vapor emission lines have been detected in protoplanetary disks ( e.g. , * ? ? ? * ; * ? ? ? * ) , and the position of snow line is inferred from the modeling @xcite . on the other hand , observations of water ice distribution in the disk are limited at this moment . crystalline h@xmath1o ice emission features at 44 and 62@xmath0 m have been found for several herbig ae / be stars @xcite , but the limited angular resolution in far - infrared wavelengths hampers us to obtain ice distribution . while near - infrared ( nir ) water ice absorption toward edge - on disks is reported @xcite , the ice absorption is formed at somewhere through line of sight , thus it is still not straightforward to derive its radial distribution in the disk . @xcite proposed a new observational way to investigate the radial distribution of ice in face - on disks . they showed that ice absorption should also be imprinted in the light scattered by icy grains and that multi - wavelength imaging in nir wavebands , including h@xmath1o band at 3.1@xmath0 m , is a useful tool to constrain the ice distribution in the disk . @xcite applied this method to the circumstellar disk around a herbig fe star hd 142527 , and showed that the water ice grains present in a disk surface at a radial distance of 140 au . on the other hand , @xcite calculated the stability and distribution of water ice grains in the disk surface considering the photodesorption ( photosputtering ) process by uv irradiation . they showed that the water ice grains can be rapidly destroyed at the disk surface around a / b type stars due to uv photo desorption processes . although @xcite already detected the water ice grains in the disk surface around f - type star hd142527 , it would be interesting to observe the water ice grains in the disk surface around a / b type stars to check the prediction by @xcite . in this paper , we showed the observations of water ice grains in the disk surface around herbig be star hd100546 , and discuss the presence / stability of water ice grains in the disk surface . part of our data was already published in @xcite focusing the planet candidate @xcite direct imaging observations of the herbig be star hd 100546 using k band filter ( central wavelength @xmath2 = 2.20@xmath0 m , and width @xmath3 = 0.33@xmath0 m ) , h@xmath1o ice filter ( @xmath2 = 3.06@xmath0 m , @xmath3 = 0.15@xmath0 m ) and l band filter ( @xmath2 = 3.78@xmath0 m , @xmath3 = 0.70@xmath0 m ) were performed using the nici ( near infrared coronagraphic imager ; * ? ? ? * ) on the gemini south telescope on march 31 , 2012 . we fixed the instrument rotator during the observations of both object and psf ( point - spread function ) reference stars to fix the pupil and to obtain the stable psf patterns . a full width at half maximum ( fwhm ) of 0.10@xmath4 was achieved at all the wavelength using the instrument ao system . the central region close to the central star was saturated , however , the outer part ( r @xmath5 0.22@xmath4 ) is not saturated and can be used for disk observations . the total exposure times were 1672 s , 3192 s , and 2128 s for k , h@xmath1o ice , and l , respectively . as a psf reference star , we observed hr 4977 ( a0v ) just before / after hd 100546 . hd 105116 ( k band ) and bs4638 ( h@xmath1o ice and l band ) was observed as a photometric standard star . the mean flux density for bs 4638 at 3.06@xmath0 m was estimated by scaling the kurucz s stellar model atmosphere ( @xmath6 = 19500 k , @xmath7 = 3.95 , solar metallicity ) to match the flux density in k @xcite . the calculated flux density was 5.87 jy at 3.06@xmath0 m . observation parameters are summarized in table [ obssummary ] . the images were first processed using the iraf packages for dark subtraction , flat - fielding with sky flats , bad pixel correction , and sky subtraction . since the stellar halo was very bright , psf subtraction was required to investigate the faint structure near the central star . the reference psf was chosen to match the psf of hd 100546 for each frame with careful visual inspection to determine whether the circular bright halo of the central point source was well suppressed after psf subtraction . the reference psf was made by combining the adopted reference star images . the flux scaling of the reference psf was performed so that no region had negative intensity after the subtraction , in particular , just outer part of central radius ( r @xmath8 ) . the reference psf was shifted to match the central position and subtracted from each frame of hd 100546 . the each subtracted frame was rotated to match the north direction . then , the final psf subtracted image was made by combining the rotated object frames . in order to estimate the systematic uncertainty of the surface brightness due to the psf subtraction process , we changed the scaling of the psf before subtraction , and measured the acceptable range of the scaling factor . the systematic uncertainty of the surface brightness was measured to be 20% depending on the position . this systematic uncertainty was typically larger than the statistical error derived from the standard deviation of the best psf - subtracted object frames . the final psf subtracted images of hd100546 disk is shown in figure [ obsimages ] . in figure [ obsimages ] , an extended disk structure is detected in all three bands . especially in l band , a dark lane is seen in the south - west direction from the star , showing the typical inclined flaring disk morphology . the north - east side is facing to us , while the south - west scattered light beyond the dark lane can be the scattered light from the other side of the disk surface . this morphology is consistent with the previous studies @xcite . using these three color images , we extracted the scattered light spectra of different region of the protoplanetary disk . since the position angle ( pa ) of this disk major axis is 145@xmath9 @xcite , we set the 0.162@xmath4(9 pixel ) square region along with major and minor axis at a distance of 0.360@xmath4 , 0.522@xmath4 , 0.684@xmath4 , 0.846@xmath4 , and 1.008@xmath4 from the central star , which is shown in figure [ fig : extract ] . the extracted spectra of the each region are shown in figure [ fig : spectra ] . in almost all the regions , relatively shallow 3@xmath0 m absorption feature is present in their spectra likely due to water ice grains , indicating that the water ice grains present in the disk surface . the shallowness of this ice absorption feature can be due to the loss of ice grains at the disk surface . @xcite claimed that the water ice grains can be quickly destroyed at the disk surface around a / b type stars due to its strong uv photodesorption . to assess their prediction , quantitative comparison with the model prediction is required . since we are interested in the water ice distribution in the disk and it is necessary to quantify the absorption feature depth for comparison with the disk model , we will use the water ice absorption optical depth @xmath10 following the convention . note that the absorption feature in the scattered light spectra is not a pure absorption but rather an albedo effect ( see @xcite ) , however , the optical depth @xmath10 is so often used as an indicator of the depth of the feature , thus we use it for descriptive purposes . the optical depth is derived from the following formula as usual . where @xmath15 and @xmath16 are the observed surface brightness at k and l filters , respectively , while @xmath17,@xmath18 , and @xmath19 are the central wavelengths of k , h@xmath1o and l filters , respectively . as the ice absorption depth becomes deeper , the optical depth @xmath10 value becomes larger . we derived the @xmath10 at each extracted disk regions , and these values ranged from 0 to 2 . the @xmath10 map is shown in fig.[fig : h2otaumap ] . in this figure , the central region ( @xmath200.22 ) is masked due to the saturation problem and outer region ( [email protected] ) is also masked because of the low signal to noise ratio . in general , south - western(sw ) region shows relatively high @xmath10 value ( @xmath21 ) , which coincides with the dark - lane seen in l disk image . this can be qualitatively understood that some scattered light of this region may come from the backside of the disk , which suffers more extinction than that from the frontside of the disk . other than the sw region , a possible trend could be recognized that the inner region shows lower optical depth ( @xmath22 ) which might imply the decrease of the ice grains toward the central star as expected , although the patchy structure in the @xmath10 map hampers us to make a solid conclusion . since the @xmath10 map shows significant asymmetry along the disk minor axis ( sw - ne ) and scattered light intensity depends on the scattering geometry , discussion on the scattered light spectra along the disk minor axis is more complicated than that along the disk major axis ( se - nw ) . thus we focus on a comparison with the model along the disk major axis . the radial distributions of @xmath10 along with the disk major axis are summarized in table[measuresummary ] and shown in fig.[fig : h2otau ] . as already described , the error is dominated by systematic error , not the statistical one , and is estimated 20% of the surface brightness at each wavebands . to discuss whether the observed absorption feature is consistent with the model predictions , we derived the expected @xmath10 value based on the disk model calculations by @xcite who included the effect of photodesorption of water ice grains by uv photons from the central star . our calculation model consists of two parts : one is the disk structure calculation ( model part 1 ) in which the density and temperature distributions in the disk , as well as the snow line , are obtained , and the other is the radiative transfer calculation ( model part 2 ) that simulates observations . in the model part 1 , we obtain the location of the snow line , which is primarily determined by two reasons ; the thermal sublimation and the photodesorption of water ice particles . in order to evaluate the thermal sublimation of ice particles , we obtain the temperature and the gas density distributions in the disk as follows . the temperature in the disk is determined by the energy balance between heating and cooling . heating sources for the disk include the radiation from the central star illuminating the surface of the disk and the viscous heating due to the disk accretion . cooling process is the radiative transfer , which finally emits energy from the disk to outer space by means of radiation . the surface density distribution is provided as a model parameter , and the gas density distribution along the vertical direction with respect to the disk is determined so that the hydrostatic equilibrium is achieved . the temperature in the disk and the shape of the disk surface affect each other , because the angle between the direction of the light from the central star and the disk surface determines the radiative energy received by the disk surface , and the inclination of the disk surface is a function of the temperature distribution . thus , the temperature distribution and the gas density distribution , which is related to the disk surface , should be calculated consistently . using the temperature and the gas density , we can evaluate the vapor pressure and the saturated vapor pressure of water , which are used to determine the snow line . on the other hand , we calculate the radiative transfer of uv radiation from the central star to the disk and obtain the uv flux exposed to ice particles in the disk , which gives the photodesorption rate . finally , we can see if ice particles can be present stably with the vapor pressure ( contributing to the condensation ) , the saturated vapor pressure ( contributing to the sublimation ) , and the photodesorption rate , and we obtain the location of the snow line in the disk . for our calculations , we use physical parameters for hd 100546 given by @xcite , which are listed in table [ modelparameter ] . we adopt the mass fractions of silicate and water to the disk gas of 0.0043 and 0.0094 , respectively @xcite . we also assume that they form spherical dust particles , that ice particles contain pure ice and silicate grains are composed of only silicate , that radii of dust particles range from 0.025 @xmath0 m to 2.5 @xmath0 m , and that their size distributions follow the power - law with the index of -3.5 . absorption and scattering coefficients of dust particles are given by @xcite . to make a comparison with observation ( model part 2 ) , we carry out monte carlo radiative transfer calculations . deviations of obtained results seen in fig . [ fig : h2otau ] are caused by statistical errors intrinsically related to monte carlo simulations . in the calculations , we take into account the anisotropic scattering and polarization produced by dust particles . the scattering matrices of silicate and ice particles are obtained using the bhmie code by @xcite with the complex refractive indices by @xcite . the radial profiles of the @xmath10 of disk models with or without photodesorption along with the disk major axis are also shown in fig.[fig : h2otau ] . a model without photodesorption effect shows deeper water ice absorption feature ( larger @xmath10 ) than the observations , while models with photodesorption effect show relatively shallow absorption ( smaller @xmath10 ) . this is due to the destruction of water ice grains at the disk surface via photodesorption by strong irradiation of uv photons from the central star @xcite . although the observed @xmath10 values match with both disk models with / without photodesorption effect , the model with photo desorption effect seems slightly better match with the observations at least for nw region . it would be interesting to note that the water vapor is reported to be depleted in the disk atmosphere of herbig aebe stars by the observations of water vapor lines @xcite . a plausible explanation is due to photodissociation of water by uv photons in the disk atmosphere . although photodissociation effect is not included in the model of @xcite , they discussed that the photodesorption process is much important for the water ice stability , while photodissociation effect is crucial for water vapor destruction in the disk surface . in any cases , the uv photons seem to play important role on the survival of both water ice and gas in the disk surface around herbig ae / be stars . it is apparent that our data do not have a high signal - to - noise ratio enough to distinguish the models with / without photodesorption process . this is because the systematic error dominates over the total error . further observations with better photometric accuracy are strongly desired . since these observations shown here employed techniques similar to so - called lucky imaging technique , improvement of systematic error is principally limited . however , when we make use of the polarimetric differential imaging ( pdi ) and/or spectroscopy , the systematic error can be significantly reduced and it will change the situation dramatically . thus the l - band pdi and/or spectroscopy is promising for the advance of this observations . furthermore , other effects on the depth of water ice absorption , such as grain size , grain shape , grain structure , ice / rock ratio ( abundance ) , dust settling , turbulent mixing , and so on , should be investigated in the future theoretical studies to comprehensively understand the water ice distribution in the protoplanetary disks . we are grateful to all of the staff members of the subaru telescope , gemini telescope , and optical coatings japan for the production of narrow band filters . we also thank anonymous referee for their useful comments . we appreciate dr . tom hayward on his kind support during the observations and installation of our h@xmath1o ice filter to nici . mh was supported by jsps / mext kakenhi ( grant - in - aid for young scientists b : 21740141 , grant - in - aid for scientific research on innovative areas : 26108512 ) . mt is partly supported by the jsps fund ( no . 22000005 ) . fukagawa , m. , tamura , m. , itoh , y. , kudo , t. , imaeda , y. , oasa , y. , hayashi , s. s. , & hayashi , m. 2006 , , 636 , l153 honda , m. , et al . 2010 , , 718 , l199 hayashi , c. , nakazawa , k. , & nakagawa , y. 1985 , protostars and planets ii , 1100 brauer , f. , henning , t. , & dullemond , c. p. 2008 , , 487 , l1 morbidelli , a. , chambers , j. , lunine , j. i. , et al . 2000 , meteoritics and planetary science , 35 , 1309 carr , j. s. , & najita , j. r. 2008 , science , 319 , 1504 salyk , c. , pontoppidan , k. m. , blake , g. a. , et al . 2008 , , 676 , l49 malfait , k. , waelkens , c. , bouwman , j. , de koter , a. , & waters , l. b. f. m. 1999 , , 345 , 181 meeus , g. , waters , l. b. f. m. , bouwman , j. , et al . 2001 , , 365 , 476 pontoppidan , k. m. , dullemond , c. p. , van dishoeck , e. f. , et al . 2005 , , 622 , 463 terada , h. , tokunaga , a. t. , kobayashi , n. , et al . 2007 , , 667 , 303 inoue , a. k. , honda , m. , nakamoto , t. , & oka , a. 2008 , , 60 , 557 oka , a. , inoue , a. k. , nakamoto , t. , & honda , m. 2012 , , 747 , 138 honda , m. , inoue , a. k. , fukagawa , m. , et al . 2009 , , 690 , l110 chun , m. , toomey , d. , wahhaj , z. , et al . 2008 , , 7015 , 70151v allen , d. a. , & cragg , t. a. 1983 , , 203 , 777 carter , b. s. , & meadows , v. s. 1995 , , 276 , 734 van boekel , r. , min , m. , waters , l. b. f. m. , et al . 2005 , , 437 , 189 quanz , s. p. , schmid , h. m. , geissler , k. , et al . 2011 , , 738 , 23 mulders , g. d. , min , m. , dominik , c. , debes , j. h. , & schneider , g. 2013 , , 549 , aa112 mulders , g. d. , waters , l. b. f. m. , dominik , c. , et al . 2011 , , 531 , aa93 soubiran , c. , le campion , j .- f . , cayrel de strobel , g. , & caillo , a. 2010 , , 515 , aa111 currie , t. , muto , t. , kudo , t. , et al . 2014 , , 796 , ll30 miyake , k. , & nakagawa , y. 1993 , icarus , 106 , 20 bohren , c. f. , & huffman , d. r. 1983 , new york : wiley , 1983 , avenhaus , h. , quanz , s. p. , meyer , m. r. , et al . 2014 , , 790 , 56 fedele , d. , pascucci , i. , brittain , s. , et al . 2011 , , 732 , 106 zhang , k. , pontoppidan , k. m. , salyk , c. , & blake , g. a. 2013 , , 766 , 82 quanz , s. p. , amara , a. , meyer , m. r. , et al . 2013 , , 766 , ll1 ccccc hd100546 & k & 1672 & + hr4977 & k & 456 & psf reference + hd105116 & k & 152 & photometric reference + hd100546 & h@xmath1o ice & 3192 & + hr4977 & h@xmath1o ice & 2280 & psf reference + bs4638 & h@xmath1o ice & 152 & photometric reference + hd100546 & l & 2128 & + hr4977 & l & 532 & psf reference + bs4638 & l & 152 & photometric reference rrrrr + 104 & [email protected] & [email protected] & [email protected] & [email protected] + 87 & [email protected] & [email protected] & 65@xmath2313 & [email protected] + 70 & 60@xmath2312 & [email protected] & 111@xmath2322 & [email protected] + 54 & 126@xmath2325 & 71@xmath2314 & 210@xmath2342 & [email protected] + 37 & 304@xmath2361 & 223@xmath2345 & 485@xmath2397 & [email protected] + + 104 & [email protected] & [email protected] & [email protected] & [email protected] + 87 & [email protected] & [email protected] & 63@xmath2313 & 0.76@xmath23 0.39 + 70 & 52@xmath2310 & [email protected] & 109@xmath2322 & 0.74@xmath23 0.39 + 54 & 106@xmath2321 & 92@xmath2318 & 219@xmath2344 & 0.59@xmath23 0.39 + 37 & 250@xmath2350 & 237@xmath2347 & 506@xmath23100 & 0.50@xmath23 0.39 lc stellar effective temperature @xmath24 & 10500 k + stellar mass @xmath25 & 2.4 @xmath26 + stellar luminosity @xmath27 & 36 @xmath28 + stellar radius @xmath29 & 1.8 @xmath30 + distance d & 103 pc + disk inner radius @xmath31 & 0.5 au + disk outer radius @xmath32 & 500 . au + disk inclination & 45@xmath9 + surface density at 1au & 45 g/@xmath33 + surface density power - law index q & 1

we made near infrared multicolor imaging observations of a disk around herbig be star hd100546 using gemini / nici . k ( 2.2@xmath0 m ) , h@xmath1o ice ( 3.06@xmath0 m ) , and l(3.8@xmath0 m ) disk images were obtained and we found the 3.1@xmath0 m absorption feature in the scattered light spectrum , likely due to water ice grains at the disk surface . we compared the observed depth of the ice absorption feature with the disk model based on @xcite including water ice photodesorption effect by stellar uv photons . the observed absorption depth can be explained by the both disk models with / without photodesorption effect within the measurement accuracy , but slightly favors the model with photodesorption effects , implying that the uv photons play an important role on the survival / destruction of ice grains at the herbig ae / be disk surface . further improvement on the accuracy of the observations of the water ice absorption depth is needed to constrain the disk models .

You are an expert at summarizing long articles. Proceed to summarize the following text: let @xmath2 be the set of all square matrices of order @xmath3 with entries in @xmath4 and @xmath5 be the group of all invertible matrices of @xmath2 . a map @xmath6 is called an affine map if there exist @xmath7 and @xmath8 such that @xmath9 , @xmath10 . we denote @xmath11 , we call @xmath12 the _ linear part _ of @xmath13 . the map @xmath13 is invertible if @xmath14 . denote by @xmath15 the vector space of all affine maps on @xmath1 and @xmath16 the group of all invertible affine maps of @xmath17 . let @xmath0 be an abelian affine sub - semigroup of @xmath18 . for a vector @xmath19 , we consider the orbit of @xmath0 through @xmath20 : @xmath21 . denote by @xmath22 the closure of a subset @xmath23 . the semigroup @xmath0 is called _ hypercyclic _ if there exists a vector @xmath24 such that @xmath25 . for an account of results and bibliography on hypercyclicity , we refer to the book @xcite by bayart and matheron . we refer the reader to the recent book @xcite and @xcite for a thorough account on hypercyclicity . costakis and manoussos in @xcite localize the concept of hypercyclicity using j - sets . by analogy , we generalize this notion to affine case as follow : suppose that @xmath0 is generated by @xmath26 affines maps @xmath27 @xmath28 then for @xmath29 , we define the extended limit set j@xmath30 to be the set of @xmath31 for which there exists a sequence of vectors @xmath32 with @xmath33 and sequences of non - negative integers @xmath34 for @xmath35 with @xmath36 such that @xmath37 note that condition ( 1.1 ) is equivalent to having at least one of the sequences @xmath38 for @xmath39 containing a strictly increasing subsequence tending to @xmath40 . we say that @xmath0 is _ locally hypercyclic _ if there exists a vector @xmath41 such that j@xmath42 . so , the question to investigate is the following : when an abelian sub - semigroup of @xmath17 can be hypercyclic ? the main purpose of this paper is twofold : firstly , we give a general characterization of the above question for any abelian _ sub - semigroup _ of @xmath17 using j - sets . secondly , we generalize the results proved in @xcite , by a.ayadi and h.marzougi , for linear semigroups , which answers negatively , the question raised in the paper of costakis and manoussos @xcite : is it true that a locally hypercyclic abelian semigroup @xmath43 generated by matrices @xmath44 is hypercyclic whenever j@xmath45 for a finite set of @xmath10 whose vector space is equal @xmath1 ? similarly for @xmath46 . denote by @xmath47 the canonical basis of @xmath1 . let @xmath48 . let introduce the following notations and definitions . denote by : + let @xmath49 be fixed , denote by : + @xmath50 , @xmath51 and @xmath52 . + @xmath53 the canonical basis of @xmath54 and @xmath55 the identity matrix of @xmath56 . + for each @xmath57 , denote by : + @xmath58 the set of matrices over @xmath4 of the form @xmath59 + @xmath60 the group of matrices of the form ( [ eq1 ] ) with @xmath61 . + let @xmath62 and @xmath63 such that @xmath64 in particular , @xmath65 . write + @xmath66 in particular if @xmath67 , then @xmath68 and @xmath69 . + @xmath70 . + @xmath71 where @xmath72 , for @xmath73 . so @xmath74 . + @xmath75 the second projection defined by @xmath76 . + define the map @xmath77 + @xmath78 we have the following composition formula @xmath79 then @xmath80 is an injective homomorphism of groups . write + @xmath81 , it is an abelian subgroup of @xmath82 . + let consider the normal form of @xmath83 : by proposition [ p:2 ] , there exists a @xmath84 and a partition @xmath85 of @xmath86 such that @xmath87 . for such a choice of matrix @xmath88 , we let + @xmath89 . so @xmath90 , since @xmath91 . + @xmath92 . we have @xmath93 . + @xmath94 . + denote by @xmath95 , @xmath96 and @xmath97 , then @xmath98 for such choise of the matrix @xmath99 , we can write @xmath100,\ \ \mathrm{with}\ \ \ q\in gl(n , { \mathbb{c}}).\ ] ] we have @xmath101 , @xmath102 and @xmath103 . we have @xmath104 is an abelian semigroup of @xmath105 . our principal results are the following : [ t:1 ] let @xmath0 be a finitely generated abelian semigroup of affine maps on @xmath1 . if @xmath106 for some @xmath107 then @xmath108 . [ c:1 ] under the hypothesis of theorem [ t:1 ] , the following are equivalent : * @xmath0 is hypercyclic . * * @xmath110 . [ c:2 ] under the hypothesis of theorem [ t:1 ] , set @xmath111 . if @xmath0 is not hypercyclic then @xmath112 , ( @xmath113 ) where @xmath114 are @xmath0-invariant affine subspaces of @xmath1 with dimension @xmath115 . [ p:2 ] let @xmath0 be an abelian sub - semigroup of @xmath116 and @xmath81 . then there exists @xmath117 such that @xmath118 is a sub - semigroup of @xmath119 , for some @xmath65 and @xmath120 . [ p:002]@xmath121@xcite , proposition 2.1@xmath122 let @xmath0 be an abelian subgroup of @xmath123 and @xmath81 . then there exists @xmath117 such that @xmath118 is a subgroup of @xmath124 , for some @xmath65 and @xmath120 . suppose first , @xmath125 . let @xmath126 be the group generated by @xmath83 . then @xmath126 is abelian and by proposition [ p:002 ] , there exists a @xmath117 such that @xmath127 is an abelian subgroup of @xmath128 , for some @xmath129 and @xmath130 . in particular , @xmath131 . suppose now , @xmath132 . for every @xmath133 , there exists @xmath134 such that @xmath135 ( one can take @xmath136 non eigenvalue of @xmath12 ) . write @xmath137 be the group generated by @xmath138 . then @xmath137 is an abelian subsemigroup of @xmath82 . hence by above , there exists a @xmath139 such that @xmath140 , for some @xmath130 . as @xmath141 then @xmath142 . this proves the proposition . let @xmath143 be the semigroup generated by @xmath83 and @xmath144 . then @xmath143 is an abelian sub - semigroup of @xmath145 . by proposition [ p:2 ] , there exists @xmath91 such that @xmath118 is a sub - semigroup of @xmath146 for some @xmath65 and @xmath120 and this also implies that @xmath147 is a sub - semigroup of @xmath146 . + + @xmath152 is obvious since @xmath153 by construction . + @xmath154 let @xmath31 and @xmath155 be a sequence in @xmath143 such that @xmath156 . one can write @xmath157 , with @xmath158 and @xmath159 , thus @xmath160 , so @xmath161 . therefore , @xmath162 . hence , @xmath163 . + @xmath164 since @xmath165 and for every @xmath166 , @xmath167 , we get @xmath168 hence @xmath169 . therefore , @xmath178 and sequences of non - negative integers @xmath34 for @xmath39 such that @xmath179 hence @xmath180 . + @xmath181 since @xmath182 , then there exists a sequence of vectors @xmath183 with @xmath184 and sequences of non - negative integers @xmath34 for @xmath185 with @xmath186 such that @xmath187 suppose that @xmath106 with @xmath107 . by proposition [ p:2 ] , we can assume that @xmath194 and @xmath195 . by lemma [ l:1 ] , we have @xmath196 . then by theorem [ t:5 ] , @xmath197 . it follows by lemma [ ll1l:9 ] , that @xmath198 . with @xmath211^{t},\ x_{k}\in\{0\}\times\mathbb{c}^{n_{k}-1},\ x_{i}\in\mathbb{c}^{n_{i } } , \ \mathrm{if}\ i\neq k\right\}$ ] . it follows that @xmath212 with @xmath213 is an affine space with dimension @xmath115 .

we give a characterization of hypercyclic using ( locally hypercyclic ) of semigroup @xmath0 of affine maps of @xmath1 . we prove the existence of a @xmath0-invariant open subset of @xmath1 in which any locally hypercyclic orbit is dense in @xmath1 .

You are an expert at summarizing long articles. Proceed to summarize the following text: proteins are essential building blocks of living organisms . they function as catalyst , structural elements , chemical signals , receptors , etc . the molecular mechanism of protein functions are closely related to their structures . the study of structure - function relationship is the holy grail of biophysics and has attracted enormous effort in the past few decades . the understanding of such a relationship enables us to predict protein functions from structure or amino acid sequence or both , which remains major challenge in molecular biology . intensive experimental investigation has been carried out to explore the interactions among proteins or proteins with other biomolecules , e.g. , dnas and/or rnas . in particular , the understanding of protein - drug interactions is of premier importance to human health . a wide variety of theoretical and computational approaches has been proposed to understand the protein structure - function relationship . one class of approaches is biophysical . from the point of view of biophysics , protein structure , function , dynamics and transport are , in general , dictated by protein interactions . quantum mechanics ( qm ) is based on the fundamental principle , and offers the most accurate description of interactions among electrons , photons , atoms and even molecules . although qm methods have unveiled many underlying mechanisms of reaction kinetics and enzymatic activities , they typically are computationally too expensive to do for large biomolecules . based on classic physical laws , molecular mechanics ( mm ) @xcite can , in combination with fitted parameters , simulate the physical movement of atoms or molecules for relatively large biomolecular systems like proteins quite precisely . however , it can be computationally intractable for macromoelcular systems involving realistic biological time scales . many time - independent methods like normal mode analysis ( nma ) @xcite , elastic network model ( enm ) @xcite , graph theory @xcite and flexibility - rigidity index ( fri ) @xcite are proposed to capture features of large biomolecules . variational multiscale methods @xcite are another class of approaches that combine atomistic description with continuum approximations . there are well developed servers for predicting protein functions based on three - dimensional ( 3d ) structures @xcite or models from the homology modeling ( here homology is in biological sense ) of amino acid sequence if 3d structure is not yet available @xcite . another class of important approaches , bioinformatical methods , plays a unique role for the understanding of the structure - function relationship . these data - driven predictions are based on similarity analysis . the essential idea is that proteins with similar sequences or structures may share similar functions . also , based on sequential or structural similarity , proteins can be classified into many different groups . once the sequence or structure of a novel protein is identified , its function can be predicted by assigning it to the group of proteins that share similarities to a good extent . however , the degree of similarity depends on the criteria used to measure similarity or difference . many measurements are used to describe similarity between two protein samples . typical approaches use either sequence or physical information , or both . among them , sequence alignment can describe how closely the two proteins are related . protein blast @xcite , clustalw2 @xcite , and other software packages can preform global or local sequence alignments . based on sequence alignments , various scoring methods can provide the description of protein similarity @xcite . additionally , sequence features such as sequence length and occurrence percentage of a specific amino acid can also be employed to compare proteins . many sequence based features can be derived from the position - specific scoring matrix ( pssm ) @xcite . moreover , structural information provides an efficient description of protein similarity as well . structure alignment methods include rigid , flexible and other methods . the combination of different structure alignment methods and different measurements such as root - mean - square deviation ( rmsd ) and z - score gives rise to various ways to quantify the similarity among proteins . as per structure information , different physical properties such as surface area , volume , free energy , flexible - rigidity index ( fri ) @xcite , curvature @xcite , electrostatics @xcite etc . can be calculated . a continuum model , poisson boltzmann ( pb ) equation delivers quite accurate estimation for electrostatics of biomolecules . there are many efficient and accurate pb solvers including pbeq @xcite , mibpb @xcite , etc . together with physical properties , one can also extract geometrical properties from structure information . these properties include coordinates of atoms , connections between atoms such as covalent bonds and hydrogen bonds , molecular surfaces @xcite and curvatures @xcite . these various approaches reveal information of different scales from local atom arrangement to global architecture . physical and geometrical properties described above add different perspective to analyze protein similarities . due to the advance in bioscience and biotechnology , biomolecular structure date sets are growing at an unprecedented rate . for example , the http://www.rcsb.org/pdb/home/home.do[protein data bank ( pdb ) ] has accumulated more than a hundred thousand biomolecular structures . the prediction of the protein structure - function relationship from such huge amount of data can be extremely challenging . additionally , an eve - growing number of physical or sequence features are evaluated for each data set or amino - acid residue , which adds to the complexity of the data - driven prediction . to automatically analyze excessively large data sets in molecular biology , many machine learning methods have been developed @xcite . these methods are mainly utilized for the classification , regression , comparison and clustering of biomolecular data . clustering is an unsupervised learning method which divides a set of inputs into groups without knowing the groups beforehand . this method can unveil hidden patterns in the data set . classification is a supervised learning method , in which , a classifier is trained on a given training set and used to do prediction for new observations . it assigns observation to one of several pre - determined categories based on knowledge from training data set in which the label of observations is known . popular methods for classification include support vector machine ( svm ) @xcite , artificial neural network ( ann ) @xcite , deep learning @xcite , etc . in classification , each observation in training the set has a feature vector that describes the observation from various perspectives and a label that indicates to which group the observation belongs . a model trained on the training set indicates to which group a new observation belongs with feature vector and unknown label . to improve the speed of classification and reduce effect from irrelevant features , many feature selection procedures have been proposed @xcite . machine learning approach are successfully used for protein hot spot prediction @xcite . the data - driven analysis of the protein structure - function relationship is compromised by the fact that same protein may have different conformations which possess different properties or delivers different functions . for instance , hemoglobins have taut form with low affinity to oxygen and relaxed form with high affinity to oxygen ; and ion channels often have open and close states . different conformations of a given protein may only have minor differences in their local geometric configurations . these conformations share the same sequence and may have very similar physical properties . however , their minor structural differences might lead to dramatically different functions . therefore , apart from the conventional physical and sequence information , geometric and topological information can also play an important role in understanding the protein structure - function relationship . indeed , geometric information has been extensively used in the protein exploration . in contrast , topological information has been hardly employed in studying the protein structure - function relationship . in general , geometric approaches are frequently inundated with too much geometric detail and are often prohibitively expensive for most realistic biomolecular systems , while traditional topological methods often incur in too much reduction of the original geometric and physical information . persistent homology , a new branch of applied topology , is able to bridge traditional geometry and topology . it creates a variety of topologies of a given object by varying a filtration parameter , such as a radius or a level set function . in the past decade , persistent homology has been developed as a new multiscale representation of topological features . the 0-th dimensional version was originally introduced for computer vision applications under the name size function " @xcite and the idea was also studied by robins @xcite . the persistent homology theory was formulated , together with an algorithm given , by edelsbrunner et al . @xcite , and a more general theory was developed by zomorodian and carlsson @xcite . there has since been significant theoretical development @xcite , as well as various computational algorithms @xcite . often , persistent homology can be visualized through barcodes @xcite , in which various horizontal line segments or bars are the homology generators which survive over filtration scales . persistence diagrams are another equivalent representation @xcite . computational homology and persistent homology have been applied to a variety of domains , including image analysis @xcite , chaotic dynamics verification @xcite , sensor network @xcite , complex network @xcite , data analysis @xcite , shape recognition @xcite and computational biology @xcite . compared with traditional computational topology @xcite and/or computational homology , persistent homology inherently has an additional dimension , the filtration parameter , which can be utilized to embed some crucial geometric or quantitative information into the topological invariants . the importance of retaining geometric information in topological analysis has been recognized @xcite , and topology has been advocated as a new approach for tackling big data sets @xcite . recently , we have introduced persistent homology for mathematical modeling and prediction of nano particles , proteins and other biomolecules @xcite . we have proposed molecular topological fingerprint ( mtf ) to reveal topology - function relationships in protein folding and protein flexibility @xcite . we have employed persistent homology to predict the curvature energies of fullerene isomers @xcite , and analyze the stability of protein folding @xcite . more recently , we have introduced resolution based persistent topology @xcite . most recently , we have developed new multidimensional persistence , a topic that has attracted much attention in the past few years @xcite , to better bridge geometry and traditional topology and achieve better characterization of biomolecular data @xcite . we have also introduced the use of topological fingerprint for resolving ill - posed inverse problems in cryo - em structure determination @xcite . the objective of the present work is to explore the utility of mtfs for protein classification and analysis . we construct feature vectors based on mtfs to describe unique topological properties of protein in different scales , states and/or conformations . these topological feature vectors are further used in conjugation with the svm algorithm for the classification of proteins . we validate the proposed mtf - svm strategy by distinguishing different protein conformations , proteins with different local secondary structures , and proteins from different superfamilies or families . the performance of proposed topological method is demonstrated by a number of realistic applications , including protein binding analysis , ion channel study , etc . the rest of the paper is organized as following . section [ sec : methods ] is devoted to the mathematical foundations for persistent homology and machine learning methods . we present a brief description of simplex and simplicial complex followed by basic concept of homology , filtration , and persistence in section [ persistenthomology ] . three different methods to get simplicial complex , vietoris - rips complex , alpha complex , and ech complex are discussed . we use a sequence of graphs of channel proteins to illustrate the growth of a vietoris - rips complex and corresponding barcode representation of topological persistence . in section [ svm+roc ] , fundamental concept of support vector machine is discussed . an introduction of transformation of the original optimization problem is given . a measurement for the performance of classification model known as receiver operating characteristic is described . section [ feature+preprocessing ] is devoted to the description of features used in the classification and pre - processing of topological feature vectors . in section [ sec : numerical ] , four test cases are shown . case 1 and case 2 examine the performance of the topological fingerprint based classification methods in distinguishing different conformations of same proteins . in case 1 , we use the structure of the m2 channel of influenza a virus with and without an inhibitor . in case 2 , we employ the structure of hemoglobin in taut form and relaxed form . case 3 validates the proposed topological methods in capturing the difference between local secondary structures . in this study , proteins are divided into three groups , all alpha protein , all beta protein , and alpha+beta protein . in case 4 , the ability of the present method for distinguishing different protein families is examined . this paper ends with some concluding remarks . this section presents a brief review of persistent homology theory and illustrates its use in proteins . a brief description of machine learning methods is also given . the topological feature selection and construction from biomolecular data are described in details . * simplex * a @xmath0-simplex denoted by @xmath1 is a convex hull of @xmath2 vertices which is represented by a set of points @xmath3 where @xmath4 is a set of affinely independent points . geometrically , a @xmath5-@xmath6 is a line segment , a @xmath7-simplex is a triangle , a @xmath8-simplex is a tetrahedron , and a @xmath9-simplex is a @xmath10-cell ( a four dimensional object bounded by five tetrahedrons ) . a @xmath11face of the @xmath0-simplex is defined as a convex hull formed from a subset consisting @xmath12 vertices . * simplicial complex * a simplicial complex @xmath13 is a finite collection of simplices satisfying two conditions . first , faces of a simplex in @xmath13 are also in @xmath13 ; second , intersection of any two simplices in @xmath13 is a face of both the simplices . the highest dimension of simplices in @xmath13 determines dimension of @xmath13 . * homology * for a simplicial complex @xmath13 , a @xmath0-chain is a formal sum of the form @xmath14 $ ] , where @xmath15 $ ] is oriented @xmath0-simplex from @xmath13 . for simplicity , we choose @xmath16 . all these @xmath0-chains on @xmath13 form an abelian group , called chain group and denoted as @xmath17 . a boundary operator @xmath18 over a @xmath0-simplex @xmath1 is defined as , @xmath19,\ ] ] where @xmath20 $ ] denotes the face obtained by deleting the @xmath21th vertex in the simplex . the boundary operator induces a boundary homomorphism @xmath22 . an very important property of the boundary operator is that the composition operator @xmath23 is a zero map , @xmath24+\sum_{j > i}(-1)^i(-1)^{j-1}[u_0, ... ,\widehat{u_j}, ... \widehat{u_i}, ... u_k ] \\ & = 0 \end{aligned}\ ] ] a sequence of chain groups connected by boundary operation form a chain complex , @xmath25{}}c_n(\mathcal{k})\xrightarrow{\makebox[.27in]{$\partial_n$}}c_{n-1}(\mathcal{k})\xrightarrow{\makebox[.27in]{$\partial_{n-1}$}}\cdots\xrightarrow{\makebox[.27in]{$\partial_1$}}c_0(\mathcal{k } ) \xrightarrow{\makebox[.27in]{$\partial_0$}}0.\ ] ] the equation @xmath26 is equivalent to the inclusion @xmath27 , when @xmath28 and @xmath29 denotes image and kernel . elements of @xmath30 are called @xmath0th cycle group , and denoted as @xmath31=ker@xmath18 . elements of @xmath32 are called @xmath0th boundary group , and denoted as @xmath33=im@xmath34 . a @xmath0th homology group is defined as the quotient group of @xmath31 and @xmath33 . @xmath35 the @xmath0th betti number of simplicial complex @xmath13 is the rank of @xmath36 , @xmath37 betti number @xmath38 is finite number , since @xmath39 . betti numbers computed from a homology group are used to describe the corresponding space . generally speaking , the betti numbers @xmath40 , @xmath41 and @xmath42 are numbers of connected components , tunnels , and cavities , respectively . * filtration and persistence * a filtration of a simplicial complex @xmath13 is a nested sequence of subcomplexes of @xmath13 . @xmath43 with a filtration of simplicial complex @xmath13 , topological attributes can be generated for each member in the sequence by deriving the homology group of each simplicial complex . the topological features that are long lasting through the filtration sequence are more likely to capture significant property of the object . intuitively , non - boundary cycles that are not mapped into boundaries too fast along the filtration are considered to be possibly involved in major features or persistence . equipped with a proper derivation of filtration and a wise choice of threshold to define persistence , it is practicable to filter out topological noises and acquire attributes of interest . the @xmath44-persistent @xmath0th homology group of @xmath45 is defined as @xmath46 where @xmath47 and @xmath48 . the consequent @xmath44-persistent @xmath0th betti number is @xmath49 . a well chosen @xmath44 promises reasonable elimination of topological noise . * vietoris - rips complex * based on a metric space @xmath50 and a given cutoff distance @xmath51 , an abstract simplicial complex can be built . if two points in @xmath50 have a distance shorter than the given distance @xmath51 , an edge is formed between these two points . consequently , simplices of different dimensions are formed and a simplicial complex is built . for a point cloud data , natural metric space based on euclidean distance or other metric spaces based on alternative definition of distance can be used to build a vietoris - rips complex . for example , any correlation matrix can be used directly to form a vietoris - rips complex . figure [ fig : ex1 ] illustrates growth of vietoris - rips complex along with increment of @xmath51 over the point set of @xmath52 atoms from m2 chimera channel . there are many ways of constructing complex other than vietoris - rips complex , including alpha complex , cech complex , cw complex , etc . in the present work , we used vietoris - rips complex in part because of its intuitive nature and in part because of the moderate size of the systems we studied . the computational topology package javaplex@xcite was used for computation of persistent homology . the results were represented in the form of barcodes @xcite . figure [ fig : ex2 ] illustrates barcodes computed from a point cloud data extracted from @xmath53 atoms of protein i d 2ljc . svm is a machine learning method that can be applied to classification and regression problems . it computes a hyperplane which maximizes margin between positive and negative training sets . in this work , classification svm type 1 , also known as c - support vector classification ( c - svc ) @xcite is used . for the problem of classification , with pre - determined classes , a classifier is trained on a data set with the description of samples and their classes and it predicts the class of a new observation . the input for svm is a set of samples . each sample has a feature vector that describes the properties of the sample and a label that implies to which class the sample belongs . given the input which is the training set , svm will generate a hyperplane in the feature space or higher dimensional spaces depending on which kernel it uses that separates the classes . for two - class svm , it looks for a hyperplane @xmath54 that separates the classes . the determination of the coefficients @xmath55 and @xmath56 breaks down to a constrained optimization problem as @xmath57 subject to @xmath58 where @xmath59 denotes the feature vector of the @xmath21th sample , @xmath60 is the label of the @xmath21th sample which takes value of either @xmath5 or @xmath61 , and @xmath62 is a penalty coefficient for misclassified points . to handle linearly inseparable data , one maps the data into a higher dimensional space as @xmath63 with @xmath64 . since in the optimization problem and in scoring function of the classifier , the operator used is dot product , @xmath65 does not need to be explicitly found . a decaying kernel @xmath66 function is used to represent @xmath67 . commonly used kernel functions include linear function : @xmath68 , polynomial : @xmath69 , radial basis functions ( rbfs ) such as gaussian @xmath70 . in fact , the admissible kernels of fleibility - rigidity index ( fri ) @xcite work too . in this work , the gaussian kernel is used and a 5-fold cross validation was applied to search for optimized training parameters for problems with large amount of samples . to solve the optimization problem , the original problem is transformed into the corresponding lagrange dual problem . for a contained optimization problem @xmath71 the lagrange function of this problem is defined as @xmath72 where @xmath73 and @xmath74 are lagrange multipliers . the lagrange dual problem is defined as @xmath75 where @xmath76 . the lagrange function of the original optimization problem ( [ eq : opt-1 ] ) is formulated as @xmath77 tthe corresponding dual problem with karush - kuhn - tucker conditions is defined as @xmath78 the dual problem can be solved with sequential minimal optimization ( smo ) method @xcite . roc is a plot that visualizes the performance of a binary classifier @xcite . a binary classifier uses a threshold value to decide the prediction label of an entry . in testing process , we define true positive rate ( tpr ) and false positive rate ( fpr ) for the testing set . @xmath79 an roc space is a two dimensional space defined by points with @xmath80 coordinate representing fpr and @xmath81 coordinate representing tpr . in the prediction process of a binary classifier , a score is assigned to a sample by the classifier . a test sample may be labeled as positive or negative with different threshold value used by the classifier . corresponding to a certain threshold value , there is a pair of fpr and tpr values which is a point in the roc space . all such points will fall in the box @xmath82\times[0,1]$ ] . points above the diagonal line @xmath83 are considered as good predictors and those below the line are considered as poor predictors . if a point is below the diagonal line , the predictor can be inverted to be a good predictor . for points that are close to the diagonal line , they are considered to act similarly to random guess which implies a relatively useless predictor . roc curve is obtained by plotting fpr and tpr as continuous functions of threshold value . the area between roc curve and @xmath80 axis represents probability that the classifier assigns higher score to a randomly chosen positive sample than to a randomly chosen negative sample if positive is set to have higher score than negative . the area under the curve ( auc ) of roc is a measure of classifier quality . intuitively , a higher auc implies a better classifier . in this work , algebraic topology is employed to discriminate proteins . specifically , we compute mtfs through the filtration process of protein structural data . mtfs bear the persistence of topological invariants during the filtration and are ideally suited for protein classification . to implement our topological approach in the svm algorithm , we construct protein feature vectors by using mtfs . we select distinguishing protein features from mtfs . these features can be both long lasting and short lasting betti 0 , betti 1 , and betti 2 intervals . table [ tab : features ] lists topological features used for classification . detailed explanation of these features is discussed . the length and location value of bars are in the unit of angstrom ( ) for protein data .

protein function and dynamics are closely related to its sequence and structure . however prediction of protein function and dynamics from its sequence and structure is still a fundamental challenge in molecular biology . protein classification , which is typically done through measuring the similarity between proteins based on protein sequence or physical information , serves as a crucial step toward the understanding of protein function and dynamics . persistent homology is a new branch of algebraic topology that has found its success in the topological data analysis in a variety of disciplines , including molecular biology . the present work explores the potential of using persistent homology as an independent tool for protein classification . to this end , we propose a molecular topological fingerprint based support vector machine ( mtf - svm ) classifier . specifically , we construct machine learning feature vectors solely from protein topological fingerprints , which are topological invariants generated during the filtration process . to validate the present mtf - svm approach , we consider four types of problems . first , we study protein - drug binding by using the m2 channel protein of influenza a virus . we achieve 96% accuracy in discriminating drug bound and unbound m2 channels . additionally , we examine the use of mtf - svm for the classification of hemoglobin molecules in their relaxed and taut forms and obtain about 80% accuracy . the identification of all alpha , all beta , and alpha - beta protein domains is carried out in our next study using 900 proteins . we have found a 85% success in this identification . finally , we apply the present technique to 55 classification tasks of protein superfamilies over 1357 samples . an average accuracy of 82% is attained . the present study establishes computational topology as an independent and effective alternative for protein classification . key words : persistent homology , machine learning , protein classification , topological fingerprint . * running title : topological protein classification *

You are an expert at summarizing long articles. Proceed to summarize the following text: star - forming disc galaxies like the milky way must accrete @xmath0 1 @xmath2 of fresh gas each year ( see * ? ? ? * and references therein ) and have built their discs gradually over the last 10 gyr ( e.g. * ? ? ? a central question is the origin of the accreted gas and how this gas reaches the thin disc whitin which the process of star formation takes place . the virial - temperature corona , in which disc galaxies are embedded , is the only reservoir of baryons capable of sustaining an accretion rate of @xmath3 for a hubble time . we present the results of a set of grid - based hydrodynamical simulations supporting the idea that the gas needed by the disc to form stars is drawn from this corona . coronae of disc galaxies are similar in many respects to the hot atmospheres of giant elliptical galaxies and galaxy clusters , but with lower gas temperature and density ( e.g. * ? ? ? * ; * ? ? ? * ) . as the hot gas of these more massive systems , the coronal gas is unlikely to fragment into clouds via thermal instability @xcite , but it is expected to cool monolithically and feed the central black hole rather than produce an extended cold disc in which stars can form . however , if the gas needed to feed star formation has to be drawn from the corona , a mechanism that makes the hot gas accrete onto the disc must be at work . there is abundant evidence that star formation in galaxies like the milky way powers a galactic fountain : ejection of gas from the mid - plane by supernova explosions @xcite . through the fountain a significant fraction ( from 10 to 25 % ) of the whole hicontent of the galaxy is carried into its halo ( see * ? ? ? * and references therein ) . in this work , supported by several lines of argument , we hypothesise that the transfer of gas from the corona to the star - forming disc is effected by the hiclouds ejected by the galactic fountain @xcite . our hydrodynamical simulations suggest that the gas accretion proceeds through the following steps : ( i ) stripping of gas from fountain clouds by the corona as a result of the kelvin - helmholtz instability , ( ii ) mixing of the ( high metallicity ) stripped gas with a comparable amount of coronal gas in the turbulent wake of the clouds ; as a consequence of the mixing , the gas cooling time of the coronal gas is reduced to a value lower than the cloud s flight time , ( iii ) formation of knots of cold gas that accrete onto the disc in a dynamical time . the stripped gas leads to condensation of coronal gas only if the mass - loss rate exceeds a critical value @xmath4 , determined by the physical properties of the cloud and the corona . in addition , dimensional analysis suggests that the actual mass - loss rate must lie close to @xmath4 . in view of the proposed scenario and of these considerations the aims of the simulations are two - fold : ( i ) to provide an estimate of the actual mass - loss rate @xmath5 for comparison with @xmath4 , ( ii ) to determine the critical ambient pressure ( dependent on metallicity ) above which the mass of cool gas increases with time through condensation in the wake . in figure [ nocool ] the evolution of the mass of gas at @xmath6 k ( accreted gas ) , for a simulation with physical conditions representative of galactic coron , is shown ( see caption for details ) . in particular , in the left panel of the figure , the mass loss of a cloud is displayed and the agreement between the simulation and the analytical prediction is remarkable . from the right panel of figure [ nocool ] it has been possible to derive an estimate of the global accretion rate ( @xmath7 ) which is of the same order as that required to feed star formation . by comparing the figures in the two panels it is apparent that the cooled mass of coronal gas is comparable to the mass lost by the cloud . in other words , when the cooling is switched on the evaporation of some cloud mass produces an accretion of roughly the same amount of coronal mass . we have used hydrodynamical simulations to check whether the interaction between the galactic - fountain clouds , powered by a star - forming disc , and the virial - temperature corona , in which disc galaxies are embedded , could lead the coronal gas to cool promptly in the clouds wake and accrete onto the disc to feed star formation . the results of our analysis , described in detail in @xcite , can be summarized as follows : * the interaction between the galactic fountain and the hot corona causes the fountain clouds to be stripped of some of their gas . the simulations provide a reliable estimate of the mass - loss rate and confirm that its value lies close to @xmath4 , in agreement with analytical expectations ; * the condensation of coronal gas in the cloud s wake prevails over evaporation if the metallicity and/or pressure of the corona are high enough . in our simulations this happens for likely parameters of the coron in disc galaxies , condensation is present also for densities as low as @xmath8 , provided that [ fe / h ] @xmath9 ; * the derived global accretion rate is of the same order as that required to feed star formation . therefore the condensation of the hot corona seems a viable mechanism to sustain the star formation rate in star - forming galaxies . we used cpu time assigned under the inaf - cineca agreement 2008 - 2010 . f. marinacci gratefully acknowledges the support from the marco polo program by university of bologna and the hospitality of the rudolf peierls centre for theoretical physics .

star - forming disc galaxies such as the milky way need to accrete @xmath0 1 @xmath1 of gas each year to sustain their star formation . this gas accretion is likely to come from the cooling of the hot corona , however it is still not clear how this process can take place . we present simulations supporting the idea that this cooling and the subsequent accretion are caused by the passage of cold galactic - fountain clouds through the hot corona . the kelvin - helmholtz instability strips gas from these clouds and the stripped gas causes coronal gas to condense in the cloud s wake . for likely parameters of the galactic corona and of typical fountain clouds we obtain a global accretion rate of the order of that required to feed the star formation . address = dipartimento di astronomia , universit di bologna , via ranzani 1 , 40127 bologna , italy . address = rudolf peierls centre for theoretical physics , oxford university , keble road , oxford ox1 3np , uk . address = dipartimento di astronomia , universit di bologna , via ranzani 1 , 40127 bologna , italy . address = dipartimento di astronomia , universit di bologna , via ranzani 1 , 40127 bologna , italy . address = dipartimento di astronomia , universit di bologna , via ranzani 1 , 40127 bologna , italy . address = inaf - osservatorio astronomico di bologna , via ranzani 1 , 40127 , bologna , italy .

You are an expert at summarizing long articles. Proceed to summarize the following text: proton capture reactions at a very low temperature play an important role in nucleosynthesis process . most importantly , in explosive nucleosynthesis ( _ e.g. _ an x - ray burst ) , the rapid proton capture ( @xmath0 ) process is responsible for the production of proton - rich isotopes upto mass 100 region . in nature , the proton capture reactions , important for nucleosynthesis , usually involve certain nuclei as targets which are not available on earth or can not be produced in terrestrial laboratories with our present day technology . therefore , theory remains the sole guide to extract the physics . in our present work , we have studied the endpoint of the @xmath0 process in a microscopic approach using a new phenomenological mass formula @xcite . in a similar work , schatz _ et al . _ @xcite calculated the endpoint of @xmath0 process using mass values from a finite range droplet model ( frdm)@xcite calculation and proton capture rates from hauser - feshbach code non - smoker@xcite . we will show that the results of our calculation are different from their observations . in the present work , we will concentrate only on x - ray burst scenarios , which have typical timescale of 100 seconds and a peak proton flux density of the order of @xmath1 . this type of burst provides a highly proton - rich environment around the peak temperatures 1 - 2 gk . we try to look at different models of the x - ray burster and find out the endpoint of the @xmath0 process nucleosynthesis . when an x - ray burst takes place , a proton - rich high temperature environment , which triggers the @xmath0 process , is created . the process passes through nuclei near the proton drip line , not available on earth . in regions far from the stability valley , rates derived from phenomenological calculations may not represent the reality very well , leading to considerable uncertainty in the process . very often , the reaction rates are varied by a large factor to study their effects . on the other hand , in a microscopic calculation , uncertainty in reaction rates can be reduced and therefore , this approach is expected to give a more accurate result for unknown mass regions . in a previous work@xcite , we have shown that the rates may vary at most by a factor less than two when the cross - section values range over four orders of magnitude . a microscopic calculation has been performed to evaluate proton capture rates for the nuclei involve in the @xmath0 process in the present work . we use the spherical optical model to calculate the rates of the relevant reactions . as most of the nuclei involved in the process lie around the drip line , experimental density information are not available . hence , theoretical density profiles have been calculated using relativistic mean field ( rmf ) theory . in the present work , we use the fsu gold lagrangian density@xcite and solve the rmf equations in the spherical approximation for the nuclei involved in the @xmath0 process . this lagrangian density , containing additional nonlinear terms for the vector isoscalar meson self interaction and an interaction between the isoscalar vector and the isovector vector mesons , has been found to be very useful in describing nuclear properties throughout the mass table [ see _ e.g. _ bhattacharya @xcite and references therein ] . the microscopic optical model potential for capture reactions are obtained using effective interactions derived from the nuclear matter calculation in local density approximation , _ i.e. _ by substituting the nuclear matter density with the density distribution of the finite nucleus . in the present work , we have constructed the potential by folding the density dependent m3y ( ddm3y)@xcite interaction with densities from rmf approach . this interaction was extracted from a finite - range energy - independent @xmath2-matrix element of the reid potential by adding a zero - range energy - dependent pseudo - potential and a density - dependent factor . the interaction at the point @xmath3 is , thus , given by @xmath4 where @xmath5 is the incident energy and @xmath6 , the nuclear density . the @xmath7 interaction is given by @xmath8 for @xmath9 in @xmath10 , and @xmath11 is the zero range pseudo potential , @xmath12 the density dependent factor @xmath13 has been chosen of the form @xmath14 from the work by chaudhuri@xcite where the constants were obtained from a nuclear matter calculation as @xmath15 = 2.07 and @xmath16 = 1.624 @xmath17 . we have used this form in our calculation without changing any of the above parameters . we have also fixed the various parameters and prescriptions in the hauser - feshbach calculation for the relevant mass region by comparing our results to the experimental low energy proton capture cross sections for these nuclei . our method of calculation for mean field and proton capture rates has been described in our earlier works@xcite in detail . the computer code talys@xcite has been used for rate calculation . binding energy of nuclear ground state is one of the most important inputs in the study of astrophysical reactions . experimental measurements are very difficult to perform in nuclei far from the stability line . therefore , one has to take recourse to theoretical predictions . though we have used a mean field calculation to extract the nuclear density profiles , no rmf calculation has been able to achieve a prediction of mass values with accuracy sufficient for studying the proton drip line . in fact , even skyrme hartree - fock calculations can predict the mass values with an root mean square ( rms ) error slightly less than 0.6 mev only . thus , in the present work , we have obtained the mass values from a newly developed mass formula@xcite . it uses a purely phenomenological form with empirically fitted parameters and predicts the known mass values of 2140 nuclei with an rms error of 0.376 mev . in view of the success of the formula to predict the proton dripline and @xmath0 process upto mass 80 region@xcite and to predict the peaks in @xmath9 process@xcite quite well , it will be interesting to see the effect of this mass formula on @xmath0 process beyond mass 80 region and to the endpoint of the @xmath0 process . in an x - ray burst environment , a nucleus ( @xmath18 ) may capture a proton to form the nucleus ( @xmath19 ) . however , this process has to compete with its inverse , _ i.e. _ photodisintegration by emitting a proton at high temperature [ @xmath20 reaction ] . a negative or a small positive value of the proton separation energy implies that the inverse reaction dominates and the @xmath0 process stalls at that point , the so - called waiting point . therefore , only a two proton capture process can bridge the waiting point nuclei . the bridging mechanism has been discussed in standard text books ( for example the book by illiadis@xcite ) . for the @xmath21 reaction , the rate @xmath22 of the inverse process @xmath23 is related to the the proton capture rate by the reciprocity theorem and is of the form @xcite @xmath24 in the unit of @xmath25 for @xmath21 process . the forward reaction rate , denoted by @xmath26 , is expressed in @xmath27 . the temperature @xmath28 is in gk ( @xmath29k ) and @xmath30 is the ground state q - value of the @xmath31 reaction expressed in mev . the normalized partition functions , @xmath32 and @xmath33 , have been obtained from rauscher .@xcite for protons , we use the standard values , @xmath34 and @xmath35 . as evident from the above expression , the q - value , appearing in the exponent , plays a vital role in the whole process . apart from the above processes , a nucleus can decay by emitting beta particles while , for higher mass isotopes , @xmath36-decay is another probable channel . in this work , the measured half life values for @xmath16-decay have been taken from the compilation by audi _ et al._@xcite except in the case of @xmath37as , which is taken from the experimental measurements by lpez _ et al._@xcite in absence of experimental data , half life values have been taken from the calculation by mller _ et al._@xcite both for @xmath16- and @xmath36- decay . taking into account all the above processes , we have constructed a network extended upto a=110 region to study the nucleosynthesis path and relative abundances of the elements at any instant . the @xmath0 process paths are shown in fig . 1 and fig . 2 for temperatures 1.2 and 1.5 gk , respectively , using a constant proton flux density of @xmath38 , a proton fraction of 0.7 and 100 seconds burst duration . here , black lines indicate the path along which the major portion of the total flux flow whereas gray lines indicate the minor paths . the filled boxes in figures indicate the waiting points . as evident from these figures , @xmath0 process paths depend on the temperature of the environment to some extent . for example , at t= 1.2 gk , major portions of the @xmath0 process flux at the waiting point nucleus @xmath39ge convert to @xmath40se by two - proton capture process . less than 1% of the total flux flows through the @xmath16-decay channel and follow the paths showed by gray lines in fig . 1 . in contrast , at temperature t= 1.5 gk ( in fig . 2 ) , the probability of two - proton capture of @xmath39ge gets reduced and the @xmath0 process path bifurcates from the waiting point nucleus almost in an equal proportion . this suggests that , at 1.2 gk , proton capture by @xmath39ge dominates over its inverse process , i.e. photodisintegration . as the temperature increases , contributions of photodisintegration process increases and therefore a large fraction of the total abundance chooses another path through more stable nuclei such as @xmath37ge , @xmath40as , etc . similarly , near other waiting points ( @xmath41 @xmath42kr , @xmath43sr etc . ) the abundance flow pattern changes with changing temperature . it is also evident from above the figures that , beyond mass 80 , temperature change can hardly affect the scenario . as shown in fig . 1 and fig . 2 , above a=100 region , the @xmath0 process continues through proton capture by in isotopes and @xmath16-decay of sn isotopes . here , @xmath44 in captures a proton to form @xmath45sn which completely decays to @xmath45 in as @xmath46sb is proton unbound . in turn , @xmath45 in undergoes proton capture and further exhibits @xmath16-decay . the process of proton capture followed by consecutive @xmath16-decay continues and relative abundances of nuclei decrease as one proceeds towards higher mass region . ultimately , less than 0.001% of the total flux can reach @xmath47 in , according to our present calculation . according to the calculations of schatz _ et al._@xcite , a significant portion of the @xmath48sn captures a proton to form @xmath47sb , as @xmath47sb has a sufficient positive proton separation energy ( 0.59 mev from frdm@xcite calculation ) . another proton capture leads to @xmath49te , which instantly undergoes @xmath36-decay to @xmath50sn . thus , the @xmath0 process ends in the snsbte cycle . in contrast , our calculation does not go through the snsbte cycle . it is clear that the snsbte cycle may not occur under two scenarios . firstly , if @xmath47sb isotope be very loosely bound ( proton separation energy 0.119 mev according to the mass formula@xcite ) , any @xmath47sb that is formed by a proton capture instantly reverts back to @xmath48sn . second scenario occurs when the proton capture rates are too small for @xmath48sn to initiate a proton capture process . in a previous work@xcite , we have shown that a small fluctuation in the mass values of waiting point nuclei in mass 60 - 80 region may affect the effective half life and thus , proper knowledge of ground state binding energy is necessary to understand the bridging phenomena of a waiting point nucleus below mass 80 region . however , as we move towards the higher mass region , we find that small variations in binding energy do not affect the @xmath0 process path significantly . taking the rms error into account for the proton separation energy of @xmath47sb @xmath51mev , we have checked whether the proton capture on @xmath48sn can dominate over its inverse process . we find that the fraction of the initial flux entering into the snsbte - cycle is negligibly small . we have repeated our calculation with the proton separation energy of @xmath47sb isotope from a recent experiment by elomma _ et al._@xcite , _ viz . _ 0.428(0.008)mev . we find that our results remain almost invariant . repeating the entire calculation with ground state binding energies from frdm@xcite calculation or the duflo - zuker@xcite mass formula do not alter this conclusion . hence , we see that for a reasonable variation of mass values the @xmath0 process fails to enter into the snsbte - cycle for an x - ray burst of 100 seconds duration . if we consider an x - ray burst having duration greater than 150 seconds , a very small fraction of total flux , less than 0.001% of the original , enters the cycle . cm schatz _ et al._@xcite have used capture rates different from ours . in fig . 3 , we have plotted proton capture rates with temperature for the isotopes @xmath48sn and @xmath47sb respectively . the different values have been indicated as follows : pres - present work and non - smoker - non - smoker@xcite results used by schatz _ et al._@xcite in case of @xmath48sn nuclei , it is evident from fig . 3 that proton capture rates from our calculation are approximately @xmath52 times smaller than the rates from non - smoker calculation . it seems , the difference is mainly due to the fact that they have used the form of the interaction from jeukenne _ et al._@xcite which is different from our case . for better understanding , we have repeated the entire calculation using reaction rates from hauser - feshbach code non - smoker@xcite . in this case , considerable fraction of the initial flux ( @xmath53 0.1% ) enters into the snsbte cycle . thus , we may conclude that the proton capture rates , and not the mass values , determine the endpoint of @xmath0 process . in our opinion , our approach in fixing the parameters in the reaction calculation by fitting known reaction rates , and extrapolating the calculation to unknown reactions , may be relied upon to predict the endpoint correctly . there are many x - ray burster models in literature with various density - temperature profiles . our present goal is to study how different temperature - density profiles can affect the relative abundances of the nuclei produced by @xmath0 process and the endpoint . in fig . 4 , we have plotted the relative abundances of elements versus mass number at 100th second of the burst in various models , as described below . -0.4 cm -0.35 cm in the first case ( model - i ) , we assume a constant density - temperature framework with temperature @xmath54 gk , density @xmath55 and proton fraction value 0.7 . model - ii@xcite describes a situation where ignition takes place at a constant density of @xmath56 and the burst reaches a peak temperature @xmath57 = 1.9 gk after 4 seconds while the cooling phase lasts for approximately 200 seconds . in another example , taken from the book by illiadis@xcite ( model - iii ) , nuclear burning starts with temperature and density values of @xmath58 gk and @xmath55 , respectively . after 4 seconds , a maximum temperature of @xmath57 = 1.36 gk and a minimum density of @xmath59 are achieved . after 100 seconds , the temperature drops to @xmath60 gk and the density increases to @xmath61 . for all the above models , it is assumed that the x - ray burst environment is sufficiently proton - rich to maintain the number 0.7 as the constant proton fraction . as a significant amount of proton flux is consumed during the thermonuclear reaction , another situation may arise , where the proton fraction decreases gradually with time . such a situation is given in model - iv where the proton fraction decreases to 0.16 after 100 seconds with temperature - density profile same as that of the model - iii . it is evident from fig . 4 that abundance peaks in all cases are obtained at mass values of 72 , 76 and 80 , as a result of the existence of waiting point nuclei @xmath42kr , @xmath43sr and @xmath62zr , respectively . other peaks ( for example , peaks at mass 85 , 93 and 95 for isotopes @xmath63mo , @xmath64pd , @xmath65cd , respectively ) suggest that the @xmath0 process flux gets accumulated at these points due to very small positive or negative proton separation energies of those isotopes . we have considered the mass number in each case for which the flux amount drops below 1% of the initial flux . it is evident from fig . 4 that for model - i , a @xmath66 93 is the region above which the @xmath0 process flux fall below the range of our interest , whereas for model - ii , the region is around a @xmath66 95 . other abundance peaks are observed at masses @xmath6797 and 101 for the isotopes @xmath68cd and @xmath45sn respectively , though , the fractions of the total flux accumulated at those isotopes are less than 0.1% of the initial value . in case of models iii and iv , the locations to be studied are around a @xmath66 93 and 91 respectively . from above observations , it can be concluded that , for various density - temperature profiles , the @xmath0 process flux falls below a significant amount near mass 90 - 95 . the observations from fig . 4 suggest that the end points of the @xmath0 process has a rather weak dependence on different x - ray burster models . the location above which the @xmath0 process flux falls below an insignificant amount is calculated using the microscopic optical model utilizing the densities from the rmf approach and with a new phenomenological mass formula . present result is compared with result obtained from another existing work@xcite and the reason for these indifferences between the results are discussed . our results do not significantly depend on the mass models . for different x - ray burster models , endpoints are calculated . this work has been carried out with financial assistance of the ugc sponsored drs programme of the department of physics of the university of calcutta . chirashree lahiri acknowledges the grant of a fellowship awarded by the ugc . 99 g. gangopadhyay , _ int . j. mod _ e * 20 * ( 2011 ) 179 . h. schatz _ ( 2001 ) 3471 . p. mller , j. r. nix , and k. l. kratz , ( 1997 ) 131 . t. rauscher and f .- k . thielmann , ( 2000 ) 1 . t. rauscher and f .- k . thielmann , ( 2001 ) 47 . c. lahiri and g. gangopadhyay , _ phys . _ c * 84 * ( 2011 ) 057601 . b. g. todd - rutel and j. piekarewicz , ( 2005 ) 122501 . m. bhattacharya and g. gangopadhyay , c * 77 * ( 2008 ) 047302 . m. bhattacharya , g. gangopadhyay and s. roy , c * 85 * ( 2012 ) 034312 . a.m. kobos , b.a . brown , r. lindsay , g.r . and satchler , a * 425 * ( 1984 ) 205 . chaudhuri , a * 449 * ( 1986 ) 243 . chaudhuri , a * 459 * ( 1986 ) 417 . c. lahiri and g. gangopadhyay , a * 47 * ( 2011 ) 87 . a. j. koning , s. hilaire , and m. duijvestijn , proceedings of the international conference on nuclear data for science and technology , april 2227 , 2007 , nice , france , edited by o. bersillon , f. gunsing , e. bauge , r. jacqmin , and s. leray ( edp sciences , paris , 2008 ) , pp . c. lahiri and g. gangopadhyay , _ int _ e * 20 * ( 2011 ) 2417 . c. lahiri and g. gangopadhyay , to appear in _ int . _ e ( 2012 ) . c. illiadis , _ nuclear physics of the stars _ ( wiley - vch verlag gmbh , weinheim , 2007 ) . t. rauscher and f - k thielemann , ( 2000 ) 1 . g. audi , o. bersillon , j. blachot and a.h . wapstra , _ nucl . phys . _ a * 729 * ( 2003 ) 3 . lpez jimnez _ _ , _ phys . rev . _ c * 66 * ( 2002 ) 025803 . v.- v. elomaa _ ( 2009 ) 252501 . j. duflo and a.p . zuker , _ phys . _ c * 52 * ( 1995 ) r23 . j. jeukenne , a. lejeune , and c. mahaux , c * 16 * ( 1977 ) 80 .

densities from relativistic mean field calculations are applied to construct the optical potential and , hence calculate the endpoint of the rapid proton capture ( @xmath0 ) process . mass values are taken from a new phenomenological mass formula . endpoints are calculated for different temperature - density profiles of various x - ray bursters . we find that the @xmath0 process can produce significant quantities of nuclei upto around mass 95 . our results differ from existing works to some extent .

You are an expert at summarizing long articles. Proceed to summarize the following text: molecules such as co or hcn have been commonly used as tracers of molecular gas in high - redshift galaxies . however , recent observations with the _ herschel space observatory _ @xcite have shown strong spectroscopic signatures from other light hydrides , such as water , h@xmath3o@xmath4 , or hf , in nearby active galaxies ( e.g. , @xcite ) . these lines are blocked by the earth s atmosphere , but can be observed , redshifted , in distant galaxies using the current millimeter and submillimeter facilities . for example , @xcite have recently reported a detection of water in j090302 - 014127b ( sdp.17b ) at @xmath5 . one of the exciting recent results from hifi @xcite is the detection of widespread absorption in the fundamental @xmath2 rotational transition of hydrogen fluoride toward galactic sources @xcite . fluorine is the only atom that reacts exothermically with @xcite . the product of this reaction , hf , is thus easily formed in regions where is present and its very strong chemical bond makes this molecule relatively insensitive to uv photodissociation . as a result , hf is the main reservoir of fluorine in the interstellar medium ( ism ) , with a fractional abundance of @xmath6 relative to typically measured in diffuse molecular clouds within the galaxy @xcite . interstellar hf was first detected by @xcite with the infrared space observatory ( iso ) . the @xmath7 rotational transition was observed in absorption toward sagittarius b2 , at a low spectral resolution using the long - wavelength spectrometer ( lws ) . the hifi instrument allows for the first time observations of the fundamental rotational transition of hf at 1.232476 thz to be carried out , at high spectral resolution . given the very large einstein a coefficient ( @xmath8 ps . ; critical density @xmath9 ) , this transition is generally observed in absorption against dust continuum background . only extremely dense regions with strong ir radiation field could possibly generate enough collisional or radiative excitation to yield an hf feature with a positive frequency - integrated flux . the hifi observations corroborate the theoretical prediction that hf will be the dominant reservoir of interstellar fluorine under a wide range of interstellar conditions . the hf @xmath2 transition promises to be a excellent probe of the kinematics of , and depletion within , absorbing material along the line of sight toward bright continuum sources , and one that is uncomplicated by the collisionally - excited line emission that is usually present in the spectra of other gas tracers . as suggested by @xcite , redshifted hf @xmath2 absorption may thus prove to be an excellent tracer of the interstellar medium in the high - redshift universe , although only the gas reservoir in front of a bright continuum background can be studied by means of the hf absorption spectroscopy . water is another interstellar molecule of key importance in astrophysical environments , being strongly depleted on dust grains in cold gas , but abundant in warm regions influenced by energetic process associated with star formation ( see @xcite and references therein ) . the excited @xmath0 transition of p- , with a lower level energy of 137 k , has a frequency of 1.228788 thz and can be observed simultaneously with the @xmath2 transition of hf in high - redshift systems . consequently , we have searched for the hf @xmath2 and @xmath0 transitions , redshifted down to 251 ghz , in apm 082791 + 5255 using the iram plateau de bure interferometer . the broad absorption line ( bal ) quasar apm 082791 + 5255 at _ _ z__=3.9118 , with a true bolometric luminosity of @xmath10 l@xmath11 , is one of the most luminous objects in the universe @xcite . co lines up to @xmath12 have been detected using the iram 30-m telescope . iram pdbi high spatial resolution observations of the co @xmath13 and @xmath14 lines , and of the 1.4 mm dust continuum have been presented by @xcite . the line fluxes in the co ladder and the dust continuum fluxes are well fit by a two - component model that invokes a `` cold '' component at 65 k with a high density of @xmath15(h@xmath3 ) = @xmath16 @xmath17 , and a `` warm '' , @xmath18 k , component with a density of @xmath19 @xmath20 . wei et al . argue that the molecular lines and the dust continuum emission arise from a very compact ( @xmath21 pc ) , highly gravitationally magnified ( @xmath22 ) region surrounding the central agn . part of the difference relative to other high-_z _ qsos may therefore be due to the configuration of the gravitational lens , which gives us a high - magnification zoom right into the central 200-pc radius of apm 08279 + 5255 where ir pumping plays a significant role for the excitation of the molecular lines . high - angular resolution ( @xmath23 ) vla observations of the co @xmath2 emission in apm 08297 + 5255 @xcite reveal that the molecular emission originates in two compact peaks separated by 04 and is virtually co - spatial with the optical / near infrared continuum emission of the central active galactic nucleus ( agn ) . this morphological similarity again indicates that the molecular gas is located in a compact region , close to the agn . @xcite present a revised gravitational lens model of apm 08297 + 5255 , which indicates a magnification by only a factor of 4 , in contrast to much higher magnification factors of 100 suggested in earlier studies . their model suggests that the co emission originates from a 550 pc radius circumnuclear disk viewed at an inclination angle of 25 , or nearly face - on . the total molecular mass is then @xmath24 m. @xcite first pointed out the importance of infrared pumping for the excitation of hcn in apm 08279 + 5255 . subsequent observations of @xcite reveal surprisingly strong @xmath25 emission of hcn , hnc , and in the host galaxy , providing additional evidence that these transitions are not collisionally excited . @xcite argue that the high rotational lines of hcn can be explained by infrared pumping at moderate opacities in a 220 k warm gas and dust component . these findings are consistent with the overall picture in which the bulk of the gas and dust is situated in a compact , nuclear starburst , where both the agn and star formation contribute to the heating . prior to the observations reported here , water had not been detected in apm 08279 + 5255 . however , @xcite give an upper limit of 0.7 jykms@xmath26 ( @xmath27 ) for the ground state @xmath28 ortho- line . observations of apm 08279 + 5255 presented here were carried out on 2010 june 22 , september 2122 , and december 15 , using the plateau de bure interferometer . visibilities were obtained in the cd set of configurations of the six - element array in june and december and with a four - element subarray in september , totalling 4.9 hr of on - source observations . data reduction and calibration were carried out using the gildas software package in the standard antenna based mode . the passband calibration was measured on 3c454.3 , and amplitude and phase calibration were made on 0749 + 540 , 0836 + 716 and 0917 + 449 . the absolute flux calibration , performed using mwc349 as the primary calibrator ( 2.55 jy at 250 ghz ) , is accurate to within 10% . point source sensitivities of 4.5mjybeam@xmath26 were obtained in channels of 20mhz , consistent with the measured system temperatures ( 200300 k ) . the conversion factor from flux density to brightness temperature in the @xmath29 ( pa=22 ) synthesized beam is 9.1 k(jybeam@xmath26)@xmath26 . , hf @xmath2 , and hcn @xmath30 , assuming z=3.9118 . velocity scale is with respect to the hf @xmath2 frequency . hf and hcn lines are not detected . ( bottom ) distribution of the velocity - integrated para- @xmath0 line intensity in apm 08279 + 5255 ( red contours ) superposed on a grayscale image of the 1.23 thz ( rest frame ) dust continuum emission . contour levels are -2 , 2 , 3 , 4 , 5 , 6 , 7 times the rms of 0.8 jybeam@xmath26 . white symbols mark the locations of sources a and b @xcite . synthesized beam is show as a white ellipse in the lower - left corner . ] figure [ fig : image ] ( upper panel ) shows a spectrum of apm 08279 + 5255 near the rest frame frequency of the hf @xmath2 transition , integrated over the pdbi image , which also covers frequencies of the para- @xmath0 and hcn @xmath30 transitions , in addition to hf . the continuum is detected with a high snr . the integrated flux density , computed from emission free channels , is @xmath31 mjy , consistent with the previous measurements of the source sed @xcite . no hf absorption is seen , with a 3@xmath32 upper limit of 1.5 jy , assuming a fwhm line width of 500 , as implied by earlier co observations . the integrated line and continuum fluxes given above impose a 3@xmath32 upper limit of 0.092 for the velocity - averaged hf @xmath33 optical depth ( velocity - integrated optical depth @xmath34 ) . the corresponding column density of cold hf in front of the continuum source can then be computed using eq . ( 3 ) of @xcite to be @xmath35 . given the typical galactic hf/ abundance ratio of @xmath6 , this value would imply an average column density @xmath36 lying in front of the continuum source in apm 08279 + 5255 . this value is three orders of magnitude below the beam averaged column density inferred from the dust continuum flux observed toward the source . m and 680 pc , respectively , for the dust mass and magnified radius for apm 08279 + 5255 , where @xmath37 is the lens magnification . these values imply an average column density of @xmath38 for an assumed dust - to - gas mass ratio of 100 . approximately 50% of this gas should be in front of the continuum source . ] the para- @xmath0 line is clearly detected with the integrated line flux density of @xmath39 jy . a gaussian fit gives a line width of @xmath40 , consistent with that of co. figure [ fig : image ] ( lower panel ) shows spatial distribution of the para- @xmath0 emission ( red contours ) superposed on a grayscale image of the dust continuum . the line and continuum emission peak toward sources a and b of @xcite . the small offset between the and continuum emission is not significant at the spatial resolution of the present observations . implication for water excitation in apm 08279 + 5255 are discussed below . some excess emission above the continuum level is seen near the frequency of the hcn @xmath30 line , however , the result does not constitute a detection at the sensitivity limit of the present observations . in modeling the water line flux observed from apm 08279 + 5255 , we have computed the co and h@xmath3o line luminosities expected for an isothermal , constant density medium . we solved the equations of statistical equilibrium for the h@xmath3o and co level populations , making the large velocity gradient ( lvg ) approximation and treating the effects of radiative trapping with an escape probability method . we adopted the rate coefficients of @xcite and @xcite , respectively , for the excitation of co and h@xmath3o in collisions with h@xmath3 , and we assumed an ortho - to - para ratio ( opr ) of 3 for both h@xmath3 and h@xmath3o . following @xcite , we neglect any effects of dust extinction upon the emergent co and line fluxes ; although the dust optical depths at thz frequencies raise the possibility that such effects could be significant , their importance depends strongly on the geometry of the source and the spatial relationship between the warm dust and the molecular emission region . in galactic hot cores with column densities comparable to that in apm 08279 + 5255 , the para- @xmath0 line can be seen with net - emission flux ( orion kl , ngc6334i ) , or in absorption ( sagittarius b2 ) , depending on the specific source geometry . a mixture of such regions may contribute to the observed spectrum of apm 08279 + 5525 , leading to partial cancellation of the emergent line flux . using the relative strengths of the multiple co transitions observed by @xcite to constrain the gas temperature , density , and velocity gradient , we thereby obtained as best fit parameters the values @xmath41 k , @xmath42 , and @xmath43 , respectively . these parameters are very close to those obtained previously by @xcite in their single component model for the co emission detected from this source . adopting the same parameters for the water emitting region , we have computed the para - h@xmath3o @xmath44/co @xmath45 line flux ratio as a function of the assumed @xmath46 abundance ratio . in the case of h@xmath3o , the pumping of rotational transitions by far - infrared continuum radiation can strongly affect the predicted line fluxes . the importance of radiative pumping in this source has been discussed previously by @xcite for the case of hcn , although , in that case , pumping takes place through a low - lying _ vibrational _ band . pumping through pure _ rotational _ transitions is relatively much more important for an asymmetric top molecule like water , because such molecules possess a more complex energy level structure than the simple ladder shown by spinless linear or diatomic molecules ( such as hcn and co ) ; furthermore , the lowest vibrational band of water lies at a considerably shorter wavelength ( @xmath47 ) than that of hcn ( @xmath48 ) , where the continuum radiation is considerably weaker . the dominance of radiative pumping in rotational transitions of water vapor was also discussed by @xcite , in their recent analysis of the water line emission observed by _ herschel _ toward the starburst galaxy mrk 231 . under conditions where radiative pumping is dominant , the water line fluxes are almost independent of the gas temperature and density . following @xcite , we define @xmath49 as the sky covering factor of the infrared continuum source at the location of the molecular emission region ; the mean intensity is then given by modified blackbody of the form @xmath50 , where @xmath51 is the planck function and @xmath52 . in figure [ fig : model ] ( upper left panel ) , we present the predicted para - h@xmath3o @xmath44 / co @xmath53 line flux ratio , as a function of @xmath46 and for several different values of @xmath49 : 0 ( red ) , 0.1 ( magenta ) , 0.25 ( blue ) , 0.5 ( black ) , and 1.0 ( green ) . the results shown in figure [ fig : model ] clearly indicate the dominant role of radiative pumping ; a detailed analysis indicates the importance of the @xmath54 ( rest frame 2.97 thz ) transition in directly pumping the @xmath55 state of para - water , along with the @xmath56 ( rest frame 6.45 thz ) transition , which pumps the @xmath57 state ; the latter can decay subsequently to @xmath55 . dotted horizontal lines indicate the para - h@xmath3o @xmath44 / co @xmath53 ratio measured in apm08279 + 5255 and its uncertainty . the upper right panel of figure [ fig : model ] shows entirely analogous results for the @xmath28 transition of ortho - water , with the horizontal dotted line indicating the 3 @xmath32 upper limit obtained by @xcite . a comparison of the results shown in the upper panels of figure [ fig : model ] indicates that a limited range of parameters is permitted by the measured value of the @xmath58 line flux and the upper limit on @xmath59 . for the h@xmath3o opr of 3 assumed here , acceptable fits are obtained for @xmath60 and @xmath46 in the range @xmath61 , although the range of acceptable parameters would obviously broaden if opr values smaller than 3 were permitted . in the regime of interest , the expected @xmath0 line fluxes depend only weakly upon the h@xmath3o abundance , the most important pumping transitions being optically - thick ; thus , the exact range of acceptable values for @xmath46 depends strongly upon our estimate of the likely error in the measured line flux . nevertheless , the @xmath46 ratio inferred for an assumed opr of 3 is apparently smaller than that typically measured ( @xmath62 ) in hot core regions within our galaxy ( e.g. @xcite ) . observations of additional transitions will be needed to constrain the water opr and abundance better . the lower panels of figure [ fig : model ] present results for several other transitions that are potentially detectable from ground - based observatories , some of which are more strongly dependent upon the water abundance . the results shown in these panels were all obtained for @xmath63 and opr=3 , and the labels indicate the rest frequencies and in parentheses redshifted frequencies in ghz . our results for transitions of para- and ortho - water appear , respectively , in the left and right panels . we note that our lvg solution for the emission in apm 08279 + 5525 not only matches that of @xcite , using independent data , but the deduced velocity gradient @xmath64 for the emitting region is close to the expected virial value , as defined for example by eq . ( 5 ) of @xcite , @xmath65 ( i.e. the dense gas emitting in the line emission is near virial equillibrium ) . the absence of detectable hf @xmath2 absorption in apm 08279 + 5255 is unexpected , given the low column density of hf required to produce measurable absorption . an important caveat in this analysis is the assumption of a galactic hf/ ratio that , in turn , is related to the elemental abundance of fluorine . fluorine nucleosynthesis and thus the evolution of the fluorine abundance in cosmic time remains poorly understood , with production in agb stars ( e.g. @xcite ) , in wolf - rayet stars @xcite and neutrino - induced nucleosynthesis in type ii supernovae @xcite all proposed as possible mechanisms . while the face - on geometry of apm 08279 + 5255 is not favorable for absorption studies , the lack of hf absorption is still puzzling and may be indicative of a lower fluorine abundance in this source compared with the galactic ism . nevertheless , hf absorption may still prove to be a good tracer of in high - redshift sources and additional observations of objects with different geometries , over a wide range of redshifts , are urgently needed . our lvg models indicate that the para- @xmath44 transition in apm 8279 + 5255 is predominantly radiatively pumped . @xcite has reached similar conclusions regarding water excitation in j090302 - 014127b ( sdp.17b ) at @xmath5 . the para-@xmath44 line intensity in apm 8279 + 5525 is sensitive to the details of the excitation model . consequently , observations of this single transition do not provide a good estimate of the water abundance . however , our lvg models suggests that many additional water lines should be detectable with the current millimeter - wave facilities . the transitions that are expected to be the strongest ( see fig . 2 ) are : @xmath66 ( rest frame frequency 1162.2 ghz ) , @xmath67 ( 1207.6 ghz ) , @xmath68 ( 752.0 ghz ) , as well as two very high - energy transitions : @xmath69 ( 1794.8 ghz ) , and @xmath70 ( 1410.6 ghz ) , which are sensitive to the gas density . with multi - line observations , the excitation conditions and the water abundance will be much better constrained . this excitation scenario can further be tested with observations of the 2.97 thz pumping transition , which is expected to appear in absorption . based on observations carried out with the iram plateau de bure interferometer . iram is supported by insu / cnrs ( france ) , mpg ( germany ) and ign ( spain ) . this research has been supported by the national science foundation grant ast-0540882 to the caltech submillimeter observatory . we thank pierre cox for allocating director s discretionary time to allow these observations to be carried out and an anonymous referee for constructive and helpful comments .

we report a detection of the excited @xmath0 rotational transition of para- in apm 08279 + 5255 using the iram plateau de bure interferometer . at @xmath1 , this is the highest - redshift detection of interstellar water to date . from lvg modeling , we conclude that this transition is predominantly radiatively pumped and on its own does not provide a good estimate of the water abundance . however , additional water transitions are predicted to be detectable in this source , which would lead to an improved excitation model . we also present a sensitive upper limit for the hf @xmath2 absorption toward apm 08279 + 5255 . while the face - on geometry of this source is not favorable for absorption studies , the lack of hf absorption is still puzzling and may be indicative of a lower fluorine abundance at @xmath1 compared with the galactic ism .

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Proceed to summarize the following text: the problem of motion , one of the cardinal problems of einstein s gravitation theory , has received continuous attention over the years . the early , classic works of lorentz - droste , eddington - clark , einstein - infeld - hoffmann , fock , papapetrou and others led to a good understanding of the equations of motion of @xmath7 bodies at the first post - newtonian ( 1pn ) approximationpn order refers to the terms of order @xmath8 in the equations of motion . ] e.g. _ , @xcite for a general review of the problem of motion ) . in the 1970 s , an important series of works by a japanese group @xcite led to a nearly complete control of the problem of motion at the second post - newtonian ( 2pn ) approximation . then , in the early 80 s , motivated by the observation of secular orbital effects in the hulse - taylor binary pulsar psr1913 + 16 , several groups solved the two - body problem at the 2.5pn level ( while completing on the way the derivation of the 2pn dynamics ) ( for more recent work on the 2.5pn dynamics see @xcite ) . in the late 90 s , motivated by the aim of deriving high - accuracy templates for the data analysis of the upcoming international network of interferometric gravitational - wave detectors , two groups embarked on the derivation of the equations of motion at the third post - newtonian ( 3pn ) level . one group used the arnowitt - deser - misner ( adm ) hamiltonian approach and worked in a corresponding adm - type coordinate system . another group used a direct post - newtonian iteration of the equations of motion in harmonic coordinates . the end results of these two approaches have been proved to be physically equivalent @xcite . however , both approaches , even after exploiting all symmetries and pushing their hadamard - regularization - based methods to the maximum of their possibilities , left undetermined _ one and only one _ dimensionless parameter : @xmath9 in the adm approach and @xmath4 in the harmonic - coordinates one . the unknown parameters in both approaches are related by @xmath10 as was deduced from the comparison between the invariant energy functions for circular orbits in the two approaches @xcite , and from two independent proofs of the equivalence between the two formalisms for general orbits @xcite . the appearance of one ( and only one ) unknown parameter in the equations of motion is quite striking ; it is related with the choice of the regularization method used to cure the self - field divergencies of point particles . both lines of works and @xcite regularized the self - field divergencies by some version of the hadamard regularization method . the second line of work defined an extended version of the hadamard regularization @xcite , which permitted a self - consistent derivation of the 3pn equations of motion , but its use still allowed for the presence of arbitrary parameters in the final equations . on the other hand , the hadamard regularization also yielded some arbitrary parameters in the gravitational radiation field of point - particle binaries at the 3pn order , the most important of which being the parameter @xmath11 entering the binary s energy flux @xcite . let us notice that the regularization ( when dealing with point particles ) and the renormalization ( needed when dealing either with point particles or with extended bodies ) of self - field effects has recurrently plagued the general relativistic problem of motion . even at the 1pn level , early works often contained incorrect treatments of self - field effects ( see , _ e.g. _ , section 6.14 of @xcite for a review ) . at the 2pn level , the self - field divergencies are more severe than at the 1pn level . for instance , they caused ref . @xcite to incorrectly evaluate the `` static '' ( _ i.e. _ , velocity - independent ) part of the 2pn two - body hamiltonian . the first correct and complete evaluation of the 2pn dynamics has been obtained by using the riesz analytical continuation method @xcite . ( see @xcite for a detailed discussion of the evaluation of the static 2pn two - body hamiltonian . ) in brief , the riesz analytical continuation method consists of replacing the problematic delta - function stress - energy tensor of a set of point particles @xmath12 , @xmath13^{-1/2 } \ , \delta^{(4 ) } ( x^{\lambda } - y_a^{\lambda } ( s_a))\ , , \label{tdelta4}\ ] ] [ where @xmath14 , @xmath15 by an auxiliary , smoother source @xmath16^{-1/2 } \ , z_a^{(4 ) } ( x^{\lambda } - y_a^{\lambda } ( s_a))\ , . \label{triesz}\ ] ] [ actually , in the implementation of @xcite , one works with @xmath17 . ] in eq . ( [ triesz ] ) the four - dimensional delta function entering eq . ( [ tdelta4 ] ) has been replaced by the lorentzian is the lorentzian version of the euclidean kernel @xmath18 discussed in appendix [ formulae ] . ] four - dimensional riesz kernel @xmath19 , which depends on the complex number @xmath20 . when the real part of @xmath20 is large enough the source @xmath21 is an ordinary function of @xmath22 , which is smooth enough to lead to a well - defined iteration of the harmonically relaxed einstein field equations , involving no divergent integrals linked to the behavior of the integrands when @xmath23 . one then analytically continues @xmath20 down to @xmath24 , where the kernel @xmath19 tends to @xmath25 . the important point is that it has been shown @xcite that all the integrals appearing in the 2pn equations of motion are meromorphic functions of @xmath20 which admit a smooth continuation at @xmath26 ( without poles ) . it was also shown there that the formal construction based on ( [ triesz ] ) does generate , at the 2.5pn level , the metric and equations of motion of @xmath7 `` compact '' bodies ( _ i.e. _ , bodies with radii comparable to their schwarzschild radii ) . the riesz analytic continuation method just sketched works within a normal 4-dimensional space - time ( as recalled by the superscript @xmath27 in ( [ triesz ] ) ) . however , it was mentioned in @xcite that the same final result ( at the 2.5pn level ) is obtained by replacing @xmath19 by @xmath28 , _ i.e. _ , by formally considering delta - function sources in a space - time of complex dimension @xmath29 . in other words , at the 2.5pn level , the riesz analytic continuation method is equivalent to the _ dimensional regularization _ method . however , it was also mentioned at the time @xcite that the generalization of riesz analytic continuation beyond the 2.5pn level did not look straightforward because of the appearance of poles , proportional to @xmath30 , at the 3pn level ( when using harmonic coordinates ) . recently , damour , jaranowski and schfer @xcite showed how to use dimensional regularization within the adm canonical formalism . they found that the reduced hamiltonian describing the dynamics of two point masses in space - time dimension @xmath31 was _ finite _ ( no pole part ) as @xmath32 . they also found that the unique 3pn hamiltonian defined by the analytic continuation of @xmath1 towards 3 had two properties : ( i ) the velocity - dependent terms had the unique structure compatible with global poincar invariance , , originally introduced in the adm approach @xcite , takes the unique value @xmath33 . this value was obtained in @xcite using the result for the binary energy function in the case of circular orbits , as calculated in the harmonic - coordinates formalism , and also directly from the requirement of poincar invariance in the adm formalism @xcite . ] and ( ii ) the velocity - independent ( `` static '' ) terms led to an unambiguous determination of the unknown adm parameter @xmath9 , namely @xmath34 the fact that the dimensionally regularized 3pn adm hamiltonian ends up being globally poincar invariant is a confirmation of the consistency of dimensional regularization , because this symmetry is not at all manifest within the adm approach which uses a space - plus - time split from the start . by contrast , the global poincar symmetry is manifest in harmonic coordinates , and indeed the 3pn harmonic - coordinates equations of motion derived in @xcite were found to be manifestly poincar invariant . in the present paper , we shall show how to implement dimensional regularization ( henceforth often abbreviated as `` dim . reg . '' or even `` dr '' ) in the computation of the equations of motion in harmonic coordinates , _ i.e. _ , following the same iterative post - newtonian formalism as in refs . @xcite . similarly to the adm calculation of ref . @xcite , our strategy will essentially consist of computing the _ difference _ between the @xmath1-dimensional result and the 3-dimensional one @xcite corresponding to hadamard regularization . this difference is computed in the form of a laurent expansion in @xmath35 , where @xmath1 denotes the spatial dimension . the main reason for computing the @xmath36-expansion of the difference is that it depends only on the singular behavior of various metric coefficients in the vicinity of the point particles , so that the functions involved in the delicate divergent integrals can all be computed in @xmath1 dimensions in the form of local expansions in powers of @xmath37 or @xmath38 ( where @xmath39 ; @xmath40 , @xmath41 , denoting the locations of the two point masses ) . dimensional regularization as we use it here can then be seen as a powerful argument for completing the 3-dimensional hadamard - regularization results of @xcite and fixing the value of the unknown parameter . we leave to future work the task of an exact calculation of the @xmath1-dimensional equations of motion , instead of the calculation of the first few terms in a laurent expansion in @xmath36 around @xmath42 , as done here . the first step towards such a calculation is taken in appendix [ littleg ] , where we give the explicit expression of the basic quadratically non - linear green function @xmath43 in @xmath1 dimensions . the detailed way of computing the difference between dim . reg . and hadamard s reg . will turn out to be significantly more intricate than in the adm case . this added complexity has several sources . a first source of complexity is that the harmonic - gauge @xmath1-dimensional calculation will be seen to contain ( as anticipated long ago @xcite ) poles , proportional to @xmath44 , by contrast to the adm calculation which is finite as @xmath32 . a second source of complexity is that the end results @xcite for the 3-dimensional 3pn equations of motion have been derived using systematically an _ extended _ version of the hadamard regularization method , incorporating both a generalized theory of singular pseudo - functions and their associated ( generalized ) distributional derivatives @xcite , and an improved definition of the finite part as @xmath45 , say @xmath46_1 $ ] , of a singular function @xmath47 , designed so as to respect the global poincar symmetry of the problem @xcite . we shall then find it technically convenient to subtract the various contributions to the end results of @xcite which arose because of the specific use of the extended hadamard regularization methods of @xcite before considering the difference with the @xmath1-dimensional result . a third source of added complexity ( with respect to the adm case derived in @xcite and used ( in its @xmath1-dimensional generalization ) in @xcite was written , on purpose , in a way which does not involve any hidden distributional terms ( the only delta - function contributions it contains being explicit contact terms @xmath48 ) . this allowed one to estimate the difference between the @xmath1-dimensional hamiltonian @xmath49 and the hadamard - regularized 3-dimensional one hr@xmath50 $ ] without having to worry about distributional derivatives . however , as a check on the consistency of dim . , the authors of @xcite did perform another calculation of @xmath51 based on a starting form of the hamiltonian which involved hidden distributional terms , with the same final result . ] ) comes from the presence in the harmonic - gauge integrals we shall evaluate of `` hidden - distributional '' terms in the integrands . by hidden distributional terms we mean terms proportional to the second spatial derivatives of the poisson kernel @xmath52 , or to the fourth spatial derivatives of the iterated poisson kernel @xmath53 . such terms , @xmath54 or @xmath55 , considered as schwartz distributional derivatives @xcite , contain pieces proportional to the delta function @xmath56 , which need to be treated with care . the generalized distributional derivative defined in @xcite , and used to compute the end results of @xcite , led to an improved way , compared to the normal schwartz distributional derivative , of evaluating contributions coming from the product of a singular function and a derivative of the type @xmath57 or @xmath58 , and more generally of any derivatives of singular functions in a certain class . we shall find it convenient to subtract these additional non - schwartzian contributions to the 3pn equations of motion before applying dimensional regularization . however , we shall note at the end that dim . reg . automatically incorporates all of these non - schwartzian contributions . a fourth , but minor , source of complexity concerns the dependence of the end results of @xcite for the 3pn acceleration of the first particle ( label @xmath59 ) , say @xmath60 , on two arbitrary length scales @xmath61 and @xmath62 , and on the `` ambiguity '' parameter @xmath4 . explicitly , we define @xmath63 \equiv \text{r.h.s . of eq.~(7.16 ) in ref . \cite{blanchet:2000ub}}\ , . \label{a1bf}\ ] ] here the acceleration is considered as a function of the two masses @xmath64 and @xmath65 , the relative distance @xmath66 ( where @xmath67 is the unit vector directed from particle 2 to particle 1 ) , the two coordinate velocities @xmath68 and @xmath69 , and also , as emphasized in ( [ a1bf ] ) , the parameter @xmath4 as well as two regularization length scales @xmath70 and @xmath71 . the latter length scales enter the equations of motion at the 3pn level through the logarithms @xmath72 and @xmath73 . they come from the regularization as the field point @xmath74 tends to @xmath75 or @xmath76 of poisson - type integrals ( see section [ hadamardpoisson ] below ) . the length scales @xmath70 , @xmath71 are `` pure gauge '' in the sense that they can be removed by the effect induced on the world - lines of a coordinate transformation of the bulk metric @xcite . on the other hand , the dimensionless parameter @xmath4 entering the final result ( [ a1bf ] ) corresponds to genuine physical effects . it was introduced by requiring that the 3pn equations of motion admit a conserved energy ( and more generally be derivable from a lagrangian ) . this extra requirement imposed _ two relations _ between the two length scales @xmath61 , @xmath62 and two other length scales @xmath77 , @xmath78 entering originally into the formalism , namely the constants @xmath77 and @xmath78 parametrizing the hadamard partie finie of an integral as defined by eq . ( [ pf ] ) below . these relations were found to be of the form @xmath79 where the so introduced _ single _ dimensionless parameter @xmath4 has been proved to be a purely numerical coefficient ( independent of the two masses ) . when estimating the difference between dim . reg . and hadamard reg . it will be convenient to insert eq . ( [ lnr2s2 ] ) into ( [ a1bf ] ) and to reexpress the acceleration of particle 1 in terms of the _ original _ regularization length scales entering the hadamard regularization of @xmath80 , which were in fact @xmath61 and @xmath78 . thus we can consider alternatively @xmath81 \equiv \mathbf{a}_1^\text{bf } \bigl[\lambda ; r_1 ' , r'_2 ( s_2 , \lambda)\bigr]~~\text{and $ 1\leftrightarrow 2$}\ , , \label{foncta1bf}\ ] ] where the regularization constants are subject to the constraints ( [ lnr2s2 ] ) [ we will then check that the @xmath4-dependence on the r.h.s . of ( [ foncta1bf ] ) disappears when using eq . ( [ lnr2s2 ] ) to replace @xmath62 as a function of @xmath78 and @xmath4 ] . our strategy will consist of _ two steps_. the _ first step _ consists of subtracting all the extra contributions to eq . ( [ a1bf ] ) , or equivalently eq . ( [ foncta1bf ] ) , which were specific consequences of the extended hadamard regularization defined in @xcite . as we shall detail below , there are _ seven _ such extra contributions @xmath83 , @xmath84 . essentially , subtracting these contributions boils down to estimating the value of @xmath80 that would be obtained by using a `` pure '' hadamard regularization , together with schwartz distributional derivatives . such a `` pure hadamard - schwartz '' ( phs ) acceleration was in fact essentially the result of the first stage of the calculation of @xmath80 , as reported in the ( unpublished ) thesis @xcite . it is given by @xmath85=\mathbf{a}_1^\text{bf } [ r'_1 , s_2]-\sum_{a=1}^7\delta^a\mathbf{a}_1\ , . \label{accph}\ ] ] the _ second step _ of our method consists of evaluating the laurent expansion , in powers of @xmath86 , of the _ difference _ between the @xmath1-dimensional and the pure hadamard - schwartz ( 3-dimensional ) computations of the acceleration @xmath80 . we shall see that this difference makes a contribution only when a term generates a _ pole _ @xmath87 , in which case dim . reg . adds an extra contribution , made of the pole and the finite part associated with the pole [ we consistently neglect all terms @xmath88 . one must then be especially wary of combinations of terms whose pole parts finally cancel ( `` cancelled poles '' ) but whose dimensionally regularized finite parts generally do not , and must be evaluated with care . we denote the above defined difference @xmath89\equiv\mathcal{d}\mathbf{a}_1 \bigl[\varepsilon,\ell_0;\lambda ; r_1 ' , r'_2\bigr]\ , . \label{deltaacc}\ ] ] it depends both on the hadamard regularization scales @xmath70 and @xmath78 ( or equivalently on @xmath4 and @xmath70 , @xmath71 ) and on the regularizing parameters of dimensional regularization , namely @xmath36 and the characteristic length @xmath90 associated with dim . reg . and introduced in eq . ( [ l0 ] ) below . we shall explain in detail below the techniques we have used to compute @xmath91 ( see section [ difference ] ) . finally , our main result will be the explicit computation of the @xmath36-expansion of the dim . acceleration as @xmath92 = \mathbf{a}_1^\text{phs } [ r'_1 , s_2 ] + \mathcal{d}\mathbf{a}_1 [ \varepsilon , \ell_0 ; r'_1 , s_2]\ , . \label{a1dimreg}\ ] ] with this result in hands , we shall prove ( in section [ renormalise ] ) two theorems . * * theorem 1**__the pole part @xmath93 of the dimensionally - regularized acceleration ( [ a1dimreg ] ) , as well as of the metric field @xmath94 outside the particles , can be re - absorbed ( _ i.e. _ , renormalized away ) into some shifts of the two `` bare '' world - lines : @xmath95 , with , say , @xmath96 ( `` minimal subtraction '' ; ms ) , so that the result , expressed in terms of the `` dressed '' quantities , is finite when @xmath97 . _ _ the situation in harmonic coordinates is to be contrasted with the calculation in adm - type coordinates within the hamiltonian formalism @xcite , where it was shown that all pole parts directly cancel out in the total 3pn hamiltonian ( no shifts of the world - lines were needed ) . the central result of the paper is then as follows . * * theorem 2**__the `` renormalized '' ( finite ) dimensionally - regularized acceleration is physically equivalent to the extended - hadamard - regularized acceleration ( end result of ref . @xcite ) , in the sense that there exist some shift vectors @xmath98 and @xmath99 , such that @xmath100 = \lim_{\varepsilon\rightarrow 0 } \ , \bigl[\mathbf{a}_1^\mathrm{dr } [ \varepsilon , \ell_0 ] + \delta_{\bm{\xi } ( \varepsilon , \ell_0 ; r'_1 , r'_2 ) } \ , \mathbf{a}_1 \bigr ] \label{eta}\ ] ] ( where @xmath101 denotes the effect of the shifts on the acceleration below ) . ] ) , if and only if the heretofore unknown parameter @xmath4 entering the harmonic - coordinates equations of motion takes the value @xmath102 _ _ the precise shifts @xmath103 needed in theorem 2 involve not only a pole contribution @xmath93 , which defines the `` minimal '' ( ms ) shifts considered in theorem 1 , but also a finite contribution when @xmath104 . their explicit expressions read : @xmath105 \mathbf{a}_{n1}~~\text{and}~~1\leftrightarrow 2\,,\ ] ] where @xmath106 is the usual newton s constant [ see eq . ( [ l0 ] ) below ] , @xmath107 denotes the acceleration of the particle 1 ( in @xmath1 dimensions ) at the newtonian level , and @xmath108 depends on the euler constant @xmath109 . [ the detailed proofs of theorems 1 and 2 will consist of our investigations expounded in the successive sections of the paper , and will be completed at the end of sections [ shift ] and [ lambdadetermined ] respectively , taking also into account the results of section [ kinetic ] . ] notice that an alternative way of presenting our central result is to say that , in fact , each choice of specific renormalization prescription ( within dim . reg . ) , such as `` minimal subtraction '' as assumed in theorem 1 for conceptual simplicity , below to use a modified minimal subtraction that we shall denote @xmath110 . ] leads to renormalized equations of motion which depend only on the dim . characteristic length scale @xmath90 through the logarithm @xmath111 , and that any of these renormalized equations of motion are physically equivalent to the final results of @xcite . in particular , this means , as we shall see below , that each choice of renormalization prescription within dim . reg . determines the two regularization length scales @xmath61 , @xmath62 entering eq . ( [ a1bf ] ) . of course , what is important is not the particular values these constants can take in a particular renormalization scheme [ indeed @xmath61 and @xmath62 are simply `` gauge '' constants which can anyway be removed by a coordinate transformation ] , but the fact that the different renormalization prescriptions yield equations of motion falling into the `` parametric '' class ( _ i.e. _ , parametrized by @xmath61 and @xmath62 ) of equations of motion obtained in @xcite . an alternative way to phrase the result ( [ eta])-([lambda ] ) , is to combine eqs . ( [ accph ] ) and ( [ a1dimreg ] ) in order to arrive at @xmath112 + \delta_{\bm{\xi } ( \varepsilon , \ell_0 ; r'_1 , r'_2 ) } \ , \mathbf{a}_1 \bigr ] = \sum_{a=1}^7\delta^a\mathbf{a}_1\ , . \label{equiveta}\ ] ] under this form one sees that the sum of the additional terms @xmath113 differs by a mere shift , _ when and only when _ @xmath4 takes the value ( [ lambda ] ) , from the specific contribution @xmath91 we shall evaluate in this paper , which comes directly from dimensional regularization . therefore one can say that , when @xmath114 , the extended - hadamard regularization @xcite is in fact ( physically ) equivalent to dimensional regularization . however the extended - hadamard regularization is incomplete , both because it is unable to determine @xmath4 , and also because it necessitates some `` external '' requirements such as the imposition of the link ( [ lnr2s2 ] ) in order to ensure the existence of a conserved energy and in fact of the ten first integrals linked to the poincar group . by contrast dim . reg . succeeds automatically ( without extra inputs ) in guaranteeing the existence of the ten conserved integrals of the poincar group , as already found in ref . @xcite . in view of the necessary link ( [ lambdaomegas ] ) provided by the equivalence between the adm - hamiltonian and the harmonic - coordinates equations of motion , our result ( [ lambda ] ) is in perfect agreement with the previous result ( [ omegas ] ) obtained in @xcite . is a complicated rational fraction while @xmath9 is so simple this is because @xmath9 was introduced precisely to measure the amount of ambiguities of certain integrals , and that the adm hamiltonian reported in @xcite was put in a minimally ambiguous form , already in three dimensions , for which an _ a posteriori _ look at the `` ambiguities '' discussed in the appendix a of @xcite already showed that @xmath6 . by contrast , @xmath4 has been introduced as the only possible unknown constant in the link between the four arbitrary scales @xmath115 ( which has _ a priori _ nothing to do with ambiguities of integrals ) , in a framework where the use of the extended hadamard regularization makes in fact the calculation to be unambiguous . ] our result is also in agreement with the recent finding of itoh and futamase @xcite ( see also @xcite ) , who derived the 3pn equations of motion in harmonic gauge using a `` surface - integral '' approach , aimed at describing _ extended _ relativistic compact binary systems in the strong - field point particle limit . the surface - integral approach of refs . @xcite is interesting because , like the matching method used at 2.5pn order in @xcite , it is based on the physical notion of extended compact bodies . in this respect , we recall that the matching method used in @xcite showed that the internal structure ( love numbers ) of the constituent bodies would start influencing the equations of motion of ( non - spinning ) compact bodies only at the 5pn level . effacement property _ strongly suggests that it is possible to model , in a physically preferred manner , two compact bodies as being two point - like particles , described by two masses and two world - lines , up to the 4.5pn level included . it remains , however , to prove that the dimensional regularization of delta - function sources does yield the physically unique equations of motion of two compact bodies up to the 4.5 pn order . the work @xcite proved it at the 2.5 pn level , and the agreement of the present results with those of @xcite indicates that this is also true at the 3pn level . besides the independent confirmation of the value of @xmath9 or @xmath4 , let us also mention that our work provides a confirmation of the _ consistency _ of dim . reg . , because our explicit calculations [ which involved combinations of hundreds of laurent expansions of the form @xmath116 are entirely different from the ones of @xcite : we use harmonic coordinates ( instead of adm - type ones ) , we work at the level of the equations of motion ( instead of the hamiltonian ) , we use a different form of einstein s field equations and we solve them by a different iteration scheme . finally , from a practical point of view our confirmation of the value of @xmath9 or@xmath4 allows one to use the full 3pn accuracy in the analytical computation of the dynamics of the last orbits of binary orbits @xcite . it remains , however , the task of computing , using dimensional regularization , the parameter @xmath11 entering the 3.5pn gravitational energy flux @xcite to be able to have full 3.5pn accuracy in the computation of the gravitational waveforms emitted by inspiralling compact binaries ( see , _ e.g. _ , @xcite and references therein ) . the organization of the paper is as follows . in section [ fieldeq ] we derive our basic 3pn solution of the field equations for general fluid sources in @xmath1 spatial dimensions , using @xmath1-dimensional generalizations of the elementary potentials introduced in ref . section [ hadamard ] collects all the additional terms included in @xcite which are due specifically to the extended hadamard regularization , and derives the pure hadamard - schwartz ( phs ) contribution to the equations of motion . the differences between the dimensional and phs regularizations for all the potentials and their gradients are computed in section [ difference ] . then the dim . equations of motion are obtained in section [ results ] , where we comment also on their interpretation in terms of space - time diagrams . section [ renormalise ] is devoted to the renormalization of the dim . equations by means of suitable shifts of the particles world - lines , and to the equivalence with the end results of @xcite when eq . ( [ lambda ] ) holds . at this stage , the proofs of theorems 1 and 2 stated above are finally completed . we end the paper with some conclusions ( section [ conclusions ] ) and three appendices . appendix [ expfieldeq ] provides further material on the @xmath1-dimensional metric and geodesic equation , appendix [ formulae ] gives a compendium of useful formulae for working in @xmath1 dimensions , and appendix [ littleg ] generalizes the well - known quadratic - order elementary kernel @xmath117 to @xmath1 dimensions . the latter calculation of the @xmath1-dimensional kernel @xmath118 is not directly employed in the present paper , but represents a first step in obtaining the equations of motion in any dimension @xmath1 ( not necessarily of the form @xmath119 ) . this section is devoted to the field equations of general relativity in @xmath120 space - time dimensions , and to the geodesic equation describing the motion of point particles . we use the sign conventions of ref . @xcite , and in particular our metric signature is mostly @xmath121 . space - time indices are denoted by greek letters , and spatial indices by latin letters ( @xmath122 run from 1 to @xmath1 ) . a summation is understood for any pair of repeated indices . we work in the harmonic gauge , which is such that @xmath123 as usual , @xmath124 denotes the inverse metric and @xmath125 the christoffel symbols . using this gauge condition , one can easily prove that the ricci tensor reads _ in any dimension _ @xmath126 where a comma denotes partial derivation . note that the spatial dimension @xmath1 does not appear explicitly in this expression , whereas some @xmath1-dependent coefficients do appear when expressing the ricci tensor in terms of the so - called `` gothic '' metric @xmath127 [ see eq . ( [ riccigoth ] ) in appendix [ expfieldeq ] below ] . in any dimension , the einstein field equations read @xmath128 where @xmath129 denotes the matter stress - energy tensor , given by the functional derivative @xmath130 of the matter action @xmath131 with respect to the metric tensor . _ by definition _ , @xmath132 denotes the constant involved in eq . ( [ einstein ] ) , which shows that its dimension is such that @xmath133 where @xmath106 is the usual newton constant ( in 3 spatial dimensions ) and @xmath90 is an arbitrary length scale . this scale will be involved in our dimensionally regularized results below , but we will finally show that the physical observables do not depend on it . as is well known , the combination of eq . ( [ einstein ] ) with its trace allows us to rewrite it as @xmath134 in which the spatial dimension @xmath1 now appears explicitly . we wish to expand in powers of @xmath135 the field equations resulting from ( [ ricciharm ] ) and ( [ ricci ] ) . the basic idea is to introduce a sequence of `` elementary potentials '' , @xmath136 , @xmath137 , @xmath138 , which allow one to parametrize conveniently the successive post - minkowskian contributions to the metric @xmath139 . for instance , at the first post - minkowskian order it is convenient to parametrize the metric as @xmath140 where the so - introduced elementary potentials @xmath136 and @xmath137 satisfy equations of the form @xmath141 where @xmath142 denotes the flat dalembertian and where , _ by definition _ , the sources @xmath143 and @xmath144 are linear combinations of the contravariant components @xmath129 of the stress - energy tensor of the matter . let us underline that the factor @xmath145 in these equations is a _ choice_. we could of course introduce here a functional dependence on the spatial dimension @xmath1 , for instance by replacing the factor @xmath146 by the surface of the unit @xmath147-dimensional sphere [ see eq . ( [ omegad1 ] ) in appendix [ formulae ] ] , but this would only complicate the intermediate expressions without changing our final result . the matter sources @xmath143 and @xmath144 _ defined _ by eqs . ( [ glowest ] ) , ( [ vlowest ] ) read ( in @xmath1 spatial dimensions ) : @xmath148 the definition for @xmath149 has been added for future use . note that @xmath144 and @xmath149 take the same forms as usual in 3 dimensions ( see eqs . ( 3.9 ) of ref . @xcite ) , but that the definition of @xmath143 involves an explicit dependence on @xmath1 . conversely , the first and third of these equations allow us to express @xmath150 in terms of the above matter sources : @xmath151 . a simple consequence of the expression of @xmath143 is that the @xmath1-dimensional _ newtonian potential _ generated by a mass @xmath152 located at @xmath40 reads explicitly @xmath153 where the factor @xmath154 comes from @xmath143 ( _ i.e. _ , from einstein s equations ) , while the factor @xmath155 comes from the expression of the green function of the laplacian in @xmath1 dimensions ( see eq . ( [ green ] ) below and appendix [ formulae ] ) . we give below the simplest forms of the metric and of the potential equations that we could obtain . we will explain afterwards which rules we followed to simplify them . let us first define the useful combination @xmath156 then the metric components can be written in a rather compact form : [ metric ] @xmath157 + \frac{8\hat r_i}{c^5 } + \frac{16}{c^7}\left[\hat y_i+\frac{1}{2}\hat w_{ij}v_j\right]\right\ } + { \mathcal{o}}\left(\frac{1}{c^9}\right),\nonumber\\ \label{g0i}\\ g_{ij}&=&e^{\frac{2\mathcal{v}}{(d-2)c^2}}\left\{\delta_{ij } + \frac{4}{c^4 } \hat w_{ij } + \frac{16}{c^6 } \left[\hat z_{ij}-v_i v_j + \frac{1}{2(d-2)}\ , \delta_{ij } v_k v_k \right]\right\ } + { \mathcal{o}}\left(\frac{1}{c^8}\right ) . \label{gij}\end{aligned}\ ] ] the various elementary potentials @xmath136 , @xmath137 , @xmath158 , @xmath159 , @xmath160 , @xmath161 , @xmath162 , @xmath163 and @xmath164 introduced in these definitions are @xmath1-dimensional analogues of those used in eqs . ( 3.24 ) of ref . actually , an extra potential is needed for @xmath165 , and it has been denoted @xmath158 in eq . ( [ calv ] ) above . we give in appendix [ expfieldeq ] the explicit expansion of this metric in powers of @xmath135 , as well as its inverse @xmath166 and its determinant @xmath167 , which can be useful for future works . note that the first post - newtonian order of the spatial metric , @xmath168 + { \mathcal{o}}(1/ c^4)$ ] , explicitly depends on @xmath1 contrary to our choice ( [ glowest ] ) for @xmath169 . this dissymmetry between @xmath169 and @xmath170 is imposed by the field equations ( [ ricci ] ) . the successive post - newtonian truncations of the field equations ( [ ricciharm])-([ricci ] ) give us the sources for these various potentials . the equations for @xmath171 and @xmath172 have already been written in eqs . ( [ vlowest ] ) above . we get for the remaining potentials : [ potentialeq ] @xmath173\nonumber\\ & & + \hat w_{ij}\ , \partial_{ij}v + 2 v_i\,\partial_t\partial_i v + \frac{1}{2}\left(\frac{d-1}{d-2}\right ) v \partial^2_t v \nonumber\\ & & + \frac{d(d-1)}{4(d-2)^2}\left(\partial_t v\right)^2 -2 \partial_i v_j\ , \partial_j v_i\ , \label{dalx}\\ \box\hat z_{ij}&=&-\frac{4\pi g}{d-2}\ , v\left(\sigma_{ij } -\delta_{ij}\,\frac{\sigma_{kk}}{d-2}\right ) -\frac{d-1}{d-2}\ , \partial_t v_{(i}\ , \partial_{j)}v + \partial_i v_k\ , \partial_j v_k\nonumber\\ & & + \partial_k v_i\ , \partial_k v_j -2 \partial_k v_{(i}\ , \partial_{j)}v_k -\frac{\delta_{ij}}{d-2}\ , \partial_k v_m \left(\partial_k v_m -\partial_m v_k\right)\nonumber\\ & & -\frac{d(d-1)}{8(d-2)^3}\ , \delta_{ij}\left(\partial_t v\right)^2 + \frac{(d-1)(d-3)}{2(d-2)^2}\ , \partial_{(i } v\partial_{j ) } k\ , , \label{dalzij}\\ \box\hat y_i&=&-4\pi g \biggl[-\frac{1}{2}\left(\frac{d-1}{d-2}\right)\sigma\hat r_i -\frac{(5-d)(d-1)}{4(d-2)^2}\ , \sigma v v_i + \frac{1}{2}\ , \sigma_k\hat w_{ik } + \frac{1}{2}\sigma_{ik } v_k\nonumber\\ & & \hphantom{-4\pi g \biggl[}+\frac{1}{2(d-2)}\ , \sigma_{kk}v_i -\frac{d-3}{(d-2)^2}\ , \sigma_i \left(v^2 + \frac{5-d}{2}\ , k\right)\biggr ] \nonumber\\ & & + \hat w_{kl}\ , \partial_{kl } v_i -\frac{1}{2}\left(\frac{d-1}{d-2}\right ) \partial_t\hat w_{ik}\ , \partial_k v + \partial_i\hat w_{kl}\ , \partial_k v_l -\partial_k\hat w_{il}\ , \partial_l v_k \nonumber\\ & & -\frac{d-1}{d-2}\ , \partial_k v \partial_i \hat r_k -\frac{d(d-1)}{4 ( d-2)^2}\ , v_k\ , \partial_i v \partial_k v -\frac{d(d-1)^2}{8 ( d-2)^3}\ , v\partial_t v\partial_i v \nonumber\\ & & -\frac{1}{2}\left(\frac{d-1}{d-2}\right)^2 v \partial_k v \partial_k v_i + \frac{1}{2}\left(\frac{d-1}{d-2}\right ) v \partial^2_tv_i + 2 v_k\ , \partial_k\partial_t v_i \nonumber\\ & & + \frac{(d-1)(d-3)}{(d-2)^2}\ , \partial_k k \partial_i v_k + \frac{d(d-1)(d-3)}{4(d-2)^3 } \left(\partial_t v\partial_i k + \partial_i v\partial_t k\right ) , \label{dalyi}\\ \box\hat t&=&-4\pi g\biggl[\frac{1}{2(d-1)}\ , \sigma_{ij } \hat w_{ij } + \frac{5-d}{4(d-2)^2}\ , v^2\sigma_{ii } + \frac{1}{d-2}\ , \sigma v_i v_i -\frac{1}{2}\left(\frac{d-3}{d-2}\right)\sigma\hat x\nonumber\\ & & \hphantom{-4\pi g\biggl[}-\frac{1}{12}\left(\frac{d-3}{d-2}\right)^3 \sigma v^3 -\frac{1}{2}\left(\frac{d-3}{d-2}\right)^3 \sigma v k + \frac{(5-d)(d-3)}{2(d-1)(d-2)}\ , \sigma_i v_i v\nonumber\\ & & \hphantom{-4\pi g\biggl[}+\frac{d-3}{d-1}\ , \sigma_i\hat r_i -\frac{d-3}{2(d-2)^2}\ , \sigma_{ii } k\biggr ] + \hat z_{ij}\ , \partial_{ij}v + \hat r_i\ , \partial_t\partial_i v\nonumber\\ & & -2 \partial_i v_j\ , \partial_j\hat r_i -\partial_i v_j\ , \partial_t\hat w_{ij } + \frac{1}{2}\left(\frac{d-1}{d-2}\right ) v v_i\ , \partial_t\partial_i v + \frac{d-1}{d-2}\ , v_i\ , \partial_j v_i\ , \partial_j v\nonumber\\ & & + \frac{d(d-1)}{4(d-2)^2}\ , v_i\ , \partial_t v\ , \partial_i v + \frac{1}{8}\left(\frac{d-1}{d-2}\right)^2 v^2\partial^2_t v + \frac{d(d-1)^2}{8(d-2)^3}\ , v\left(\partial_t v\right)^2\nonumber\\ & & -\frac{1}{2}\left(\partial_t v_i\right)^2 -\frac{(d-1)(d-3)}{4(d-2)^2}\ , v \partial^2_t k -\frac{d(d-1)(d-3)}{4(d-2)^3}\ , \partial_t v\ , \partial_t k\nonumber\\ & & -\frac{(d-1)(d-3)}{4(d-2)^2}\ , k \partial^2_t v -\frac{d-3}{d-2}\ , v_i\ , \partial_t\partial_i k -\frac{1}{2}\left(\frac{d-3}{d-2}\right)\hat w_{ij}\ , \partial_{ij } k\ , . \label{dalt}\end{aligned}\ ] ] in eq . ( [ dalzij ] ) , parentheses around indices mean their symmetrization , _ i.e. _ , @xmath174 . for @xmath42 , the above set of equations ( [ potentialeq ] ) reduces to eqs . ( 3.26 ) and ( 3.27 ) of ref . the order of the terms and their writing has been chosen to be as close as possible to this reference . the harmonic gauge conditions ( [ harmgauge ] ) impose the following differential identities between the potentials : [ gaugeidentities ] @xmath175\nonumber\\ & & \hphantom{\partial_t\biggl\{}+\frac{2}{c^4 } \left(\frac{d-1}{d-2}\right)\biggl[\hat x + \frac{d-2}{d-1}\,\hat z -\frac{d-3}{d-1}\,v_k v_k + \frac{1}{2}\,v\hat w\nonumber\\ & & \hphantom{\partial_t\biggl\{+\frac{2}{c^4 } \left(\frac{d-1}{d-2}\right)\biggl[}+\frac{(d-1)^2}{6(d-2)}\ , v^3 -\frac{(d-1)(d-3)}{(d-2)^2}\ , v k \biggr ] \biggr\}\nonumber\\ & & + \partial_i\biggl\{v_i + \frac{2}{c^2}\left[\hat r_i + \frac{1}{2}\left(\frac{d-1}{d-2}\right ) v v_i\right]\nonumber\\ & & \hphantom{+\partial_i\biggl\{}+\frac{4}{c^4}\biggl[\hat y_i -\frac{1}{2}\ , \hat w_{ij } v_j + \frac{1}{2}\ , \hat w v_i + \frac{1}{2}\left(\frac{d-1}{d-2}\right ) v \hat r_i + \frac{1}{4}\left(\frac{d-1}{d-2}\right)^2 v^2 v_i\nonumber\\ & & \hphantom{+\partial_i\biggl\{+\frac{4}{c^4}\biggl[}- \frac{(d-1)(d-3)}{2 ( d-2)^2}\ , k v_i \biggr ] \biggr\ } = { \mathcal{o}}\left(\frac{1}{c^6}\right ) , \label{divi}\\ g^{\mu\nu}\gamma^i_{\mu\nu } = 0&\rightarrow&\partial_t\left\ { v_i + \frac{2}{c^2}\left [ \hat r_i + \frac{1}{2}\left(\frac{d-1}{d-2}\right)v v_i \right]\right\}\nonumber\\ & & + \partial_j\left\ { \hat w_{ij } -\frac{1}{2}\ , \hat w \delta_{ij } + \frac{4}{c^2}\left[\hat z_{ij } -\frac{1}{2}\ , \hat z \delta_{ij}\right ] \right\ } = { \mathcal{o}}\left(\frac{1}{c^4}\right ) , \label{djwij}\end{aligned}\ ] ] @xmath176 where @xmath177 and @xmath178 denote the traces of potentials @xmath159 and @xmath162 . for @xmath42 , these identities reduce to eqs . ( 3.28 ) of ref . @xcite . in this paper we shall check ( see sections [ iteratedeinstein ] and [ poles ] ) that all the dimensionally - regularized potentials we use obey , at the indicated accuracy , the differential identities ( [ gaugeidentities ] ) equivalent to the harmonic gauge conditions . in order to simplify as much as possible the above equations ( [ potentialeq ] ) for the potentials , we used the following rules : 1 . we used the harmonic gauge condition ( [ divi ] ) to replace everywhere @xmath179 in terms of @xmath180 and higher post - newtonian order terms , and the gauge condition ( [ djwij ] ) to replace @xmath181 in terms of @xmath182 and @xmath183 [ our knowledge of the higher order terms @xmath184 in eq . ( [ djwij ] ) was actually not necessary for the simplification of eqs . ( [ potentialeq ] ) ] . we also used the lowest order terms of eqs . ( [ gaugeidentities ] ) to simplify their own higher order contributions . if the source of a potential @xmath185 contained a double ( contracted ) gradient of the form @xmath186 , where @xmath20 and @xmath187 were two lower - order potentials , we got rid of the double gradient by defining another potential @xmath188 . we could then write its equation as @xmath189 , in which @xmath190 and @xmath191 were replaced by their own explicit sources . the contribution proportional to @xmath192 was then transferred into the source of a higher order potential . 3 . at order @xmath193 , equation ( [ ricciharm])-([ricci ] ) for @xmath194 ( _ i.e. _ , for @xmath195 ) contains the term @xmath196 , that we introduced in the source of potential @xmath161 , eq . ( [ dalx ] ) . in all other equations involving the same source @xmath196 , we used @xmath197 to eliminate it , instead of reintroducing it in the sources of other potentials . this is the reason why @xmath161 is involved in the spatial metric @xmath170 too at order @xmath193 [ _ via _ the exponential of @xmath198 in eq . ( [ gij ] ) ] , and why @xmath199 appears again in @xmath169 at order @xmath200 . see the expanded form of the metric ( [ metricexp ] ) in appendix [ expfieldeq ] . 4 . in the equation for @xmath194 at order @xmath200 , we _ chose _ to eliminate a source proportional to @xmath201 , by including an all - integrated term @xmath202 in the definition of @xmath169 , eq . ( [ g00 ] ) . on the other hand , we could not eliminate at the same time the source term proportional to @xmath203 in @xmath195 , although it involves a double ( contracted ) gradient too . this is the reason why such a term appears in eq . ( [ dalt ] ) . the above simplification rules have been applied systematically with a single exception . indeed , eq . ( [ dalzij ] ) for @xmath204 involves double ( contracted ) gradients @xmath205 and @xmath206 . therefore , the application of rule ( ii ) would have yielded another potential @xmath207 such that no double gradient appears in its source ( but extra compact sources @xmath208 and @xmath209 would have been involved ) . although this modified potential @xmath210 actually simplifies slightly some equations ( but not all of them ) , we have chosen to use @xmath162 which is the direct @xmath1-dimensional analogue of the potential written in eq . ( 3.27c ) of ref . indeed , as explained in the following sections , the 3-dimensional results of this reference will be necessary for our @xmath1-dimensional calculations , and it is more convenient to keep the same notation . notice also that after the above simplifications , the resulting metric involves only potentials which are at most cubically non - linear ( like for the term @xmath196 in the potential @xmath161 using the terminology of section [ dimregstat ] below ) . there is no need to introduce any quartically non - linear elementary potential because it turns out that it is possible to `` integrate directly '' all of them ( at the 3pn level ) in terms of other potentials . the only quartic contributions are the terms composed of @xmath211 and @xmath212 in the metric component @xmath169 [ see eq . ( [ g00exp ] ) in appendix [ expfieldeq ] ] . the fact that there are no intrinsically quartic potentials at the 3pn order made the closed - form calculation in @xcite possible . we shall comment more on this interesting fact in section [ iteratedeinstein ] . let us now apply the general potential parametrization of the metric defined above to the specific case of ( monopolar ) point particles , _ i.e. _ , to the action @xmath213 the stress - energy tensor @xmath214 \ , \delta \ , s_\text{matter } / \delta \ , g_{\mu\nu } ( x)$ ] deduced from this action reads @xmath215^{-\frac{1}{2 } } \ , \delta^{(d+1 ) } ( x^{\lambda } - y_a^{\lambda } ( s_a))\ , , \label{tmunu}\ ] ] where @xmath216 is ( @xmath217 times ) the proper time along the world - line of the @xmath218 particle and where @xmath219 is the dirac density in @xmath120 dimensions ( @xmath220 ) . here , we take advantage of the fact ( emphasized in @xcite ) that dim . respects the basic properties of the algebraic and differential calculus : associativity , commutativity and distributivity of point - wise addition and multiplication , leibniz s rule , schwarz s rule @xmath221 , integration by parts , etc . in addition , the post - newtonian expansion of @xmath222 yields `` @xmath1-dimensional functions '' which are formally as smooth as wished ( by taking the real part of @xmath1 small enough ) in the vicinity of the world - lines : see for instance eq . ( [ newton ] ) . this allows us to work with self - gravitating point particles essentially as if they were _ test _ particles . for instance , we can use @xmath223 \ , \delta^{(d ) } \ , ( \mathbf{x}-\mathbf{y}_a ) = f [ g_{\mu\nu } ( y_a ) ] \ , \delta^{(d ) } \ , ( \mathbf{x}-\mathbf{y}_a)$ ] . in particular , the @xmath224-evaluated determinant factor @xmath225^{-\frac{1}{2}}$ ] in ( [ tmunu ] ) came from the field - point dependent factor @xmath226^{-\frac{1}{2}}$ ] in the definition of @xmath227 . similarly , the usual derivation of the equations of motion of a test particle formally generalizes to the case of self - gravitating point particles in @xmath1 dimensions . one then finds that the equations of motion of point particles can equivalently be written as @xmath228 or as the usual geodesic equations . the latter can be written either in covariant form @xmath229 ( @xmath230 ) , _ i.e. _ , explicitly @xmath231 \ , \frac{dy_a^{\mu}}{ds_a } \ , \frac{dy_a^{\nu}}{ds_a } = 0 \ , , \label{geod2}\ ] ] where @xmath232 as usual , or in the explicit form corresponding to using the coordinate time @xmath233 as parameter along the world - lines , which is easily derived from the covariant expression with a lower index , @xmath234 . like in 3 dimensions , cf . ( 3.32)-(3.33 ) of ref . @xcite , it can thus be put in the form @xmath235 where @xmath236 @xmath237 denoting the coordinate velocity . let us emphasize again that in @xmath1 dimensions , all the non - linear functions of @xmath238 and @xmath239 that will enter our calculation of ( [ eqgeod])-([pifi ] ) can be treated as in the @xmath240 evaluation of smooth functions of @xmath241 . for instance , denoting for simplicity @xmath242 , the newtonian approximation , say @xmath243 , of the basic scalar potential @xmath244 , reads , in the vicinity of @xmath245 , @xmath246 where @xmath247 is ( in any @xmath1 ) an indefinitely differentiable function of @xmath241 near @xmath75 . analytically continuing @xmath1 to sufficiently `` low '' ( and even with negative real part , if needed ) values , we see not only that @xmath248_{\mathbf{x } = \mathbf{y}_1 } = u_2 ( \mathbf{y}_1)$ ] , but that @xmath249_{\mathbf{x } = \mathbf{y}_1 } = ( u_2 ( \mathbf{y}_1))^n$ ] , and , _ e.g. _ , @xmath250_{\mathbf{x } = \mathbf{y}_1 } = ( u_2 ( \mathbf{y}_1))^p \ , \partial_i \ , u_2 ( \mathbf{y}_1)$ ] , etc . although the expressions ( [ pifi ] ) do not depend explicitly on the dimension @xmath1 , the metric ( [ metric ] ) does depend on it , and therefore the post - newtonian expansions of eqs . ( [ pifi ] ) involve many @xmath1-dependent coefficients . we give their full expressions in appendix [ expfieldeq ] , eqs . ( [ exppi])-([expfi ] ) , but we quote below only their newtonian orders and the very few terms which will contribute to the poles @xmath251 in our dimensionally regularized calculations : @xmath252 the acceleration @xmath253 can thus be written as @xmath254 + \frac{4}{c^4}\biggl[\partial_i\hat x + 2\frac{d\hat r_i}{dt}+\cdots\biggr ] + \frac{16}{c^6}\biggl[\partial_i\hat t + \frac{d\hat y_i}{dt}+\cdots\biggr ] + { \mathcal{o}}\left(\frac{1}{c^{8}}\right).\nonumber\end{aligned}\ ] ] in section [ iteratedeinstein ] we shall give flesh to the formal expressions written above by explaining by which algorithm one can compute , with the required accuracy , the explicit @xmath1-dimensional expansions near @xmath255 [ analogous to the simple case ( [ u ] ) ] of the various elementary potentials entering eq . ( [ smallaccel ] ) , and notably of the crucial ones @xmath161 , @xmath160 ( to be computed with 1pn accuracy ) and @xmath164 , @xmath163 ( to be computed at newtonian order only ) . the main aim of this section is to complete the _ first step _ of the strategy outlined in the introduction , _ i.e. _ to collect a complete list of the additional contributions to the equations of motion which are specific consequences of the use of the _ extended _ hadamard regularization methods defined in @xcite . however , to do that we need to start by recalling some material concerning the hadamard regularization in 3 dimensions , and by contrasting it with dimensional regularization . such material is needed for understanding our computation based on the `` difference '' in section [ difference ] . we shall start by recalling the definition of the `` ordinary '' hadamard regularization and complete it by defining what we shall call the `` pure '' hadamard regularization . then we shall recall the main new features of the _ extended _ hadamard regularization defined in @xcite , and collect the additional contributions to the equations of motion which are specific consequences of the use of the extended hadamard regularization ( there are seven such additional contributions ) . the phrase `` hadamard regularization '' covers two distinct concepts : ( i ) the regularization of the `` limit '' @xmath256 where @xmath257 belongs to a class @xmath258 of singular functions ( generated by the iteration of einstein s equations ) , and ( ii ) the regularization of the 3-dimensional integral @xmath259 of some function @xmath260 . the class of functions @xmath261 consists of all functions @xmath262 on @xmath263 that are smooth except at @xmath75 and @xmath76 , around which they admit laurent - type expansions in powers of @xmath37 or @xmath38 ( see section ii.a of @xcite for the precise definition of @xmath261 ) . when @xmath264 ( _ i.e. _ , around singularity 1 ) we have , @xmath265 , @xmath266 where the landau @xmath267-symbol takes its usual meaning , and the @xmath268 s denote the coefficients of the various powers of @xmath37 , which are functions of the positions and velocities of the particles , and of the unit direction @xmath269 of approach to singularity 1 . the powers of @xmath37 are relative integers , @xmath270 , bounded from below by some typically negative @xmath271 depending on the @xmath272 in question . the hadamard `` _ _ partie finie _ _ '' of the singular function @xmath272 at the location of the singular point 1 ( first meaning of hadamard regularization ) is defined as the angular average of the zeroth - order coefficient in the expansion ( [ fx ] ) . it is denoted @xmath273 , so that @xmath274 where @xmath275 denotes the usual surface element on the 2-dimensional sphere centered on 1 . we shall employ systematically a bracket notation @xmath276 for the angular average of a function of the angles ( either @xmath277 or @xmath278).-dimensional ( as we shall see later there can be no confusion about this ) . ] a distinctive feature of the hadamard partie finie ( [ f1 ] ) is its `` non - distributivity '' in the sense that @xmath279 the non - distributivity represents a crucial departure away from the simple algebraic properties of the analog of @xmath273 in dim . . which is merely @xmath280 . it is an interesting fact that in a post - newtonian expansion the non - distributivity starts playing a role only at the 3pn order ( because the functions there become singular enough ) . up to the 2pn order one can show that @xmath281 for all the functions involved in the equations of motion in harmonic coordinates @xcite . several of the problems of the hadamard self - field regularization ( in the `` ordinary '' sense ) when applied at the 3pn level ( _ e.g. _ the occurrence of the unknown constant @xmath4 ) are related to the latter non - distributivity . the second notion of hadamard _ partie finie _ ( denoted @xmath282 in the following ) is to give a meaning to the generally divergent integral @xmath283 . in this work we shall have to consider only the ultra - violet ( uv ) divergencies of the integrals , _ i.e. _ , at the locations of the two local singularities @xmath75 and @xmath76 . all functions involved at the 3pn order are such that there are no infra - red ( ir ) divergencies when @xmath284 ( this is true not only in 3 dimensions but also for any dimension @xmath285 in a neighborhood of @xmath42 ) . the hadamard partie finie of the ( uv ) divergencies is then defined as @xmath286 the description of this formula in words is as follows . one first excises two _ spherical _ balls @xmath287 and @xmath288 surrounding the two singularities ( each one having the same radius @xmath289 ) , and one computes the integral on the volume external to these balls , _ i.e. _ , @xmath290 first term in eq . ( [ pf ] ) . that integral tends to infinity when @xmath291 , but we can subtract from it its purely divergent part , which is given by the additional terms in ( [ pf ] ) ( which obviously are to be duplicated when there are 2 singularities , _ cf . _ the symbol @xmath292 ) . the limit @xmath293 then exists ( by definition ) and defines hadamard s partie finie . notice the crucial dependence of the partie finie on two constants @xmath77 and @xmath78 entering the log - terms . these constants have the dimension of length . we shall say that @xmath77 is the regularization length scale associated to the hadamard regularization of the divergencies near @xmath255 ( similarly for @xmath78 ) . note also that the hadamard partie finie does not depend ( modulo changing the values of @xmath77 and @xmath78 ) on the _ shape _ of the regularization volumes @xmath294 and @xmath295 , above chosen as simple spherical balls ( see the discussion in ref . @xcite ) . an important consequence of the definition ( [ pf ] ) is that , in general , the integral of a gradient @xmath296 is not zero , because the surface integrals surrounding the singularities become infinite when the surface areas tend to zero , and may possess a finite part . we find ( see eq . ( 3.4 ) in @xcite ) @xmath297 for a general @xmath298 the r.h.s . is typically non - zero . this fact shows that the application of the ordinary hadamard regularization in the post - newtonian iteration has to be supplemented by a notion of distributional derivatives , in order to ensure that the integrals of gradients are zero like in the case of regular functions . notice that the constants @xmath77 and @xmath78 disappear from the result ( [ pfdiv ] ) . [ we shall also see the need , within dim . reg . , to consider some derivatives in the sense of distribution theory . ] let us apply the definition ( [ pf ] ) to the integral of a compact - support or `` contact '' term , _ i.e. _ , made of the product of some @xmath272 and a dirac delta - function at the point 1 . let us formally assume that and which takes the property ( [ pfdelta ] ) . this is exactly what we do in the context of the extended hadamard regularization . ] @xmath299 which is the most natural way , within hadamard s regularization , to give a sense to such integral . now the problem with that definition is that if we want to dispose of some _ local _ meaning ( at any field point @xmath241 ) for the product of @xmath272 with the delta - function , then as a consequence of the non - distributivity we can not simply equate @xmath300 with @xmath301 , _ i.e. _ : @xmath302 indeed , if it were true that @xmath303 [ for simplicity we denote @xmath304 , then multiplying by any @xmath132 we would have @xmath305 , and by integrating over @xmath263 following the rule ( [ pfdelta ] ) this would yield @xmath281 in contradiction with the violation of distributivity ( [ nondistr ] ) . therefore , both the violation of distributivity ( [ nondistr ] ) and the consequence ( [ deltanondistr ] ) are unescapable in the ordinary hadamard regularization . the previous situation should be contrasted with the @xmath1-dimensional case for which the distributivity is always satisfied , as we have simply @xmath306 and @xmath307 finally , taking the @xmath32 limit , we see that the dim . way of regularizing a three - dimensional `` contact term '' , _ i.e. _ a term like @xmath308 , is by considering it as the @xmath309 limit of its @xmath1-dimensional analogue ( [ contactd ] ) . that is , @xmath310 \equiv \left ( \lim_{d\rightarrow 3 } f^{(d ) } ( \mathbf{y}_1 ) \right ) \delta^{(3 ) } ( \mathbf{x } - \mathbf{y}_1)\ , , \label{purecontact}\ ] ] where @xmath311 is the @xmath1-dimensional version of @xmath272 , as obtained by solving einstein s equations in @xmath1 dimensions ( using the method explained in section [ iteratedeinstein ] below ) . there are no poles in the calculation of the `` contact '' terms in any of the potentials at the 3pn order so the limit @xmath309 in eq . ( [ purecontact ] ) always exists . once again the dim . prescription ( [ purecontact ] ) owns all the good features one wishes , notably the distributivity as we have emphasized in eqs . ( [ distrd])-([contactd ] ) . in the following it will be convenient , in order to compare the present dim .- reg . calculation with the hadamard - based work @xcite , to introduce the terminology _ pure hadamard _ regularization to refer to the following `` minimal '' version of the hadamard regularization : ( a ) an integral @xmath312 , where @xmath272 is made of some product of derivatives of the non - linear potentials @xmath136 , @xmath137 , @xmath313 , is regularized by the ordinary hadamard partie finie prescription ( [ pf ] ) , without bringing in any distributional contributions ( see below for the treatment of these ) ; ( b ) the regularization of a product of potentials @xmath136 , @xmath137 , @xmath314 , @xmath313 ( and their gradients ) is assumed to be distributive , which means that the value at the singular point @xmath75 of some polynomial in @xmath136 , @xmath137 , @xmath314 , @xmath313 and their gradients , say @xmath315 $ ] , is given by the replacement rule @xmath316\bigr)_1\longrightarrow { \cal f } [ ( v)_1 , ( v_i)_1 , ( \hat{w}_{ij})_1 , ( \partial_i v)_1 , \cdots]\,;\label{calf1}\ ] ] and ( c ) a contact term , _ i.e. _ of the form @xmath300 , appearing in the calculation of the _ sources _ of the non - linear potentials , is regularized by using the rule @xmath317 \delta^{(3)}(\mathbf{x}-\mathbf{y}_1)\longrightarrow { \cal f } [ ( v)_1 , ( v_i)_1 , ( \hat{w}_{ij})_1 , \cdots]\delta^{(3 ) } ( \mathbf{x}-\mathbf{y}_1)\,,\label{calfdelta}\ ] ] ( there are no gradients of potentials in the contact terms ) . the rules ( [ calf1])-([calfdelta ] ) of the pure - hadamard regularization are formally equivalent to assuming the replacement rules @xmath318 together with @xmath319 , in the case where @xmath272 and @xmath132 are made of products of our elementary potentials and their gradients . or @xmath320 ^ 3(\partial_iv)_1 $ ] , but not , for instance , @xmath321 ^ 2 $ ] . ] the rules of the pure hadamard regularization are , however , well defined , and are not submitted ( by their very definition ) to the consequences of the _ ordinary _ hadamard regularization ( [ nondistr ] ) and ( [ deltanondistr ] ) . note also that , as done in previous computations of the 3pn adm - hamiltonian @xcite and the 3pn binary s energy flux @xcite , one can formally use ( [ calf1])-([calfdelta ] ) at the price of adding a limited number of arbitrary parameters ( considered as unknown ) . let us give some reminders of the way we apply the considerations of section [ purehadamard ] to the computation of hadamard - regularized potentials having the form of poisson or poisson - like integrals . let us first discuss the prescription one has taken in @xmath42 to define the `` value at @xmath335 '' of a ( singular ) poisson potential @xmath336 . in @xmath42 , the poisson integral @xmath336 , at some field point @xmath337 , of some singular source function @xmath262 in the class @xmath338 is defined in the sense of the partie - finie integral ( [ pf ] ) , namely @xmath339 where @xmath77 and @xmath78 are the two constants introduced in eq . ( [ pf ] ) . at first sight we could think that a good choice for defining the pure hadamard value @xmath340_{\mathbf{x } ' = \mathbf{y}_1}$ ] is simply to replace @xmath341 in ( [ px ] ) , _ i.e. _ , @xmath342 however , the work on the 3pn equations of motion @xcite suggested that the definition ( [ py1 ] ) is not acceptable : it did not seem to be able to yield equations of motion compatible with basic physical properties such as energy conservation . the choice adopted in @xcite is to define the regularized `` value at @xmath335 '' of the function @xmath336 by taking the hadamard partie finie in the singular limit @xmath343 . notice first that @xmath336 does not belong ( in general ) to the class @xmath261 because the poisson integral will generate some _ logarithms _ of @xmath70 in its expansion when @xmath344 . thus , we shall have , rather than an expansion of type ( [ fx ] ) , @xmath345+o({r_1'}^n)\ , , \label{expandpx}\ ] ] where the coefficients @xmath346 and @xmath347 depend on the angles @xmath348 , and also on the constants @xmath77 and @xmath78 , in such a way that when combining together the terms in ( [ expandpx ] ) the constant @xmath70 always appears in `` adimensionalized '' form like in @xmath349 . then we define the hadamard partie finie at point 1 exactly in the same way as in eq . ( [ f1 ] ) , except that we now include a contribution linked to the ( divergent ) logarithm of @xmath70 , which is possibly present into the zeroth - order power of @xmath70 . more precisely , we define @xmath350 where we introduced a _ new regularization length scale _ denoted @xmath61 , which can be seen as some `` small '' but finite cut - off length scale [ so that @xmath351 in eq . ( [ p1def ] ) is a finite , but `` large '' cut - off dependent contribution ] . we shall see later that the dependence on @xmath61 disappears ( as it should ) when adding to @xmath352 the difference @xmath353 . to compute the partie finie one must apply the definition ( [ p1def ] ) to the poisson integral ( [ px ] ) , which involves evaluating correctly the angular integration therein . the result , proved in theorem 3 of @xcite , is @xmath354\bigl<\mathop{f}_1{}_{-2}\bigr>\ , . \label{p1}\ ] ] we recover in the first term the value of the potential at the point 1 : @xmath355 , given by eq . ( [ py1 ] ) . the supplementary term makes the partie finie to differ from the `` nave '' guess @xmath355 in a way which was found to play a significant role in the computations of @xcite . the apparent dependence of the result ( [ p1 ] ) on the scale @xmath77 is illusory . the @xmath77-dependence of the r.h.s . of eq . ( [ p1 ] ) cancels between the first and the second term , so the result depends only on the constants @xmath70 and @xmath78 , and we have in fact the following simpler rewriting of ( [ p1 ] ) , @xmath356 similarly the regularization performed at point 2 will depend on @xmath71 and @xmath77 , so that the binary s point - particle dynamics in hadamard s regularization depends on four ( _ a priori _ independent ) length scales @xmath70 , @xmath78 and @xmath71 , @xmath77 . the explicit expression of the result ( [ p1 ] ) is readily obtained from the definition of the partie - finie integral ( [ pf ] ) . we find ( see the details in ref . @xcite ) @xmath357 \bigl<\mathop{f}_1{}_{-2}\bigr>\nonumber\\ & -&\sum_{\ell=0}^{+\infty}\frac{(-)^\ell}{\ell!}\partial_l \left(\frac{1}{r_{12}}\right)\left[\sum_{p+\ell+3<0 } \frac{s^{p+\ell+3}}{p+\ell+3}\bigl < n_2^l\mathop{f}_2{}_p\bigr > + \ln\left(\frac{s}{s_2}\right)\bigl < n_2^l \mathop{f}_2{}_{-\ell-3}\bigr>\right]\biggr\}.\nonumber\\ \label{p1result}\end{aligned}\ ] ] note that the terms corresponding to singularity 2 involve the multipolar expansion around the point @xmath76 of the factor @xmath358 present into the integrand . ] employing our usual notation where capital letters denote multi - indices : @xmath359 , and , for instance , @xmath360 . the expansion is symmetric - trace - free ( stf ) because @xmath361 . here @xmath362 is a short - hand for @xmath363 partial derivatives @xmath364 of @xmath365 . the multipole expansion in @xmath1 dimensions ( also stf ) is given by eq . ( [ r1ofr2 ] ) below . ] because we work at the level of the equations of motion , many of the terms we shall need in this paper are in the form of the _ gradient _ of a poisson - like potential . for the gradient we have a formula analogous to ( [ p1 ] ) and given by eq . ( 5.17a ) of @xcite , namely @xmath366 where we have taken into account ( in the rewriting of the second line ) the always correct fact that the constant @xmath77 cancels out and gets `` replaced '' by @xmath70 . notice that in ( [ dip1 ] ) there is no additional term to the partie finie integral similar to the last term in ( [ p1 ] ) . the corresponding explicit expression is @xmath367\biggr\}. \nonumber\\ \label{dip1result}\end{aligned}\ ] ] finally we must also treat the more general case of potentials in the form of retarded integrals [ see eqs . ( [ potentialeq ] ) ] , but because we shall have to consider ( in section [ diff ] ) only the _ difference _ between the dimensional and hadamard regularizations , it will turn out that in fact the first - order retardation ( 1pn relative order ) is sufficient for this purpose . actually , in this paper we are not interested in radiation - reaction effects , so we shall use the symmetric ( half - retarded plus half - advanced ) integral . at the 1pn order we thus have to evaluate @xmath368 where @xmath336 is given by ( [ px ] ) , and where @xmath369 denotes ( two times ) the double or `` twice - iterated '' poisson integral of the second - time derivative , still endowed with a prescription of taking the hadamard partie finie , namely @xmath370 in the case of @xmath369 the results concerning the partie finie at point 1 were given by eqs . ( 5.16 ) and ( 5.17b ) of @xcite , [ q1]@xmath371 where the @xmath372 s denote the analogues of the coefficients @xmath373 , parametrizing the expansion of @xmath272 when @xmath374 , but corresponding to the double time - derivative @xmath375 instead of @xmath272 . [ in the following we shall not need the explicit forms of the results ( [ q1 ] ) . ] let us clarify an important point concerning the treatment of the repeated time derivative @xmath376 in eqs . ( [ q1 ] ) . as we are talking here about hadamard - regularized integrals ( which excise small balls around both @xmath75 and @xmath76 ) , the value of @xmath377 can be simply taken in the sense of ordinary functions , _ i.e. _ , without including eventual `` distributional '' contributions proportional to @xmath378 or @xmath379 and their derivatives . however , we know that such terms are necessary for the consistency of the calculation . this is why we must also include somewhere in our formalism the _ difference _ between the evaluation of these distributional terms in @xmath1 dimensions , and the specific distributional contributions issued from the generalized framework used in @xcite . this difference will be included in section [ additional ] below , among the complete list of additional contributions specifically related to the use of the extended regularization approach we shall now describe . the `` _ _ extended - hadamard _ _ '' regularization , proposed in refs . @xcite , tackles the particular properties of the ordinary hadamard regularization , notably the non - distributivity of eqs . ( [ nondistr ] ) and ( [ deltanondistr ] ) , and the fact that the integral of a gradient is not zero [ eq . ( [ pfdiv ] ) ] . these properties are implemented within a theory of pseudo - functions , _ viz _ linear forms defined on the set of singular functions @xmath261 . the use of pseudo - functions in this context enables one to give a precise meaning to the object @xmath380 needed in the computation of the contact terms , and which is otherwise ill - defined in distribution theory . furthermore the use of some generalized versions of distributional derivatives permits a systematic treatment of integrals and a natural implementation of the property that the integral of a gradient is always zero . in this paper we shall content ourselves with recalling the principle of the extended - hadamard regularization , and with presenting its `` ready - to - use '' consequences . to any @xmath298 we associate the `` partie finie '' pseudo - function @xmath381 , which is the linear form on @xmath261 defined by the duality bracket @xmath382 which means that the action of @xmath381 on any @xmath383 is the partie - finie integral , as given by ( [ pf ] ) , of the ordinary product . the pseudo - function @xmath381 reduces to a distribution in the ordinary sense of schwartz @xcite when restricted to the usual set @xmath384 of smooth functions with compact support on @xmath263 . the product of pseudo - functions coincides , by definition , with the ordinary point - wise product , namely @xmath385 . in the class of pseudo - functions constructed in ref . @xcite , the `` dirac - delta '' pseudo - function @xmath386 is defined by @xmath387 where @xmath273 denotes hadamard s partie finie ( [ f1 ] ) . this definition , which obviously yields a natural extension of the dirac function @xmath388 in the context of hadamard s regularization , leads also to new objects which have no equivalent in distribution theory , the most important one being the pseudo - function @xmath389 which played a crucial role in @xcite for the calculation of the compact - support parts of potentials as well as the purely distributional parts of derivatives . it is given by @xmath390 where one should be reminded that it is in general not allowed to replace the r.h.s . by the product of regularizations : @xmath391 . in the actual computation @xcite the pseudo - function @xmath389 acts always on smooth functions with compact support ( @xmath392 ) , in which case it reduces to a distribution in the ordinary sense , which was shown to admit the `` intrinsic '' form @xmath393 here @xmath394 denotes a multi - index composed of @xmath363 multipolar indices @xmath395 , @xmath396 means a product of @xmath363 partial derivatives @xmath397 , and @xmath398 a product of @xmath363 unit vectors ( we do not write the @xmath363 summation symbols , from 1 to 3 , over the indices composing @xmath399 ) . notice that the sum in eq . ( [ pffdelta ] ) is finite because @xmath272 admits some maximal order of divergency when @xmath400 . now we discover that the `` monopole '' term in the latter multipolar sum , having @xmath401 , is nothing but @xmath402 which is exactly the result we would get following the pure - hadamard regularization rule ( [ calfdelta ] ) . [ indeed , as we are considering here only the contact terms entering the source terms for the 3pn - level nonlinear potentials , the `` ordinary '' hadamard regularization @xmath273 coincides with the `` pure '' hadamard regularization ( [ calfdelta ] ) . ] the sum of the other terms then define what we can call some non - distributive contributions because their appearance is the direct consequence of the violation of distributivity ( [ nondistr ] ) . thus , @xmath403 in the work @xcite care has been taken of all such non - distributivity terms . consider for instance the poisson integral of a compact - support term @xmath389 ( say , proportional to the matter source densities @xmath143 , @xmath144 or @xmath149 ) . using ( [ pffdelta ] ) the poisson integral reads ) we assume that @xmath74 is distinct from the 2 singularities @xmath75 and @xmath76 ; see @xcite for more details . ] @xmath404 [ where @xmath405 should be better written @xmath406 . evaluating now the partie finie ( [ f1 ] ) at both singular points [ _ i.e. _ , when @xmath374 and @xmath407 we obtain [ poissonfdeltaab]@xmath408 the result ( [ poissonfdelta1 ] ) is in agreement with the pure - hadamard regularization ; however eq . ( [ poissonfdelta2 ] ) does involve some extra terms with respect to the pure - hadamard calculation , since the latter is easily seen to simply yield @xmath409 , which is nothing but the `` monopolar '' term @xmath401 of the multipolar sum in the r.h.s . of ( [ poissonfdelta2 ] ) . therefore we decompose ( [ poissonfdelta2 ] ) as @xmath410 the second ingredient of the extended - hadamard regularization concerns the treatment of partial derivatives in some ( extended ) distributional sense . essentially , one requires @xcite that the derivative reduces to the ordinary derivative in the case of regular functions , and is such that one can integrate by parts any integrals . the latter property ( valid for the spatial derivative ) translates into @xmath411 this rule contains the standard definition of the distributional derivative @xcite as a particular case . it implies the important property that the integral of a divergence is zero . let us pose @xmath412\ , , \label{derivedistr}\ ] ] where @xmath413 denotes the derivative of @xmath272 viewed as an `` ordinary '' pseudo - function , and @xmath414 $ ] represents the purely distributional part of the spatial derivative ( with support concentrated on @xmath75 or @xmath76 ) . looking for explicit solutions of the basic relation ( [ intparts ] ) we have found @xcite , with the help of eq . ( [ pfdiv ] ) : @xmath415 = 4\pi \ , \text{pf } \biggl ( n_1^i \biggl [ \frac{1}{2 } \ , r_1 \ , \mathop{f}_1{}_{-1}+\sum_{k\geq 0 } \frac{1}{r_1^k } \ , \mathop{f}_1{}_{-2-k}\biggl ] \delta_1 \biggr ) + 1 \leftrightarrow 2 \ , . \label{distrpart}\ ] ] notice that @xmath414 $ ] depends only on the _ singular _ coefficients of @xmath272 ( coefficients of negative powers of @xmath37 in the expansion of @xmath272 ) . the derivative operator defined by eqs . ( [ derivedistr])-([distrpart ] ) does not represent the unique solution of ( [ intparts ] ) , but it has been checked during the calculation @xcite that using another possible solution results in _ physically equivalent _ equations of motion at 3pn order ( _ i.e. _ , reducing to each other by a gauge transformation ) . concerning multiple derivatives we dispose of the general formula @xmath416=\sum_{k=1}^\ell\partial_{i_1\dots i_{k-1 } } \text{d}_{i_{k}}[\partial_{i_{k+1}\dots i_\ell}f]\ , , \label{multdistr}\ ] ] giving the distributional term associated with the @xmath363-th spatial derivative , @xmath417\equiv\partial_l\text{pf}f -\text{pf}\partial_l f$ ] ( where @xmath418 ) , in terms of the single derivative @xmath414 $ ] . as an example , to treat the second - derivative of the newtonian potential , @xmath419 where @xmath420 , one uses @xmath421=-\frac{4\pi}{3}\ , \text{pf}\left(\delta^{ij}+\frac{15}{2}\hat{n}_1^{ij}\right)\delta_1 \ , \label{ex}\ ] ] where @xmath422 . therefore the extended distributional derivative differs in general from the usual schwartz s derivative [ _ cf . _ the second term in ( [ ex ] ) ] . [ this is unavoidable if one wants to respect the basic rule of integration by parts ( [ intparts ] ) for general functions in the class @xmath261 . ] notice also that we do find a distributional term in the case of the first derivative : @xmath423=2\pi\ , \text{pf}\left(r_1 n_1^i \delta_1\right)$ ] . we recall also ( for future use ) the case of the partial time - derivative , @xmath424 $ ] , whose distributional term is given by ( following ref . @xcite ) @xmath425=v_1^i\mathop{\text{d}}_1{}_i[f ] + v_2^i\mathop{\text{d}}_2{}_i[f]\ , , \label{dtf}\ ] ] in terms of the partial derivatives with respect to the _ source _ points @xmath75 and @xmath76 , namely @xmath426 $ ] and @xmath427 $ ] . the explicit expression reads @xmath425 = -4\pi \ , \text{pf } \biggl ( ( n_1v_1)\biggl [ \frac{1}{2 } \ , r_1 \ , \mathop{f}_1{}_{-1}+\sum_{k\geq 0 } \frac{1}{r_1^k } \ , \mathop{f}_1{}_{-2-k}\biggl ] \delta_1 \biggr ) + 1 \leftrightarrow 2\ , , \label{distrtpart}\ ] ] where @xmath428 denotes the ordinary scalar product [ notice the over - all sign difference with respect to eq . ( [ distrpart ] ) ] . multiple time - derivatives can be treated accordingly to eq . ( [ multdistr ] ) . for instance , @xmath429 = \text{d}_t[\partial_tf]+\partial_t\text{d}_t[f]\ , . \label{dtt}\ ] ] following the regularization @xcite all the distributional terms [ of type @xmath389 and @xmath430 issued from the latter distributional derivatives are to be treated when computing the potentials according to the extended contact term definitions of eqs . ( [ poissonfdelta1])-([poissonfdelta2 ] ) . finally let us turn to the extension of the hadamard regularization ( introduced in @xcite ) concerning the definition of a new operation of regularization , denoted @xmath46_1 $ ] , consisting of performing the hadamard regularization @xmath273 within the spatial hypersurface that is geometrically orthogonal ( in a minkowskian sense ) to the four - velocity of the particle 1 . the regularization @xmath46_1 $ ] differs from @xmath273 by a series of relativistic corrections calculated in @xcite . together with the other improvements of the extended - hadamard regularization , it resulted in equations of motion in harmonic - coordinates which are manifestly lorentz invariant at the 3pn order @xcite . here we give a formula , sufficient for the present purpose , for expressing @xmath46_1 $ ] in terms of the basic regularization @xmath273 , defined by ( [ f1 ] ) , at the 1pn order : @xmath431_1=\biggl(f+\frac{1}{c^2}(\mathbf{r}_1.\mathbf{v}_1 ) \bigl[\partial_tf+\frac{1}{2}v^i_1\partial_i f\bigr]\biggr)_1+\mathcal{o}\left(\frac{1}{c^4}\right ) . \label{lorentzian}\ ] ] the first term is simply @xmath273 , while the other terms define a set of relativistic corrections required for ensuring the lorentz invariance of the final equations in hadamard s regularization . hence , we decompose ( [ lorentzian ] ) into @xmath431_1=(f)_1~+~\text{``lorentz '' contributions}\ , . \label{lorentzian'}\ ] ] after the reminders of the last subsections , we are now in position to explain the origin of all the contributions [ included in the final result ( [ a1bf ] ) ] which were due to the specific use of the extended - hadamard regularization . actually , we shall list here the contributions due to the use of the full prescriptions of @xcite with respect to those that would follow from using what we shall call a `` _ _ pure hadamard - schwartz _ _ '' ( phs ) regularization . by this we mean : ( 1 ) treating the contact terms of all the non - linear potentials @xmath136 , @xmath137 , @xmath314 , @xmath313 as in ( [ calfdelta ] ) [ we have checked that for all the potentials involved this is equivalent to ( [ purecontact ] ) ] ; ( 2 ) treating the distributional part of an integrand such as @xmath432 in the normal schwartz s distributional way , for instance ) . ] @xmath433 and evaluating the contact term generated by the delta function in the `` pure hadamard '' way ( [ calfdelta ] ) ; ( 3 ) regularize any three - dimensional integral by the ordinary hadamard prescription ( [ pf ] ) ; and , finally , ( 4 ) using systematically , in the last stage of the calculation where one replaces the metric into the geodesic equations , the pure hadamard replacement rule appearing in ( [ calf1 ] ) [ for instance , we write @xmath320 ^ 3(\partial_iv)_1 $ ] , creating therefore a net difference with respect to the ordinary and/or extended hadamard regularizations for which @xmath434 ^ 3(\partial_iv)_1 $ ] ] . the usefulness of the definition of such a phs regularization is to `` localize '' the additional contributions brought by dim . reg . to the occurrence of poles @xmath93 ( or `` cancelled '' poles ) in @xmath1 dimensions . our complete list of additional contributions contains seven items . first of all there are four `` non - distributivity '' contributions of the type given by eq . ( [ poissonfdelta2nondistr ] ) : 1 . the so - called `` self '' terms , for which the delta - function in @xmath389 comes from the purely distributional part of the distributional derivative given by ( [ distrpart ] ) . the self terms were derived in eq . ( 6.20 ) in @xcite ; they read explicitly @xmath435 , \label{deltaself0}\ ] ] where @xmath436 denotes the usual scalar product between @xmath67 and @xmath69 , and where @xmath437 . the expression ( [ deltaself0 ] ) can be rewritten in a simpler way as ( where we denote for simplicity @xmath438 ) @xmath439 2 . the so - called `` leibniz '' terms , which are additional contributions due to the extended distributional derivative , taking into account the violation of the leibniz rule when performing some simplifications of the non - linear potentials at the 3pn order ( see the explanations in section iii b in @xcite ) . the leibniz terms were written in eq . ( 6.19 ) in @xcite , and read @xmath440 we emphasize that the contributions ( [ deltaself ] ) and ( [ deltaleibniz ] ) represent some additive effects of the use of the distributional derivative introduced in ref . @xcite , when compared to the effect of the schwartz derivative in the phs regularization . note that both eqs . ( [ deltaself ] ) and ( [ deltaleibniz ] ) depend on the choice of distributional derivative , and we have given them here in the case of the `` particular '' derivative defined by ( [ distrpart ] ) . 3 . a special non - distributivity in the compact - support potential @xmath136 when it is computed at the 3pn order . in this case the @xmath389 comes simply from the compact - support point - particle source @xmath143 of the potential . ( 4.17 ) of @xcite gives for that term @xmath441 4 . a contribution coming from the compact - support part of the potential @xmath442 parametrizing the metric at the 3pn order , and derived at the end of section iv a in @xcite : @xmath443 besides the non - distributivity of the type ( [ poissonfdelta2nondistr ] ) , we have also the more `` direct '' non - distributivity due to the fact that the pure hadamard prescription for the regularization of the value of an expression `` at @xmath75 '' , eq . ( [ calf1 ] ) , differs from the ordinary and/or extended hadamard ones [ see for instance eq . ( [ u4had ] ) ] . it plays a role only in the last stage of the computation of the 3pn equations of motion , once we substitute all the potentials computed at the right pn order into the geodesic equations . we thus have 1 . a `` direct '' non - distributivity contribution , which can be called non - distributivity in the equations of motion ( eom ) , and given by eq . ( 6.34 ) in @xcite , @xmath444n_{12}^i-\frac{779}{420}\frac{g^3m_1 ^ 2m_2}{c^6}\ , v_{12}^{jk}\,\partial_{ijk}\left(\frac{1}{r_{12}}\right ) , \label{deltaeom}\ ] ] where @xmath445 and @xmath446 . this term involves some combinations of masses different from those in eqs . ( [ deltaself])-([deltat ] ) . note that because @xmath391 the non - distributivity in the eom depends on which prescription has been chosen for the stress - energy tensor of point - particles . ( [ deltaeom ] ) corresponds to the particular prescription advocated in section v of @xcite . however it was checked in @xcite that different prescriptions yield physically equivalent equations of motion . the next correction brought about by the extended - hadamard regularization is the one due to the regularization @xmath46_1 $ ] , performed in the lorentzian rest frame of the particle . in practice the effect of such `` lorentzian '' regularization boils down to applying eq . ( [ lorentzian ] ) . it turned out that the only new contribution of this type came from the regularization of the potential @xmath447 at the 1pn order [ and also when deriving the result for @xmath448 , which is galilean - invariant , in eq . ( [ deltaeom ] ) ] , leading to 1 . the so - called `` lorentz '' contribution to the acceleration , given by eq . ( 5.35 ) in @xcite as @xmath449 \partial_{ijk}\left(\frac{1}{r_{12}}\right ) . \label{deltalorentz}\ ] ] this term has been crucial for ensuring the lorentz invariance of the final 3pn equations of motion in @xcite . finally , one must also take care of one additional contribution ( with respect to the `` phs '' definitions ) due to the non - schwartzian way of treating distributional derivatives . we have already mentioned two contributions coming from this origin : ( i ) and ( ii ) above . actually , there is a third one with the same origin and which comes from our computation ( see section [ difference ] below ) of the `` difference '' between the dimensional and hadamard regularizations of retarded potentials , namely the crucial potentials @xmath161 and @xmath450 which must both be expanded to 1pn fractional accuracy . more precisely , this contribution is due to the repeated time - derivative operator @xmath451 coming when expanding the time - symmetric green function of the dalembertian as @xmath452 . we shall explicitly exhibit in eqs . ( [ drdirresult ] ) below the way these derivatives enter our calculation of the difference . for technical reasons the time - derivative @xmath451 must be kept _ inside _ the integrals , so it has to be considered in a distributional sense , and we have therefore to take into account the different ways of treating the distributional derivatives in both regularizations . in the extended - hadamard regularization the distributional terms are given by @xmath453 $ ] which is shown in eqs . ( [ dtf])-([dtt ] ) , and when they enter the source of some poisson - type integral they are evaluated according to eqs . ( [ poissonfdeltaab ] ) . on the other hand , in one uses the ordinary schwartz derivative ( in @xmath1 dimensions ) which is described in section [ distrderiv ] . in this case the double time - derivative @xmath451 is computed with the help of the gelfand - shilov formulae ( [ gelfand])-([gelfandt ] ) below . when examining the difference between the contact terms in @xmath453 $ ] and those issued from @xmath454 , we find that only the source for the 1pn potential @xmath161 ( or rather for the combination @xmath455 which enters the equations of motion ) contributes . this gives : 1 . the following `` time - derivative '' contribution to the acceleration , @xmath456 this term was part of the final result of @xcite . however it is not mentioned in @xcite because this reference never tried to compare the results of the extended distributional derivative with those given by the ordinary schwartz derivative in 3 dimensions , except in those cases , _ items _ ( i ) and ( ii ) above , for which the schwartz derivative yielded in fact some ill - defined ( formally infinite ) expressions in 3 dimensions . [ the latter expressions turn out to be rigorously zero when computed in dimensional regularization . ] in summary , there are in all _ seven _ different terms , ( i)(vii ) , which are specifically due to the extended version of the hadamard regularization . the `` pure - hadamard - schwartz '' equations of motion are then obtained from the end result of @xcite , _ i.e. _ , @xmath60 given by eq . ( 7.16 ) of @xcite , by subtracting these terms . therefore we define ( see also section [ purehadstat ] below ) @xmath457 and _ idem _ with @xmath292 for the other particle . in this section we come to the core of our technique for evaluating the difference between the @xmath1-dimensional equations of motion and their pure - hadamard - schwartz expressions , defined above and given in practice by eq . ( [ defph ] ) . let us start by indicating how we solved ( with sufficient accuracy ) einstein s field equations in @xmath1 dimensions . one writes the post - minkowskian expansion of einstein s equations in the guise of explicit formulae for the elementary potentials @xmath458 , as given in section [ fieldeq ] . note that it is crucial to take into account the explicit @xmath1-dependence of the coefficients entering these equations . the first step of the formalism is to get sufficiently accurate explicit expressions for the basic linear potentials @xmath136 and @xmath137 . as we do not need to consider here radiation reaction effects ( which do not mix with the uv divergencies arising at the 3pn level ) it is enough to solve eqs . ( [ vlowest ] ) by means of the pn expansion of the time - symmetric green function . for instance , we have @xmath459 from eq . ( [ tmunu ] ) we see that the source @xmath143 reads @xmath460 + 1 \leftrightarrow 2\,,\ ] ] where @xmath461 note the presence of many `` contact '' evaluations of field quantities in @xmath143 . such terms are unambiguously defined in dimensional regularization . they are computed by successive iterations ( _ e.g. _ to get @xmath462 to 1pn fractional accuracy we need to have already computed @xmath463 to order @xmath464 included ) . those evaluations do not give rise to pole terms in @xmath143 , up to the 3pn accuracy . hence , as we said above , we can consider that their @xmath32 limits define a certain ( three - dimensional ) way of estimating contact terms , that we have checked to be in full agreement with the `` pure - hadamard '' prescription defined in the previous section . coming now to the spatial dependence of the scalar potential @xmath136 we get from eq . ( [ gsigma ] ) @xmath465 + \cdots + 1 \leftrightarrow 2\ , , \label{vexpand}\ ] ] where we introduced the following elementary solutions @xmath466 , @xmath467 , etc . , whose explicit forms are [ u1v1]@xmath468 where @xmath469 is related to the usual eulerian @xmath470-function by adopted here is related by @xmath471 to the constant @xmath472 chosen in @xcite . our present choice is motivated by the easy - to - remember fact that @xmath473 . ] @xmath474 inserting the explicit expression ( [ vexpand ] ) of @xmath136 into , say , the non - linear terms in the r.h.s . of eq . ( [ dalwij ] ) , yields a dalembert equation for the non - linear potential @xmath159 with a `` source function '' which is the sum of some contact terms @xmath475 and of an extended non - linear source @xmath476 which belongs to the @xmath1-dimensional analogue of the class @xmath258 , say @xmath477 . more precisely , at each stage of the iteration we find inhomogeneous wave equations of the type @xmath478 where the extended source function @xmath476 is regular everywhere except at the points @xmath75 and @xmath76 , in the vicinity of which it admits an expansion of the general form ( @xmath479 ) @xmath480 where @xmath481 and @xmath482 are relative integers ( @xmath483 ) , whose values are limited by some @xmath271 , @xmath484 and @xmath485 as indicated . the expansion ( [ fd ] ) differs from the corresponding expansion in 3 dimensions , as given in ( [ fx ] ) , by the appearance of integer powers of @xmath486 where @xmath35 . the coefficients @xmath487 depend on the unit vector @xmath277 in @xmath1 dimensions , on the positions and coordinate velocities of the particles , and also on the characteristic length scale @xmath90 of dimensional regularization . because @xmath488 when @xmath309 we necessarily have the constraint ( @xmath489 ) @xmath490 the iteration continues by inverting the wave operator by means of the time - symmetric expansion ( [ gsigma ] ) . the basic terms of this expansion which will turn out to be crucial for our 3pn calculation based on the _ difference _ are in fact the first two terms . focussing on the terms generated by the extended source @xmath476 ( rather than the simpler contact terms ) we can write the @xmath1-dimensional analogue of ( [ rpq ] ) as @xmath491 = p^{(d)}(\mathbf{x } ' ) + \frac{1}{2c^2 } q^{(d ) } ( \mathbf{x } ' ) + \mathcal{o}\left(\frac{1}{c^4}\right ) , \label{rd}\ ] ] where the @xmath1-dimensional poisson integral of @xmath311 reads @xmath492\equiv -\frac{\tilde{k}}{4\pi } \int\frac{d^d\mathbf{x}}{\vert\mathbf{x}-\mathbf{x}'\vert^{d-2 } } f^{(d)}(\mathbf{x})\ , . \label{pdx}\ ] ] we have used the fact , already mentioned above , that the @xmath1-dimensional elementary solution of the laplacian reads @xmath493 ( see appendix [ formulae ] for a proof of eq . ( [ green ] ) and for other useful formulae valid in @xmath1 dim . ) , while the 1pn term is given by @xmath494= -\frac{\tilde{k}}{4\pi(4-d ) } \int d^d\mathbf{x}\,\vert\mathbf{x}-\mathbf{x}'\vert^{4-d } \partial_t^2f^{(d)}(\mathbf{x})\ , . \label{qdx}\ ] ] note the important point that in @xmath1 dimensions , as in 3 dimensions , the time - derivative operator @xmath451 present in the integrand of ( [ qdx ] ) is to be considered in the sense of distributions ( see further discussion in section [ distrderiv ] below ) . an important technical aspect of the @xmath1-dimensional pn iteration of the elementary potentials @xmath495 is the existence of the generalization ( [ green ] ) of the usual green s function for the laplace equation , as well as of its higher pn analogues @xmath496 , allowing one to explicitly compute the spatial dependence of the _ linear _ potentials @xmath136 , @xmath137 and @xmath158 for instance . however , starting with @xmath314 we need to poisson - integrate _ non - linear _ sources , such as @xmath497 . in three dimensions , these non - linear contributions are reducible to the knowledge of the basic non - linear potential @xmath498 , such that @xmath499 . we have succeeded in explicitly computing the @xmath1-dimensional analogue of the @xmath498 potential , namely @xmath500 our result is reported in appendix [ littleg ] . as indicated there , if we wished to explicitly compute some of the higher pn potentials needed to write the closed form of the non - linear sources relevant to the 3pn equations of motion , we should extend the calculation of the potential @xmath118 to the potentials @xmath501 and @xmath502 of appendix [ littleg ] . luckily , it is not needed to use a closed - form expression for any of the non - linear potentials . indeed , similarly to what was used long ago @xcite when discussing the iteration generated by riesz - type sources , eq . ( [ triesz ] ) , one can control the uv _ singular _ part of @xmath503 from the knowledge of the uv singular part of its non - linear source @xmath262 . for any small enough value of @xmath86 . ] more precisely , in the vicinity say of @xmath75 , at each iteration stage we can decompose the source in @xmath504 where the _ singular _ part @xmath505 ( with respect to @xmath75 ) is a sum of terms of the form eq . ( [ fd ] ) , which are not ( in the limit @xmath32 ) smooth functions of @xmath506 , and where the _ regular _ part @xmath507 is a smooth @xmath508 function of @xmath506 . [ the simplest example of this decomposition is eq . ( [ u ] ) with , near point @xmath75 , @xmath509 and @xmath510 . ] if , for concreteness , we then consider @xmath511 , the above decomposition entails a corresponding decomposition of @xmath512 , and it is easy to see that @xmath513 . from this result , we can uniquely determine @xmath514 from @xmath515 using , _ e.g. _ , the formula ( [ intform ] ) in appendix [ formulae ] . this local procedure does not allow one to compute the regular part of the poisson potential in @xmath1 dimensions . fortunately , thanks to particular simplifications that occur in the structure of einstein s field equations , the knowledge of @xmath516 in @xmath1 dimensions for the complicated non - linear sources is not needed . indeed , one can see on our explicit solution of einstein s field equations at 3pn given in section [ fieldeq ] that there are no `` quartically non - linear '' source terms of the form , say , @xmath517 or @xmath518 for @xmath169 at the 3pn order ( see fig . [ fig5 ] below ) . as explained in section [ dimregstat ] below , a nice way to understand the origin of the poles @xmath519 appearing in the 3pn equations of motion is to use a diagrammatic representation . a pole can arise in @xmath80 only when three propagator lines ( including the extra one coming from @xmath520 when solving @xmath521 non - linear source ) can all shrink towards the first world - line . if terms of the type above ( _ e.g. _ @xmath522 ) were present in the source one could have a diagram where the three shrinking propagators come from @xmath520 , @xmath523 and @xmath524 . then @xmath525 $ ] would remain as an external attachment to this diagram ( and would then fork into two `` feet '' on the second world - line ) . in view of the pole @xmath526 ( with @xmath35 ) arising from the triplet of shrinking propagators , one would need to know @xmath527 $ ] up to @xmath36 accuracy , _ i.e. _ , @xmath527 = \text{reg}^{(3 ) } [ \hat{w}_{ij } ( \mathbf{x } ) ] + \varepsilon \ , \hat{w}'_{ij } ( \mathbf{x } ) + { \cal o } ( \varepsilon^2)$ ] [ in which @xmath528 is defined by this expansion ] . if such a term had been present we would have needed to use the full @xmath1-dimensional , globally determined @xmath498-potential given in appendix [ littleg ] to determine @xmath529 , which would have entered the final , renormalized equations of motion . however , because all such terms are absent at the 3pn order , the only external attachments to the dangerous shrinking diagrams are simple lines , such for instance as the lines ending on @xmath530 in figs . [ fig2]d , [ fig3]b or [ fig4]b presented below . such lines do need to be evaluated to accuracy @xmath36 , but this is easy because they represent linear potentials such as @xmath136 or @xmath137 which are known in dimension @xmath1 _ via _ eq . ( [ vexpand ] ) . in conclusion , the algorithm we use to solve , with sufficient accuracy , einstein s equations in @xmath1 dimensions consists of : ( 1 ) starting from the fully @xmath1-dimensional expressions for the linear potentials @xmath136 , @xmath137 ( and more generally for the parts of the non - linear potentials with delta - function sources ) ; ( 2 ) determining the local expansions , near @xmath40 , of the singular parts of the non - linear potentials by inverting @xmath531 _ via _ formulae ( [ intform ] ) of appendix [ formulae ] ; ( 3 ) completing @xmath532 by adding to @xmath533 the limit when @xmath534 of @xmath535 , namely @xmath536 which is known from the previous work on the 3pn equations of motion in 3-dimensions @xcite . note that we denote by @xmath536 a formal @xmath1-dimensional function , @xmath537 , the explicit expression of which in terms of @xmath37 , @xmath277 , etc . coincides with its 3-dimensional counterpart . for instance @xmath538 denotes the usual regular part of @xmath539 , obtained by subtracting from @xmath540 the two three - dimensional locally singular expansions of @xmath541 around @xmath75 and @xmath76 as given by the @xmath32 limit of eq . ( [ eqd9 ] ) and its @xmath542 analog . after this double subtraction , @xmath538 is considered as a function in @xmath543 , and we can use as approximation to @xmath544 the explicit expression @xmath545 . more generally , in our calculations we use as approximation to @xmath546 [ which symbolizes here the non - linear potentials @xmath547 and @xmath548 at newtonian order , and @xmath314 at the 1pn order ] the expression @xmath549 . evidently , the subtraction of the singular part needs to be performed only up to some finite order in @xmath550 and @xmath551 . we have checked the choice we made of @xmath7 in each calculation by doing two separate calculations for the values @xmath7 and @xmath552 , and checking that the corresponding final results are the same . we performed also direct checks of the independence of the final results on the precise @xmath1-dimensional extensions of the `` regular '' part of the non - linear potentials , such as @xmath553 [ in which @xmath554 is defined by this expansion ] . we systematically added in all our non - linear potentials @xmath547 , @xmath313 , @xmath314 some smooth contributions to @xmath536 vanishing with @xmath36 , _ i.e. _ , some substitutes for the actual @xmath554 . these `` substitutes '' were determined in such a way that ( i ) they are homogeneous solutions of the dalembertian equation at the required post - newtonian order , ( ii ) the differential identities obeyed by the potentials in @xmath1 dimensions , eqs . ( [ divi])-([djwij ] ) , are indeed satisfied up to the order @xmath36 , and with the required precision @xmath7 in powers ) has been done only in the vicinity of the two particles . ] of @xmath37 or @xmath38 . and we checked that our final results are totally insensitive to the introduction of such substitutes for the function @xmath554 . finally , when evaluating the equations of motion , as given by eq . ( [ smallaccel ] ) , we must evaluate the value at @xmath555 of many terms given either by poisson integrals of the form ( [ pdx ] ) or their 1pn generalizations ( [ qdx ] ) . this is quite easy to do in dim . reg . , because the nice properties of analytic continuation allow simply to get @xmath556_{\mathbf{x } ' = \mathbf{y}_1}$ ] ( say ) by replacing @xmath74 by @xmath75 in the explicit integral form ( [ pdx ] ) . finally , we simply have for the values at @xmath335 of the potentials , [ pqd ] @xmath557 as well as for their spatial gradients , [ dpqd]@xmath558 as said above , the main technical step of our strategy will then consist of computing the _ difference _ between such @xmath1-dimensional poisson - type potentials ( [ pqd ] ) or ( [ dpqd ] ) , and their `` pure hadamard - schwartz '' 3-dimensional counterparts , which were already obtained in section [ hadamardpoisson ] . we denote the difference between the prescriptions of dimensional and `` pure hadamard - schwartz '' regularizations by means of the script letter @xmath384 . given the results @xmath352 and @xmath559 of the two regularizations [ respectively obtained in eqs . ( [ p1 ] ) and ( [ pd ] ) ] we pose @xmath560 that is , @xmath561 is what we shall have to _ add _ to the pure hadamard - schwartz result ( [ defph ] ) in order to get the correct @xmath1-dimensional result . note that , in this paper , we shall only compute the first two terms , @xmath562 , of the laurent expansion of @xmath561 when @xmath563 . this is the information we shall need to fix the value of the parameter @xmath4 . we leave to future work an eventual computation of the @xmath1-dimensional equations of motion as an exact function of the complex number @xmath1 . similarly to the evaluation of the difference @xmath564 $ ] in ref . @xcite , the difference ( [ dp1 ] ) can be obtained by splitting the @xmath1-dimensional integral ( [ pd ] ) into three volumes , two spherical balls @xmath565 and @xmath566 of radius @xmath289 and centered on the two singularities , and the external volume @xmath567 . when @xmath309 ( with fixed @xmath289 ) , @xmath565 and @xmath566 tend to the regularization volumes @xmath287 and @xmath288 we introduced in eq . ( [ p1result ] ) . consider first , for a given value @xmath568 , the external integral , over @xmath569 . [ if wished , two balls with different radii could be used , with the same result . ] since the integrand is regular on this domain , it is clear that the external integral reduces in the limit @xmath104 to the one in 3 dimensions that is part of the hadamard regularization ( [ p1result ] ) . so we can write ( for any @xmath568 ) @xmath570 and we see that when computing the difference @xmath561 the exterior contributions will cancel out modulo @xmath571 . thus we obtain , after this preliminary step [ following eq . ( [ p1result ] ) ] , @xmath572 \bigl<\mathop{f}_1{}_{-2}\bigr>\nonumber\\ & & + \sum_{\ell\geq 0}\frac{(-)^\ell}{\ell!}\partial_l \left(\frac{1}{r_{12}}\right)\left[\sum_{p\leq -\ell-4}\frac{s^{p+\ell+3}}{p+\ell+3}\bigl < n_2^l \mathop{f}_2{}_p\bigr>+\ln\left(\frac{s}{s_2 } \right)\bigl < n_2^l\mathop{f}_2{}_{-\ell-3}\bigr>\right]\biggr\ } \nonumber\\ & & + \mathcal{o}(\varepsilon)\ , . \label{dp1result}\end{aligned}\ ] ] see section iv of ref . @xcite for a careful justification of the formal interversions of limits @xmath573 and @xmath563 that we shall do here . the point is that in order to obtain the difference @xmath561 we do not need the expression of @xmath311 for an arbitrary source point @xmath574 but only in the vicinity of the two singularities : indeed the two local integrals over @xmath565 and @xmath566 in eq . ( [ dp1result ] ) can be computed by replacing @xmath311 by its expansions when @xmath400 and @xmath575 respectively . we substitute the @xmath37-expansion eq . ( [ fd ] ) into the local integral over @xmath565 , and integrate that expansion term by term . this readily leads to @xmath576 where we still use the bracket notation to denote the angular average , but now performed in @xmath1 dimensions , _ i.e. _ , @xmath577 here @xmath578is the solid angle element around the direction @xmath277 , and @xmath579 is the volume of the unit sphere with @xmath580 dimensions ( see appendix [ formulae ] for more discussion ) . to derive ( [ int1 ] ) we used the following relation linking @xmath469 and @xmath581 , @xmath582 concerning the other local integral , over @xmath566 , things are a little bit more involved because we need to perform a multipolar re - expansion of the factor @xmath583 present in that integral around the point @xmath76 . writing down this multipole expansion presents no problem , and in symmetric - trace - free ( stf ) form it reads in @xmath1 dimensions ( in the sense of functions ) . see appendix [ formulae ] for a compendium of @xmath1-dimensional formulae on stf expansions . see also eq . ( [ eqd6 ] ) in appendix [ littleg ] . ] @xmath584 the multipole expansion being then correctly taken into account , we obtain @xmath585 as we can see , simple poles @xmath87 will occur into our two local integrals , as determined by ( [ int1 ] ) and ( [ int2 ] ) , only for the `` critical '' values @xmath586 and @xmath587 respectively . next we replace the explicit expressions ( [ int1 ] ) and ( [ int2 ] ) into the formula ( [ dp1result ] ) we had for the `` difference '' . as expected we find that the divergencies when @xmath293 , some value @xmath588 being given , cancel out between eqs . ( [ int1])-([int2 ] ) and the remaining terms in ( [ dp1result ] ) , so that the result is finite for any @xmath588 . furthermore , we find that if we neglect terms of order @xmath571 , the only contributions which remain are the ones coming from the poles ( and their associated finite part ) , _ i.e. _ , for the latter critical values @xmath586 in the case of singularity 1 and @xmath587 in the case of singularity 2 . the other contributions in ( [ int1 ] ) and ( [ int2 ] ) have a finite limit when @xmath104 which is therefore cancelled by the corresponding terms in hadamard s regularization . as a result we obtain the following closed - form expression for the difference , which will constitute the basis of all the practical calculations of the present paper , @xmath589\right)\bigl<\mathop { f}_1{}_{-2,q}^{(\varepsilon)}\bigr>\nonumber\\ & -&\frac{1}{\varepsilon ( 1+\varepsilon)}\sum_{q_0\leq q\leq q_1}\left(\frac{1}{q+1}+\varepsilon\ln s_2\right ) \sum_{\ell=0}^{+\infty}\frac{(-)^\ell}{\ell!}\partial_l \left(\frac{1}{r_{12}^{1+\varepsilon}}\right)\bigl < n_2^l\mathop{f}_2{}_{-\ell-3,q}^{(\varepsilon)}\bigr>\nonumber\\ & + & \mathcal{o}(\varepsilon)\ , . \label{dp1total}\end{aligned}\ ] ] notice that ( [ dp1total ] ) depends on the two `` constants '' @xmath590 and @xmath591 . as we shall check these @xmath592 and @xmath591 will exactly cancel out the same constants present in the `` pure - hadamard '' calculation , so that the dimensionally regularized acceleration will be finally free of the constants @xmath70 and @xmath78 . note also that the coefficients @xmath487 and @xmath593 in @xmath1 dimensions depend on the length scale @xmath90 associated with dimensional regularization [ see eq . ( [ l0 ] ) ] . taking this dependence into account one can verify that @xmath70 and @xmath78 in ( [ dp1total ] ) appear only in the combinations @xmath594 and @xmath595 . let us give also ( without proof ) the formula for the difference between the _ gradients _ of potentials , _ i.e. _ , @xmath596 the formula is readily obtained by the same method as before , and we have @xmath597 formulae ( [ dp1total ] ) and ( [ ddip1total ] ) correspond to the difference of poisson integrals . but we have already discussed that we shall need also the difference of inverse dalembertian integrals at the 1pn order . to express as simply as possible the 1pn - accurate generalizations of eqs . ( [ dp1total ] ) and ( [ ddip1total ] ) , let us define two _ functionals _ @xmath598 and @xmath599 which are such that their actions on any @xmath1-dimensional function @xmath311 is given by the r.h.s.s of eqs . ( [ dp1total ] ) and ( [ ddip1total ] ) , _ i.e. _ , so that [ calffi ] @xmath600\ , , \label{calf}\\ \mathcal{d}\partial_ip(1)&=&\mathcal{h}_i\left[f^{(d)}\right]\ , . \label{calfi}\end{aligned}\ ] ] the difference of 1pn - retarded potentials and gradients of potentials is denoted [ drdir ] @xmath601 where in 3 dimensions the potential @xmath602 is defined by eq . ( [ rpq ] ) and the regularized values @xmath603 and @xmath604 follow from ( [ p1 ] ) , ( [ dip1 ] ) , ( [ q1 ] ) , and where in @xmath1 dimensions @xmath605 and @xmath606 are given by ( [ rd ] ) , ( [ pqd ] ) , ( [ dpqd ] ) . with this notation we now have our result , which will be stated without proof , that the difference in the case of such 1pn - expanded potentials reads in terms of the above defined functionals @xmath598 and @xmath599 as [ drdirresult ] @xmath607 -\frac{3}{4c^2}\bigl<\mathop{k}_1{}_{-4}\bigr > + \mathcal{o}\left(\frac{1}{c^4}\right ) , \label{dr1result}\\ \mathcal{d}\partial_ir(1)&=&\mathcal{h}_i\left[f^{(d ) } -\frac{r_1 ^ 2}{2c^2(d-2)}\partial_t^2f^{(d)}\right ] -\frac{1}{4c^2}\bigl < n_1^i\mathop{k}_1{}_{-3}\bigr > + \mathcal{o}\left(\frac{1}{c^4}\right ) . \label{ddir1result}\end{aligned}\ ] ] these formulae involve some `` effective '' functions which are to be inserted into the functional brackets of @xmath598 and @xmath599 . beware of the fact that the effective functions are not the same in the cases of a potential and the gradient of that potential . note the presence , besides the main terms @xmath608 $ ] and @xmath609 $ ] , of some extra terms , purely of order 1pn , in eqs . ( [ drdirresult ] ) . these terms are made of the average of some coefficients @xmath372 of the powers @xmath610 in the expansion when @xmath400 of the _ second - time - derivative _ of @xmath272 , namely @xmath375 . they do not seem to admit a simple interpretation . they are important to get the final correct result . let us end this section by explaining in more details how we dealt with distributional derivatives in @xmath1 dimensions . first , it is clear that if we were dealing with @xmath1-dimensional integrals of the type @xmath611 where @xmath612 is some ( formally ) everywhere smooth function of @xmath537 , with fast enough decay at infinity , and where @xmath613 is the elementary newtonian potential in @xmath1 dimensions [ see eqs . ( [ u1 ] ) above ] , we should , in a straightforward @xmath1-continuation of schwartz distributional derivatives , consider that @xmath614 contains , besides an `` ordinary '' singular function @xmath615 ( treated as a pseudo - function in the sense of schwartz ) , a distributional part proportional to @xmath616 . in other words , we would write [ du1v1]@xmath617 where the indication `` ord '' refers to the `` ordinary '' ( pseudo - function ) part of the repeated derivative . we have also added the corresponding result for the fourth derivatives of the `` less singular '' kernel @xmath618 , eqs . ( [ v1 ] ) . note that the decompositions above of @xmath619 or @xmath620 into `` ordinary '' and `` distributional '' pieces arise because of our working in ( @xmath1-dimensional ) @xmath241-space , and of explicitly computing some derivatives , say as @xmath621 \ , r_1^{n-2}$ ] . if we were working in the ( @xmath1-dimensional ) fourier - transform space @xmath622 ( which is where dimensional continuation is most clearly defined @xcite ) , the corresponding decomposition would be simply algebraic : _ e.g. _ @xmath623 , where @xmath624 denotes the stf part of @xmath625 . the decompositions ( [ du1v1 ] ) are clearly needed when dealing with simple integrals of the type ( [ i ] ) ( with a smooth @xmath626 ) to ensure consistency with the requirement that one may integrate by parts ( which is one of the defining properties of dim . @xcite ) , and we shall therefore employ it , when applicable . on the other hand , most of the singular integrals that we have to deal with look like ( [ i ] ) but contain a _ singular _ function @xmath627 , of the type of eq . ( [ fd ] ) . it is , however , a very simplifying feature of dim . reg . that when considering integrals like ( [ i ] ) with some _ singular _ @xmath626 we can simply ignore any distributional contributions @xmath628 or its derivatives . indeed , as long as the integer @xmath482 in the powers @xmath629 present in ( [ fd ] ) is different from zero ( which is precisely the case of all delicate terms involving several propagators shrinking towards a particle world - line ) , the `` singular '' expansion ( [ fd ] ) can be considered , in dim . , as defining a sufficiently smooth function [ by taking both @xmath630 and @xmath7 large enough in ( [ fd ] ) ] which _ vanishes _ , as well as its derivatives , at @xmath255 . therefore , all the `` dangerous '' terms of the form @xmath631 unambiguously vanish in dim . let us now consider the consequences of this fact for the time derivatives occurring in expansions such as eqs . ( [ drdirresult ] ) . the distributional time - derivatives , acting in our present example on @xmath632 or @xmath633 , _ i.e. _ , on functions of @xmath634 , can be treated in a simple way from the rule @xmath635 applicable to the purely distributional part of the derivative . for instance we can write [ dtu1v1]@xmath636 we have checked using these formulae that all the @xmath1-dimensional terms coming from second - order derivatives of potentials , taken in the distributional sense ( for instance the term @xmath637 in the source of the @xmath638-potential , but also fourth - order derivatives acting on @xmath633 . ] ) yield the _ same _ purely distributional contributions , in the limit @xmath563 , as the ones that would be computed using what we called above a `` pure schwartz '' , three - dimensional computation of such contributions [ to `` smooth '' integrals ( [ i ] ) ] . on the other hand , the extended version of distributional derivatives introduced in @xcite does yield some specific additional contributions , two of which were already mentioned in @xcite and are reported in the _ items _ ( i ) and ( ii ) of section [ additional ] above , and a third one ( also included in @xcite ) which comes in connection with the second time - derivatives in our formulae for the difference , eqs . ( [ drdirresult ] ) . let us indicate here that the distributional second - time - derivatives in @xmath1 dimensions have been obtained by using the following ( generalizations of ) gelfand - shilov formulae @xcite , valid for general functions @xmath639 admitting some expansions of the type ( [ fd ] ) : namely , for the spatial derivative , @xmath640 where @xmath641 is the @xmath363-th partial derivative of the @xmath1-dimensional dirac delta - function at the point 1 ( @xmath359 ) and where the angular average is performed over the @xmath147-dimensional sphere having total volume @xmath581 ; and , concerning the time derivative , @xmath642 from the latter formula one deduces the second time - derivative in a way similar to eqs . ( [ dtt ] ) . we have indicated in the _ item _ ( vii ) of section [ additional ] the correction it leads to when comparing with the extended - hadamard prescription for the second time - derivative , and we have subtracted it from @xmath60 to define the pure hadamard - schwartz result ( [ defph ] ) . therefore , we consistently do not need to include such an effect into the differences @xmath643 discussed here . finally we are now in position to obtain the supplement of acceleration @xmath91 induced by dimensional regularization , which is composed of the sum of all the differences of potentials and their gradients computed by means of the generals formulae of ( [ dp1total ] ) , ( [ ddip1total ] ) and ( [ drdirresult ] ) . the term @xmath91 when added to the `` pure - hadamard - schwartz '' acceleration defined by ( [ defph ] ) , gives our result for the dimensionally regularized ( `` dr '' ) acceleration @xmath644 more details on the practical computation of @xmath645 ( which parts of the potentials contribute ; what is the diagrammatic picture ) will be given in section [ dimregstat ] . the preceding section has explained the method we used to compute the dimensionally regularized equations of motion as the sum ( @xmath646 ; considered modulo 2 ) @xmath647 = \mathbf{a}_a^\text{phs } [ r'_a , s_{a+1 } ] + { \mathcal d } \mathbf{a}_a [ r'_a , s_{a+1 } ; \varepsilon , \ell_0]\,,\ ] ] where the label `` phs '' refers to the `` pure hadamard - schwartz '' definition of the acceleration ( _ i.e. _ , the `` raw '' result of @xcite , after subtraction of the additional contributions quoted in section [ additional ] above , eq . ( [ defph ] ) ) , and where @xmath648 is the difference induced when using dimensional continuation as regularization method , instead of hadamard s one . a first check on our results will be that , as indicated in ( [ eq5.1 ] ) , the four regularization parameters ( with dimension of length ) , @xmath649 , that enter the hadamard method must cancel between @xmath650 and @xmath648 to leave a result for the dimensionally regularized accelerations @xmath651 which depends only on the two regularization parameters of dimensional continuation : @xmath35 and the basic length scale @xmath90 entering newton s constant in @xmath1 dimensions , @xmath652 , where we recall that @xmath106 denotes the usual three - dimensional newton constant . the dimensionally regularized acceleration ( [ eq5.1 ] ) has the structure @xmath653 & = & \mathbf{a}_{\text{n}a } [ \mathbf{y}_{12 } ] + \mathbf{a}_{1\text{pn}a } [ \mathbf{y}_{12 } , \mathbf{v}_1 , \mathbf{v}_2 ] \nonumber \\ & + & \mathbf{a}_{2\text{pn}a } [ \mathbf{y}_{12 } , \mathbf{v}_1 , \mathbf{v}_2 ] + \mathbf{a}_{2.5\text{pn}a } [ \mathbf{y}_{12 } , \mathbf{v}_1 , \mathbf{v}_2 ] + \mathbf{a}_{3\text{pn}a } [ \mathbf{y}_{12 } , \mathbf{v}_1 , \mathbf{v}_2]\,,\end{aligned}\ ] ] where we denote @xmath654 . the 3pn term ( which is the only one to have a pole at @xmath655 ) has a tensor structure of the form ( say for the first particle , @xmath59 ) @xmath656 where , as usual , @xmath657 denotes the unit vector directed from particle 2 to particle 1 . the scalar coefficients @xmath20 , @xmath658 , @xmath659 entering the equation of motion of @xmath75 can be decomposed in powers of the masses , say [ eq5.4 ] @xmath660 where @xmath661 and @xmath662 are natural integers , with the restrictions indicated . note that , in eqs . ( [ eq5.4 ] ) , we have conventionally factored out an integer power of the `` full '' ( @xmath1-dimensional ) gravitational constant @xmath132 , and a corresponding integer power of @xmath663 . this creates a mismatch between the usual 3-dimensional dimension of , say , @xmath664 and the dimension of @xmath665 . using @xmath652 one sees that it is the combination @xmath666 which has the same dimension as @xmath664 . alternatively said , the ensuing fact that @xmath667 has the same dimension as @xmath664 implies , as indicated in eqs . ( [ eq5.4 ] ) , a dependence of @xmath668 on @xmath669 when @xmath563 . notice also that in eq . ( [ eq5.3 ] ) we have introduced separate notations for the coefficient of @xmath68 and that of @xmath69 . actually , the poincar invariance of the equations of motion imposes the restriction @xmath670 so that the last two terms in eq . ( [ eq5.3 ] ) are proportional to the relative velocity @xmath671 . [ note , however , that @xmath658 is not a function of @xmath672 only ; it depends both on @xmath68 and @xmath69 . ] because the calculation of the separate contributions @xmath650 and @xmath673 to the equations of notion breaks the over - all poincar invariance of the formalism , our computation of the separate pieces @xmath650 and @xmath673 will involve partial contributions to @xmath658 and @xmath659 that do not coincide . it is only at the end of the calculation that the equality @xmath674 will be satisfied , so that finally @xmath675 most of the coefficients @xmath668 , @xmath676 , @xmath677 entering the 3pn acceleration are well behaved when @xmath678 , in the sense that their evaluation never involves any poles @xmath93 . by this we mean that whatever be the ( reasonable ) way of decomposing the integral giving a coefficient in separate contributions , the latter contributions do not involve poles @xmath93 . the subset of coefficients whose evaluation involves poles coincides with the set of `` delicate '' coefficients in the hadamard regularization , namely the nine coefficients contributing to terms of the following form in the acceleration of the first particle : @xmath679 \mathbf{n}_{12 } \nonumber\\ & & \qquad + \frac{g^3 \ , m_1 ^ 2 \ , m_2}{c^6 \ , r_{12}^4 } \bigl [ c_{21 } ( \mathbf{v}_1 , \mathbf{v}_2 ) \mathbf{n}_{12 } \ , + c'_{21 } ( \mathbf{v}_1 , \mathbf{v}_2 ) \mathbf{v}_1 - c''_{21 } ( \mathbf{v}_1 , \mathbf{v}_2)\ , \mathbf{v}_2\bigr ] \nonumber\\ & & \qquad + \frac{g^3 \ , m_2 ^ 3}{c^6 \ , r_{12}^4 } \bigl[c_{03 } ( \mathbf{v}_1 , \mathbf{v}_2 ) \ , \mathbf{n}_{12 } + c'_{03 } ( \mathbf{v}_1 , \mathbf{v}_2)\ , \mathbf{v}_1 - c''_{03 } ( \mathbf{v}_1 , \mathbf{v}_2)\ , \mathbf{v}_2 \bigr]\,.\end{aligned}\ ] ] the first three terms in eq . ( [ eq5.5 ] ) do not depend on velocities and will be referred to as the _ static _ delicate contributions , by contrast to the _ kinetic _ delicate contributions involving the velocity - dependent coefficients @xmath680 , @xmath681 , @xmath682 , @xmath683 , @xmath684 , and @xmath685 ( they depend on @xmath68 , @xmath69 and also on @xmath67 ) . correspondingly to the decomposition ( [ eq5.1 ] ) of the equations of motion , the dimensionally regularized static contributions , unambiguously obtained from the test - mass limit @xmath686 . ] @xmath687 , @xmath688 , @xmath689 to the acceleration of the first particle can be written as the sum ( @xmath690 , @xmath691 , @xmath692 ) @xmath693 = c_{mn}^\text{phs } [ r'_1 , s_2 ] + { \mathcal d } c_{mn } [ r'_1 , s_2 , \varepsilon]\,.\ ] ] in this subsection , we discuss the explicit evaluation of the pure hadamard - schwartz static coefficients @xmath694 . as explained in the previous section , the phs static contributions @xmath695 $ ] are obtained from the results reported in @xcite by undoing two things . first , the `` bf '' results reported there for @xmath80 ( eq . ( 7.16 ) of @xcite ) were expressed in terms of the three parameters @xmath61 , @xmath62 and @xmath4 , instead of the two pure hadamard parameters @xmath61 and @xmath78 more relevant for the present purpose . the introduction of the parameter @xmath4 was motivated by requiring that the full set of equations of motion ( which _ a priori _ depended on four independent regularizing parameters @xmath61 , @xmath62 , @xmath77 , @xmath78 ) admit a conserved energy . this led to the link , eqs . ( 7.9 ) in @xcite,$ ] given by ( [ distrpart ] ) above . another derivative , the `` correct '' one , was also considered in @xcite and shown to yield equivalent equations of motion . the pure hadamard result does not depend on this choice because we shall subtract below the specific additional contributions coming from the distributional derivative @xmath414 $ ] . ] @xmath696 when inserting ( [ eq5.8 ] ) in the expression of @xmath697 $ ] we find , as it should be , that the result simplifies to an expression depending only on the two pure hadamard parameters @xmath61 and @xmath78 . this leads to the following net results from @xcite , [ eq5.9]@xmath698 & = & - \frac{3187}{1260 } + \frac{44}{3 } \ln \left ( \frac{r_{12}}{r'_1 } \right ) , \\ c_{22}^\text{bf } [ r'_1 , s_2 ] & = & \frac{34763}{210 } - \frac{41}{16 } \ , \pi^2\ , , \\ c_{13}^\text{bf } [ r'_1 , s_2 ] & = & \frac{1565}{9 } - \frac{41}{16 } \ , \pi^2 - \frac{44}{3 } \ln \left ( \frac{r_{12}}{s_2 } \right).\end{aligned}\ ] ] second , ref . @xcite obtained their results for the equations of motion by adding to the pure hadamard - schwartz contributions 7 additional corrections , imposed by their extended - hadamard regularization and explained in section [ additional ] above : see the _ items _ ( i)-(vii ) there . note that these various corrections affect the `` delicate '' contributions to @xmath80 , in general both the static and kinetic ones , but only five of them contribute to the static part . these are the self term ( [ deltaself ] ) , the leibniz term ( [ deltaleibniz ] ) , the @xmath136-correction given by ( [ deltav ] ) , the eom non - distributivity ( [ deltaeom ] ) , and the distributional time - derivative one ( [ deltatime ] ) . following eq . ( [ defph ] ) , and focussing on the static contributions , we now _ subtract _ these static terms from the result ( [ eq5.9 ] ) in order to get the looked - for pure hadamard contributions : [ eq5.14]@xmath699 & = & c_{31}^\text{bf } [ r'_1 , s_2 ] - \frac{779}{210}\ , , \\ c_{22}^\text{phs } [ r'_1 , s_2 ] & = & c_{22}^\text{bf } [ r'_1 , s_2 ] + \frac{97}{210}\ , , \\ c_{13}^\text{phs } [ r'_1 , s_2 ] & = & c_{13}^\text{bf } [ r'_1 , s_2 ] - 5 + \frac{88}{9 } - \frac{151}{9 } + \frac{2}{15}\,,\end{aligned}\ ] ] _ i.e. _ , explicitly , [ eq5.15]@xmath699 & = & - \frac{1123}{180 } + \frac{44}{3 } \ln \left ( \frac{r_{12}}{r'_1 } \right ) , \\ c_{22}^\text{phs } [ r'_1 , s_2 ] & = & 166 - \frac{41}{16 } \ , \pi^2\,,\\ c_{13}^\text{phs } [ r'_1 , s_2 ] & = & \frac{7291}{45 } - \frac{41}{16 } \ , \pi^2 - \frac{44}{3 } \ln \left ( \frac{r_{12}}{s_2 } \right).\end{aligned}\ ] ] note in passing that though the coefficient @xmath700 does not contain regularization logarithms , its evaluation involves many intermediate logarithmic divergencies that cancel in the final result . such `` cancelled logs '' lead to as much ambiguity in the final result than uncancelled ones that explicitly depend on an arbitrary regularization scale such as @xmath61 or @xmath78 in @xmath701 or @xmath702 . we now turn to the evaluation of the `` dim .- reg . minus pure - hadamard '' differences @xmath703 in eq . ( [ eq5.7 ] ) , coming from the differences @xmath704 in eq . ( [ eq5.1 ] ) . we start from the @xmath1-dimensional expression for the acceleration @xmath80 [ see eq . ( [ smallaccel ] ) for a short - hand form ] , which is itself expressed in terms of the @xmath1-dimensional elementary potentials @xmath136 , @xmath137 , @xmath158 , @xmath159 , @xmath450 , @xmath161 , @xmath705 , @xmath163 and @xmath164 defined in section [ fieldeq ] . each elementary potential can be naturally decomposed in a `` compact '' ( or , equivalently , `` contact '' ) piece ( whose source is compact , _ i.e. _ , involves the basic delta - function sources @xmath143 , @xmath144 , @xmath149 ) and a `` non - compact '' one ( whose source is non - linearly generated and extends all over space ) . the potentials @xmath136 , @xmath137 and @xmath158 are purely `` compact '' , @xmath706 , @xmath707 , @xmath708 , while all the other potentials admit a decomposition of the form @xmath709 , etc . for instance , the `` compact '' part of @xmath159 is defined by @xmath710 while its `` non - compact '' part is defined by @xmath711 a more complicated example is the potential @xmath712 with @xmath713\,,\ ] ] and @xmath714 the @xmath715 contribution can be further decomposed into the piece of @xmath716 whose source is quadratic in compact potentials , namely @xmath717 and its `` cubically non - compact '' piece given by @xmath718 to get a feeling of the actual evaluation of the difference @xmath719 let us consider a specific contribution to @xmath80 , say the term @xmath720 \equiv \frac{4}{c^4 } \ , ( \partial_i \ , \hat x)_{\mathbf{x } = \mathbf{y}_1}\,.\ ] ] it can be decomposed into : ( i ) its `` compact '' piece @xmath721 $ ] , ( ii ) its `` quadratically non - compact '' one @xmath722 $ ] , ( iii ) its `` cubically non - compact '' part @xmath723 $ ] . it is sometimes convenient to think of the various contributions to @xmath80 in terms of space - time diagrams . if we represent the basic delta - function sources [ proportional to @xmath724 and @xmath725 as two world - lines and each propagator @xmath520 as a dotted line , a `` compact '' contribution to @xmath80 will be represented by one of the diagrams in fig . [ fig1 ] . . the dotted line represents @xmath520 , the cross represents the field point @xmath241 ( here taken on the first worldline ) , and the bullet represents either a source point or ( in the figures below ) an intermediate nonlinear vertex . ] for instance , fig . [ fig1]a can represent a term @xmath726 in @xmath727 in which the ( compact ) source @xmath143 of @xmath136 is proportional to @xmath728 and involves no further powers of the masses , while fig . [ fig1]b represents a _ self - action _ term , @xmath78 ) the self - action diagrams , such as fig . [ fig1]b or fig . [ fig1]d , are the first divergencies that one encounters and must then renormalize away , dimensional regularization has the technically useful property of setting all of these diagrams to zero . indeed , when using a time - symmetric propagator @xmath729 these diagrams are seen to involve the coinciding - point limits of @xmath730 , which vanish when @xmath731 by dimensional continuation in @xmath1 . ] in @xmath732 with source proportional to @xmath733 . by contrast , [ fig1]c might correspond to another term in @xmath732 where the compact source @xmath143 is concentrated at @xmath76 , @xmath734 , and where a part of the `` effective mass '' , @xmath735 contains , besides the overall factor @xmath65 , another factor @xmath64 . as all the sources of @xmath736 contain , besides some `` basic '' @xmath737 , a potential ( @xmath136 , @xmath137 , @xmath738 or @xmath158 ) , the diagrams contained in @xmath739 $ ] will be at least of the form of fig . [ fig1]c , [ fig1]d , [ fig1]e , [ fig1]f , or will involve a more complicated mass - dependence . the quadratically non - compact terms @xmath740 $ ] will then contain diagrams of the type of fig . [ fig2 ] , while the cubically non - compact term @xmath741 $ ] contains many subdiagrams of the type sketched in fig . [ fig3 ] . . ] . ] the particular term ( [ eq5.20 ] ) that we considered contains only diagrams of the general type of figs . [ fig1 ] , [ fig2 ] or [ fig3 ] . note , however , that there are also more non - linear contributions to @xmath727 , such as some terms in @xmath742 = \frac{16}{c^6 } \ , ( \partial_i \ , \hat t)_1\,,\ ] ] corresponding to diagrams of the type sketched in fig . [ fig4 ] . . ] similarly to the diagrams fig . [ fig1]c or fig . [ fig2]d , all the diagrams above can be modified by the presence of additional lines propagating directly between the two world - lines and corresponding to `` potential '' modifications of compact - support sources . as underlined in section [ iteratedeinstein ] above , the 3pn equations of motion do _ not _ involve `` quartically non - linear '' contributions corresponding to diagrams such as those of fig . terms like @xmath743 or @xmath744 are of this form , and they do occur in the 3pn acceleration @xmath80 , but since they involve double contracted gradients , it was possible to integrate them away thanks to rule ( ii ) of section [ fieldeq ] ; see eq . ( [ expfi ] ) in appendix [ expfieldeq ] below . on the other hand , terms of the form @xmath745 or @xmath746 do not occur at the 3pn order @xmath747 , although they are of the third post-_minkowskian _ order @xmath748 . at the 3pn order . ] drawing diagrams often helps to highlight the nature of the uv singularities contained in the integrals they represent . as a rule of thumb , the `` delicate '' diagrams , that might involve poles , or cancelled poles , when @xmath563 ( corresponding to logarithms , or cancelled logarithms , in @xmath42 ) are characterized by the presence of a subdiagram containing three propagator lines that can simultaneously shrink to zero size , as a subset of vertices coalesce together on one of the two world - lines . examples of such uv dangerous diagrams are fig . [ fig2]d and fig . [ fig3]b [ for vertices coalescing towards @xmath749 or fig . [ fig3]d and fig . [ fig4]d [ for vertices coalescing on the second world - line ] . the former diagrams can give poles proportional to @xmath750 ( with some velocity dependence , or some extra mass dependence due to an extra line propagating between the two world - lines ) , while the latter can give poles proportional to @xmath751 ( possibly with some extra velocity or mass dependence ) . the reason why three simultaneously shrinking propagators can yield poles as @xmath678 is easy to see in the approximation where the relativistic propagators @xmath520 are replaced by non - relativistic ones @xmath752 . indeed , when three such propagators shrink simultaneously , the overall integral contains a subintegral of the form @xmath753 . on the other hand , beyond our obtaining a heuristic feeling of what are the origins of the poles in @xmath80 , we did not use a diagrammatic technique for evaluating the equations of motion . [ note , however , that a generalization of the ( 2pn level ) work @xcite would lead to a diagrammatic technique for evaluating the fokker lagrangian of two point masses . ] our actual computations used the techniques elaborated in the previous sections . we evaluated the contributions to the difference @xmath754 coming from all the terms in the expression for @xmath80 deduced from ( [ eqgeod ] ) together with the complete expanded forms ( [ exppi])-([expfi ] ) . however , as expected from various arguments diagrammatic analysis , existence of ( possibly cancelled ) logarithms in the corresponding @xmath42 evaluation most of the terms lead to a vanishing difference @xmath754 . the only terms that give non - vanishing contributions to @xmath704 are the four terms given in eq . ( [ smallaccel ] ) , @xmath755 & = & \frac{4}{c^4 } \ , ( \partial_i \ , \hat x)_1\ , , \quad a_1^i [ \hat t ] = \frac{16}{c^6 } \ , ( \partial_i \ , \hat t)_1 \ , , \nonumber\\ a_1^i [ \hat r_i ] & = & \frac{8}{c^4 } \ , \frac{d}{dt } \ , ( \hat r_i)_1\ , , \quad a_1^i [ \hat y_i ] = \frac{16}{c^6 } \ , \frac{d}{dt } \ , ( \hat y_i)_1\ , . \label{eq5.23}\end{aligned}\ ] ] note that , for the contributions associated to @xmath161 and @xmath450 , one needs a 1pn - accurate treatment of both their respective sources and the propagator @xmath520 . apart from the compact support terms in the sources for @xmath161 , @xmath164 , @xmath450 and @xmath163 which lead to zero difference , most of the non - compact terms do lead to some non - vanishing contributions to the difference of acceleration @xmath704 . we give in tables [ table1]-[table4 ] the contributions to @xmath703 associated to the various individual source terms of the `` delicate '' potentials @xmath161 , @xmath164 , @xmath450 and @xmath163 , which were displayed in section [ fieldeq ] , eqs . ( [ potentialeq ] ) [ of course , we limit ourselves to non - compact source terms ] . in these tables , we use the simplifying notation @xmath756 where @xmath757 denotes the euler constant . .static contributions of @xmath758 . all the results are presented modulo some neglected terms @xmath571 . the `` principal '' part of a term corresponds to the term @xmath759 in the 1pn symmetric propagator @xmath760 , while the `` retarded '' part corresponds to the purely 1pn piece @xmath761 . the `` extra term '' refers to the last term in the r.h.s . ( [ ddir1result ] ) . note that , in view of eqs . ( [ eq5.23 ] ) , one must multiply the results by a factor 4 in order to get the contributions to the coefficients @xmath762 in the equations of motion . [ cols="^,^,^,^",options="header " , ] summing up the separate non - vanishing contributions displayed in tables [ table1]-[table4 ] , we get the following total differences [ eq5.28]@xmath763 finally , adding eqs . ( [ eq5.28 ] ) to the pure hadamard - schwartz result ( [ eq5.15 ] ) , we get the dimensionally regularized static contributions to @xmath80 : [ eq5.29 ] @xmath764 as expected the two hadamard regularization length scales @xmath61 and @xmath78 have cancelled between @xmath694 and @xmath765 to leave a result which depends only on the dim . regularization parameter @xmath86 . one might be surprised by the presence in @xmath766 of terms @xmath767 compared to corresponding terms @xmath768 in @xmath694 , and by the absence of any adimensionalizing length scale in these logarithms of @xmath769 . these two properties can be understood when one remembers from the discussion above that the coefficients which have the same physical dimension as @xmath770 are the combinations @xmath771 . in the present case , this means that @xmath772 are dimensionless . it is easy to see , thanks to the pole terms @xmath773 and the expansion @xmath774 , that the combinations @xmath775 and @xmath776 do indeed depend only on the dimensionless quantities @xmath36 and @xmath777 . the first computation of the dimensional continuation of the 3pn gravitational interaction of point masses was done in adm coordinates and resulted into a _ finite _ ( _ i.e. _ , without @xmath778 poles ) answer @xcite . our task in analyzing the physical meaning of the harmonic - coordinates result ( [ eq5.29 ] ) is to interpret the presence of @xmath778 poles in it . for this we have to remember that , as in quantum field theory ( qft ) , dimensional continuation is a _ regularization _ method which , like all regularization methods , transforms truly infinite results , say containing @xmath779 , into finite , but `` large '' ones , which depend on some cut - off parameter , _ e.g. _ @xmath780 or @xmath781 . any regularization must be followed by a _ renormalization _ process which allows one to absorb the cut - off dependent terms in some of the basic _ bare parameters _ of the theory . in order to have a clearer understanding of the poles in the ( static ) equations of motion ( [ eq5.29 ] ) [ we shall prove below that our discussion extends to the full , velocity - dependent equations of motion ] , we need to analyze the presence of poles in the `` bulk '' metric , _ i.e. _ , the metric @xmath782 evaluated at a generic field point @xmath241 , away from the two world - lines . indeed , if we were considering the gravitational field generated by regular ( _ i.e. _ , non point - like ) sources , a complete physical description of their gravitational effects would necessitate the simultaneous consideration of the bulk metric and of the equations of motion of the ( extended ) sources . similarly , in the present formal study of two - point - like sources , we need to consider both the equations of motion @xmath783 and the bulk metric @xmath784 . it is here that the diagrammatic representation introduced above plays a useful role in highlighting the structure of divergencies in the equations of motion and in the bulk metric . indeed , it is clear that the divergent diagrams of the equations of motion of the first particle , where the @xmath778 pole is due to the presence of a subdivergence induced by three propagators shrinking onto the second world - line ( such as in fig . [ fig3]d or fig . [ fig4]d ) , will correspond to similar @xmath778 poles in the bulk metric , for the corresponding `` bulk diagrams '' where the special point marked by a cross in the diagrams above ( denoting the coincidence @xmath245 ) is detached from the first world - line to end at an arbitrary point in the bulk , as indicated in fig . [ fig6 ] . , is detached from the first worldline . ] evidently , in addition to such diagrams as fig . [ fig6]a and [ fig6]b which will contain ( at least ) a factor @xmath751 , there will exist `` mirror diagrams '' , containing a factor @xmath785 , and obtained by exchanging the labels 1 and 2 . on the other hand , note that the bulk poles @xmath786 of the type of fig . [ fig6]c and [ fig6]d do not ( necessarily ) correspond to poles in the equations of motion of @xmath75 because their coincidence limits @xmath45 induce diagrams of the type of fig . [ fig3]a or [ fig4]a containing four shrinking propagators instead of three . [ though such diagrams would exhibit worse divergences in a dimensionful cut - off regularization schemes , they are generally non dangerous in dimensional regularization because the integral @xmath787 has no pole as @xmath563 . ] a careful analysis of the possible presence of poles in the various potentials @xmath788 we use to parametrize the bulk metric ( aided by the structure of potentially dangerous terms sketched in fig . [ fig6 ] ) shows that , at the 3pn approximation , in @xmath169 , @xmath789 in @xmath790 , and @xmath791 in @xmath170 . ] such poles can only be present in the 1pn - level expansion of @xmath161 and in the newtonian - level approximation of @xmath164 . drawing on the results of @xcite and @xcite we can also see that all velocity - dependent terms in the poles present in @xmath792 and @xmath164 ( _ i.e. _ , the terms proportional to @xmath793 or @xmath794 ) exactly cancel in the combination @xmath795 that matters for the bulk metric . [ this shows up , for instance , in eq . ( 7.1 ) of @xcite which implies that the divergencies linked to the second world - line , characterized by the presence of @xmath591 , do not depend on velocities . this shows up also in the absence of poles in the velocity - dependent contributions to @xmath80 proportional to @xmath751 , see eqs . ( [ eq5.76c])-([eq5.76d ] ) below . ] we are therefore left with evaluating the poles present in the _ static limit _ @xmath796 of @xmath797 and @xmath164 . clearly , from fig . [ fig6 ] , the poles in @xmath792 and @xmath164 will come only from cubically non - compact ( @xmath798 ) sources . finally , as we are only interested in the pole part we can neglect the @xmath1-dependence of the coefficients in the sources of @xmath161 and @xmath164 ( which we indicate by using a symbol @xmath799 ) . thus , these poles can only come from [ eq5.31ab ] @xmath800 \nonumber\\ & \simeq & \box^{-1 } \bigl[\partial_{ij } \ , v \ , \box^{-1 } \bigl(- \partial_i \ , v \partial_j \ , v\bigr)\bigr ] \ , , \label{eq5.30}\\ \hat t^{cnc}_\text{static } & \simeq & \box^{-1 } \biggl [ \frac{1}{2 } \ , v^2 \ , \partial_t^2 \ , v + \partial_{ij } \ , v\ , \hat{z}_{ij}^{nc } \biggr ] \nonumber \\ & \simeq & \box^{-1 } \biggl[\frac{1}{2 } \ , v^2 \ , \partial_t^2 \ , v + \partial_{ij } \ , v \ , \box^{-1 } \bigl(- 2\partial_i \ , v \partial_t \ , v_j \bigr ) \biggr]\ , . \label{eq5.31}\end{aligned}\ ] ] the static poles ( involving factors @xmath785 or @xmath751 ) in eqs . ( [ eq5.31ab ] ) are then obtained by : ( 1 ) considering sources involving three times @xmath801 or three times @xmath802 ( where @xmath803 denotes the piece @xmath804 in @xmath136 ) , ( 2 ) evaluating the time derivatives in the static limit , using for instance @xmath805 and ( 3 ) expanding up to the required accuracy the ( time - symmetric ) propagators according to @xmath806 . as an example among the simplest terms , let us consider the @xmath785 contribution coming from the first term on the r.h.s . of eq . ( [ eq5.31 ] ) , @xmath807_\text{static } = - \frac{1}{2 } \ , \delta^{-1 } [ v_1 ^ 2 \ , a_1^j \ , \partial_j \ , v_1 ] = - \frac{1}{6 } \ , a_1^j \ , , \delta^{-1 } [ v_1 ^ 3]\,.\ ] ] using @xmath808 and @xmath809 $ ] , one finds that the pole part of ( [ eq5.33 ] ) reads @xmath810 similarly an analysis of the second source term in eq . ( [ eq5.31 ] ) yields @xmath811 so that the full ( static ) contribution of @xmath164 is @xmath812 the analysis of the pole part in the static limit of @xmath161 , eq . ( [ eq5.30 ] ) , is more intricate because one must expand to 1pn accuracy both @xmath813 and the propagator @xmath520 . this yields @xmath814 let us now consider the improved @xmath136-potential ( [ calv ] ) that makes up the essential part of @xmath169 , @xmath815 such that @xmath816 $ ] , see eq . ( [ g00 ] ) . combining the results above , we find that the only @xmath778 poles in the bulk metric @xmath817 show up in @xmath169 at the 3pn level and are ( when expressed in terms of the improved potential @xmath198 , and after cancellation of @xmath818 terms between @xmath161 and @xmath164 of the following static form @xmath819\,,\ ] ] where @xmath820 , @xmath821,@xmath822 s are finite when @xmath104 . to understand better the structure of result ( [ eq5.39 ] ) let us introduce the notation @xmath823 where @xmath106 is the 3-dimensional newton constant and @xmath824 the @xmath1-dimensional acceleration of @xmath825 . [ this definition ensures that @xmath826 has the physical dimension of a length . ] in terms of the definition ( [ eq5.40 ] ) , the result ( [ eq5.39 ] ) can be equivalently written as @xmath827 + \frac{1}{c^2 } \mathcal{v}_2 + \frac{1}{c^4 } \mathcal{v}_4 + \frac{1}{c^6 } \mathcal{v}_6\,,\ ] ] where the pole part is entirely contained in the terms proportional to @xmath828 and @xmath829 [ @xmath830 here differs from @xmath822 in eq . ( [ eq5.39 ] ) by some finite corrections when @xmath104 ] . the fact that poles appear only in @xmath198 , at order @xmath791 , implies that there are no divergencies in the harmonic gauge conditions ( [ gaugeidentities ] ) in the bulk . indeed , ( [ divi ] ) needs @xmath198 at order @xmath832 only , and ( [ djwij ] ) at newtonian order only . result ( [ eq5.41 ] ) indicates a simple way of renormalizing away the poles present in the bulk metric . indeed , the logic up to now has been to describe in the simplest possible manner a gravitationally interacting two - particle system , parametrized by the following _ bare _ parameters : @xmath833 , @xmath834 , @xmath835 , @xmath836 , @xmath837 , considered in everywhere - harmonic coordinates , @xmath838 . in particular , the internal structure of each particle has been , up to now , entirely described by a monopolar stress - energy distribution , _ i.e. _ , @xmath839 . in other words , we have set to zero any higher multipolar structure . ( [ eq5.41 ] ) is most simply interpreted by saying that the non - linear interactions ( see fig . [ fig6 ] ) dress each particle by a cloud of gravitational energy which generates , at the 3pn order , a divergent _ dipole _ in the newtonian - like potential . therefore , to get a net , finite bulk gravitational field we must endow each initial particle by an infinite , bare dipole , corresponding to a counterterm @xmath840 , which will cancel the non - linearly generated one ( [ eq5.41 ] ) . an equivalent , but technically simpler way of endowing each particle by a bare structure able to cancel the dipolar pole terms in ( [ eq5.41 ] ) is simply to say that the central _ bare _ world - lines used in our derivations up to now , henceforth denoted as @xmath841 , can be decomposed in a finite _ renormalized _ part @xmath842 and a formally infinite shift @xmath843 involving a pole @xmath93 , @xmath844 the gravitational potential of two point particles [ @xmath845 is then @xmath846 \\ & + & \frac{1}{c^2 } \ , \mathcal{v}_2 ( \mathbf{x } , \mathbf{y}_a^\text{ren } + \bm{\xi}_a ) + \frac{1}{c^4 } \ , \mathcal{v}_4 ( \mathbf{x } , \mathbf{y}_a^\text{ren } + \bm{\xi}_a ) + \frac{1}{c^6 } \ , \mathcal{v}_6 ( \mathbf{x } , \mathbf{y}_a^\text{ren } + \bm{\xi}_a)\ , . \nonumber\end{aligned}\ ] ] assuming that the vector @xmath843 is of 3pn order [ _ i.e. _ , @xmath847 , we can rewrite eq . ( [ eq5.430 ] ) as @xmath848 \\ & + & \frac{1}{c^2 } \ , \mathcal{v}_2 ( \mathbf{x } , \mathbf{y}_a^\text{ren } ) + \frac{1}{c^4 } \ , \mathcal{v}_4 ( \mathbf{x } , \mathbf{y}_a^\text{ren } ) + \frac{1}{c^6 } \ , \mathcal{v}_6 ( \mathbf{x } , \mathbf{y}_a^\text{ren } ) + \mathcal{o}\left(\frac{1}{c^8}\right ) , \nonumber\end{aligned}\ ] ] which makes it clear that the potential will be finite ( at 3pn accuracy ) when @xmath563 if we choose @xmath849 where by @xmath850 we mean a term finite when @xmath563 and of the 3pn order . we shall henceforth refer to @xmath843 in eq . ( [ eq5.44 ] ) as a _ shift _ of the @xmath218 world - line . the reasoning above shows that the introduction of such shifts , at the 3pn order and having the pole structure ( [ eq5.40 ] ) , is _ necessary _ to renormalize away the poles present in the _ bulk metric_. it remains to show that these shifts are also _ sufficient _ to renormalize away the poles present in the _ equations of motion_. the effect of 3pn - level shifts @xmath843 on the equations of motion is easy to obtain . indeed , the equations of motion we computed above concern the original , _ bare _ world - lines @xmath841 . for the first particle , they had the structure ( in dimensional regularization ) @xmath851 where @xmath852 . here @xmath853 denotes the dimensionally continued newtonian - level acceleration , @xmath854 where by a slight abuse of notation we pose @xmath855 , where @xmath856 denotes the @xmath1-dimensional gravitational constant , and where the @xmath1-dependent correcting factors @xmath857 tend to 1 as @xmath563 , but will play a significant role below . when inserting the redefinitions ( [ eq5.42 ] ) into ( [ eq5.45 ] ) one easily finds that the _ renormalized _ equations of motion , _ i.e. _ , the equations for @xmath842 , read [ using only @xmath858 at this stage ] @xmath859 where @xmath860 with @xmath861 let us note that the effect on the equations of motion of a ( 3pn ) shift of the world - lines , eq . ( [ eq5.50 ] ) , is technically identical to the effect on the equations of motion of the restriction to the world - lines of a 3pn - level coordinate transformation , say @xmath862 and @xmath863 . indeed , a coordinate transformation has two effects : ( i ) it changes the bulk metric by @xmath864 , where @xmath865 denotes the lie derivative along @xmath866 , and ( ii ) it induces a shift of the world - lines @xmath867 ( plus non - linear terms in @xmath866 ) , where we denote the coordinate change at the location of the @xmath868-th particle by @xmath869_{\mathbf{x } = \mathbf{y}_a}$ ] . because of the diffeomorphism invariance of the total action , the effect ( i ) does not change the action , , ensures that it does not contribute to first order in @xmath866 . ] so that the net effect of a coordinate transformation on the equations of motion reduces to the effect ( [ eq5.50 ] ) of the following shift induced on the world - lines : @xmath870 the coordinate transformations considered in @xcite , see eq . ( 6.11 ) there , were of the type @xmath871 , where the @xmath872 s are some coefficients , so that we see that the latter induced shift reduces at the 3pn order to the ( regularization of the ) purely spatial coordinate transformation evaluated on the world - line : @xmath873 we have checked that the formula given by eq . ( 6.15 ) in @xcite for the coordinate transformation of the acceleration of the particle 1 gives exactly the same result as the one computed from the effect of the shift ( [ eq5.50 ] ) . [ the agreement extends to @xmath1 dimensions if we consider the straightforward extension of the latter coordinate transformation @xmath866 to @xmath1 dimensions , namely @xmath874 . ] note that the coordinate transformations @xmath875 were considered in @xcite only in terms of their effects , eqs . ( [ xia])-([xia ] ) , on the equations of motion . this was sufficient to prove , for instance , that the two constants @xmath70 and @xmath71 in the 3pn equations of motion are not physical , because they can be gauged away in 3 dimensions and therefore will never appear in any physical result . however , we remark that the extension to @xmath1 dimensions of the coordinate transformation @xmath875 of the _ bulk _ metric , say @xmath874 ( with coefficients @xmath876 , as needed to remove the poles in the equations of motion ) , does not lead to a bulk metric free of poles . indeed , assuming @xmath876 , we see that the pole in the spatial coordinate transformation @xmath877 would then induce a pole in the spatial components of the metric , @xmath878 , but this is inadmissible because we have proved above eqs . ( [ eq5.31ab ] ) that , at 3pn order , only the time - time component of the bulk metric contained a pole . a bulk coordinate transformation of the type above can then remove the poles in the time - time component of the bulk metric only at the price of creating a pole in the initially pole - free spatial metric . we shall leave to future work a complete clarification of the possibility of using , within our dim . context , a coordinate transformation to induce the shifts ( [ eq5.44 ] ) . for the time being , what is important is that our introduction above of shifts of the world - lines ( _ a priori _ unconnected to any coordinate transformation ) is a consistent way of renormalizing away the poles in the metric , and that its effect on the equations of motion , eq . ( [ eq5.50 ] ) , is identical to the transformations of the acceleration obtained in ref . @xcite . it remains now to show that the same world - line shifts ( [ eq5.44 ] ) that renormalize away the poles in the bulk metric , eq . ( [ eq5.43 ] ) , do renormalize away also the poles present in the original bare equations of motion [ see ( [ eq5.29 ] ) for the static contributions and ( [ eq5.76 ] ) below for the kinetic ones ] . for this purpose let us consider a shift of the more general form @xmath879 where @xmath880 represents a certain numerical coefficient depending on @xmath1 , and where @xmath824 denotes the @xmath1-dimensional acceleration of @xmath825 given by its newtonian approximation ( [ eq5.46 ] ) [ but , for notational simplicity , we henceforth drop the label @xmath7 on such accelerations entering 3pn effects ] . inserting ( [ eq5.51 ] ) into ( [ eq5.50 ] ) yields ( for the index @xmath59 ) @xmath881\,,\ ] ] where @xmath882 and @xmath883 [ and also , as before , @xmath884 . before further evaluating ( [ eq5.52 ] ) by inserting the explicit expression ( [ eq5.46 ] ) for the acceleration , we shall consider some simple but important consequences of the structure ( [ eq5.52 ] ) . as recalled in the introduction , previous work on the 3pn equations of motion in _ harmonic coordinates _ has shown that these equations necessarily belonged to a three - parameter class of equations of motion , say @xmath885 the dimensionless parameter @xmath4 could not be determined by the previous work in harmonic coordinates . however , comparison with the work in adm coordinates , has shown @xcite that , _ if there were consistency _ between the two calculations one should have the following link between @xmath4 and the corresponding adm `` static ambiguity '' parameter @xmath9 , @xmath886 if dimensional regularization is a fully consistent regularization scheme for classical perturbative gravity , we then expect that the dim . determination of @xmath9 in adm coordinates @xcite , namely @xmath887 , should lead to a dim . reg . direct determination of @xmath4 ( in harmonic coordinates ) of @xmath888 . we will turn to this verification in a moment . the two other parameters , denoted above @xmath61 , @xmath62 , entering the general `` parametric '' harmonic equations of motion ( [ eq5.53 ] ) have the dimension of length and have the character of gauge parameters . indeed , they can be chosen at will ( except that one can not set them to zero ) by the effect of shifts of the world - line , induced for instance [ but not necessarily , _ cf . _ a discussion in section [ shift ] above ] by some gauge transformations . in the way they were originally introduced @xcite , the two parameters @xmath70 and @xmath71 can be interpreted as some infinitesimal radial distances used as cut - offs when the field point tends towards the two singularities @xmath75 and @xmath76 . therefore in principle @xmath592 and @xmath889 should initially be thought of as being ( formally ) infinite . however , it is trivial to show that by a ( formally infinite ) gauge transformation , involving the logarithmic ratios @xmath890 and @xmath891 , where @xmath892 and @xmath893 denote any two _ finite _ length scales , one can replace everywhere @xmath70 , @xmath71 by the finite scales @xmath892 , @xmath893 . by this process it is therefore possible to identify the two sets of scales and thereby to think of the scales @xmath70 , @xmath71 as being in fact finite , as implicitly done in ref . @xcite . in the language of renormalization theory , the original ( infinitesimal ) scales @xmath70 and @xmath71 would be referred to as _ hadamard - regularization _ scales entering the computation of divergent poisson integrals [ see section [ hadamardpoisson ] above ] , while the ( finite ) scales @xmath892 and @xmath893 would be referred to as the arbitrary _ renormalization _ scales entering the final , renormalized harmonic - coordinates equations of motion . in the present paper , in order to remain close to the notation used in @xcite , we shall keep the notation @xmath70 and @xmath71 , but interpret them as arbitrary finite constants , which means that we shall _ identify _ them with the finite renormalization length scales @xmath892 and @xmath893 . in other words , the scales @xmath70 , @xmath71 used in the present section should in principle be distinguished from the scales @xmath70 , @xmath71 used in section [ hadamardpoisson ] above . [ remember , in this respect , that the regularization scales @xmath70 , @xmath71 have disappeared when computing the dim . equations of motion . ] with our notation , and still focussing on the static contributions to the equations of motion , the `` parametric '' equations of motion ( [ eq5.53 ] ) imply the following structure for the static coefficients @xmath894 : [ cstaticlambda ] @xmath895 it will be convenient to replace the parameter @xmath4 by the parameter @xmath9 , using ( [ eq5.54 ] ) as a defining one - to - one map between @xmath4 and @xmath9 . with this change of notation the static coefficients become [ cstaticomega ] @xmath896 note that there are two combinations of the three coefficients @xmath897 which do not depend on @xmath669 , namely @xmath898 , and the combination @xmath899 , or even better the combination @xmath900 which depends neither on @xmath669 nor on @xmath9 ( or @xmath4 ) , and which contains , like for @xmath898 , simpler looking rational numbers . we now come back to the effect of the general shift ( [ eq5.51 ] ) on the dim . equations of motion . let us first focus on the static terms . we recall that the ( dim . _ renormalized _ equations of motion had necessarily the form ( [ eq5.49 ] ) . by projecting the latter equation along the static terms @xmath894 , with @xmath901 [ recalling eq . ( [ eq5.5 ] ) ] , it will induce a result for the _ renormalized _ static coefficients of the form @xmath902 where the @xmath903 s are the static coefficients corresponding to @xmath904 , eq . ( [ eq5.50 ] ) . when choosing @xmath905 of the form ( [ eq5.51 ] ) , we see from eq . ( [ eq5.52 ] ) that @xmath906 is simply obtained by projecting the first term on the r.h.s . of ( [ eq5.52 ] ) . remembering that @xmath907 and @xmath908 , we see that the latter term contains the factor @xmath909 . therefore , without doing any further calculation , we see that the shifts @xmath910 have the special properties : @xmath911 and @xmath912 . in other words , a shift of the world - lines of the type ( [ eq5.51 ] ) leaves invariant both @xmath700 and the combination @xmath913 ( as well therefore as the combination @xmath914 considered above ) . as a consequence , we can compute without effort from our previous regularized ( but unrenormalized ) dim . reg . results ( [ eq5.29 ] ) the following two combinations of the @xmath915-_renormalized _ static coefficients : @xmath916 by comparing ( [ eq5.63 ] ) with eq . ( [ eq5.59 ] ) we discover that our present calculation using dimensional regularization in harmonic coordinates necessarily implies that @xmath917 this nicely confirms the previous determination of @xmath9 by a dim . reg . calculation in adm - type coordinates @xcite . we think that our present harmonic - coordinates dim . result calculation is important in proving the consistency of dimensional regularization , and thereby in confirming the physical significance of the result ( [ eq5.65 ] ) . a recent calculation @xcite has also confirmed independently the result ( [ eq5.65 ] ) by means of a completely different method based on surface integrals , and aimed at describing compact ( strongly - gravitating ) objects . by comparing ( [ eq5.64 ] ) with ( [ eq5.61 ] ) , we further see that @xmath918}\,.\ ] ] contrary to ( [ eq5.65 ] ) which represents the determination of a physical parameter ( having an invariant meaning ) , the result ( [ eq5.66 ] ) has no invariant physical significance . ( [ eq5.66 ] ) is simply a consequence of our particular choice for the shift vector ( [ eq5.51 ] ) , in which we assumed that @xmath880 is a purely numerical coefficient , independent on any properties indexed by the particles labels 1 and 2 . in summary the particular shift ( [ eq5.51 ] ) yields some equations of motion which are physically equivalent to a subclass of the general equations of motion considered in @xcite , characterized by the constraint ( [ eq5.66 ] ) . next we relate the common length scale ( [ eq5.66 ] ) to the basic length scale @xmath90 entering dimensional regularization . for doing this we need to fully specify the value of the shift , _ i.e. _ , to choose a specific coefficient @xmath919 in eq . ( [ eq5.51 ] ) . we already know from eq . ( [ eq5.40 ] ) that the coefficient @xmath880 in eq . ( [ eq5.51 ] ) must tend to @xmath920 when @xmath32 , if the @xmath921-shift is to remove the poles in the bulk metric . as in quantum field theory ( qft ) we could then define the _ minimal subtraction _ ( ms ) shift as @xmath922 however , as is well - known in qft , such a ms subtraction has the unpleasant feature of leaving some logarithms of @xmath923 and the euler constant in the renormalized results . these numbers come from the expansion of the gamma function and the associated dimension - dependent powers of @xmath923 entering the @xmath1-dimensional green function . in our context , these numbers showed up in eq . ( [ eq5.29 ] ) in the guise of the combination @xmath924 like in qft , this leads us to consider the following _ modified minimal subtraction _ ( @xmath925 ) shift , @xmath926 which differs from the ms shift ( [ eq5.67 ] ) by the explicit factor of @xmath927 it contains , and by the use of the @xmath1-dimensional ( newtonian ) acceleration given by eq . ( [ eq5.46 ] ) . the inclusion of two explicit powers of @xmath928 in the coefficient @xmath880 entering eq . ( [ eq5.51 ] ) , _ i.e. _ the definition @xmath929 , means , when remembering that @xmath824 , eq . ( [ eq5.46 ] ) , contains one power of @xmath928 , that the static terms in eq . ( [ eq5.52 ] ) will have four powers of @xmath928 and the kinetic terms three . the overall factor @xmath930 in the static terms is natural because these terms are of order @xmath931 and the @xmath241-space gravitational propagator in @xmath1 dimensions always includes the combination @xmath932 . finally , using the fact that the expansion of @xmath933 near @xmath42 reads @xmath934 it is easy to see that the @xmath925 shift defined by ( [ eq5.68 ] ) will cancel the @xmath935 in the bare dim . results of eq . ( [ eq5.29 ] ) . finally , we find that the evaluation of eq . ( [ eq5.52 ] ) for the specific @xmath925 shift , given by @xmath936 , yields for the @xmath925-renormalized static coefficients [ eq5.70abc]@xmath937 where the reader can note that the @xmath938 entering the bare dim . result ( [ eq5.29 ] ) have been transformed into @xmath939 through the @xmath36-expansion of the factor @xmath940 present in @xmath904 . we already discussed above the comparison of two simple combinations of the dim . results ( [ eq5.70abc ] ) with the hadamard - regularization results ( [ cstaticlambda ] ) . it is easy to see that the remaining independent combination , say ( [ eq5.70 ] ) , is fully consistent with its counterpart ( [ eq5.55 ] ) , and allows one to relate the basic renormalization length scale @xmath90 entering dim . reg . to the common length scale ( [ eq5.66 ] ) entering the general equations of motion of @xcite : @xmath941 evidently , the precise values one gets for @xmath61 and @xmath62 depend on the precise choice of the compensating shift . let us now remark that in fact one can recover exactly , provided of course that the crucial result ( [ eq5.65 ] ) holds , the general `` dissymmetric '' class of equations of motion of @xcite , _ i.e. _ , the general parametric result ( [ cstaticlambda ] ) or ( [ cstaticomega ] ) with @xmath942 . for this purpose it suffices to consider a slightly more general shift than the one assumed in the @xmath925 regularization . namely , consider a shift of the same form as ( [ eq5.51 ] ) , but in which one allows the @xmath1-dependent coefficient @xmath880 to depend on the label of the particle in question , that is @xmath943 where now @xmath944 and @xmath945 are allowed to be different from each other . the most general way of parametrizing such dissymmetric @xmath946 [ however constrained by @xmath947 is @xmath948\ , , \label{ead}\ ] ] with two independent numerical coefficients @xmath949 . it is then easily checked that the shift ( [ shifteta ] ) defined by the particular choice @xmath950 transforms the dim . equations of motion into the general ( @xmath951-dependent ) family of solutions obtained in @xcite . if we suppose that the constraints ( [ eq5.73 ] ) hold , then @xmath952 and we recover the shift assumed in the @xmath925 regularization . on the other hand , note than one can reach even more general classes of renormalized harmonic equations of motion in dim . reg . ( as one could have also done in hadamard regularization ) . indeed , we could use the freedom indicated in eq . ( [ eq5.44 ] ) of adding _ arbitrary _ finite parts to the shifts . anyway , the result that the shift ( [ shifteta])-([rhoa ] ) gives equivalence between the dim . reg . and the extended - hadamard 3pn accelerations [ we check in section [ kinetic ] below that the kinetic terms work also ] , constitutes the main result of the present paper ( theorem 2 in the introduction ) . the two approaches we have discussed here are of course equivalent : choosing some dim . basic length scale @xmath90 and some specific , simplifying dim . reg . shift ( such as the @xmath925 one ) , and then determining the values of the scales @xmath951 for which the dim . results match with the hadamard reg . ones ; or arbitrarily choosing some hadamard scales @xmath951 and then determining the corresponding general dissymmetric dim . reg . shift ( [ shifteta])-([rhoa ] ) , in terms of the chosen @xmath951 s . what is important is that we have checked that the _ three _ renormalized dim . static coefficients ( [ eq5.29 ] ) are fully compatible with the _ three _ extended - hadamard reg . static coefficients ( [ cstaticlambda ] ) or ( [ cstaticomega ] ) , and that their comparison yields _ one and only one _ physical result , namely : @xmath114 . up to now we have verified the following aspects of the consistency of a dim . treatment of the 3pn dynamics of two point particles : ( 1 ) consistency between the shift ( [ eq5.40 ] ) needed to renormalize the bulk metric and the shift ( [ eq5.68 ] ) [ or ( [ shifteta])-([ead ] ) ] needed to renormalize the equations of motion ; ( 2 ) consistency between the three finite , renormalized dim . static coefficients ( [ eq5.70abc ] ) and the general three - dimensional ones ( [ cstaticlambda ] ) , @xcite ; and ( 3 ) consistency between the present dim . reg . value of @xmath4 and the previously derived dim . value of @xmath9 in the adm - hamiltonian @xcite . it remains , however , to check that the velocity - dependent terms in the renormalized dim . equations of motion do agree with their analogs in the harmonic - coordinates equations of motion of @xcite . this will in particular prove that the dim . equations of motion are lorentz invariant . in the notation of eq . ( [ eq5.5 ] ) above , we need to consider the values of the velocity - dependent coefficients @xmath953 , @xmath954 , @xmath955 , @xmath956 , @xmath957 , and @xmath958 . in ref . @xcite , they were shown to take the following parametric forms , which actually depend only on the regularization scale @xmath61 but not on @xmath62 nor @xmath4 : [ eq5.78 ] @xmath959 \ln\left(\frac{r_{12}}{r'_1}\right ) + \frac{48197}{840}\ , v_1 ^ 2 -\frac{36227}{420 } ( v_1 v_2 ) + \frac{36227}{840}\ , v_2 ^ 2 \nonumber\\ & & - \frac{45887}{168 } ( n_{12}v_1)^2 + \frac{24025}{42 } ( n_{12}v_1)(n_{12}v_2 ) -\frac{10469}{42 } ( n_{12}v_2)^2 \,,\qquad\label{eq5.78a}\\ c'^{\text{bf}}_{21}(r'_1 ) = c''^{\text{bf}}_{21}(r'_1)&= & -44 ( n_{12}v_{12 } ) \ln\left(\frac{r_{12}}{r'_1}\right ) + \frac{31397}{420}(n_{12}v_1 ) - \frac{36227}{420 } ( n_{12}v_2 ) \,,\label{eq5.788b}\\ c_{03}^{\text{bf}}&=&18 ( v_1 v_2 ) - 9\ , v_2 ^ 2 - ( n_{12}v_1)^2 + 2(n_{12}v_1)(n_{12}v_2 ) + \frac{43}{2}(n_{12}v_2)^2\,,\label{eq5.78c}\\ c'^{\text{bf}}_{03 } = c''^{\text{bf}}_{03}&=&4(n_{12}v_1 ) + 5 ( n_{12}v_2)\ , . \label{eq5.78d}\end{aligned}\ ] ] as explained in eq . ( [ eq5.1 ] ) , the dim . expressions of these coefficients can be computed as the sum of pure hadamard - schwartz ( phs ) contributions , @xmath694 and @xmath960 , and the `` @xmath961 '' differences , @xmath703 , @xmath962 . the calculation of the phs contributions has been explained in section [ additional ] above , and we get the following results from eqs . ( [ defph ] ) and ( [ eq5.78 ] ) : [ eq5.74 ] @xmath963 \ln\left(\frac{r_{12}}{r'_1 } \right ) + \frac{10639}{168}\ , v_1 ^ 2 - \frac{5879}{60 } ( v_1 v_2 ) + \frac{5843}{120}\ , v_2 ^ 2 \nonumber\\ & & - \frac{50885}{168 } ( n_{12}v_1)^2 + \frac{1892}{3 } ( n_{12}v_1)(n_{12}v_2 ) - \frac{3325}{12 } ( n_{12}v_2)^2 \,,\label{eq5.74a}\\ c'^{\text{phs}}_{21}&=&- 44 ( n_{12}v_{12 } ) \ln\left(\frac{r_{12}}{r'_1}\right ) + \frac{7279}{84 } ( n_{12}v_1 ) - \frac{5879}{60 } ( n_{12}v_2 ) \,,\label{eq5.74b}\\ c''^{\text{phs}}_{21}&=&- 44 ( n_{12}v_{12 } ) \ln\left(\frac{r_{12}}{r'_1}\right ) + \frac{5189}{60 } ( n_{12}v_1 ) - \frac{5843}{60 } ( n_{12}v_2 ) \,,\label{eq5.74c}\\ c_{03}^\text{phs}&=&18 ( v_1 v_2 ) - \frac{64}{7}\ , v_2 ^ 2 - ( n_{12}v_1)^2 + 2 ( n_{12}v_1)(n_{12}v_2 ) + \frac{311}{14}(n_{12}v_2)^2\,,\label{eq5.74d}\\ c'^{\text{phs}}_{03}&=&4(n_{12}v_1 ) + 5 ( n_{12}v_2)\,,\label{eq5.74e}\\ c''^{\text{phs}}_{03}&=&4(n_{12}v_1 ) + \frac{37}{7 } ( n_{12}v_2 ) \,.\label{eq5.74f}\end{aligned}\ ] ] @xmath176 secondly , the method explained in sections [ diff ] and [ dimregstat ] for computing the differences @xmath964 is found ( after doing calculations similar to those reported in the tables of `` static '' contributions above ) to lead to [ eq5.75 ] @xmath965 \left[v_{12}^2 -5 ( n_{12}v_{12})^2 \right ] + \frac{499}{42}\ , v_1 ^ 2 - \frac{359}{15}(v_1 v_2 ) \nonumber\\ & & + \frac{184}{15}\ , v_2 ^ 2 - \frac{2957}{42 } ( n_{12}v_1)^2 + \frac{425}{3 } ( n_{12}v_1 ) ( n_{12}v_2 ) - \frac{217}{3 } ( n_{12}v_2)^2 + { \mathcal o } ( \varepsilon)\,,\qquad \label{eq5.75a}\\ { \mathcal d } c'_{21}&=&\left [ \frac{22}{\varepsilon } -33\ln\left(\overline q\ , r'^{4/3}_1 r_{12}^{2/3}\right ) \right](n_{12}v_{12 } ) + \frac{499}{21}(n_{12}v_1 ) - \frac{359}{15}(n_{12}v_2 ) + { \mathcal o } ( \varepsilon)\ , , \label{eq5.75b}\\ { \mathcal d } c''_{21}&=&\left [ \frac{22}{\varepsilon } -33\ln\left(\overline q\ , r'^{4/3}_1 r_{12}^{2/3}\right ) \right](n_{12}v_{12 } ) + \frac{359}{15}(n_{12}v_1 ) - \frac{368}{15}(n_{12}v_2 ) + { \mathcal o } ( \varepsilon)\ , , \label{eq5.75c}\\ { \mathcal d } c_{03}&=&\frac{1}{7}\ , v_2 ^ 2 - \frac{5}{7 } ( n_{12}v_2)^2\,,\label{eq5.75d}\\ { \mathcal d } c'_{03}&=&0\,,\label{eq5.75e}\\ { \mathcal d } c''_{03}&=&-\frac{2}{7}(n_{12}v_2 ) \,.\label{eq5.75f}\end{aligned}\ ] ] @xmath176 together with eqs . ( [ eq5.28 ] ) above , these equations give the full difference between the dimensionally regularized and the pure hadamard accelerations , and they constitute the main new input of the present work . the bare dim . results , @xmath966 , read therefore [ eq5.76 ] @xmath967 \left[v_{12}^2 - 5 ( n_{12}v_{12})^2\right ] + \frac{1805}{24}\ , v_1 ^ 2 - \frac{1463}{12 } ( v_1 v_2 ) \nonumber\\ & & + \frac{1463}{24}\ , v_2 ^ 2 -\frac{8959}{24 } ( n_{12}v_1)^2 + \frac{2317}{3 } ( n_{12}v_1 ) ( n_{12}v_2 ) - \frac{4193}{12 } ( n_{12}v_2)^2 \,,\qquad\label{eq5.76a}\\ c'^{\text{dr}}_{21 } = c''^{\text{dr}}_{21}&=&\left [ \frac{22}{\varepsilon } - 33 \ln\left(\overline q\ , r_{12}^2\right ) \right](n_{12}v_{12 } ) + \frac{1325}{12}(n_{12}v_1 ) - \frac{1463}{12 } ( n_{12}v_2 ) \,,\label{eq5.76b}\\ c_{03}^\text{dr}&=&18 ( v_1 v_2 ) - 9\ , v_2 ^ 2 - ( n_{12}v_1)^2 + 2(n_{12}v_1)(n_{12}v_2 ) + \frac{43}{2}(n_{12}v_2)^2\,,\label{eq5.76c}\\ c'^{\text{dr}}_{03 } = c''^{\text{dr}}_{03}&=&4(n_{12}v_1 ) + 5 ( n_{12}v_2)\,.\label{eq5.76d}\end{aligned}\ ] ] @xmath176 note that in the final result the equality between @xmath968 and @xmath969 , _ i.e. _ , between @xmath658 and @xmath659 in eq . ( [ eq5.3 ] ) , is recovered . the bare dim . kinetic coefficients ( [ eq5.76 ] ) contain poles @xmath93 but do not depend anymore of the arbitrary hadamard regularization scale @xmath61 which appeared in ( [ eq5.74 ] ) . as in the case discussed above of the static coefficients the previous kinetic coefficients do not involve any adimensionalizing length scales in the logarithms of @xmath663 they contain . this is consistent with the fact that it is the combinations @xmath970 and @xmath971 which have the same physical dimension as their @xmath42 counterparts . finally , given a specific choice of shift , say the @xmath925 one , eq . ( [ eq5.68 ] ) , the _ kinetic coefficients are obtained by adding to ( [ eq5.76 ] ) the velocity - dependent part of the effect of the shift , _ i.e. _ , the second term on the r.h.s . of eq . ( [ eq5.52 ] ) [ with , say , @xmath972 . our final @xmath973 results for the renormalized kinetic coefficients are found to be [ eq5.77 ] @xmath974 \ln\left(\frac{r_{12}}{\ell_0}\right ) + \frac{1321}{24}\ , v_1 ^ 2 -\frac{979}{12 } ( v_1 v_2 ) \nonumber\\ & & + \frac{979}{24}\ , v_2 ^ 2 - \frac{6275}{24 } ( n_{12}v_1)^2 + \frac{1646}{3 } ( n_{12}v_1)(n_{12}v_2 ) -\frac{2851}{12 } ( n_{12}v_2)^2 \,,\qquad\label{eq5.77a}\\ c'^{\overline{\text{ms}}}_{21 } = c''^{\overline{\text{ms}}}_{21}&= & -44 ( n_{12}v_{12 } ) \ln\left(\frac{r_{12}}{\ell_0}\right ) + \frac{841}{12}(n_{12}v_1 ) - \frac{979}{12 } ( n_{12}v_2 ) \,,\label{eq5.77b}\\ c_{03}^{\overline{\text{ms}}}&=&18 ( v_1 v_2 ) - 9\ , v_2 ^ 2 - ( n_{12}v_1)^2 + 2(n_{12}v_1)(n_{12}v_2 ) + \frac{43}{2}(n_{12}v_2)^2\,,\label{eq5.77c}\\ c'^{\overline{\text{ms}}}_{03 } = c''^{\overline{\text{ms}}}_{03}&=&4(n_{12}v_1 ) + 5 ( n_{12}v_2 ) \,.\label{eq5.77d}\end{aligned}\ ] ] when comparing these results with the ones of ref . @xcite , eqs . ( [ eq5.78 ] ) above , one remarkably finds that our previously derived link ( [ eq5.73 ] ) is necessary and sufficient for ensuring the full compatibility between the renormalized dim . results and the corresponding hadamard reg note that the rational coefficients entering the dim . results are often simpler than the coefficients entering the equations of motion of @xcite . the results eq . ( [ eq5.77 ] ) complete our check of the full consistency of the dim . evaluation of the 3pn equations of motion , and the proof of theorems 1 and 2 stated in the introduction . we have used dimensional regularization ( _ i.e. _ analytic continuation in the spatial dimension @xmath1 ) to determine the spacetime metric and the equations of motion ( eom ) in _ harmonic coordinates _ , of two , gravitationally interacting , point masses , at the third post - newtonian ( 3pn ) order of general relativity . our starting point consisted in writing the 3pn - accurate metric @xmath975 in terms of a certain number of `` elementary potentials '' @xmath976 , satisfying a hierarchy of inhomogeneous dalembert equations of the form @xmath977 . the sources of the latter equations contain both `` compact '' terms , _ i.e. _ , in the present case _ contact _ terms of the form @xmath978 \delta^{(d)}(\mathbf{x}-\mathbf{y}_1)$ ] , and nonlinearly generated `` non compact '' terms of the typical form , say , @xmath979 . this representation of the 3pn metric , as well as the associated iterative way of solving for the potentials [ using the time - symmetric green s function @xmath980 is a direct generalization of the one used in ref . @xcite . however , it has been crucial for our work to determine ( in section [ fieldeq ] ) the dependence upon the dimension @xmath1 of the coefficients appearing in this representation , as well as the @xmath1-dependence of the kernels expressing the operators @xmath759 and @xmath981 in @xmath241-space . by studying the structure of the iterative solution for the metric , and that of the corresponding eom ( which are conveniently pictured by means of diagrams , see figs . [ fig1]-[fig8 ] ) , we determined , in the form of a laurent expansion in @xmath35 , the pole part of the metric @xmath94 , and the pole and finite parts of the eom , namely @xmath982 where @xmath983 and @xmath984 . [ see , however , appendix [ littleg ] where the basic quadratically non - linear kernel @xmath43 is computed in any @xmath1 dimensions , not necessarily close to 3 . ] our calculations relied heavily on previous work in @xmath42 @xcite , and were technically implemented in two steps ( at least for the determination of the eom , which are more delicate ; the determination of the pole part of the metric uses only the second step ) . \(i ) a first step consisted of subtracting from the final , published results for the eom @xcite , seven contributions that were specific consequences of the use of an extension of the hadamard regularization method @xcite ( which included an extension of the schwartz notion of distributional derivative ) . the result of this first step is referred to as the `` pure hadamard - schwartz '' evaluation of the eom . \(ii ) the second step was the evaluation of the _ difference _ between the dimensional regularization of each contribution to the eom ( written in terms of the iterative solutions for the various potentials @xmath976 ) , and the corresponding `` pure hadamard - schwartz '' contribution obtained in the first step . this difference is obtained , similarly to the method used in @xcite , by splitting the @xmath1-dimensional integral into several pieces , and by carefully analyzing the terms due to the neighborhoods of the two singular points @xmath985 ( including possible @xmath1-dimensional distributional contributions ) . concerning the `` bulk '' metric @xmath94 , at a field point away from the singular particle world - lines @xmath986 , we derived only the pole part , that is the coefficient of @xmath778 in the laurent expansion of @xmath987 . we found that at the 3pn order only the time - time component of the metric @xmath988 contained a pole [ see eq . ( [ eq5.39 ] ) ] . for the eom we derived both the pole part and the finite part , _ i.e. _ @xmath989 . the parts of the eom for which the regularization was delicate are given by the nine coefficients @xmath990 defined in eq . ( [ eq5.5 ] ) . our complete results for the dimensionally - regularized values of these nine `` delicate '' coefficients are given in eqs . ( [ eq5.29 ] ) and ( [ eq5.76 ] ) . we proved that the pole parts of both the metric and the eom can be `` renormalized away '' by suitable _ shifts of the world - lines _ of the form @xmath991 , where @xmath841 is the original world - line on which are initially concentrated the @xmath992-function sources representing the point masses , where the shifts @xmath993 are of the 3pn order , and where the eom of the renormalized world - line @xmath842 is _ finite _ as @xmath104 . the general form of the needed shifts is given by eq . ( [ eq5.44 ] ) with ( [ eq5.40 ] ) . the renormalized eom corresponding to the `` modified minimal subtraction '' scheme ( [ eq5.68 ] ) are given by eqs . ( [ eq5.70abc ] ) and ( [ eq5.77 ] ) . the finite renormalized 3pn - accurate eom obtained by using the general ( two - parameter ) renormalization shift ( [ shifteta])-([rhoa ] ) were shown to be _ equivalent _ to the final ( three - parameter ) eom of @xcite if and only if the hadamard - undetermined dimensionless parameter @xmath4 which entered the latter equations takes the unique value @xmath994 . this value is in agreement with the result of a previous dimensional - regularization determination of the arnowitt - deser - misner hamiltonian ( in adm - like coordinates ) @xcite , which led to the unique determination of the adm analogue of @xmath4 , namely @xmath6 . the value for @xmath4 or @xmath9 is also in agreement with the recent work @xcite which derived the 3pn equations of motion in harmonic gauge using a surface - integral approach . our result provides an important check of the consistency of dimensional regularization because our calculations are very different from the ones of @xcite , notably we use a different coordinate system and a different method for iterating einstein s field equations . however , the applicability of our general approach to higher post - newtonian orders remains unexplored . finally , the present work opens the way to a dimensional - regularization determination of the several unknown dimensionless parameters that were shown to enter the hadamard - regularization of the 3pn binary s energy flux ( in harmonic coordinates ) @xcite . the completion of the 3pn energy flux is urgent in view of its importance in determining the gravitational waveforms emitted by inspiralling black hole binaries , which are primary targets for the international network of interferometric gravitational wave detectors ligo / virgo / geo . most of the algebraic calculations reported in this paper were done with the help of the software _ mathematica_. t.d . would like to thank the kavli institute for theoretical physics for hospitality ( under the partial support of the national science foundation grant no . phy99 - 07949 ) while this work was completed . we give in this appendix several expanded expressions which are too lengthy to be included in the body of the article . the expanded form of the metric ( [ metric ] ) is easier to compare with the literature , and notably with eqs . ( 3.24 ) of ref . @xcite : [ metricexp ] @xmath995 + \frac{8}{c^6}\left[\hat x + v_i v_i + \frac{1}{6}\ , v^3 + \left(\frac{d-3}{d-2}\right ) v k\right]\nonumber\\ & & + \frac{32}{c^8}\left [ \hat t -\frac{1}{2}\ , v \hat x + \hat r_i v_i -\frac{1}{2}\ , v v_i v_i -\frac{1}{48}\ , v^4 + \frac{1}{4}\left(\frac{d-3}{d-2}\right ) k v^2 -\frac{1}{4}\left(\frac{d-3}{d-2}\right)^2k^2\right]\nonumber\\ & & + { \mathcal{o}}\left(\frac{1}{c^{10}}\right ) , \label{g00exp}\\ g_{0i}&=&-\frac{4}{c^3}\ , v_i -\frac{8}{c^5}\left[\hat r_i -\frac{1}{2}\left(\frac{d-3}{d-2}\right)v v_i\right ] -\frac{16}{c^7}\biggl [ \hat y_i+\frac{1}{2}\ , \hat w_{ij}v_j\nonumber\\ & & + \frac{1}{4}\left(1+\frac{1}{(d-2)^2}\right)v^2 v_i -\frac{1}{2}\left(\frac{d-3}{d-2}\right ) v \hat r_i + \frac{1}{2}\left(\frac{d-3}{d-2}\right)^2 k v_i\biggr ] + { \mathcal{o}}\left(\frac{1}{c^{9}}\right ) , \label{g0iexp}\\ g_{ij}&=&\delta_{ij}\biggl\ { 1 + \frac{2}{(d-2)c^2}\ , v + \frac{2}{(d-2)^2 c^4}\left[v^2 -2 ( d-3 ) k\right]\nonumber\\ & & + \frac{8}{c^6}\left [ \frac{\hat x}{d-2 } + \frac{v_k v_k}{d-2 } + \frac{v^3}{6(d-2)^3 } -\frac{(d-3)}{(d-2)^3}\ , v k\right ] \biggr\}\nonumber\\ & & + \frac{4}{c^4}\ , \hat w_{ij } + \frac{16}{c^6}\left [ \hat z_{ij } + \frac{v\hat w_{ij}}{2(d-2 ) } -v_i v_j\right ] + { \mathcal{o}}\left(\frac{1}{c^8}\right ) . \label{gijexp}\end{aligned}\ ] ] @xmath176 the inverse metric is such that @xmath996 in @xmath120 space - time dimensions . in terms of the modified newtonian potential @xmath198 defined in eq . ( [ calv ] ) above , it reads : [ invmetric ] @xmath997 + \frac{8\hat r_i}{c^5 } + \frac{16}{c^7}\left[\hat y_i -\frac{1}{2}\hat w_{ij}v_j\right]\right\ } + { \mathcal{o}}\left(\frac{1}{c^9}\right),\nonumber\\ \label{gup0i}\\ g^{ij}&=&e^{-\frac{2\mathcal{v}}{(d-2)c^2}}\left\{\delta_{ij } -\frac{4}{c^4 } \hat w_{ij } -\frac{16}{c^6 } \left[\hat z_{ij } + \frac{1}{2(d-2)}\ , \delta_{ij } v_k v_k \right]\right\ } + { \mathcal{o}}\left(\frac{1}{c^8}\right ) . \label{gupij}\end{aligned}\ ] ] note the change of signs in the exponentials [ with respect to the covariant metric ( [ metric ] ) ] , in front of @xmath998 in eq . ( [ gup0i ] ) , as well as for the @xmath999 and @xmath193 terms in eq . ( [ gupij ] ) . note also that the @xmath1000 contribution to @xmath170 has disappeared in the inverse spatial metric @xmath1001 . the full post - newtonian expansion of this inverse metric reads : [ invmetricexp ] @xmath1002 -\frac{8}{c^6}\left[\hat x -v_i v_i + \frac{v^3}{6 } -\left(\frac{d-3}{d-2}\right ) v k\right]\nonumber\\ & & -\frac{32}{c^8}\left [ \hat t + \frac{1}{2}\ , v \hat x -\hat r_i v_i -\frac{1}{2}\ , v v_i v_i + \frac{v^4}{48 } -\frac{1}{4}\left(\frac{d-3}{d-2}\right ) k v^2 + \frac{1}{4}\left(\frac{d-3}{d-2}\right)^2k^2\right]\nonumber\\ & & + { \mathcal{o}}\left(\frac{1}{c^{10}}\right ) , \label{gup00exp}\\ g^{0i}&=&-\frac{4}{c^3}\ , v_i -\frac{8}{c^5}\left[\hat r_i + \frac{1}{2}\left(\frac{d-3}{d-2}\right)v v_i\right ] -\frac{16}{c^7}\biggl [ \hat y_i-\frac{1}{2}\ , \hat w_{ij}v_j\nonumber\\ & & + \frac{1}{4}\left(1+\frac{1}{(d-2)^2}\right)v^2 v_i + \frac{1}{2}\left(\frac{d-3}{d-2}\right ) v \hat r_i -\frac{1}{2}\left(\frac{d-3}{d-2}\right)^2 k v_i\biggr ] + { \mathcal{o}}\left(\frac{1}{c^{9}}\right ) , \label{gup0iexp}\\ g^{ij}&=&\delta_{ij}\biggl\ { 1 -\frac{2}{(d-2)}\ , \frac{v}{c^2 } + \frac{2}{(d-2)^2 c^4}\left[v^2 + 2 ( d-3 ) k\right]\nonumber\\ & & -\frac{8}{c^6}\left [ \frac{\hat x}{d-2 } + \frac{v_k v_k}{d-2 } + \frac{v^3}{6(d-2)^3 } + \frac{d-3}{(d-2)^3}\ , v k\right ] \biggr\}\nonumber\\ & & -\frac{4}{c^4}\ , \hat w_{ij } -\frac{16}{c^6}\left [ \hat z_{ij } -\frac{1}{2(d-2)}\ , v\hat w_{ij}\right ] + { \mathcal{o}}\left(\frac{1}{c^8}\right ) . \label{gupijexp}\end{aligned}\ ] ] @xmath176 the determinant @xmath1003 of the metric is a useful quantity , notably to compute the `` gothic '' metric @xmath127 , which is the natural variable when using the harmonic - coordinate system . the simplest way to compute it is to use the exponential form ( [ metric ] ) of the metric , and to perform a cofactor expansion across both the first line and the first column : @xmath1004 since @xmath1005 \times\left[\delta_{ij } + { \mathcal{o}}(1/c^4)\right]$ ] , the determinant of the @xmath1006 matrix @xmath1007 reads @xmath1008 therefore , the determinant of the full metric is given by @xmath1009 where we have used the fact that @xmath1010 . note that this formula suffices to compute @xmath167 up to order @xmath200 included if one knows the spatial metric @xmath170 up to this same order . at the 3pn order , we get easily @xmath1011\nonumber\\ & & \hphantom{-e^\frac{4\mathcal{v}}{(d-2)c^2}\biggl\{}+ \frac{16}{c^6}\ , v_i v_i \biggr\ } + { \mathcal{o}}\left(\frac{1}{c^8}\right)\nonumber\\ & = & -e^\frac{4\mathcal{v}}{(d-2)c^2 } \left[1 + \frac{4}{c^4}\ , \hat w + \frac{16}{c^6}\left(\hat z + \frac{1}{d-2}\ , v_i v_i\right ) \right]+{\mathcal{o}}\left(\frac{1}{c^8}\right ) , \label{g}\end{aligned}\ ] ] where we used the expansion @xmath1012 , valid for any matrix @xmath1013 whose entries are small with respect to 1 . we can now compute the square root of this determinant , and give its full post - newtonian expansion : [ sqrtg ] @xmath1014+{\mathcal{o}}\left(\frac{1}{c^8}\right ) \label{sqrtgcompact}\\ & = & 1 + \frac{2}{(d-2)}\ , \frac{v}{c^2 } + \frac{2}{c^4}\left [ \hat w + \frac{v^2}{(d-2)^2 } -\frac{2(d-3)}{(d-2)^2}\ , k \right ] + \frac{8}{c^6}\biggl [ \hat z + \frac{v_i v_i}{d-2}\nonumber\\ & & + \frac{\hat x}{d-2 } + \frac{v\hat w}{2(d-2 ) } + \frac{v^3}{6(d-2)^3 } -\frac{d-3}{(d-2)^3}\ , v k \biggr ] + { \mathcal{o}}\left(\frac{1}{c^8}\right ) . \label{sqrtgexpanded}\end{aligned}\ ] ] the gothic metric @xmath127 can now be written easily by combining eqs . ( [ invmetric ] ) or ( [ invmetricexp ] ) with ( [ sqrtgcompact ] ) or ( [ sqrtgexpanded ] ) . we shall not display here the explicit results , since they were not directly useful for the present article . let us however quote the expression of the ricci tensor in terms of the gothic metric , in @xmath120 space - time dimensions and in any gauge : @xmath1015 as usual , a comma denotes partial derivation , and @xmath1016 is the inverse of @xmath1017 . in terms of the gothic metric , the harmonic gauge condition ( [ harmgauge ] ) takes a particularly simple form : @xmath1018 this is the reason why this gothic metric can be useful to write the field equations . note that several terms of eq . ( [ riccigoth ] ) vanish in this gauge , namely the first two ( involving second derivatives ) and those proportional to @xmath1019 in the second line . nevertheless , this expression for @xmath1020 is slightly more complicated than the one we used in section [ fieldeq ] above , eq . ( [ ricciharm ] ) , which does not depend explicitly on the spatial dimension @xmath1 . it should be noted that many equations given in the book @xcite are erroneous when @xmath165 ( _ i.e. _ , when @xmath1021 , in this book s notation ) , including eq . ( i , 14 , 30 ) in @xcite which gives the ricci tensor in terms of the gothic metric . let us end this appendix by displaying the full expansion of the geodesic equation ( [ eqgeod ] ) , or more precisely of the vectors @xmath1022 and @xmath1023 , quickly illustrated in eqs . ( [ smallexppifi ] ) . the following expressions are @xmath1-dimensional generalizations of eqs . ( 3.35 ) of ref . @xcite , and we keep the same writing and order of the terms to ease the comparison . the `` linear momentum '' @xmath1022 reads @xmath1024\nonumber\\ & & + \frac{1}{c^6}\biggl [ \frac{5}{16}\ , v^6 v^i + \frac{3(5 d -4)}{8(d-2)}\ , v v^4 v^i -\frac{3}{2}\ , v_i v^4 -6 v_j v^i v^j v^2 + \frac{(3 d -2)^2}{4(d-2)^2}\ , v^2 v^2 v^i\nonumber\\ & & + 2\hat w_{ij } v^j v^2 + 2\hat w_{jk } v^i v^j v^k -\frac{2(2d-1)}{d-2}\ , v v_i v^2 -\frac{4(2d-1)}{d-2}\ , v v_j v^i v^j -4 \hat r_i v^2\nonumber\\ & & -8 \hat r_j v^i v^j + \frac{d^3}{6(d-2)^3}\ , v^3 v^i + \frac{4 d}{d-2}\ , v_j v_j v^i + \frac{4 d}{d-2}\ , \hat w_{ij } v v^j + \frac{4 d}{d-2}\ , \hat x v^i\nonumber\\ & & + 16 \hat z_{ij } v^j -2\,\frac{d(d-2)+2}{(d-2)^2}\ , v^2 v_i -8 \hat w_{ij } v^j -\frac{8}{d-2 } v \hat r_i -16 \hat y_i\nonumber\\ & & -\frac{(3d-2)(d-3)}{(d-2)^2}\ , k v^2 v^i -\frac{2 d^2(d-3)}{(d-2)^3}\ , k v v^i + \frac{8(d-3)}{(d-2)^2}\ , k v_i \biggr ] + { \mathcal{o}}\left(\frac{1}{c^8}\right ) . \label{exppi}\end{aligned}\]]@xmath176 this @xmath1-dimensional expression actually allows us to understand better some of the numerical coefficients found for @xmath42 in ref . for instance , we find that a factor 33 comes from the expression @xmath1025 , and that a factor 49 comes from @xmath1026 . the full post - newtonian expansion of the `` force '' @xmath1023 is given by an even longer formula : @xmath1027 \nonumber\\ & & + \frac{1}{c^4}\biggl [ \frac{3 d -2}{8(d-2)}\ , \partial_i v v^4 -2 \partial_i v_j v^j v^2 + \frac{d^2}{2(d-2)^2}\ , v\partial_i v v^2 + 2 \partial_i\hat w_{jk } v^j v^k \nonumber\\ & & -\frac{4}{d-2}\left(v_j\ , \partial_i v v^j + v\partial_i v_j v^j\right ) -8 \partial_i\hat r_j v^j + \frac{1}{2}\ , v^2 \partial_i v + 8 v_j\ , \partial_i v_j \nonumber\\ & & + 4\partial_i\hat x + 2\left(\frac{d-3}{d-2}\right)\left ( k \partial_i v + v \partial_i k\right ) -\frac{d(d-3)}{(d-2)^2}\ , \partial_i k v^2 \biggr ] \nonumber\\ & & + \frac{1}{c^6}\biggl [ \frac{1}{16}\left(\frac{5d-4}{d-2}\right)v^6\partial_i v -\frac{3}{2}\ , \partial_i v_j v^j v^4 + \frac{1}{8}\left(\frac{3d-2}{d-2}\right)^2 v\partial_i v v^4 + \partial_i\hat w_{jk } v^2 v^j v^k \nonumber\\ & & -2\left(\frac{2d-1}{d-2}\right ) v_j\ , \partial_i v v^2 v^j -2\left(\frac{2d-1}{d-2}\right ) v \partial_i v_j v^2 v^j -4 \partial_i \hat r_j v^2 v^j \nonumber\\ & & + \frac{1}{4}\left(\frac{d}{d-2}\right)^3 v^2 \partial_i v v^2 + \frac{4d}{d-2}\ , v_j \partial_i v_j v^2 + \frac{2d}{d-2}\ , \hat w_{jk}\ , \partial_i v v^j v^k \nonumber\\ & & + \frac{2d}{d-2}\ , v \partial_i \hat w_{jk } v^j v^k + \frac{2d}{d-2}\ , \partial_i\hat x v^2 + 8\partial_i \hat z_{jk } v^j v^k -4\,\frac{d(d-2)+2}{(d-2)^2}\ , v_j v \partial_i v v^j \nonumber\\ & & -2\,\frac{d(d-2)+2}{(d-2)^2}\ , v^2 \partial_i v_j v^j -8 v_k \partial_i\hat w_{jk } v^j -8 \hat w_{jk } \partial_i v_k v^j -\frac{8}{d-2}\ , \hat r_j \partial_i v v^j \nonumber\\ & & -\frac{8}{d-2}\ , v\partial_i\hat r_j v^j -16 \partial_i \hat y_j v^j -\frac{1}{6}\ , v^3 \partial_i v -4 v_j v_j \partial_i v + 16\hat r_j \partial_i v_j + 16 v_j \partial_i\hat r_j \nonumber\\ & & -8 v v_j \partial_i v_j -4\hat x\partial_i v -4 v \partial_i\hat x + 16\partial_i\hat t -\frac{d^2(d-3)}{(d-2)^3}\ , k \partial_i v v^2 -2\left(\frac{d-3}{d-2}\right ) k v \partial_i v \nonumber\\ & & + \frac{8(d-3)}{(d-2)^2}\ , k \partial_i v_j v^j -\frac{(3d-2)(d-3)}{4(d-2)^2}\ , \partial_i k v^4 -\frac{d^2(d-3)}{(d-2)^3}\ , \partial_i k v v^2 \nonumber\\ & & -\frac{d-3}{d-2}\ , v^2\partial_i k + \frac{8(d-3)}{(d-2)^2}\ , \partial_i k v_j v^j -4\left(\frac{d-3}{d-2}\right)^2 k \partial_i k \biggr ] + { \mathcal{o}}\left(\frac{1}{c^8}\right ) . \label{expfi}\end{aligned}\]]@xmath176 this appendix is intended to provide a compendium of ( mostly well - known ) formulae for working in a space with @xmath1 dimensions . as usual , though we shall motivate some formulae below by writing some intermediate expressions which make complete sense only when @xmath1 is a strictly positive integer , our final formulae are to be interpreted , by complex analytic continuation , for a general complex dimension , @xmath1028 . actually one of the main sources of the power of dimensional regularization is its ability to prove many results by invoking complex analytic continuation in @xmath1 . we discuss first the volume of the sphere having @xmath580 dimensions ( _ i.e. _ , embedded into euclidean @xmath1-dimensional space ) . we separate out the infinitesimal volume element in @xmath1 dimensions into radial and angular parts , @xmath1029 where @xmath1030 denotes the radial variable ( _ i.e. _ , the euclidean norm of @xmath1031 and @xmath578 is the infinitesimal solid angle sustained by the unit sphere with @xmath580 dimensional surface . to compute the volume of the sphere , @xmath1032 , one notices that the following @xmath1-dimensional integral can be computed both in cartesian coordinates , where it reduces simply to a gaussian integral , and also , using ( [ ddx ] ) , in spherical coordinates : @xmath1033 where @xmath470 in the last equation denotes the eulerian function . this leads to the well known result @xmath1034 for instance one recovers the standard results @xmath1035 and @xmath1036 , but also @xmath1037 , which can be interpreted by remarking that the sphere with 0 dimension is actually made of two points . if we parametrize the sphere @xmath581 in @xmath580 dimensions by means of @xmath580 spherical coordinates @xmath1038 , @xmath1039 , @xmath313 , which are such that the sphere @xmath1040 in @xmath1041 dimensions is then parametrized by @xmath1039 , @xmath1042 , @xmath313 , and so on for the lower - dimensional spheres , then we find that the differential volume elements on each of the successive spheres obey the recursive relation @xmath1043 note that this implies @xmath1044 which can also be checked directly by using the explicit expression ( [ omegad1 ] ) . next we consider the dirac delta - function @xmath1045 in @xmath1 dimensions , which is formally defined , as in ordinary distribution theory @xcite , by the following linear form acting on the set @xmath384 of smooth functions @xmath1046 with compact support : @xmath1047 , @xmath1048 where the brackets refer to the action of a distribution on @xmath1049 . let us now check that the function defined by @xmath1050 [ where @xmath1051 is the radial coordinate in @xmath1 dimensions , such that @xmath1052 is the `` green s function '' of the poisson operator , namely that it obeys the distributional equation @xmath1053 for any @xmath1054 we have @xmath1055 , thus we see that @xmath1056 in the sense of functions . let us formally compute its value in the sense of distributions in @xmath241-space . [ the usual verification of ( [ deltau ] ) is done in fourier space . ] we apply the distribution @xmath1057 on some test function @xmath1049 , use the definition of the distributional derivative to shift the laplace operator from @xmath1058 to @xmath1059 , compute the value of the @xmath1-dimensional integral by removing a ball of small radius @xmath289 surrounding the origin [ say @xmath1060 , apply the fact that @xmath1061 in the exterior of @xmath1062 , use the gauss theorem to transform the result into a surface integral , and finally compute that integral by inserting the taylor expansion of @xmath1059 around the origin . the proof of eq . ( [ deltau ] ) is thus summarized in the following steps : @xmath1063 \nonumber\\ & = & \lim_{s\rightarrow 0}\int s^{d-1}d\omega_{d-1}(-n_i ) \left[u\partial_i\varphi-\partial_iu\,\varphi\right ] \nonumber\\ & = & \lim_{s\rightarrow 0}\int s^{d-1}d\omega_{d-1}(-n_i)\left[-\tilde{k}\,(2-d)s^{1-d}n_i\varphi ( \mathbf{0})\right ] \nonumber\\ & = & \omega_{d-1}\tilde{k}\,(2-d)\varphi(\mathbf{0 } ) \nonumber\\ & = & -4\pi\varphi(\mathbf{0})\ , . \label{deltauphi}\end{aligned}\ ] ] in the last step we used the relation between @xmath469 and the volume of the sphere , which is @xmath1064 from @xmath1065 one can next find the solution @xmath1066 satisfying the equation @xmath1067 ( in a distributional sense ) , namely @xmath1068 from ( [ v ] ) we can then define a whole `` hierarchy '' of higher - order functions @xmath1069 , @xmath313 satisfying the poisson equations @xmath1070 , @xmath313 in a distributional sense . however , the latter hierarchy of functions @xmath1058 , @xmath1066 , @xmath313 is better displayed using some different , more systematic notation . this leads to the famous riesz kernels , here denoted @xmath1071 , in @xmath1-dimensional euclidean space @xcite . [ these euclidean kernels differ from the minkowski kernels @xmath1072 , also introduced by riesz , and alluded to in the introduction . ] these kernels depend on a complex parameter @xmath1054 . they are defined by @xmath1073 for any @xmath1054 , and also for any @xmath1028 , the riesz kernels satisfy the recursive relations @xmath1074 furthermore , they obey also an interesting convolution relation , which reads simply , with the chosen normalization of the coefficients @xmath1075 , as @xmath1076 when @xmath1077 we recover the dirac distribution in @xmath1 dimensions , @xmath1078 ( the coefficient vanishes in this case , @xmath1079 ) , and we have @xmath1080 , @xmath1081 , @xmath313 . the convolution relation ( [ convolutiondeltaalpha ] ) is nothing but an elegant formulation of the riesz formula in @xmath1 dimensions . to check it let us consider the fourier transform of @xmath1082 in @xmath1 dimensions , @xmath1083 using ( [ ddx ] ) we can rewrite it as @xmath1084 in which the angular integration can be performed as an application of eq . ( [ recursiondomega ] ) . this yields an expression depending on the usual bessel function , ] to obtain eq . ( [ fourier ] ) we employ the integration formula @xmath1085 ] @xmath1086 where @xmath1087 . the radial integration in eq . ( [ radialang ] ) is then readily done from using the previous expression , and we obtain @xmath1088 where the factor in front of the power @xmath1089 , say @xmath1090 , is checked from the parseval theorem for the inverse fourier transform , which necessitates that @xmath1091 . finally we can check the riesz formula by going to the fourier domain , using the previous relations . the result , @xmath1092 is equivalent to eq . ( [ convolutiondeltaalpha ] ) . a set of formulae concerning symmetric - trace - free ( stf ) multipole expansions in @xmath1 dimensions is presented next , without proofs . we use the multi - index notation @xmath1093 ; more generally the notation is the same as in appendix a of @xcite . in particular @xmath1094 denotes the stf projection of @xmath1095 , @xmath1096 $ ] means the integer part of @xmath1097 , @xmath1098 denotes the ( unnormalized , minimal ) sum of @xmath1099 where the @xmath143 s are permutations of the indices such that @xmath1098 is fully symmetric in @xmath399 ( for convenience we do not normalize the latter sum , for instance @xmath1100 ) . @xmath1101}a_\ell^k\delta_{\{i_1i_2}\cdots \delta_{i_{2k-1}i_{2k}}\hat{n}_{l-2k\}}\,,\\ \hat{n}_l&=&\sum_{k=0}^{[\frac{\ell}{2 } ] } b_\ell^k\delta_{\{i_1i_2}\cdots \delta_{i_{2k-1}i_{2k}}n_{l-2k\}}\ , , \label{nl}\end{aligned}\ ] ] where the coefficients are @xmath1102 in particular ( the brackets @xmath1103 surrounding the indices mean the stf projection ) @xmath1104 spherical averages : @xmath1105 stf decomposition of a scalar function : @xmath1106 decomposition of a function @xmath1107 in terms of gegenbauer polynomials : is the coefficient of @xmath1108 in the expansion @xmath1109 the particular polynomial @xmath1110 represents an appropriate generalization of the legendre polynomial in @xmath1 dimensions [ indeed @xmath1111 . ] @xmath1112 integration formulae : @xmath1113 a very important technical fact which allowed one to compute analytically the @xmath42 equations of motion is the possibility to obtain explicitly the quadratically non linear potentials , _ i.e. _ , to evaluate in closed form the integrals appearing in the pn expansion of the cubic - vertex diagram of fig . [ fig7 ] . at the lowest approximation in the @xmath1114 expansion , the diagram of fig . [ fig7 ] leads , in @xmath42 , to the integral @xmath1115 which was ( probably ) first evaluated by fock in 1939 ( `` sur le mouvement des masses finies daprs la thorie de la gravitation einsteinienne '' @xcite ) , with the simple result @xmath1116 remembering that @xmath1117 , @xmath1118 and @xmath1119 the combination @xmath1120 entering the logarithm in eq . ( [ eqd2 ] ) is simply seen to be the perimeter of the triangle joining the three spatial points @xmath241 , @xmath75 and @xmath1121 entering the ( newtonian approximation of the ) diagram of fig . [ fig7 ] . at the @xmath1122 level of the pn expansion of fig . [ fig7 ] , there enter several new integrals which can be reduced to @xmath1123 together with @xmath1124 the explicit evaluation of the integrals ( [ eqd3 ] ) , ( [ eqd4 ] ) is also possible , as was shown in refs . @xcite,@xcite ( drawing on earlier works @xcite ) . in this appendix we shall explicitly evaluate the @xmath1-dimensional generalization of ( [ eqd1 ] ) . it will be clear , however , that our method can be rather straightforwardly generalized to the @xmath1122 diagrams contained in fig . [ fig7 ] , _ i.e. _ , to the @xmath1-dimensional generalizations of ( [ eqd3])-([eqd4 ] ) . for our present purpose it will be more convenient not to include the two factors of @xmath928 that accompany the two propagators issued from 1 and 2 in fig . we shall therefore define @xmath1125 the method we present here consists of four basic steps : ( i ) expand the integrand in series and construct a corresponding series for a _ solution @xmath1126 , ( ii ) resum the series to get an explicit line - integral form of @xmath1127 , ( iii ) compute @xmath1128 in a distributional sense to discover that it satisfies @xmath1129 where @xmath1130 is a distributional source ( localized along a line ) , and finally ( iv ) subtract @xmath1131 ( which is given by another line - integral ) from @xmath1127 to get @xmath498 as a sum of line - integrals ( which are expressible in terms of one special function of one argument ) . what is crucial in the argument is the uniqueness of the global solution ( decaying at infinity ) of any ( distributional ) poisson equation @xmath1132 when the ( distributional ) source decays fast enough ( or , at least , does not grow too fast ) at infinity . in our case , the sources @xmath143 involved will have fast - enough decay at infinity if we analytically continue @xmath1 toward large enough real parts ( say @xmath1133 > 3 $ ] ) . there are several ways of implementing our method . for instance , we could start by expanding @xmath1134 in the source of ( [ eqd5 ] ) in powers of @xmath37 , such an expansion being valid only in a neighborhood of @xmath75 . namely , we have the @xmath1-dimensional generalization of the familiar @xmath42 legendre - polynomial expansion of @xmath1135 near @xmath255 ( more precisely in the ball @xmath1136 ) @xmath1137 here , we denoted for visual clarity @xmath1138 , where @xmath1139 is a gegenbauer polynomial such that @xmath1140 is the usual @xmath42 legendre polynomial [ see also appendix [ formulae ] above ] . the quantity @xmath1141 in ( [ eqd6 ] ) denotes the cosine of the angle @xmath1142 between @xmath506 and @xmath1143 . the notation we shall use is summarized in fig . [ fig8 ] . when inserting the _ local _ expansion ( [ eqd6 ] ) into the source of ( [ eqd5 ] ) we are led to solving ( locally ) an equation of the form @xmath1144 . however , using the general formula @xmath1145 we know a particular solution of @xmath1146 , namely @xmath1147 the formulae ( [ eqd7])-([eqd8 ] ) apply to any source with fixed multipolarity @xmath1148 and a power law dependence on a radius . in particular , they apply when @xmath1149 , @xmath1150 and @xmath1151 ( because a generalized legendre polynomial is just proportional to the contraction of an stf - projected multi unit vector @xmath1152 onto a fixed `` @xmath1153 '' direction ; see appendix [ formulae ] ) . this leads to a corresponding expansion of a _ local _ solution @xmath1154 ( near @xmath75 ) of @xmath1155 of the form @xmath1156 in order to proceed further , we now need to resum the expansion ( [ eqd9 ] ) . this is done by a trick introduced , in a similar context of resummation of multipolar expansions containing extra @xmath363-dependent denominators , by ref . one introduces some radial - integration operators @xmath1157 \ , ( \mathbf{r } ) = \int_0 ^ 1 d\lambda \ , \lambda^{\alpha } \ , \phi ( \lambda \ , \mathbf{r})$ ] . for instance , in the context of ( [ eqd9 ] ) , one replaces @xmath1158 by @xmath1159 = \int_0 ^ 1 d\lambda \ , ( \lambda \ , r_1)^{\ell}$ ] or equivalently @xmath1160 by @xmath1161 . this transforms back the multipolar series appearing in ( [ eqd9 ] ) into the original `` legendre '' series entering eq . ( [ eqd6 ] ) . this allows one to write @xmath1154 as a simple line - integral : @xmath1162 here , @xmath1163 is a point on the segment joining @xmath75 to @xmath241 , located a distance @xmath1164 away from @xmath75 . it is more convenient to replace the line - integration over the dimensionful length @xmath1164 ( @xmath1165 ) into an integration over the dimensionless parameter @xmath1166 ( @xmath1167 ) . this leads to the explicit expression @xmath1168 the resummed line - integral expression ( [ eqd10 ] ) allows one to define everywhere @xmath1154 , _ including in the domain _ @xmath1169 where the original series ( [ eqd9 ] ) was _ not _ convergent . having in hands such a global definition of @xmath1154 then allows one to compute its laplacian , _ in the sense of distributions _ , and to see how it differs from @xmath1170 . the calculation of @xmath1171 is done by techniques similar to the ones used in ref . one needs to rewrite some terms in the form of @xmath1172-derivatives . for instance , several of the terms appearing in @xmath1171 can be rewritten as the line - integral @xmath1173 \nonumber \\ & & \qquad = \int_0 ^ 1 d\alpha \ , \frac{\partial}{\partial \alpha } \bigl [ \alpha \ , r_1^{2-d } \ , \vert\mathbf{y}_{\alpha } - \mathbf{y}_2 \vert^{2-d } \bigr ] = r_1^{2-d } \ , r_2^{2-d}\,,\end{aligned}\ ] ] where the last line - integral gave only the end contribution @xmath1174 corresponding to @xmath1175 . besides the terms yielding ( [ eqd12 ] ) , _ i.e. _ , the looked - for `` source '' of the complete @xmath498 , the calculation of @xmath1176 yields also the distributional source ( where @xmath1177 entered through @xmath1178 ) @xmath1179 this is conveniently transformed by introducing @xmath1180 ( with @xmath1181 ) and @xmath1182 which varies along a semi - infinite line going from @xmath76 to infinity along the direction @xmath1143 , _ i.e. _ , _ away _ from @xmath75 . this transformation allows one to rewrite ( [ eqd13 ] ) in the more transparent form @xmath1183 at this stage , we recognize in ( [ eqd15 ] ) a very simple source , namely a _ uniform _ distribution of `` mass '' along the half - line along which @xmath1184 runs . this allows one to easily compute the unique , global ( decaying at infinity ) solution of the poisson equation with source ( [ eqd15 ] ) and to subtract it from @xmath1185 to get the unique , global @xmath498 in the form of two line - integrals : @xmath1186 where @xmath1187 and @xmath1188 . in other words , ( [ eqd16 ] ) expresses @xmath498 as , essentially , the difference between the newtonian potentials generated by two uniform line distributions : a segment joining @xmath75 to @xmath241 and the half - line starting from @xmath76 in the direction away from @xmath75 . it is easily seen ( modulo the slight delicacy of the logarithmic divergence of the potential of a semi - infinite line when @xmath32 , _ i.e. _ , the occurrence of a @xmath778 pole ; see below ) that the result ( [ eqd16 ] ) yields , when @xmath32 , the well - known result ( [ eqd2 ] ) . [ actually , this was the way one of us ( t.d . ) had derived long ago for himself ( [ eqd2 ] ) , unaware of its derivations in the literature . ] the expression ( [ eqd16 ] ) has the advantage of being explicitly regular ( except at the point @xmath255 ) in the ball @xmath1136 . however , it has the default of treating dissymmetrically the two points @xmath75 and @xmath76 ( in spite of the fact that the result ( [ eqd16 ] ) for @xmath498 _ is _ , actually , symmetric under @xmath542 ) . one can derive an exchange - symmetric expression for @xmath498 by modifying the first step of our method . instead of expanding the source @xmath1170 in the neighborhood of @xmath245 , _ i.e. _ , in a series of positive powers of @xmath37 , we can expand it in the _ neighborhood of infinity _ , _ i.e. _ , in a series of negative powers of @xmath37 . such an expansion is directly related to the expansions used in @xcite , which led to the decomposition of @xmath1189 in two pieces denoted @xmath472 and @xmath1190 there , where the source of @xmath1190 was a uniform mass distribution along the segment joining @xmath75 and @xmath76 . let us briefly indicate the successive steps of this new calculation of @xmath498 . instead of the `` local '' expansion ( [ eqd6 ] ) ( valid for @xmath1136 ) , one expands @xmath1134 near infinity ( @xmath1169 ) as @xmath1191 solving term by term @xmath1192 `` near infinity '' by means of ( [ eqd8 ] ) , and transforming away the appearing @xmath363-dependent denominators by means of @xmath1193 , leads to the following resummation of @xmath1194 : @xmath1195 here , @xmath1196 is still defined by ( [ eqd11 ] ) , but the parameter @xmath1172 now varies in @xmath1197 so that ( [ eqd18 ] ) is the potential of a semi - infinite line . computing the distributional laplacian of the particular solution @xmath1194 , eq . ( [ eqd18 ] ) , leads to the presence , besides the looked - for source @xmath1170 , of an additional distributional source localized now along the segment joining @xmath75 to @xmath76 , namely @xmath1198 where @xmath1199 varies between 0 and 1 and where @xmath1200 is again defined by ( [ eqd14 ] ) . it is then easy to subtract from @xmath1194 ( which tends , in @xmath42 , to the function @xmath1201 of @xcite ) the poisson integral of the source ( [ eqd19 ] ) ( which is a uniform distribution along the segment @xmath75-@xmath76 and which tends , in @xmath42 , to the function @xmath1202 of @xcite ) to get the following alternative expression for @xmath498 , @xmath1203 or , equivalently , @xmath1204 the form ( [ eqd20])-([eqd20 ] ) still does not look quite symmetric between 1 and 2 but a moment of reflection will show that it is . the two methods above have expressed @xmath498 in terms of the newtonian potentials generated by half - lines or segments , _ i.e. _ , integrals of the type @xmath1205 where @xmath1206 varies along a straight line ( but where @xmath74 might be @xmath241 or @xmath76 ) . clearly , any such potential can be reduced ( through linear decompositions ) to the newtonian potential generated by a _ half - line_. let us then consider a generic half - line starting at the point @xmath1207 and going to infinity in the direction @xmath1208 , and let us consider the newtonian potential generated by this half - line at the origin of the coordinate system ( not located on the half - line ) . let us denote @xmath1209 , @xmath1210 , @xmath1211 , @xmath1212 ( cosine of the angle @xmath11 between the radius vector from the origin , _ i.e. _ , the `` field point '' , towards the beginning point of the half - line and the direction of the half - line , away from its beginning ) . then it is easy to find that @xmath1213 where the function @xmath1214 is given by the integral @xmath1215 the integral ( [ eqd22 ] ) converges for @xmath1216 , has a pole @xmath1217 as @xmath32 , and can be expressed in terms of hypergeometric functions , _ e.g. _ @xmath1218 $ ] . it is , however , simpler to keep the form ( [ eqd22 ] ) . reads @xmath1219 on this expression one sees clearly the occurrence of the simple pole of @xmath1214 when @xmath1220 , which is given by the `` monopolar '' term @xmath401 as @xmath1221 ] finally , using the half - line potentials ( [ eqd21 ] ) as building blocks one can write our result ( [ eqd20 ] ) in the final , @xmath292 symmetric , form @xmath1222 \nonumber \\ & - & \frac{r_1^{3-d } \ , r_2^{3-d}}{2(4-d ) } \ , \varphi ( c_{12})\,.\end{aligned}\ ] ] the quantities entering ( [ eqd23 ] ) are those defined in fig . [ fig8 ] , notably @xmath1223 , @xmath1224 , @xmath1225 , with @xmath1226 being the orthogonal distance between the field point @xmath241 and the segment joining @xmath75 to @xmath76 [ with associated argument @xmath1227 in @xmath1214 ] . note the following properties of the function @xmath1214 , the simplest way to prove ( [ eqd24 ] ) is to notice that the newtonian potential of an _ infinite _ line can be written either as twice that of two half - lines beginning at the orthogonal projection of the field point on the original line , so that @xmath1230 or as that of two other half - lines obtained by a more arbitrary cut ( under an angle @xmath1231 and @xmath1232 ) . we can verify the @xmath32 limit of eqs . ( [ eqd16 ] ) and ( [ eqd23 ] ) by using the following @xmath104 expansion of the elementary function @xmath1214 , namely @xmath1233 to obtain ( [ phic ] ) we notice that the finite part of @xmath1214 when @xmath104 , which is @xmath1234\ , , \label{phi0c}\ ] ] can be re - expressed in the form of the following sum of two convergent integrals , @xmath1235 combining the expansion ( [ phic ] ) with the basic relations we have @xmath1236 the perpendicular distance @xmath1237 is given by @xmath1238 ] associated with the triangle of fig . [ fig8 ] , our @xmath1-dimensional expressions ( [ eqd16 ] ) or ( [ eqd23 ] ) are found to admit the expansion @xmath1239 which indeed reduces to the three - dimensional result ( [ eqd2 ] ) modulo an additive constant linked to the @xmath778 pole . nice as it is to have in hand an analytic expression for the @xmath1-dimensional basic non linear potential @xmath498 , its practical utility in explicit computations of the @xmath1-dimensional equations of motion is not evident because , contrary to the 3-dimensional expression ( [ eqd2 ] ) , the expression is not _ explicitly _ regular along the @xmath1240 segment . [ the regularity of eq . ( [ eqd23 ] ) as @xmath1241 comes by compensations between the three terms in the bracket , using ( [ eqd24 ] ) . ] it would need some transforming [ using ( [ eqd24 ] ) , and/or using the other expressions derived from the previous form ( [ eqd16 ] ) , which are regular along the @xmath1240 segment , but singular somewhere else ] to write an explicit expression which is regular everywhere , except at the two isolated points @xmath75 and @xmath76 . finally , let us just mention that the method explained above can , in principle , be straightforwardly generalized to the computation of the higher post - newtonian potentials contained in the diagram of fig . [ fig7 ] . for instance , in computing the @xmath1-dimensional analog of ( [ eqd3 ] ) , say @xmath1242 , it is easy [ by iterating ( [ eqd8 ] ) ] to get the analog of ( [ eqd9 ] ) . then a more complicated radial - integration operator ( see , _ e.g. _ , @xcite ) will allow one to resum the series to get a line - integral expression for @xmath1243 or @xmath1244 . we anticipate that a somewhat more delicate application of either @xmath1245 ( to go back to @xmath498 ) or @xmath1246 ( to go back to @xmath1247 ) will yield additional line - distributed sources . it should then be a simple matter to compute the poisson , or iterated poisson , integral of these line - distributed sources . we leave an explicit study of these details to future work .

dimensional regularization is used to derive the equations of motion of two point masses in harmonic coordinates . at the third post - newtonian ( 3pn ) approximation , it is found that the dimensionally regularized equations of motion contain a pole part [ proportional to @xmath0 which diverges as the space dimension @xmath1 tends to @xmath2 . it is proven that the pole part can be renormalized away by introducing suitable shifts of the two world - lines representing the point masses , and that the same shifts renormalize away the pole part of the `` bulk '' metric tensor @xmath3 . the ensuing , finite renormalized equations of motion are then found to belong to the general parametric equations of motion derived by an extended hadamard regularization method , and to uniquely determine the 3pn ambiguity parameter @xmath4 to be : @xmath5 . this value is fully consistent with the recent determination of the equivalent 3pn `` static ambiguity '' parameter , @xmath6 , by a dimensional - regularization derivation of the hamiltonian in arnowitt - deser - misner coordinates . our work provides a new , powerful check of the consistency of the dimensional regularization method within the context of the classical gravitational interaction of point particles .

You are an expert at summarizing long articles. Proceed to summarize the following text: the recent development of artificial free - standing structures of nanometer dimensions has led to great interest in their mechanical properties . a wealth of experimental information is now available for nanowires @xcite and nanotubes@xcite , and a computational literature is developing on the subject@xcite . many of these works make use of results from the continuum theory of elasticity to analyze the behavior of nanometer structures . however , the applicability of continuum theories to nanoscale objects , where atomic - level inhomogeneities come to the fore , has yet to be explored in depth . rigorous understanding of the elastic properties of nanoscale systems is crucial in understanding their mechanical behavior and presents an intriguing theoretical challenge lying at the cross - over between the atomic level and the continuum . in the absence of an appropriate theoretical description at this cross - over , critical questions remain to be answered including the extent to which continuum theories can be pushed into the nanoregime , how to provide systematic corrections to continuum theory , what effects do different bonding arrangements have on elastic response , and what signatures in the electronic structure correlate with the mechanical properties of the overall structure ? recently , there have been a number of theoretical explorations of the impact of nanoscale structure on mechanical properties@xcite . these studies fall under two broad approaches , either the addition of surface and edge corrections to bulk continuum theories@xcite or the extraction of overall mechanical response from atomic scale interactions@xcite . the latter approach has the distinct advantage of allowing first principles understanding of how different chemical groups and bonding arrangements contribute to overall elastic response , thus opening the potential for the rational design of nanostructures with specific properties . in coarse graining from interatomic interactions to mechanical response , some works rely upon the problematic decomposition of the total system energy into a direct sum of atomic energies@xcite , which is always arbitrary and particularly inconvenient for connection with _ ab initio _ electronic structure calculations . the remaining works which attempt to build up overall response from atomic level contributions@xcite fail to account properly for the poisson effect . below we show that failure to account for this effect leads to surprisingly unphysical results . this manuscript presents the first theory for the analysis of overall mechanical response in terms of atomic - level observables which suffers from neither difficulty from the preceding paragraph . this analysis allows , for the first time , quantitative understanding of how continuum theory breaks down on the nanoscale , of how to make appropriate corrections , and of how to predict the effects of different bonding arrangements on overall elastic response . it is well known that the decomposition of overall elastic response into a sum of atomic level contributions is not unique . we show here , however , that with the additional constraint of dependence of moduli on local environment our definition of atomic level moduli becomes physically meaningful and essentially unique when coarse - grained over regions of extent comparable to the decay range of the force - constant matrix . for concreteness , in this work we focus on nanowires . however , we will also describe briefly how this work can be extended to any system with nanometer dimensions . the manuscript proceeds as follows . section [ overview ] briefly overviews the present state of the field . next , as the traditional concept of young s modulus becomes ill - defined on the nanoscale , we begin by carefully defining continuum elastic constants appropriate for nanowires in section [ sec - nr ] . we then show how to decompose these constants exactly into atomic - level contributions based on true physical observables ( rather than individual atomic energies ) using a straight - forward application of born and huang s method of long waves @xcite , resulting in a decomposition similar in spirit to those in references [ 17 ] and [ 18 ] ( section [ sec - mlw ] ) . section [ sec - fail ] demonstrates the surprising , radical breakdown of this approach when applied to nanoresonators . then , in section [ sec - mlwn ] , we identify the source of the difficulty as the poisson effect and present the first analysis of mechanical response truly applicable to nanoresonators . the manuscript then goes on to applications . section [ sec - trans ] verifies the physical meaningfulness of our newly defined quantities by verifying that they predict response to modes of strain for which they were not directly constructed . we then , in section [ sec - fte ] , use our approach to generate a new , much more accurate , relationship between experimentally accessible observables describing response to flexural and extensional strain in nanomechanical resonators . finally , section [ sec - cbames ] uses this theory to explore possible links between underlying electronic structure and local elastic response . as the introduction mentions , the literature pursues two broad categories of approach to the study of mechanical properties on the nanoscale , either surface and edge corrections to continuum theory or extraction of overall response from the underlying atomic interactions . in the former category , reference [ 13 ] , through scaling arguments and numerical examples , notes that the young s modulus for nanomechanical resonators scales as a bulk term plus surface and edge corrections . although providing insight and motivation , this work leaves completely open how one should understand these corrections from first principles . reference [ 14 ] provides a more rigorous study based on separating nanoscale systems into continuum surface and bulk regions . this latter approach allows prediction of changes in stiffness properties as one approaches nanometer length scales and has the appeal of generating physically motivated correction terms . however , it relies on the separation of a nanomechanical resonator into bulk and surface continua as an _ ansatz _ and therefore neither predicts when such a picture suffices to give an accurate description nor prescribes further corrections . references [ 15 - 18 ] , on the other hand , start from the more general atomic level description and then try to unveil physical properties from the underlying atomic description . it is important to note that these works do not deal directly with nanoscale systems but rather focus on the effects of nanoscale inhomogeneities in _ bulk _ systems . references [ 15 ] and [ 16 ] concern the elastic properties of grain boundaries . these works define atomic - level elastic moduli as the second derivative of the energy associated with each atom with respect to strain and then go on to study the behavior of such moduli near grain boundaries . the difficulty with this approach is that it requires a breakdown into individual contributions from each atom of the total energy of any system . such an atomic energy is neither observable nor uniquely defined and therefore can not serve as an appropriate basis for theoretical understanding . although references [ 17 ] and [ 18 ] work from valid physical observables , the components of the force - constant matrix , these works focus on bulk - like or mesoscopic scale systems and fail for nanoscale systems for the reasons which we describe in this work . reference [ 17 ] investigates nonlocal elastic constants on the mesoscopic scale and links them to the underlying atomic interactions . it then proceeds to define an elastic constant for each atom and studies the behavior of these quantities near surfaces and grain boundaries . reference [ 18 ] defines a bond frequency from the force - constant matrix from which it deduces the possibility of bond rupture during crack nucleation . neither of the above works properly accounts for the poisson effect , which we show in sections [ sec - mlw ] , [ sec - fail ] and [ sec - mlwn ] to play a critical role in the elasticity of nanoscale systems . moreover , straightforward generalization of these works to include this effect fail for the the same reasons as does the related approach which we describe in section [ sec - mlw ] . the prime difficulty in the application of continuum theory to objects of nanometer cross - section is the loss of the ratio of the inter - atomic spacing to the cross - sectional dimension as a small parameter . however , so long as the length of an object and the wavelength of the distortions considered both greatly exceed the inter - atomic spacing and the cross - sectional dimension , the object properly may be viewed as a one - dimensional continuum . although we focus in this work on nanowires , the generalization of the discussion below to nanoscale systems of other dimensionality such as thin plates or nanoscopic objects is straightforward . viewed as a linear continuum , the free energy per unit length @xmath0 of a nanowire is @xmath1 where @xmath2 is the linear strain of extension , @xmath3 is the radius of curvature and @xmath4 is the rate of twist of the torsion . the coupling constant @xmath5 is the _ extensional rigidity _ , @xmath6 is the _ flexural rigidity _ and @xmath7 is the _ torsional rigidity_. unlike traditional bulk continuum concepts , the free - energy function eq . ( [ eqn : free ] ) is observable in principle and thus provides an unambiguous operational definition of the rigidities . we avoid the use of traditional continuum concepts , such as the young s modulus and the cross - sectional area , because such concepts are neither uniquely nor well - defined for nanoscale systems . the rigidities in eq . ( [ eqn : free ] ) are related to the phonon frequencies through @xmath8 where @xmath9 is the frequency for either the longitudinal , transverse or rotational acoustic modes , respectively , @xmath10 is the linear atomic number density , @xmath11 is the wave vector and @xmath12 is the mass of a single atom . ( this work focuses on single species systems for simplicity . ) finally , @xmath13 is defined unambiguously as the mean rotational moment @xmath14 , where the sum ranges over all atoms in the cell , @xmath15 is the number of atoms per unit cell , the wire is assumed to run along the @xmath16-axis and the origin lies on the center line of the wire . finally we note that although the rigidities in eq . ( [ eqn : free ] ) are well - defined , certain traditional continuum relations between them do not hold . specifically , we will show below that the traditional continuum relationship between the extensional rigidity @xmath5 and the flexural rigidity @xmath6 fails on the nanoscale , similar to relations which have been recently used in the analysis of experiments@xcite . one reason for breakdown of traditional continuum relations on the nanoscale is that the continuum perspective course grains away important fluctuations which occur over distances on the order of the inter - atomic spacing . to overcome this shortcoming , we propose to coarse grain only on distances over which the underlying interatomic interactions vary , the decay length of the force - constant matrix . the straightforward approach to generate such a theory is the `` method of long waves '' developed by born and huang@xcite , which is somewhat similar to the approaches which have been used previously to defects in bulk systems@xcite . this section applies the method of long waves to nanoresonators . the next section shows how , surprisingly , this and related approaches fail in the study of systems with free surfaces and therefore , in general , can not be used to describe nanoscale systems . in section [ sec - mlwn ] , we describe how to go beyond the straightforward application of the `` method of long waves '' in order to achieve a meaningful description . we focus initially on longitudinal waves and , as noted in section [ sec - nr ] , choose our origin to lie on the center line of the wire , which we let run along the @xmath16-axis . for nanowires , the presence of surfaces breaks periodicity in the transverse directions resulting in a one - dimensional crystal with an extremely large unit cell of length @xmath17 , where @xmath15 and @xmath10 are as in section [ sec - nr ] . in all expressions below , boldfaced quantities are @xmath18-dimensional and arrowed vector quantities are three - dimensional . finally , sums with greek indices range over atoms in the unit cell . to relate the rigidities to the dynamical matrix , we begin similarly to born and huang and choose to factor the bloch phases ( @xmath19 ) out of the representation of the phonon polarization vector @xmath20 , incorporating them into the definition of the dynamical matrix @xmath21 , so that the acoustic phonon polarization vectors are periodic across the cell boundaries . this ensures a uniform description of the distribution of elastic energy along the axis of the wire . to generate a scalar equation for the phonon frequency , born and huang project the secular equation for the dynamical matrix , @xmath22 against the zeroth - order polarization vector @xmath23}$ ] . here , however , to more symmetrically represent the distribution of elastic energy , we project against the full polarization vector @xmath20 . equating the frequency @xmath9 in eq . ( [ eqn : sec ] ) with the longitudinal frequency in eq . ( [ eqn : mlw ] ) gives @xmath24^{[2]}}{{\bf u}^\dagger { \bf u } } = \sum_{s , t=0}^{1 } \frac{(-1)^{s}}{n_c } \ : { \bf u}^{\dagger[s ] } \ : { \bf d}^{[2-s - t ] } \ : { \bf u}^{[t ] } , \label{eqn : omegasq}\ ] ] where we have expanded the numerator of the rayleigh quotient to second - order in powers of @xmath25 and where the @xmath26 sub - block of @xmath27}$ ] , which couples atoms @xmath28 and @xmath29 , is @xmath30}]_{\alpha\beta}=\frac{1}{n ! } \sum_{\vec r } { \phi_{\alpha\beta}(\vec r ) \left({\hat z } \cdot(\vec r + \vec \tau_\beta - \vec \tau_\alpha ) \right)^n}.\ ] ] here , @xmath31 is a lattice vector along the @xmath32 , @xmath33 is the @xmath26 sub - block of the force - constant matrix which couples atoms @xmath28 and @xmath29 located at positions @xmath34 and @xmath35 , respectively , and @xmath36 finally , substituting eq . ( [ eqn : dn ] ) and @xmath37}]_\alpha=\hat z$ ] into eq . ( [ eqn : omegasq ] ) , allows us to express @xmath5 as a sum over atoms ( @xmath28 ) in the unit cell and all atoms ( @xmath38 ) in the system , @xmath39}_\alpha } \cdot \phi_{\alpha\beta}(\vec{r})\cdot { \vec u^{[1]}_\beta } \nonumber\\ & & + { { \delta \vec z}_{\alpha\beta } } \cdot \phi_{\alpha\beta}(\vec{r})\cdot { \vec u^{[1]}_\beta } - { \vec u^{[1]}_\alpha } \cdot \phi_{\alpha\beta}(\vec{r})\cdot { { \delta \vec z}_{\alpha\beta } } \mbox{\large\}}. \label{eqn : atmod_sf}\end{aligned}\ ] ] here , @xmath40 , @xmath41}_{\alpha}$ ] is the first - order polarization vector and we refer to the @xmath42 as the `` atomic moduli '' . the atomic moduli as currently defined in eq . ( [ eqn : atmod_sf ] ) provide a useful microscopic analysis of elastic response in bulk systems which is similar in spirit to the decompositions used previously in the study of bulk material systems @xcite . to see that eq . ( [ eqn : atmod_sf ] ) indeed decomposes the overall elastic response of bulk systems into atomic contributions coarse - grained over distances on the order of the decay - length of the force - constant matrix , we note first that for infinite bulk systems , elastic waves are planar . this implies that the first - order polarization vector ( @xmath43}_{\alpha}$ ] ) is uniform from primitive cell to primitive cell and thus depends only on the local environment of each atom . next , we note that although the strain terms ( @xmath44 ) scale linearly with distance between atoms , the terms which contribute to the final result are bounded in range by the inter - atomic interactions ( @xmath45 ) . thus , the atomic moduli depend only on the local atomic environment over distances which the decay of the force - constant matrix determines . we now demonstrate through direct calculations that the approach outlined in the previous section gives unphysical results when applied to nanoresonators . specifically , we study the behavior of @xmath46$]-oriented nanoresonators of silicon , which recent _ ab initio _ studies@xcite predict to undergo a size - dependent structural phase transition between the two structures in figure [ fig : wires ] at a cross - section of @xmath473 nm . ( the interested reader may refer to reference [ 21 ] for explicit details of the microscopic structure of these wires . ) initially , we work with the stillinger - weber inter - atomic potential@xcite , which suffices for the exploration of general nanoelastic phenomena and which allows study of cells with many thousands of atoms . later in the manuscript ( section [ sec - cbames ] ) we use the sawada tight - binding model@xcite with modifications proposed by kohyama@xcite to explore the correlation between our local approach and the underlying electronic structure . for all calculations below , we fully relax the atomic coordinates , the periodicity of the wire and , need be , the electronic structure . finally , we employ periodic boundary conditions along the @xmath16-direction . figure [ fig : ecomp]a shows that the atomic moduli @xmath42 predicted for nanowires using the straightforward approach of eq . ( [ eqn : atmod_sf ] ) are unphysical in that they depend upon the macroscopic dimensions of the system and not simply on the local environment of each atom . in particular , the atomic moduli on the surface grow linearly with the diameter of the wire and the moduli in the center of the wire fail to approach the expected bulk limit , @xmath48 , where @xmath49 is the young s modulus in bulk and @xmath50 is the number density in bulk . ( note that these two effects are interelated , as the moduli must sum to give the macroscopic value in the bulk limit . ) [ 0.375 ] for @xmath51 nanowires of varying diameter : ( a ) straightforward theory ( eq . ( [ eqn : atmod_sf ] ) ) and ( b ) new theory ( eq . ( [ eqn : atmod ] ) ) . the insets denote the approximate diameters of the wires . the value of the atomic modulus ( eq . ( [ eqn : atmod_sf ] ) or eq . ( [ eqn : atmod ] ) ) is along the ordinate and radial distance of the atom from the center line is along the abscissa.,title="fig : " ] + radial distance [ @xmath52 although one has some freedom in choosing the terms used in the perturbation expansion eqs . ( [ eqn : sec ] ) and ( [ eqn : omegasq ] ) , for example to project eq . ( [ eqn : sec ] ) against @xmath53}}$ ] instead of @xmath20 , all such expansions will lead to similar linear scaling along the surface of the wire and approach an incorrect value at the center of the wire . thus , straightforward application of the method of long waves fails to result in a local , and hence physically meaningful , description of elastic response in nanoscale systems , as will straightforward variations thereon such as those in references [ 17 ] and [ 18 ] . to cure the difficulties uncovered in the previous section , we proceed by first identifying the cause of the pathological behavior and then exploiting the freedom in eq . ( [ eqn : atmod_sf ] ) to remove this pathology . the failure of the straightforward approach arises from the fact that elastic waves in nanoresonators , or any system with free surfaces , are not strictly planar . in particular , the poisson effect , which the first order polarization vector @xmath54}$ ] contains , causes each atom to displace by an amount in direct proportion to its distance from the center line of the system . ( [ eqn : atmod_sf ] ) then leads directly to linear scaling of the atomic moduli at the surfaces of the wire . defining an atomic elastic reponse dependent soley on the local enivornment requires separation of extensive elastic effects from intensive nanoscopic effects . to seperate the extenive motion in the first order polarization vector from that of its intensive motion , we define the atomic displacements ( @xmath55 ) as the intensive nanoscopic motions , @xmath56}_{\alpha } - \left ( -\sigma_x \hat x \hat x \cdot - \sigma_y \hat y \hat y \cdot \right ) \vec \tau_\alpha , \label{eqn : decomp}\end{aligned}\ ] ] where @xmath57 are the poisson ratios there are three logical choices for the poisson ratios : some sort of local atomic definition , an overall average for the wire , or the bulk values . we choose to use bulk poisson ratios for a number of reasons . first , a locally varying definition makes it impossible to exploit the continuous rotational and translational symmetries in the dynamical matrix , which we find necessary to employ below in constructing atomic moduli with local behavior . second , only by employing the bulk ( rather than average ) poisson ratios do we find a definition which approaches the appropriate bulk value in the centers of wires of finite width . finally , we note that we always have the freedom of working with bulk poisson ratios because any motion along the surface in addition to that resulting from bulk poisson effect will not scale extensively with the diameter of the wire and can therefore be incorporated into the intensive atomic . after making the decomposition in eq . ( [ eqn : decomp ] ) , we next employ the continuous rotational and translational symmetries of the dynamical matrix to eliminate all extensive dependencies in eq . ( [ eqn : atmod_sf ] ) . appendix [ sec - app ] outlines the procedure for doing this , which then transforms eq . ( [ eqn : atmod_sf ] ) into @xmath58 where @xmath59 represents the _ total _ strain between atoms @xmath28 and @xmath29 , @xmath60 and @xmath61 renormalizes as @xmath62 where @xmath63 , is a diagonal @xmath26 matrix with elements @xmath64 , @xmath65 and @xmath66 , respectively and @xmath67 is the kronecker delta . this new construction ensures that the modulus of each atom depends only upon its local atomic environment because @xmath68 no longer includes extensive motions and , although @xmath69 still depends on relative atomic distances , the renormalized @xmath70 decays as @xmath71 does . thus , it is now the range of the force - constant matrix which controls the size of the neighborhood upon which each atomic modulus can depend . therefore , the moduli of atoms in the interior now _ must _ correspond to the expected bulk value , the moduli of the atoms on the surface now _ can not _ depend upon the extent of the system , and the resulting description is physically meaningful . figure [ fig : ecomp]b illustrates the success of eq . ( [ eqn : atmod ] ) . the fact that decomposition of elastic response into atomic level contributions is not unique raises questions as to the physical meaning of such a decomposition . the new decomposition eq . ( [ eqn : atmod ] ) is the first which remains dependent only upon local environment for systems with free surfaces . any other definition _ which respects locality _ can only redistribute portions of each atom s modulus among other atoms within a region of extent comparable to the decay of the force - constant matrix . any sum over such a region of the moduli will always be nearly the same . therefore , coarse grained over such regions , properly localized atomic moduli become physically meaningful . section [ sec - trans ] and appendix [ sec - appb ] demonstrate this explicitly by comparing the predictions for flexion from either properly localized moduli or straightforwardly defined moduli , respectively . to further demonstrate that alternate local definitions are equivalent in this coarse - grained sense , we have explored alternate local constructions . in particular , while our present construction takes care to employ the continuous symmetries of the dynamical matrix in such a way so as to respect symmetry among the @xmath72 cartesian coordinates , we have also repeated the construction while treating the @xmath73 coordinates symmetrically but not the @xmath16 coordinate and have found nearly identical results for all of the applications below . we close this section with a brief description of how the above approach extends to any system of nanometer dimensions . to be considered small in this context , a dimension must be much smaller than the wavelength of the distortions considered . this manuscript focuses on nanowires , systems with two small dimensions . for a system with one small dimension , for instance a plate with nanometer thickness , the above approach develops in the same way with the one minor change that the definition of @xmath68 ( eq . ( [ eqn : decomp ] ) involves the poisson effect only in the one small dimension . for an object with three small dimensions , for instance adiabatic loading of a nano - object , the nature of the poisson effect depends upon the mode of loading , and the system should treated as either of the above cases accordingly . finally , objects of no small dimension , and hence no poisson effect , can be described straightforwardly as bulk - like using eq . ( [ eqn : atmod_sf ] ) . to establish that our new atomic moduli are not mere convenient mathematical constructions but are physically meaningful , we now consider their transferability to phenomena not considered in their original construction . in particular , we consider flexion , where the elastic distortion is no longer homogeneous throughout the cross - section . if our atomic moduli are indeed a measure of the local elastic response , then under flexion the free energy per unit length will take the form @xmath74 where @xmath75 is a measure of the longitudinal strain which atom @xmath28 experiences . for this form to be sensible , the diameter @xmath76 of the wire must not become comparable to the range of the force - constant matrix so that the atoms which contribute to each @xmath42 all experience similar strains @xmath75 . within continuum theory , uniform flexion with radius of curvature @xmath3 corresponds to a longitudinal strain which varies linearly across the wire , @xmath77 . ( this holds to better than to two parts in @xmath78 for all wires in our study . ) this would then predict a flexural rigidity of @xmath79 figure [ fig : eandf]a shows the fractional error ( @xmath80 ) @xmath81 in predicting the flexural rigidity from eq . ( [ eqn : atmof ] ) , where @xmath6 is determined directly through numerical calculations . the figure shows that these errors are indeed quite small . note that use of the straightforward definition in eq . ( [ eqn : atmod_sf ] ) with its unusually scaling surface moduli leads to invalid predictions for flexion . ( appendix [ sec - appb ] shows this directly through scaling arguments . ) as a result , to have _ predictive _ power , definitions of atomic moduli must properly account for the poisson effect . to demonstrate that our new approach has greater predictive power than traditional continuum approaches , the figure also shows the fractional error ( @xmath82 ) in predicting the flexural rigidity when using the traditional continuum relation @xmath83 where @xmath84 defines the mean bending moment , which we define unambiguously as @xmath85 the linear behavior of the fractional error @xmath82 as a function of @xmath86 indicates that continuum theory does not properly account for surface effects , a result of the fact that flexion places a larger emphasis on the surface than does extension . the dramatic improvement from the use of eq . ( [ eqn : atmof ] ) arises because the atomic moduli place proper emphasis on the surface and on the interior , as they properly treat each atomic environment locally . the atomic moduli therefore properly account for elastic fluctuations along the cross - section of the wire which are on scales too small for traditional continuum theories to capture . finally , we note that the only appreciable error within the new framework occurs for the smallest wires ( d@xmath87 nm ) . at this point , the cross - sectional dimension becomes comparable to the range of the force - constant matrix , and eq . ( [ eqn : free_at ] ) represents an improper use of the physical concept of atomic moduli . as described above ( second from last paragraph in section [ sec - mlwn ] ) , only sums of atomic moduli over regions of extent comparable to the range of the force - constant matrix carry physical meaning . any sum sensitive to variations over shorter scales , as is eq . ( [ eqn : free_at ] ) when limited to wires narrower than the range of the force - constant matrix , can not be depended upon to lead to meaningful results . this underscores the fact that properly construed atomic moduli are not truly atomic - level quantities but a concept coarse grained over the range of the force constant matrix . as we have seen , however , this coarse - graining is on scales significantly smaller than those captured by traditional continuum theory . we now apply the above defined atomic moduli to derive a new relation for the extensional modulus in terms of the flexural modulus and other experimentally accessible observables , a relation often needed in experimental analyses @xcite . we then show that the new relation is much more accurate than the standard continuum - theory based relation currently employed in experimental analyses . finally , we employ our concept of atomic moduli to provide quantitative insight into the improvement of our new relation over the traditional continuum relation . the basis for the following analysis is the fact that eq . ( [ eqn : atmof ] ) gives a very good estimate of the true flexural modulus , as figure [ fig : eandf]a confirms . using the exact relation for @xmath5 , eqs . ( [ eqn : thermofree ] ) and ( [ eqn : atmod ] ) , and eq . ( [ eqn : atmof ] ) , one derives the following leading - order prediction @xmath88 for the extensional modulus , @xmath89 with a predicted error @xmath90 of @xmath91 + \left[\langle e_s \rangle_{x^{2 } } - \langle e_s \rangle\right ] \right\}. \label{eqn : error}\end{aligned}\ ] ] here , @xmath92 is the number of `` surface '' atoms , defined as those for which @xmath42 differs significantly from the limiting bulk value @xmath93 , @xmath94 is the inertia weighted average surface moduli @xmath95 with the sums ( @xmath96 ) ranging over `` surface '' atoms , @xmath97 is the average surface moduli @xmath98 and @xmath99 where @xmath100 implies sums over all atoms in the unit cell . this result holds for any division of the atoms into `` surface '' and `` bulk '' to the extent that each `` bulk '' atom has atomic modulus @xmath93 . figure [ fig : eandf]b shows the relative error @xmath101 , @xmath102 between the extensional modulus @xmath5 determined directly from numerical calculation and as determined from our new relation , eq . ( [ eqn : atefromf ] ) . ( note that @xmath103 exactly when the moduli prediction @xmath104 holds . ) as the relevant point of comparison , the figure also shows the relative error @xmath105 , @xmath106 associated with the traditional continuum result @xmath107 hence , eq . ( [ eqn : atefromf ] ) is much more accurate than the standard continuum result eq . ( [ eqn : cnte ] ) . to understand the improvement of the new relation , eqs . ( [ eqn : thermofree ] ) and ( [ eqn : atmof ] ) may also be combined to yield a prediction for the fractional error in the traditional continuum analysis , @xmath108 + \left[\langle e_s \rangle_{x^{2 } } - \langle e_s \rangle\right ] \right\ } , \label{eqn : errorct}\ ] ] which takes _ precisely _ the same form as eq . ( [ eqn : error ] ) except for the change in the prefactor in the first term from @xmath109 to @xmath110 . for continuous wires of homogeneous circular or regular polygonal cross - section , we have exactly @xmath111 . thus , generally we expect @xmath109 to be close to zero and @xmath112 to be close to unity . we now note that the term in the first set of square brackets ( @xmath113 $ ] ) in both eq . ( [ eqn : error ] ) and eq . ( [ eqn : errorct ] ) is an average difference between surface and bulk atoms and , therefore , is generally much larger than the term in the second set of square brackets @xmath114 $ ] , which is the difference between two differently weighted averages over the surface atoms . thus the larger term nearly vanishes in our new relation , eq . ( [ eqn : error ] ) , but not in the traditional continuum relation , eq . ( [ eqn : errorct ] ) . from this analysis , we see that the reason why the traditional continuum relation has larger errors is that it does not properly differentiate between the local surface and bulk environments . the atomic moduli also lead to a quick , intuitive argument to understand the improvement of eq . ( [ eqn : atefromf ] ) over eq . ( [ eqn : cnte ] ) . from our results we know that the surface moduli can be quite different than those of the bulk . it is also known that flexion places larger emphasis on the surface than does extension . if the average surface modulus is less / more than that of the bulk then the first term in eq . ( [ eqn : atefromf ] ) ( @xmath115 ) will underestimate / overestimate the extensional rigidity , while the second term will overestimate / underestimate it . therefore , errors will tend to cancel in the average of the two . we now explore the local atomic moduli as a link between local elastic properties and the underlying electronic structure . to do this , we have calculated the atomic moduli , eq . ( [ eqn : atmod ] ) , for both the @xmath51 and @xmath116 structures , using the sawada @xcite tight - binding model with modifications proposed by kohyama@xcite . we have studied various wires , all of which give similar results . for brevity , we here only report on wires with cross - sectional diameter @xmath117 nm . the supercell of our calculation is four bulk cubic lattice constants long in the periodic direction , and hence sampling the brillouin zone at the @xmath118 point in the electronic structure calculations is more than sufficient . the left panels of figures [ fig : tbc2x2 ] and [ fig : tb2x1 ] show the resulting atomic moduli , with values color coded so that yellow corresponds to the bulk value ( 17.1 ev / atom ) . the figure shows that deviations from this value concentrate near the surface in patterns characteristic of the structure of the wire . moduli near the surface fluctuate widely , ranging from 6 - 27 ev / atom for the @xmath51 structure and from 4 - 26 ev / atom for the @xmath119 structure . to allow comparison with the underlying electronic structure , the right panels in the figures display the valence charge densities from the tight - binding calculation projected ( integrated along the wire axis ) onto the cross - section of the wire . ( to compute the electron density from the tight - binding coefficients , we employed orbitals from a density functional calculation of the silicon atom . ) the figures display the electron densities using a color map similar to that employed for the atomic moduli . intriguingly , there is an apparent correlation between the values of the atomic moduli and the underlying electron density . in particular , large / small atomic moduli correlate with regions of large / small electron density in figure [ fig : tbc2x2 ] , indicating that charge distribution along the surface of these wires greatly affects the local elastic properties and thereby the overall elastic response , particularly to flexion which emphasizes surface effects . figure [ fig : tb2x1 ] exhibits a similar correlation , but not as pronounced . note that , in this figure , coincidence of red atomic moduli just under the surfaces of the first layer of atoms correlates with red charge densities in the same location . unlike in the classical potential case , where properly defined surface moduli are systematically lower or equal to the bulk value due to decrease in the number of bonds ( figure [ fig : ecomp]b ) , we find that in a quantum model , surface moduli may even greatly exceed the bulk value due to changes in local charge density which can enhance the mechanical strength of bonds ( figures [ fig : tbc2x2 ] and [ fig : tb2x1 ] . ) this contrast underscores both the importance of considering contributions of the electronic structure to mechanical response and the need for a definition of atomic moduli which can be computed from physical observables obtainable from electronic structure calculations . the ultimate use of atomic level moduli is to understand mechanical response . table [ table : tab1 ] compares the errors from both continuum theory and the use of atomic - level moduli in predicting mechanical response for the two wires under consideration in this section . the quantities compared exactly parallel those of the previous section . the first two columns of the table consider prediction of flexural response from continuum theory and our atomic moduli , respectively , and the second two columns consider prediction of extensional response from flexural response using either the traditional continuum relation or our new relation , respectively . the table shows that the new relations , eqs . ( [ eqn : atmof ] ) and ( [ eqn : atefromf ] ) are again very accurate . the table also shows that these predictions are superior to the corresponding continuum results , eqs . ( [ eqn : cntf ] ) and ( [ eqn : cnte ] ) , respectively . .comparison of errors between the traditional continuum theory and the atomic moduli description when predicting flexural response through eqs ( [ eqn : cntf ] ) and ( [ eqn : atmof ] ) ( first and second columns , respectively ) and when inferring extensional response from flexural response through eqs . ( [ eqn : cnte ] ) and ( [ eqn : atefromf ] ) ( third and fourth columns , respectively ) . [ cols="^,^,^,^,^,^",options="header " , ] interestingly , the continuum predictions for the @xmath51 wire are fairly reliable . the atomic moduli provide an avenue for understanding this as well . fluctuations in the moduli in the @xmath51 wires are localized and hence average out over regions of extent comparable to the decay of the force - constant matrix . moreover , in this particular case they tend to average to values close to that expected of the bulk . without meaningful fluctuations on the length scales of the decay of the force - constant matrix , we expect continuum theory to perform well for this wire . in contrast , the @xmath120 wire exhibits much more systematic variations in the moduli . the outermost surface atoms have a consistent and significantly reduced modulus , and there is also a clear significant and systematic variation in the moduli throughout the cross - section of the wire . because this second wire does exhibit meaningful fluctuation over distances comparable to the decay range of the force constant matrix , we expect traditional continuum relations to give particularly poor results , underscoring the importance of the local atomic - level moduli description . this manuscript presents the first definition of atomic - level elastic moduli for nanoscale systems which are defined in terms of physical observables , correctly sum to give the exact overall elastic response and depend only on the local environment of each atom . although these moduli are not necessarily uniquely defined , their sum over regions of extent comparable to the range of the force constant matrix is physically meaningful and may be used to make accurate predictions of mechanical response . the moduli resulting from our formulation transfer to different modes of strain and correctly account for elastic fluctuations on the nanoscale . they also lead to a quantitative understanding of when traditional continuum relations breakdown and how to properly correct them properly . specifically , we demonstrated a more accurate method for relating extensional and flexural properties . these moduli provide a clear and natural method for distinguishing _ mechanically _ between `` surface '' atoms and `` bulk '' atoms and give insight into the correlation between the local mechanical response and the underlying electronic structure . finally , these moduli allow the identification of which atomic arrangements lead to more pliant or stiffer response opening the possibility of their use as a tool to aid in the rational design of nanostructures with specific mechanical properties . segall acknowledges support of his department of physics fellowship and the department s continuing support . s. ismail - beigi acknowledges support of the mit mrsec program of the national science foundation under award number dmr 9400334 . computational support was provided by the mit xolas prototype sun cluster and the cornell center for materials research shared experimental facilities , supported through the nsf mrsec program dmr9632275 . this appendix outlines the use of rotational and translation symmetries of the force constant matrix to reformulate the ill - defined atomic moduli in eq . ( [ eqn : atmod_sf ] ) into the well - defined form in eq . ( [ eqn : atmod ] ) . from continuous translational symmetry , all force - constant matrices obey @xmath121 for any vector @xmath122 that is the constant vector for all atoms in the unit cell . moreover , the @xmath123 rotational symmetries imply @xmath124 where the @xmath125 correspond to one of the @xmath72 cartesian coordinates , @xmath126 is the @xmath127 component of @xmath128 and @xmath129 corresponds to the @xmath130 component of the position vector @xmath131 . the ill - defined atomic moduli , eqs . ( [ eqn : thermofree])-([eqn : atmod_sf ] ) , contain divergent terms which are in one of the following two forms : @xmath132 or @xmath133 all of these terms scale linearly along the surface of the wire and give rise to the linear scaling of the surface moduli with system size evident in figure [ fig : ecomp]a . from eq . ( [ eqn : tran ] ) , one can set eq . ( [ eqn : firstterm ] ) equal to @xmath134 using both eqs . ( [ eqn : tran ] ) and ( [ eqn : rot ] ) , one can set eq . ( [ eqn : secondterm ] ) equal to @xmath135\sigma_t . \label{equ : secondnew}\end{aligned}\ ] ] the above terms now only depend on relative distances over a range controlled by the decay of the force - constant matrix and therefore no longer scale linearly with the size of the system . combining the above transformations with the separation of the extensive motion from the intensive motion in the first - order polarization vector , eq . ( [ eqn : decomp ] ) , then results in the well defined form eq . ( [ eqn : atmod ] ) . we now demonstrate that although the moduli defined in eq . ( [ eqn : atmod_sf ] ) indeed sum to give the correct overall extensional response , their unphysical scaling properties lead to invalid physical predictions when used in other contexts and , therefore , that they are ill - defined . in particular , we use simple scaling arguments to prove that these ill - defined moduli give a nonnegligible error in predicting the flexural rigidity in the continuum limit . consider a circular wire whose radius @xmath3 is sufficiently large such that continuum theory applies . from figure [ fig : ecomp]a we see that the surface moduli defined by eq . ( [ eqn : atmod_sf ] ) scale linearly with @xmath3 . because of this , and in order to arrive at an analytic result for the error in predicting the flexural rigidity , we can assume , to a good approximation , that all surface moduli @xmath136 are proportional to their radial distance @xmath137 from the center of the wire , @xmath138 here , @xmath139 is the same for all surface atoms . next , we define @xmath140 as the inner radius such that all atoms at position @xmath141 are in the bulk and all atoms with positions @xmath142 are on the surface . finally , we define @xmath143 . because of the facts that the surface moduli scale linearly with the system and that the sum of all atomic moduli must equal the extensional rigidity , we conclude that the moduli in the bulk region of the wire , derived from eq . ( [ eqn : atmod_sf ] ) , @xmath144 can not equal the average value of the bulk material @xmath93 . ( figure [ fig : ecomp]a also evidences this behavior . ) the two above facts imply that the following equality must hold , @xmath145 or to leading order in @xmath146 , @xmath147 in the continuum limit , the traditional continuum relation eq . ( [ eqn : cntf ] ) holds , and therefore the fractional error in predicting the flexural rigidity from the ill - defined atomic moduli eq . ( [ eqn : atmod_sf ] ) is equal to @xmath148 or to leading order in @xmath146 , @xmath149 using eq . ( [ eqn : ext ] ) to solve for @xmath150 , eq . ( [ eqn : flx ] ) becomes @xmath151 therefore , in the continuum limit , the ill - defined moduli do not approach the correct result and thus give a prediction which is even worse than that of traditional continuum theory .

we initiate the development of a theory of the elasticity of nanoscale objects based upon new physical concepts which remain properly defined on the nanoscale . this theory provides a powerful way of understanding nanoscale elasticity in terms of local group contributions and gives insight into the breakdown of standard continuum relations . we also give two applications . in the first , we show how to use the theory to derive a new relation between the bending and stretching properties of nanomechanical resonators and to prove that it is much more accurate than the continuum - based relations currently employed in present experimental analyses . in the second , we use the new approach to link features of the underlining electronic structure to the elastic response of a silicon nanoresonator .

You are an expert at summarizing long articles. Proceed to summarize the following text: in the first paper ( manset & bastien 2000 , hereafter referred to as paper i ) , we presented numerical simulations of the periodic polarimetric variations produced by a binary star surrounded by a circumbinary envelope composed of electrons . thomson scattering was considered in an optically thin situation , along with pre- and post - scattering extinction factors which produce a time - varying ( small ) optical depth and affect the morphology of the periodic variations by adding an additional harmonic to the variations . we studied the effects that orbital inclination , optical depth , geometry of the envelope and cavity , size and eccentricity of the orbits , and non - equal mass stars had on the average polarization level and the amplitude of the polarimetric variations . we found that high polarization levels will result from a high inclination , a high optical depth , a flat envelope , or a big central cavity . polarimetric variations are more apparent for a low inclination , a high optical depth , a flat envelope , a small cavity , or an orbit which brings the stars close to the inner edge of the cavity . using the formalism developed by brown , mclean , & emslie ( 1978 , hereafter bme ) the observations , or in this case the results of the numerical simulations , are fitted with a fourier series with coefficients of orders zero , one , and two ( see equations 3 and 4 below ) . the first order coefficients from the fit ( terms in @xmath8 ) correspond to single - periodic variations ( variations seen once per orbit ) , and those of second order ( terms in @xmath9 ) to double - periodic variations ( seen twice per orbit ) . those coefficients are used to find the orbital inclination ( see equations 5 and 6 below ) . it was then shown that the bme formalism can be used to find the orbital inclination if it is @xmath10 , even though this formalism does not include variable absorption effects as in our simulations . the geometry ( flatness of the envelope , size of the central cavity ) and size of the orbit have no significant influence on the inclination found by the bme formalism . an extension of the bme formalism for eccentric orbits has been performed by brown , aspin , simmons , & mclean ( 1982 ) ; however , their case considers a localized scattering region in eccentric orbit about a star , which is not a suitable geometry for the cases we want to study here . therefore , we performed our own study of eccentric orbits . for eccentric orbits , single - periodic variations appear for eccentricities as low as 0.10 . as the eccentricity increases , these single - periodic variations dominate over the double - periodic ones . for low eccentricities , @xmath11 , the inclinations can be found with the first or second - order fourier coefficients , if @xmath12 and @xmath13 respectively . for the high eccentricities , @xmath14 , only the first - order coefficients should be used , if @xmath15 . orbital eccentricity is not the only factor that can introduce single - periodic variations : variable absorption effects , non equal mass stars , and asymmetric envelopes may also produce those . in this second paper , we investigate the effects of scattering by spherical dust grains ( mie scattering ) . in addition to studying the influence of the optical depth and true orbital inclination already considered in paper i , we also consider the effects of the composition and size of the dust grains on the level of polarization and the amplitude of the polarimetric variations . our goal is to study the periodic polarimetric variations of binary young stars surrounded by envelopes in which dust grains are responsible for the scattering and polarization , and compare polarimetric observations with our numerical simulations . as we have shown in paper i that the bme formalism is still valid beyond its original limits ( circular orbits , no extinction effects ) , now we also want to see if the bme formalism can be extended to the case of mie scattering to find the orbital inclination , even though electrons and spherical dust grains have different scattering properties , and the bme formalism has been developed for thomson scattering . the simulations presented here are akin to the simulations where single - periodic polarimetric variations , usually double - peaked , are produced by a spotted star surrounded by a dusty circumstellar disk ( see for e.g. stassun & wood 1999 ) . whereas in the latter case the polarimetric variations are produced by a pair of intrinsically polarized hot spots illuminating ( and scattering off ) the disk , here the variability is produced by the binarity of the source of light . the polarization produced by a single star surrounded by a circumstellar disk has been studied with monte carlo simulations , by , for e.g. , wood et al . the scattering model used to compute the polarization and its position angle produced by two stars in orbit at the center of an ellipsoidal envelope was presented in paper i. in addition to the parameters presented there ( geometry of the envelope , orbit characteristics , optical depth , grid size ) , here we have to choose a grain composition ( which will give a specific complex refractive index ) and the size of the grains . table [ tab - grains ] presents the characteristics of the grains studied in this paper . a uniform size distribution was used for most of our simulations , but we also studied a distribution of grain sizes . the calculations for mie scattering use the following formulas : @xmath16 @xmath17 where : + @xmath18 : optical depth in the equatorial plane of the envelope ; + @xmath19 : optical depth before scattering ( between the light source and the scatterer ) ; + @xmath20 : optical depth after scattering ( between the scatterer and the observer ) ; + @xmath21 : angle between the east west direction and the scattering plane ( which includes the light source , the scatterer and the observer ) ; + @xmath22 : scattering angle , between the light source , the scatterer and the observer ; + @xmath23 : distance between the light source and the scatterer ; + @xmath24 : volume element ( computed in relation to the grid size ) ; + @xmath25 : weight which equals 1 if there is a scatterer , 0 otherwise ; + @xmath26 : optical depth per grid step ; + @xmath27 : radius of the spherical dust grain ; + @xmath28 : wavelength of observation ; + @xmath29 : extinction cross section ; + @xmath30 and @xmath31 : van de hulst intensities . + variable absorption effects are considered by using @xmath19 and @xmath20 . depending on where the binary star is located inside the envelope , the optical depths between the source and the scatterer ( @xmath19 ) , and between the scatterer and the observer ( @xmath20 ) will vary , and with them , the polarization . the van de hulst intensities and @xmath29 are calculated with the usual formulas that can be found in van de hulst ( 1981 , chap . 9 ) and make use of the complex refractive index of the grains . the polarization @xmath32 and its position angle @xmath33 are calculated from the stokes parameters @xmath34 and @xmath35 . when the polarization is produced by scattering , the sign used for @xmath32 is related to its orientation with respect to the scattering plane ( the plane containing the source of light , the scatterer , and the observer ) . a positive polarization is perpendicular to the scattering plane , whereas a negative one is parallel to it . thomson scattering always produces positive polarization , but both positive and negative polarization are produced by mie scattering . as in paper i , a `` canonical simulation '' is a simulation which has the following parameters : an envelope with axis ratios 1.0 , 1.0 and 0.25 ( also referred to as a 25% flat envelope ) , a spherical cavity with radius 0.20 ( also referred to as a 20% cavity ) , an optical depth of 0.1 , and a grid size of 65 . in addition , calculations are made for a wavelength of @xmath36 . figure [ fig - cb_2_05_05 ] presents a schematic view of this canonical geometry . for further details on the calculations , see paper i. the simulated polarimetric curves produced by the scattering model presented above are analyzed with the bme formalism , as in paper i. for reference , we give the main formulas that are used . observations are represented as first and second harmonics of @xmath37 , where @xmath38 is the orbital phase ( @xmath39 ) : + @xmath40 the inclination can be found with the first ( equation [ eq - io1-p2 ] ) or second ( equation [ eq - io2-p2 ] ) order fourier coefficients , although it is expected for circular orbits that second order variations will dominate : + @xmath41 ^ 2 & = & \frac{(u_1+q_2)^2 + ( u_2-q_1)^2}{(u_2+q_1)^2 + ( u_1-q_2)^2 } \label{eq - io1-p2},\\ \left [ \frac{1-\cos i}{1+\cos i } \right]^4 & = & \frac{(u_3+q_4)^2 + ( u_4-q_3)^2}{(u_4+q_3)^2 + ( u_3-q_4)^2}. \label{eq - io2-p2}\end{aligned}\ ] ] in an alternative representation , the eccentricity of the ellipse in the @xmath42 plane is related to the inclination . for the @xmath43 ellipse : + @xmath44 for the @xmath45 ellipse : + @xmath46 an important difference between this bme formalism and our numerical simulations is the inclusion of variable absorption effects in our work . as in paper i , a preliminary step to the numerical calculations was to determine a suitable grid size , one with which the bme formalism would find an orbital inclination close enough to the real inclination ( say , within @xmath47 ) . using astronomical silicates grains with radii of @xmath48 , we calculated models with grid sizes ( radius ) 25 to 95 , by steps of 10 . the results of the simulations , in which no noise was introduced , were used as input data for the bme equations . the orbital inclinations found in this manner are almost identical ( within a few tenths of a degree ) to those we found for thomson scattering in paper i. as in paper i , we chose a grid size of 65 , for which the inclination is within @xmath49 of the true value , and the calculations can be performed in reasonable times . we made calculations for the canonical simulation ( 25% flat envelope , 20% central cavity , @xmath50 , grid radius=65 ) , the four grain compositions listed in table [ tab - grains ] , and five grain radii ( 0.02 , 0.05 , 0.10 , 0.2 , and @xmath51 ) , at a wavelength of @xmath36 . since the polarization produced by mie scattering depends on the size of the grain @xmath27 and the wavelength of observation @xmath28 , it is common to introduce the parameter @xmath52 ; the values of @xmath53 for the previously mentioned simulations are 0.18 , 0.45 , 0.90 , 1.8 , and 4.5 . for each simulation , we used only one grain size and one grain composition . if we compare the polarizing properties of dust grains and electrons , we note that dust grains ( irrespective of composition and size ) are less efficient polarizers than electrons . grains also produce smaller absolute polarimetric variations per particle than electrons , although both electrons and grains produce about the same @xmath54 ratios ; see table [ tab - grains - e ] . that important observation shows that for binary stars embedded in dust envelopes , periodic polarimetric variations should be harder to observe and detect . for grain sizes of 0.02 , 0.05 or @xmath55 ( @xmath56 ) , the highest polarization and polarimetric variations are found for astronomical silicates , followed by graphite and amorphous carbon , and finally dirty ice ; grains made of dirty ice are 510 times less polarizing and produce variations about 510 times smaller than grains made of astronomical silicates . for grains with @xmath57 ( @xmath58 ) , the dirty ice produces the highest polarization and variations . for @xmath59 ( @xmath60 ) , astronomical silicates , graphite , and amorphous carbon have similar polarimetric properties , and dirty ices are less efficient polarizers . the highest polarizations are produced by grains with sizes in the range @xmath61@xmath62 ( @xmath632.0 ) . for astronomical silicates , we also did simulations for a wider range of grain sizes , going from @xmath64 to @xmath65 . the results are presented in figure [ fig - p_vs_x ] , where it is seen that the maximum polarization is reached for @xmath66 . these results follow what is presented in simmons ( 1982 ) . see for example his figure 4 ( polarization as a function of @xmath53 for different grain compositions ) , where it can be seen that for silicate grains and ice , polarization depends on @xmath53 with a very different behavior for these 2 grain compositions ; a grain that is the most polarizing at one value of @xmath53 will not necessarily be the most polarizing grain at other values of @xmath53 . figure 6 in simmons ( 1982 ) shows for silicate grains in particular how a different absorption coefficient changes the behavior of the polarization as a function of @xmath53 . finally , figure 5a shows that @xmath67 , the value of @xmath53 for which the polarization is maximum , depends on both the real and imaginary parts of the refractive index ( and thus on the grain composition ) . we also made some more calculations for astronomical silicates , adding grain sizes of @xmath68 and @xmath69 . big grains ( @xmath70 ) are less efficient polarizers , as was already known ( see for example daniel 1978 , simmons 1983 ) . as shown by our simulations and by simmons ( 1982 ) , the average polarization produced by dust grains thus depends on the grain composition and size . polarization reversal ( when the polarization goes from positive , or perpendicular to the scattering plane , to negative , or parallel to the scattering plane ) seen in our simulations also occurs for the @xmath53 values presented in simmons ( 1982 ) , figure 5b . for astronomical silicates , polarization reversal occurs at a value somewhere between @xmath71 and @xmath72 , or @xmath73 ( see figure [ fig - p_vs_x ] ) , in agreement with figure 5b of simmons ( 1982 ) . for dirty ice , we do not see any polarization reversal , which follows the figure 5b of simmons ( 1982 ) , except for our biggest grains ( @xmath74 , @xmath75 ) at low inclinations ( @xmath76 ) . for the grains with large imaginary refractive index values ( graphite and amorphous carbon ) , we see from table 1 of daniels ( 1980 ) that polarization reversal is not expected at all , which is what is seen here , except in 2 cases for intermediate inclinations . we also studied a distribution of grain sizes , using the mrn size distribution @xmath77 ( mathis , rumpel , & nordsiek 1977 ) for astronomical silicates with sizes between @xmath78 and @xmath79 . the results are not very different from the case where only one grain size was considered . since all polarimetric curves have the same morphology for a given set of parameters except grain size , adding many of those curves only changes the polarization level , not the morphology . as the inclination decreases from @xmath80 ( edge - on ) to @xmath81 ( pole - on ) , the polarization decreases , in general following a theoretical law ( developed for a single star at the center of an axisymmetric envelope but nonetheless interesting for our case ) that can be found for example in brown & mclean ( 1977 ) : the residual polarization ( sum of polarization of the scattered light and the unpolarized light from the star ) scales as @xmath82 . deviations from this theoretical law are found when the optical depth is high ( @xmath83 , too high to consider single scattering only ) or the grains are big ( @xmath84 , which probably introduces higher frequency variations in the polarization as a function of the scattering angle ) . with a @xmath80 inclination , the @xmath35 parameter is null , and only the @xmath34 parameter varies . as the inclination decreases , both the @xmath34 and @xmath35 parameters vary , and more so as the inclination decreases . in general , the behavior is the same as for thomson scattering . the simulations for a binary star in a circumbinary envelope give polarimetric variations that are double sine waves ( see figure [ fig - binary_as_t01_a01 ] ) , which translate to an ellipse that is traced out twice per orbit in the @xmath42 plane . the average stokes parameter @xmath35 is zero , as is expected from a geometry oriented in the plane of the sky in a east west direction , and the position angle is usually @xmath85 , perpendicular to the projection of the envelope s main axis and scattering angle . in order to decrease the number of parameters influencing the polarization and isolate causes and their effects , we can take a look at the contribution of only one star instead of considering the combined effects of the 2 stars . in figure [ fig - primary_as_t01_a01 ] we show the polarimetric variations for the primary only , as if it were in orbit around an `` invisible '' , or much fainter , secondary star , both stars being in orbit at the center of the same circumbinary shell . even though the orbit is circular , single - periodic variations are seen in addition to the usual double - periodic ones . we have shown in paper i that variable absorption effects introduce such single - periodic variations ; but if we neglect those variable absorption effects and impose a constant optical depth in our mie simulations , those single - periodic variations persist . this is due to the asymmetric phase function of dust grains . it is also seen that the peaks of polarization are not located exactly at phases 0.25 and 0.75 , as expected for an asymmetric phase function . therefore , in addition to variable absorption effects , non equal mass stars , orbital eccentricity , and asymmetric envelopes , mie scattering can also cause single - periodic variations . like the thomson scattering results presented in paper i , the amplitude of the variations increases significantly for larger size orbits . for an orbit that comes close to the inner edge of the circumbinary disk , the polarimetric variations are more apparent . in order to see the effects of geometries different than a circumbinary ellipsoidal envelope , we made simulations for flared circumbinary disks , prolate circumbinary envelopes , and circumstellar disks . we also investigated the effects of a non - coplanar geometry where the plane of symmetry of the envelope and the orbital plane differ from one another . first , we tried a geometry similar to a flared disk , with a conic opening ( with half - angle between 10 and @xmath86 ) in an ellipsoidal envelope . as the angle of the cone increases , so does the polarization and amplitude of the variations . we also studied prolate geometries instead of oblate ones , orienting the long axis of the envelope along 3 axes ( toward the observer , in a n s and e w orientation in the plane of the sky ) . finally , we inclined the envelope by @xmath87 with respect to the orbital plane . for all those cases , polarization and amplitude of variations are affected to various degrees , but the polarimetric variations have the same morphology . for circumbinary envelopes , the polarimetric curves we have produced so far seldom have an amplitude greater than 0.10% , whereas simmons ( 1983 ) could produce amplitude of many tenths of a percent , or even of a few percents , with a geometry more favorable for high amplitude polarimetric variations : a single circumstellar envelope externally illuminated by only one star . indeed , the most interesting simulations were done with a circumstellar disk around the primary , a `` naked '' secondary which does not have any circumstellar material in its environment , and no circumbinary disk . see figure [ fig - cs_45_1_5 ] for an example of such a geometry . as expected from this more asymmetric geometry , the polarimetric variations produced have a greater amplitude , ranging from 0.1 to 0.3% . the primary star that sits in the middle of its circumstellar disk does not produce any variation , as expected , but numerical noise is present at a level of @xmath88 % in polarization . variations for the secondary star , which revolves around the primary and its disk , are a sum of double- and single - periodic variations . average polarization , amplitude of the polarimetric variations , and morphology of the variability are all interesting , but periodic polarimetric observations can potentially give an additional and very interesting parameter for a binary star : its orbital inclination , which is found by using the formalism developed by bme ( equations [ eq - io1-p2 ] and [ eq - io2-p2 ] ) . we now investigate if their formalism , which was developed for thomson scattering , circular orbits , and constant optical depth , can be extended successfully to our simulations to find the orbital inclination . for an optical depth of 0.1 , we have found that the bme formalism will work ( i.e. , true inclinations can be found from the bme analysis of the polarimetric variations if @xmath89 ) as long as there are polarimetric variations of sufficient amplitudes for the mathematical analysis ( by sufficient , we mean for example , 0.002% for simulations without stochastic noise ) . the composition or the size of the grains is not important as long as some variations are introduced . big grains ( @xmath90 ) do not produce enough variations for the bme analysis to work properly . in general , the uncertainty on the inclination found by the bme formalism ( calculated with the method of propagation of errors ) is higher for dust grains than for electrons . this is understandable since , as stated before , grains produce smaller amplitude polarization variations , which will increase the uncertainty on the inclination found . the effects of optical depths were studied for some combinations of grain composition and size , with optical depths of 0.02 , 0.05 , 0.2 , and 0.5 . as expected , the polarization increases with the optical depth , as does the amplitude of the polarimetric variations . all the simulations studied here produced polarimetric variations with sufficiently large amplitudes , so again , the bme formalism works for @xmath7 . berger & mnard ( 1997 ) have studied multiple mie scattering in circumbinary shells with optical depths up to @xmath91 , and have found that even with such a high optical depth , the estimation of inclination from the bme formalism is still very good . although the inclination found by the bme equations is not significantly affected by the geometry in the case of the flared disk geometry , it is more seriously affected for prolate geometries and non - coplanar orbital and geometric planes . for the prolate envelope inclined at @xmath92 , the two cases where the long axis of the prolate envelope is in the plane of the sky give inclinations close to @xmath92 , but for the long axis coming toward the observer , the inclination found is @xmath93 too low . for the same inclination , the envelope inclined by @xmath87 with respect to the orbital plane gives orbital inclinations of @xmath94 or @xmath95 , significantly different than the @xmath96 found for the coplanar case . for the circumstellar disks , bme can find the inclination if it is @xmath97 . the lower threshold , compared to the circumbinary case , might be due to the amplitude of the variations , which are greater for the circumstellar envelope geometry than for the circumbinary envelope geometry . to investigate the possible effects of non - circular orbits on polarimetric observations , canonical simulations were performed with orbital eccentricities of 0.1 , 0.3 , and 0.5 . it should be noted that in these simulations , the orbital semi - major axes have been adjusted according to the eccentricity ( i.e. , decreased with increasing @xmath98 ) so the stars came at about the same distance from the inner edge of the envelope , irrespective of the eccentricity of their orbit ( the maximum distance between the center of the envelope and any of the two stars was always @xmath99 ) . this was done so we could study the influence of the eccentricity without having to deal with the influence of how close the stars come to the inner edge of the circumbinary envelope . the results found for mie scattering are identical to those we have found for thomson scattering ( see paper i ) . the eccentricity has no significant influence on the average level of polarization , but does change the amplitudes of the polarimetric variations : the amplitudes decrease as the eccentricity increases . as the eccentricity increases , the @xmath43 variations that are dominant for low eccentricities give way to @xmath45 variations . moreover , higher harmonics appear ; a fit with 1@xmath28 and 2@xmath28 is not sufficient to reproduce the polarimetric curves , so 3@xmath28 and 4@xmath28 harmonics are needed . when looking at the inclinations found by the bme formalism with the second - order coefficients of the fit , we find that the true inclinations are found for the lower eccentricities ( @xmath100 ) and the highest inclinations ( @xmath89 ) . for the highest eccentricity studied here , @xmath101 , the second - order coefficients can not be used to find the orbital inclination , as the bme formalism is unable to find the true inclination . for all eccentricities , even the highest one studied here @xmath101 , the first - order coefficients can be used to find almost all inclinations . there is a numerical problem with the true inclination of 80 , for which the bme formalism can not assign the right inclination . the lowest inclinations ( @xmath102 ) are harder to find . when the periastron is changed to values other than 0 , the results of the bme analysis are slightly modified . with an eccentricity of 0.5 , the second - order and first - order coefficients now both give reasonable results for @xmath89 . in future papers , we will present polarimetric observations of binary young stars , which are objects surrounded by circumstellar material ( dust grains ) . the binaries we have selected have short periods , so we believe the geometry adopted here ( circumbinary ellipsoidal envelope with a central cavity ) is suitable for these kinds of objects . some of the observed pre - main sequence binaries show periodic polarimetric variations , although they are in general of lower amplitude and less clear than those of binary hot stars ( surrounded by electrons ) , such as wolf rayet stars ( see , for example , st - louis et al . 1988 ; drissen et al . 1989b ; robert et al . this is in agreement with our previous comparison of the polarimetric properties of electrons and dust grains . non - periodic polarimetric variations that are known to exist even for single young stars ( bastien 1982 ; drissen , bastien , & st - louis 1989a ; mnard & bastien 1992 ) produce stochastic noise in the polarimetric curves of binary young stars , sometimes hiding the low amplitude periodic variations . the small amplitude variations ( @xmath103% ) often seen in binned data can be caused by non favorable inclinations or geometry , or not enough scatterers , but we believe it is in general agreement with one of our conclusions , that dust grains produce smaller amplitude variations than electrons . some stars do have high amplitude polarimetric variations of up to 0.7% . such large amplitudes were not quite produced by our simulations , although circumstellar disks , a more favorable geometry than circumbinary disks , do produce more substantial variations , up to @xmath104% . for the observed high amplitude polarimetric variations , a circumbinary envelope might thus not be an adequate geometry and circumstellar envelope(s)/disk(s ) could be present . we have presented numerical simulations of the periodic polarimetric variations produced by a binary star placed at the center of an empty cavity in a circumbinary ellipsoidal and optically thin envelope . mie single - scattering in envelopes made of single - size and single - composition grains was considered . the orbits were circular or eccentric . the mass ratio ( and luminosity ratio ) was equal to 1.0 . these parameters are to represent short - period spectroscopic binary young stars that are embedded in a circumbinary envelope and have evacuated the central regions of this envelope due to gravitational or radiative interactions . we have shown that the absolute amplitude of the variations is smaller for mie scattering than for thomson scattering , which will render more difficult the detection of polarimetric variations in binary stars surrounded by dust grains . in fact , polarimetric observations of binary young stars that will be presented in future papers show periodic variations that are sometimes much less obvious than for hot stars , for example . this may be due in part to the scattering properties of dust grains , although other factors , such as orbital inclination or scattering geometry , can also decrease ( or increase ) the amplitude of the polarimetric variations . the average polarization produced depends on the grains composition ( through the real and imaginary parts of the refractive index ) and size , and on the wavelength of observation ; a highly polarizing grain at a given grain size will not necessarily polarize as efficiently if it is smaller or bigger . for the four grain compositions that we have studied ( astronomical silicates , graphite , amorphous carbon , and dirty ice ) , the highest polarizations are produced by grains with sizes in the range @xmath105@xmath1 ( @xmath106@xmath3 for @xmath107 ) . polarization reversal was seen for astronomical silicates ( at @xmath108 , or @xmath109 ) , but not for dirty ices , graphite , and amorphous carbon grains , which is in agreement with previous works ( simmons 1982 ; daniels 1980 ) . the periodic variations produced are in general double - periodic ( seen twice per orbit ) , although we have shown that variable absorption effects ( pre- and post - scattering factors ) , non equal mass stars , orbital eccentricity , asymmetric envelopes , and mie scattering ( on spherical dust grains , which have an asymmetric scattering function ) introduce single - periodic variations ( seen once per orbit ) . circumstellar disks are more interesting than circumbinary disks , in the sense that they produce polarimetric variations that are more readily detectable ( @xmath5% versus @xmath6% ) . other geometries ( flared disks , prolate envelopes , non - coplanar disks ) did not produce any difference in the morphology or amplitude of the variations . we have shown that the bme formalism is still valid beyond its original limits ( circular orbits , no extinction effects , thomson scattering ) , under certain circumstances . in general , the bme formalism will be able to find the orbital inclination if this inclination is @xmath110 and if the amplitude of the polarimetric variations is sufficient . the composition or the size of the grains is not important as long as some variations are introduced , but the inclinations found by the bme formalism seem to be affected by prolate and non - coplanar geometries . the second - order fourier coefficients of the fit can be used for the lower orbital eccentricities ( @xmath100 ) and the highest inclinations ( @xmath89 ) . for all eccentricities , even the highest one studied here @xmath101 , the first - order coefficients can be used to find almost all inclinations . comparisons between polarimetric observations of binary stars and these simulations will help understand these systems ( presence of circumbinary and/or circumstellar disks , geometry of the envelope , orbital eccentricity , etc . ) although the interpretations might not be unique , as many parameters can have similar effects on the polarimetric variations ( mainly the amplitude and the presence of single - periodic variations ) . in that case , other types of observations ( imaging and spectroscopy ) would nicely complement polarimetric ones . n. m. would like to thank the conseil de recherche en sciences naturelles et gnie of canada , the fonds pour la formation de chercheurs et laide la recherche of the province of qubec , the facult des etudes suprieures and the dpartement de physique of universit de montral for scholarships , and p. b. for financial support . n. m. would also like to thank f. mnard for numerous discussions . we would like to thank the conseil de recherche en sciences naturelles et gnie of canada for supporting this research . bastien , p. 1982 , , 48 , 153 berger , j .- p . , & mnard , f. 1997 , in iau symposium 182 , low mass star formation from infall to outflow , ed . f. malbet & a. castets , 201 brown , j. c. , & mclean , i. s. 1977 , , 57 , 141 brown , j. c. , mclean , i. s. , & emslie , a. g. 1978 , , 68 , 415 brown , j. c. , aspin , c. , simmons , j. f. l. , & mclean , i. s. 1982 , , 198 , 787 daniel , j .- y . 1978 , , 67 , 345 daniel , j .- y . 1980 , , 87 , 204 drissen , l. , bastien , p. , & st - louis , n. 1989a , , 97 , 814 drissen , l. , robert , c. , lamontagne , r. , moffat , a. f. j. , st - louis , n. , van weeren , n. , & van genderen , a. m. 1989b , , 343 , 426 draine , b. t. 1985 , , 57 , 587 greenberg , j. m. 1968 , in nebulae and interstellar matter , ed . b. m. middlehurst & l. h. aller ( chicago : university of chicago press ) , 221 manset , n. , & bastien , p. 2000 , , 120 , 413 ( paper i ) mathis , j. s. , rumpl , w. , & nordsiek , k. h. 1977 , , 217 , 425 mnard , f. , & bastien , p. 1992 , , 103 , 564 robert , c. , moffat , a. f. j. , bastien , p. , st - louis , n. , & drissen , l. 1990 , , 359 , 211 rouleau , f. , & martin , p. g. 1991 , , 377 , 526 simmons , j. f. l. 1982 , , 200 , 91 simmons , j. f. l. 1983 , , 205 , 153 stassun , k , & wood , k. 1999 , , 510 , 892 st - louis , n. , moffat , a. f. j. , drissen , l. , bastien , p. , & robert , c. 1988 , , 330 , 286 van de hulst , h. c. 1981 , light scattering by small particles ( new york : dover publications , inc . ) wickramasinghe , n. c. 1967 , interstellar grains , ( london : chapman and hall ltd . ) wickramasinghe , n. c. , & guillaume , c. 1965 , , 207 , 366 wood , k. , bjorkman , j. e. , whitney , b. a. , & code , a. d. 1996 , , 461 , 828 llll grain composition & @xmath111 & @xmath112 & @xmath113 + be ( amorphous carbon ) & 2.240 & 0.781 & 2.37 + graphite & 2.524 & 1.532 & 2.95 + dirty ice & 1.33 & 0.09 & 1.34 + astronomical silicate & 1.715 & 0.030 & 1.72 + lllllll @xmath114 & & electrons & astron . & graphite & amorphous & dirty ice + & & & silicates & & carbon & + & @xmath115 ( % ) & 0.794 & 0.619 & 0.299 & 0.270 & 0.189 + 90@xmath116 & @xmath117 ( % ) & 0.0063&0.0048&0.0023&0.0021&0.0015 + & @xmath118&0.0079&0.0075&0.0077&0.0078&0.0079 + & @xmath115 ( % ) & 0.593 & 0.461 & 0.222 & 0.201 & 0.140 + 60@xmath116 & @xmath117 ( % ) & 0.0079&0.0061&0.0029&0.0027&0.0018 + & @xmath118&0.0133&0.0132&0.0131&0.0134&0.0129 + & @xmath115 ( % ) & 0.395 & 0.306 & 0.147 & 0.133 & 0.093 + 45@xmath116 & @xmath117 ( % ) & 0.0095&0.0073&0.0035&0.0032&0.0022 + & @xmath118&0.0241&0.0239&0.0238&0.0241&0.0237 +

following a previous paper on thomson scattering , we present numerical simulations of the periodic polarimetric variations produced by a binary star placed at the center of an empty spherical cavity inside a circumbinary ellipsoidal and optically thin envelope made of dust grains . mie single - scattering ( on spherical dust grains ) is considered along with pre- and post - scattering extinction factors which produce a time - varying optical depth and affect the morphology of the periodic variations . the orbits are circular or eccentric . the mass ratio ( and luminosity ratio ) is equal to 1.0 . we are interested in the effects that various parameters ( grain characteristics , geometry of the envelope , orbital eccentricity , etc . ) will have on the average polarization , the amplitude of the polarimetric variations , and the morphology of the variability . we show that the absolute amplitudes of the variations are smaller for mie scattering than for thomson scattering , which makes harder the detection of polarimetric variations for binary stars surrounded by dust grains . the average polarization produced depends on the grains composition and size , and on the wavelength of observation . among the four grain types that we have studied ( astronomical silicates , graphite , amorphous carbon , and dirty ice ) , the highest polarizations are produced by grains with sizes in the range @xmath0@xmath1 ( @xmath2@xmath3 for @xmath4 ) . composition and size also determine if the polarization will be positive or negative . in general , the variations are double - periodic ( seen twice per orbit ) . in some cases , because spherical dust grains have an asymmetric scattering function , the polarimetric curves produced show single - periodic variations ( seen once per orbit ) in addition to the double - periodic ones . a mixture of grains of different sizes does not affect those conclusions . circumstellar disks produce polarimetric variations of greater amplitude ( up to @xmath5% in our simulations ) than circumbinary envelopes ( usually @xmath6% ) . other geometries ( circumbinary flared disks or prolate envelopes , and non - coplanar envelopes ) do not present particularly interesting polarimetric characteristics . another goal of these simulations is to see if the 1978 bme ( brown , mclean , & emslie ) formalism , which uses a fourier analysis of the polarimetric variations to find the orbital inclination for thomson - scattering envelopes , can still be used for mie scattering . we find that this is the case , if the amplitude of the variations is sufficient and the true inclinations is @xmath7 . for eccentric orbits , the first - order coefficients of the fourier fit , instead of second - order ones , can be used to find almost all inclinations .

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Proceed to summarize the following text: the creation and motion of large numbers of crystal lattice dislocations is the most fundamental feature of crystal plasticity . during the last half century , the physical properties of individual dislocations and their interactions with localised obstacles have been studied extensively . on the other hand , the complex collective dynamics of strongly interacting many - dislocation systems is still far from being understood . fortunately , everyday plastic deformation processes very often proceed orders of magnitude slower than the typical relaxation times of the underlying dislocation system . these conditions often permit to study the problem in a quasistatic approximation @xcite . beyond the quasistatic limit , however , much less work has been devoted to studying the dynamics of collective dislocation motions which lead to the formation of metastable configurations , and to transitions between such configurations in driven dislocation systems . however , such collective motions are crucial for understanding rapid dislocation processes which not only occur in shock loading but , in the form of dislocation avalanches , are a generic feature of the dynamics of driven dislocation systems @xcite . the first studies of dynamic relaxation processes in dislocation systems were performed by miguel et al . with the protocol of applying a constant external shear stress to well relaxed dislocation configurations @xcite . the ensuing creep relaxation was numerically shown to follow andrade s law stemming from the underlying intermittent and correlated motion of dislocation structures . the connection between the mesoscopic and macroscopic features of the process was , however , not analysed in detail . another direction was taken by the present authors who conducted systematic studies of the relaxation dynamics of initially random configurations of straight dislocations . this is an important issue since the elastic energy density @xmath0 of a random dislocation system of density @xmath1 is known to diverge with the logarithm of system size @xmath2 , @xmath3 @xcite where @xmath4 is the modulus of the dislocation burgers vector . in a well - relaxed dislocation arrangement , on the other hand , the same quantity scales like @xmath5 , i.e. , the screening length corresponds to the mean dislocation spacing @xcite . as the mean square stress is proportional to the elastic energy density , this screening also removes a logarithmic divergence of the width of the internal stress probability distribution @xcite , and of the x - ray line width @xcite . numerical experience showed that , at least in single slip geometries , the relaxation processes that lead to screened dislocation arrangements exhibit slow , power law characteristics for quantities such as the elastic energy or the average dislocation velocity @xcite . a model was proposed which relates the power - law relaxation dynamics to the gradual extinction of initial dislocation density fluctuations @xcite . the present paper presents a comprehensive numerical investigation which allows to check in detail the model predictions and complements the earlier work by extending the investigation to multiple slip geometries and to dislocation systems of non - zero net burgers vector , and by studying the influence of an external driving stress on the relaxation process . the paper is organised as follows . in the problem is defined and technical details of the simulations are presented . unfolds a scaling model of the relaxation process from a chemical analogy and uses this model to predict the evolution of simulation measurables . then gives a detailed comparison between model predictions and numerical results . the results are discussed and conclusions are drawn in . an auxiliary calculation of the elastic energy of a random dislocation wall is presented in the appendix . consider a system of @xmath6 straight edge dislocations running parallel to the @xmath7 axis of a cartesian coordinate system . let all dislocations have a common burgers vector pointing along the @xmath8 axis ( a so - called single slip geometry ) , @xmath9 , where @xmath10 is the sign of the @xmath11th dislocation . assuming overdamped glide motion with a dislocation velocity @xmath12 that is proportional to the local resolved shear stress , and zero dislocation mobility in the climb direction , the equation of motion of dislocation @xmath11 piercing the @xmath13 plane at @xmath14 can be written as @xmath15 , \qquad \tau_{\mathrm{ind}}(\bi{r } ) = g b \frac{x ( x^{2}-y^{2})}{(x^{2}+y^{2})^{2}},\ ] ] where @xmath16 denotes the dislocation glide mobility , @xmath17 $ ] where @xmath18 is the shear modulus and @xmath19 is poisson s ratio of the embedding isotropic crystal , @xmath20 denotes the resolved shear stress field induced by a positive dislocation located at the origin @xcite , and @xmath21 is a constant externally applied resolved shear stress . it is useful to introduce natural coordinates at this point which will be denoted by an apostrophe ( @xmath22 ) in the following . measuring length in units of the average dislocation dislocation distance @xmath23 ( where @xmath1 denotes the total dislocation density of dislocations including both signs and , in multiple slip geometries , including all slip systems ) , stress @xmath24 in units of @xmath25 , and plastic strain @xmath26 in units of @xmath27 leads to the relations @xmath28 where @xmath29 is the elastic energy difference between two states of the system ( energy per unit dislocation length ) . in natural coordinates takes the form @xmath30 , \cr & \tau'_{\mathrm{ind}}(\bi{r } ' ) = \frac{x ' ( x'^{2}-y'^{2})}{(x'^{2}+y'^{2})^{2 } } = \frac{\cos(\varphi ) \cos(2\varphi)}{r ' } , } \ ] ] where @xmath31 denotes the angle between the @xmath8 axis and @xmath32 . to study dislocation relaxation , a large number of discrete dislocation dynamics simulations have been performed . equations of motion were solved with the 4.5th order runge kutta fehlberg method . periodic boundary conditions were applied to a square simulation area with edges parallel to the slip planes , following the method used in @xcite . to avoid overly small timesteps during the final stages of approach of narrow dipoles ( pairs of dislocations of opposite signs ) , a small number of extremely narrow dipoles were excluded from the solution of and forced to move as if they were isolated from the rest of the system . this is justified as the far - field stresses of the dislocations in a narrow dipole cancel , while their pair interaction diverges when the dislocation dislocation distance approaches zero . as a consequence , the dynamics of narrow dipoles is effectively uncoupled from the rest of the dislocation system . we do not allow for annihilation of narrow dipoles , which is a process that is governed by the atomistics of the dislocation cores . this implies that we consider dislocation spacings to be large in comparison with the dipole annihilation distance , which is believed to be of the order of one nanometer . as typical dislocation densities in highly dislocated crystals are of the order of @xmath33 m@xmath34 , i.e. the average dislocation spacings are of the order of a hundred nanometers , this is not a severe restriction . a first set of simulations was started from random configurations of equal numbers of positive and negative dislocations . the number of simulated dislocations @xmath6 ( which defines the system size @xmath35 since @xmath36 ) varied between @xmath37 and @xmath38 . it is well known that the flow stress of single - glide dislocation systems is around @xmath39 in natural units @xcite . to allow the dislocation systems to reach mechanical equilibrium at the end of the relaxation , we restricted the applied external stresses to levels below the flow stress , using stresses between @xmath40 and @xmath41 natural units . as seen in , individual simulations showed strong avalanche - like activity during relaxation , as previously observed in @xcite . to reveal scaling properties of the relaxation process , the evolution of global parameters such as stored energy , mean absolute dislocation velocity , mean square velocity , and mean strain rate was averaged over @xmath42 to @xmath43 simulations starting from different random initial configurations . this ensemble averaging resulted in smooth ensemble averaged graphs as seen in . in addition to the relaxation of ` neutral ' arrangements of dislocations moving on a single slip system aligned parallel to the edge of a square simulation area , we considered two variants of this basic setting ( these simulations were only performed at zero external stress ) : i ) to elucidate the influence of different implementations of the boundary conditions , we performed single slip simulations with the simulation box oriented at an angle @xmath31 to the slip planes . this does not affect the short - range dislocation dislocation interactions but modifies the stress field created by the periodic images . ii ) to study the influence of a net burgers vector on the relaxation process , we investigated the limiting case of fully polarised dislocation systems ( dislocations of one sign only ) of sizes @xmath44 . to investigate the differences between single and multiple slip geometries , we also performed simulations in which dislocations of multiple slip systems were present . in these simulations we considered sets of equally populated slip systems , each containing edge dislocations with a zero net burgers vector that were initially distributed at random . the methodology of the multiple slip simulations was identical to the single slip simulations described above , with the following differences : i ) for each dislocation in the system , the complete elastic stress tensor was computed , again assuming periodic boundary conditions in a square simulation area and using the method of @xcite . the forces acting on the dislocations were calculated from the stress tensor components using the peach koehler formula @xcite . ii ) in addition to narrow dipoles , which were treated in the same way as described above , pairs of attracting dislocations on intersecting slip planes needed to be treated separately . such dislocation pairs react with each other and form a reaction product ( ` dislocation lock ' ) with burgers vector equal to the net burgers vector of the constituent dislocations . in real crystals , both the mobility of dislocation locks and the stress required for separating them into the constituent dislocations depend on their atomic core structure . for simplicity , we assumed all dislocation locks to be immobile and to possess infinite separation stress . new dislocations joining an existing lock were assumed to annihilate with the constituent dislocation of the opposite burgers vector if such a dislocation was present . if a dislocation lock converted to an ordinary dislocation through such an annihilation event , it was assumed to become mobile again . to avoid overly small timesteps during the final stages of dislocation lock formation or reaction with a new dislocation ( again due to diverging elastic interactions between the constituent dislocations ) the moving constituent dislocations were pinned when their relative distance decreased below a small predefined reaction radius which mimics the core extension of the dislocation lock . for simplicity , the stress field of a lock was calculated as the superposition of the stress fields of the pinned constituent dislocations . although the positions of the constituent dislocations are scattered over a small region of the size of the reaction radius , their net stress field is a good approximation of the stress field of the lock at larger distances . finally , we note that this stress field is of long range character since the net burgers vector of a dislocation lock is not zero . in this section a simple scaling argument is used to establish the asymptotic kinetics of a bimolecular combination reaction . we then adapt the fundamental ideas behind this argument to the relaxation of dislocation systems as described in the previous section . based on the adapted model , predictions are made for the evolution of several physical quantities which can be directly obtained from the simulations described in . these predictions are compared in detail to the numerical results in . consider the direct combination reaction @xmath45 starting from a random , balanced configuration of the two reactants @xmath46 where @xmath47 denotes the spatial average of the concentration field @xmath48 . note that the initial concentrations of molecules @xmath49 and @xmath50 are equal only in an average sense because of the thermally induced random positions of the individual molecules . these concentration fluctuations need to be taken into account when modelling the reaction kinetics . for situations where long - distance transport of particles occurs by free brownian motion , a simple scaling argument which captures the essential physics was given by ovchinnikov @xcite ( which the reader is advised to consult for further details ) . broadly speaking , this involves the subsequent dominance of two consecutive mechanisms directly corresponding to the different length scales inherent in the system : i ) the reaction of those molecules that do not need long distance motion to find a reaction partner , followed by ii ) the long range brownian motion and subsequent reaction of excess reactants remaining as a result of initial concentration fluctuations . as discussed in @xcite , the first stage of reaction is controlled by a bimolecular reaction rate @xmath51 which depends on the short - range interactions between the reaction partners . this stage can be discussed in the classical approximation @xmath52 solving for initial conditions yields @xmath53 because of thermal concentration fluctuations , not all of the molecules @xmath49 and @xmath50 can be consumed during stage 1 . spatial fluctuations of the reactant concentrations lead to local excess of molecules of one type . as these excess molecules can not find a local partner , they need to migrate via long range brownian motion , leading to a second kinetic stage controlled by long - range diffusion . to characterise the concentration fluctuations in question we observe that , on scales larger than the range of the ` contact interactions ' which govern the first stage of the reaction kinetics , the positions of molecules are statistically independent . as a consequence , the initial numbers of molecules @xmath49 or @xmath50 in a sufficiently large volume @xmath54 are poisson distributed . this implies that the mean numbers of excess molecules fulfil the relations @xcite @xmath55 where @xmath56 denotes spatial averaging over a large statistically homogeneous system or , equivalently , ensemble averaging over a large number of statistically independent and equivalent realizations . to further discuss the reaction kinetics during stage 2 , we confine ourselves to the limiting case of an infinite bimolecular reaction rate @xmath57 . this choice does not affect the generality of the discussion , it only affects the moment of the crossover between stage 1 and stage 2 kinetics . we introduce the concentration difference @xmath58 as @xmath59 in a hypothetical initial state before the reaction has been ` switched on ' at @xmath60 , @xmath61 and @xmath62 are statistically independent and @xmath58 has the initial statistical properties @xcite @xmath63 to proceed , we note that for @xmath57 molecules @xmath49 and @xmath50 can not coexist for @xmath64 . therefore , @xmath58 gives a complete characterisation of the concentration map for @xmath65 : for @xmath66 , @xmath67 and @xmath68 and for @xmath69 , @xmath70 and @xmath71 . because there exists no other physical length scale in the system , the size of the regions characterised by @xmath72 and @xmath73 is determined by the diffusion length @xmath74 ( for simplicity , the same diffusion constant @xmath75 is assumed for both kinds of reactants ) . consider now the volume referring to the diffusion length at time @xmath76 , @xmath77 . one can suppose that for lengths larger than @xmath78 the fluctuations of @xmath58 are still not affected by diffusion ; therefore , one can write that @xmath79 in some regions , @xmath80 and in others , @xmath81 ; hence , on average @xmath82 meaning a slower kinetics than in stage 1 . ( for a more detailed derivation see @xcite ) . note that is only applicable within certain time limits @xmath83 $ ] . for instance , for @xmath57 , @xmath84 is equal to the time the diffusion length @xmath78 needs to exceed the average intermolecular distance . @xmath85 is determined by the time needed by @xmath78 to reach the system size @xmath86 , independent of the value of @xmath51 . now we consider the relaxation of initially random dislocation configurations with the same number of positive and negative dislocations ( @xmath87 ) following the equations of motion . although not immediately evident , this relaxation process has strong phenomenological similarities to the kinetics of the chemical reaction described in . to elucidate the analogy , the relaxation process will be envisaged as a gradual screening of the long range elastic stress fields of individual dislocations through the formation of dislocation dislocation correlations @xcite . we first envisage ` neutral ' arrangements where dislocations of both signs are present in equal amounts . in a screened dislocation arrangement , the excess of one sign over the other has been eliminated on scales above a few dislocation spacings . any dislocation arrangement where excess dislocations are completely eliminated can be envisaged as an assembly of dipoles where each dislocation has exactly one partner of opposite sign within a distance of the order of @xmath23 , and this picture will be used in the following argument . hence , we envisage the relaxation process as the gradual formation of a large number of dislocation dipoles consisting of dislocations of opposite signs , i.e. , as a bimolecular reaction process analogous to the chemical reaction . the principal difference between the two processes lies in the dynamics of individual particles : for the dislocation system , dislocation glide motion is driven by dislocation dislocation interactions which scale like @xmath88 , whereas in case of the chemical reaction we are dealing with diffusive brownian motion of the reactants . despite this difference , we may again construct a two - stage model for the dislocation relaxation process : i ) in stage 1 adjacent opposite sign dislocations form dipoles ; ii ) in stage 2 initial fluctuations in the excess dislocation density gradually die out from shorter towards longer length scales as excess dislocations which did not find a dipole partner in stage 1 undergo long range glide motion . the process terminates once the length scale on which fluctuations have been eliminated reaches the system size . to understand the time evolution of the system , we again consider the evolution of the typical length scale @xmath78 below which density fluctuations have already died out . to this end we consider that i ) similar to the chemical case , areas of size @xmath89 typically contain mobile excess dislocations ( which have still not ` reacted ' into dipoles ) with only one or the other sign and that ii ) dislocation dipoles give rise only to short range stress fields with a @xmath90 decay . as the typical distance between opposite sign dislocations which try to find each other is proportional to @xmath91 , the typical driving stress towards dipole formation scales as @xmath92 . ( recall that variables with an apostrophe ( @xmath22 ) are measured in natural units . ) the dislocation velocity is proportional to @xmath93 , and therefore the characteristic time for eliminating excess dislocations on scale @xmath94 scales as @xmath95 . hence we find that @xmath96 incidentally , this result is very similar to the evolution law of @xmath78 for brownian motion , @xmath97 . by supposing that the mobile dislocations inherit the initial concentration fluctuations we find that at time @xmath98 , regions of size @xmath99 contain about @xmath94 excess dislocations of one or the other sign , and @xmath100 dislocations in total ( see the chemical model in ) . thus , the fraction of non - paired dislocations is estimated to decrease in time as @xmath101 following the chemical model in , it is straightforward to predict the time interval @xmath102 $ ] during which the above argument is expected to hold . the start time @xmath103 is characterised by @xmath2 reaching the dislocation dislocation distance @xmath23 ( @xmath104 in natural units ) and the process is finished when @xmath2 reaches the system size @xmath86 . with this leads to @xmath105 in natural coordinates where @xmath6 denotes the total number of dislocations in the system . the fraction of ` non - paired ' dislocations is not a convenient quantity for comparing the scaling model with dislocation dynamics simulations , as the definition of ` dislocation pairs ' in a multipolar dislocation arrangement may be ambiguous . instead , we consider the evolution of the @xmath106th moment @xmath107 of the dislocation velocity and of the excess elastic energy , both of which can be determined from the simulations in a straightforward manner . to obtain scaling estimates for the relaxation of these quantities , we assume that all dislocation dipoles are at rest and only the excess dislocations move . furthermore , we assume that the motion of excess dislocations is not hindered by the dipoles already formed ( this can be rationalised with the short range of the dipole stress field ) . the velocity of excess dislocations scales as @xmath108 , leading to @xmath109 where and have been used . the dynamics of dislocations is assumed to be overdamped . hence , the work that is done by the internal stresses in driving the system is completely dissipated : the amount of dissipated energy exactly matches the reduction in elastic energy . the energy dissipated per unit time by a moving dislocation @xmath11 scales like @xmath110 ( the peach koehler force acting on the dislocation is proportional to the stress ) , and consequently the time evolution of @xmath111 can be expressed as @xmath112 since the motion is overdamped , the dislocation velocity is proportional to the acting stress . in natural units and for @xmath113 $ ] , the ensemble averaged elastic energy thus evolves like @xmath114 where was used to estimate the time evolution of the second velocity moment . the scaling argument in the previous section is based on the formation of dipoles consisting of edge dislocations of opposite signs . at first glance , such an argument seems to be completely inapplicable to systems where only dislocations of the same sign are present . dipole formation is clearly impossible in such systems . instead , a most conspicuous feature in the relaxation of single - sign edge dislocation systems is the formation of walls containing many dislocations of the same sign that are aligned in the direction perpendicular to the slip plane @xcite . accordingly , theoretical arguments have focused on parameters characterising the ` condensation ' of dislocations into walls @xcite . however , even though dislocation wall formation is a most conspicuous feature , wall formation alone can _ not _ produce a screened dislocation arrangement . the authors of @xcite evaluate the driving force for wall formation by assuming periodic spacing of dislocations along a wall and using classical results found e.g. in @xcite . if we follow this line of reasoning and note that the energy of a dislocation in a wall decreases with decreasing dislocation spacing , the minimum - energy structure for the system at hand would be a single system - spanning wall . however , for an initially random dislocation system the @xmath115 positions are independent random variables and it is not easy to see how a periodic arrangement could form in the absence of dislocation climb . we calculate the energy of a random wall in the appendix and show that forming such a wall does not produce any energy reduction with respect to the initial random 2d arrangement . how then can an arrangement of dislocations of the same sign be screened ? the answer was provided by wilkens @xcite who demonstrated that a screened dislocation arrangement can be constructed by eliminating dislocation density fluctuations above a certain scale . to this end , he proposed a construction where the crystal cross section is tiled into a grid of cells of size @xmath116 , and the same number of dislocations is randomly distributed within each cell . this construction , which eliminates all density fluctuations on scales above the cell size , leads to an arrangement where the screening radius coincides with the cell size , @xmath117 . taking the wilkens construction as a well - screened reference state offers a surprising outlook on the relaxation of initially random systems of same - sign dislocations . with respect to this reference state , the initial random arrangement contains density fluctuations on all scales which may be either positive ( @xmath118 , positive excess ) or negative ( @xmath119 , negative excess ) . to achieve screening , dislocations must migrate from regions of positive to regions of negative excess , and this process is governed by the long - range stress fields associated with the presence of ( positive or negative ) excess dislocations . in other words , the kinetics of the process follows from exactly the same scaling argument as used in the previous section : we are dealing with the stress - driven elimination of excess dislocation densities , with the only difference that the excess is now not of positive over negative dislocations , but of the local dislocation density over the average one . if the above argument is correct , the relaxation kinetics of same - sign dislocation systems should be characterised by a slow power - law stage which has the same characteristics as the relaxation of neutral dislocation systems as discussed in the previous sections . we demonstrate in the next section that this is indeed the case . the numerically determined evolution of the elastic energy is displayed in . the zero value of the energy was chosen to correspond to the final relaxed state of the system . as seen on the figure , the evolution of the elastic energy @xmath120 per dislocation can be fitted satisfactorily with the prediction in , @xmath121 . power - law relaxation occurs from times @xmath122 onwards , in good agreement with the model prediction @xmath123 in . the final equilibrium value of the elastic energy is not a priori known but was fitted to the data such as to achieve a maximum extension of the linear scaling regime . unfortunately this precludes determination of the second critical time @xmath124 from these results . the presence or absence of an external stress below the flow stress of the relaxed dislocation system seems to have negligible influence on the evolution of the elastic energy . note that it is possible to collapse the curves for different system sizes @xmath125 by normalising the graphs with @xmath126 , as done in . this observation is consistent with the fact that the elastic energy of the initial random dislocation system is of the order of @xmath127 while the energy of the final relaxed state is of the order of @xmath128 @xcite . for estimating the value of @xmath129 we used that the range of dislocation pair correlations in mechanical equilibrium is of the order of the mean dislocation dislocation distance @xmath23 @xcite . therefore , the elastic energy difference per dislocation between the initial and final states is of the order of @xmath130 despite this relation connecting the initial and final states of the system , the numerical finding that the @xmath131 curves for different system sizes can be collapsed on their entire course by normalising them with @xmath126 is not trivial , as the agreement extends also to the relaxation kinetics and characteristic crossover time @xmath132 . and external stress values @xmath133 . ] the numerically calculated evolution of the mean square velocity @xmath134 is displayed in for zero applied stress and different system sizes . due to the connection between the mean square velocity and the elastic energy of the system , it is not surprising that similar statements apply here as for the evolution of the elastic energy . as seen in the figure , the model prediction @xmath135 in fits the data well from @xmath122 . due to the fact that @xmath136 is proportional to the time derivative of the elastic energy , its graphs are much noisier than those obtained for the energy , preventing again the detection of the supposed upper critical time @xmath124 . as for the energy , size effects can be scaled out with a normalisation factor @xmath137 which is a direct consequence of and . a final analogy to the evolution of the elastic energy is that external stresses have only negligible influence on the evolution of the mean square velocity . for this reason , simulations with non - zero external stresses were omitted from . for different system sizes @xmath6 at zero external stress . ] in the evolution of the mean absolute velocity @xmath138 can be seen for different system sizes . again , a power law time dependence can be observed from @xmath122 although an exponent @xmath139 gives a better fit than the theoretically predicted @xmath140 expected according to equation . one may argue that slowly moving dislocation dipoles play a bigger role in this case , as their small velocities contribute more strongly to @xmath141 than to @xmath142 . therefore , the gradually increasing number of slowly moving dislocations might be responsible for the reduced relaxation exponent . what makes this figure very interesting is the possibility to estimate values of @xmath124 . it was found that @xmath143 gives a good approximation , in line with the model prediction @xmath144 in . it was also observed that normalisation with @xmath145 collapses the graphs referring to different system sizes @xmath125 in the region of small @xmath146 . this is consistent with the relations for the elastic energy and the mean square velocity . finally , as in case of the energy and the mean square velocity , the evolution of the mean absolute velocity is not changed by the presence of external stresses below the macroscopic flow stress . for different system sizes @xmath6 at zero external stress ] another numerically measurable quantity is the plastic strain rate , defined as @xmath147 in the following the evolution of @xmath148 is studied for applied shear stresses @xmath149 below the macroscopic flow stress for the present dislocation geometry . from the data of miguel and co - workers @xcite , this is estimated to be @xmath150 in natural units . as it was demonstrated in , external stresses in this range do not appreciably change the evolution of the elastic energy . we study the relaxation of the strain rate mainly in order to assess , by comparing with the work of miguel et al . , the relevance of different initial conditions on the creep behaviour of dislocation systems . shows the numerically determined evolution of the plastic strain rate @xmath148 for different system sizes and external stress values . as can be seen , the plastic strain rate scales roughly in proportion with the external stress . the relaxation does not follow any discernible power law but is roughly exponential . this is in marked contrast with the findings of miguel et al . @xcite who for well relaxed initial configurations demonstrate an andrade - type power - law decay , @xmath151 . the discrepancy points to the crucial importance of initial conditions for relaxation processes in dislocation systems a factor which is also borne out by the history dependence of creep relaxation processes that was demonstrated by miguel et al . @xcite . normalised with the external stress @xmath133 for different system sizes @xmath6 and external stress values @xmath133 . ] in this section we investigate the influence of different ways of implementing the periodic boundary conditions by tilting the angle between the edges of the simulation box and the trace of the slip planes . simulations with different tilt angles @xmath31 are physically equivalent except for the spatial arrangement of the periodic images of each dislocation ( see ) . this arrangement affects the dislocation dislocation interactions on scales comparable to the simulation box size . also , the interaction energy of each dislocation with its periodic images affects the initial elastic energy of the system , which is smallest for @xmath152 and has a maximum for @xmath153 . ( left ) and @xmath153 ( right ) . the latter configuration has a higher elastic energy as the nearest neighbours of each dislocation are in an energetically unfavourable configuration.,scaledwidth=80.0% ] the influence of simulation box orientation on the relaxation process is illustrated in . the absolute values of the squared velocity ( or equivalently the energy dissipation rate ) are higher for @xmath153 than for @xmath154 . however , both curves differ only by a constant factor ( the ratio of the initial excess energies ) , while the dynamics of the relaxation processes is otherwise identical . in systems with a single slip geometry with the simulation box oriented at different angles @xmath31 to the slip planes . ] for systems of dislocations of the same sign we have numerically evaluated the evolution of the mean square velocity @xmath155 ( or , equivalently , of the energy dissipation rate ) and of the mean absolute velocity @xmath138 . all calculations were performed at zero external stress since any applied stress would induce a sustained drift motion of the dislocation arrangement . results are shown in and together with the fit functions obtained for neutral dislocation arrangements ( see and ) . it is evident that the relaxation of same - sign dislocation systems follows the same scaling laws that have been observed for systems containing equal numbers of dislocations of both signs . this provides strong support for our basic conjecture that relaxation is governed by the stress - driven elimination of excess dislocations in a process that progresses from small to large scales . the processes occurring on short scales , on the other hand , are evidently different for the two systems ( dipole formation vs. formation of walls ) . this is reflected by the fact that the relaxation process in the single - sign dislocation systems shows an initial size dependence which is not present in neutral dislocation systems ( see and for comparison ) . for different system sizes @xmath6 for dislocations of the same sign . ] for different system sizes @xmath6 for dislocations of the same sign . ] compares the relaxation of dislocation systems in single , double and triple slip . as seen in the figure , at long times the relaxation in multiple slip geometries accelerates in comparison with the relaxation in single slip . this is consistent with the idea that in multiple slip geometries relaxation processes proceed through the formation of dislocation locks and the annihilation of mobile dislocations at these locks . the long range stress fields associated with dislocation locks and the removal of dislocations accelerate the relaxation as seen in the figure . one of the main assumptions behind our scaling model , namely that the motion of mobile dislocations is governed mainly by their mutual interaction , no longer holds in multiple slip geometries . therefore , the scaling model can not be applied to these situations . indeed , as seen in , no power law relaxation regime can be detected in the multiple slip simulations . in dislocation systems with single and multiple slip geometries . in the figure , @xmath1 means the total density of dislocations on all slip systems . the last two curves were shifted downwards to help comparison . ] the present paper discusses the relaxation of initially random arrangements of straight , parallel edge dislocations . following a phenomenological analogy with the kinetics of bimolecular reactions @xcite the relaxation process can be divided into three consecutive stages . stage 1 is characterised by rapid rearrangements of neighbouring dislocations , leading to the formation of dipoles , multipoles , and/or short wall segments . stage 2 hosts the gradual extinction of initial fluctuations in the burgers vector density on ever increasing length scales through the long range transport of excess dislocations . this stage gives rise to characteristic power - law relaxation dynamics . the relaxation process terminates in stage 3 when the characteristic fluctuation length reaches the system size . our considerations focus on the power - law relaxation dynamics in stage 2 . in section 3 , we formulated a scaling theory for this process by making the analogy with a bimolecular reaction . to this end , we considered a highly simplified picture where pair interactions between positive and negative excess dislocations lead to long - range dislocation transport resulting in the formation of dislocation dipoles . this led to predictions for the evolution of the elastic energy and the first two moments of the dislocation velocity . these predictions were then compared to ensemble averages of discrete dislocation dynamics simulations , and convincing agreement was found for single slip geometries . for multiple slip geometries , however , the persistent long range stress fields of dislocation locks accelerate the relaxation process , to which the scaling model can no longer be applied . the actual dislocation processes in many - dislocation systems are much more complex than the simplified picture underlying our scaling arguments . instead of long - range transport of excess dislocations and dipole formation , we see complex rearrangements resulting in dislocation dipoles , multipoles , and walls . in addition , it is well known that dislocations have a propensity to form large - scale heterogeneous patterns consisting of dislocation - rich and dislocation depleted regions . in the following we briefly discuss how these complex static and dynamic features fit into the idealised picture we used in the previous sections . we have developed our scaling argument for dipole formation which can be considered a bimolecular reaction between positive and negative dislocations . however , actual dislocation arrangements are much more complex . it is therefore important to emphasise that the core of the argument is the elimination of large - scale fluctuations in the excess burgers vector density , and _ not _ the resulting arrangement of nearby dislocations . for the long - time asymptotics of the relaxation process , which is governed by the elimination of fluctuations on larger and larger scales , the small - scale arrangement of dislocations is virtually irrelevant at least as long as the local features ( dipoles , multipoles , short walls and combinations of all these ) do not give rise to long - range stresses . according to the present argument , a well - screened dislocation arrangement is one in which burgers vector fluctuations have been eliminated on all scales . if we are dealing with a neutral dislocation system ( equal numbers of positive and negative dislocations ) this means that the net burgers vector is zero in each small volume , i.e. , for each dislocation we can find exactly one partner of opposite sign nearby . this motivates the dipolar picture even if the actual dislocation arrangements may be more complex . an alternative mechanism for creating well - screened dislocation arrangements is the formation of system - spanning walls of dislocations of the same sign . periodic arrangement of edge dislocations of one sign into a wall perpendicular to the glide plane removes the logarithmic divergence of the dislocation energy and introduces a screening length that is proportional to the dislocation spacing along the wall @xcite . walls are conspicuous both in simulations and in many experimentally observed dislocation microstructures . at first glance , wall formation mechanism seems to be completely at odds with the mechanism discussed in the present paper : formation of walls of same - sign dislocations increases , rather than reduces , the burgers vector density fluctuations . however , a closer investigation reveals that wall formation by itself is not a screening mechanism at all . forming a wall of randomly spaced dislocations does not reduce the energy in comparison with a random 2d dislocation arrangement ( see appendix ) . instead , the screening effect is contingent on the equal spacing of dislocations , i.e. on suppressing fluctuations of the burgers vector density along the wall direction . if we start from a random dislocation arrangement this is not easy to obtain : either the dislocations must have climb degrees of freedom ( which we do not consider in the present study ) , or dislocation motions that lead to the formation of multiple walls must be correlated over large distances such as to ensure that each wall collects only those dislocations that fit into an evenly spaced pattern . in the latter case , we are again dealing with the suppression of burgers vector density fluctuations on all scales above the wall spacing , and the long time asymptotics of this process is expected to obey our scaling theory . this is confirmed by the simulations , which however also demonstrate that the short - time behaviour is different for neutral dislocation systems where the local arrangement of dislocations is characterised by dipolar and multipolar patterns , and for single - sign dislocation systems where the local arrangement of dislocations is characterised by walls ( compare and ) . our scaling argument considers the stress - driven long - range transport of excess dislocations . the picture underlying the argument is schematically shown in ( top ) : a positive excess dislocation at a is attracted by a negative excess dislocation at b and the two recombine by long - range motion which is not affected by the stress field of the dislocations in between a and b. the figure also indicates that this idealisation may not be feasible when we are dealing with multipolar arrangements rather than isolated narrow dipoles : in that case the mutual interaction of the excess dislocation may be much weaker than their interaction with other dislocations ` on the way ' . as a consequence , recombination is much more likely to occur by a collective rearrangement as shown in ( bottom ) . how does this affect our scaling argument ? the total driving force for the process is the same in both cases . however , a collective rearrangement on scale @xmath2 is likely to involve @xmath156 dislocations [ @xmath157 in ( bottom ) ] . hence the driving force per dislocation is reduced by a factor of the order of @xmath6 . however , the same is true for the characteristic distance that has to be covered by each dislocation ( @xmath2 in the case of direct transport , @xmath158 in the case of collective rearrangement ) . as we assume that the dislocation velocity is proportional to the driving force , it follows that the characteristic time scale for eliminating the excess dislocation is the same in both cases , and our scaling argument remains valid . it is a well known phenomenon that dislocation microstructures forming during plastic deformation form heterogeneous patterns consisting of regions of high and low dislocation density , with characteristic lengths that are large in comparison with the dislocation spacing . on a conceptual level , possible mechanisms underlying this patterning were discussed by nabarro @xcite who pointed out that it may be energetically favourable to ` segregate ' the dislocation microstructure into areas of high and low density : if we are dealing with a well - screened dislocation arrangement , the energy density scales like @xmath159 . in this case it can be easily shown that it is energetically favourable to increase @xmath1 in some regions and decrease it proportionally in others . such ` energetically driven ' dislocation patterning could be a reason for the formation of dislocation - dense and dislocation - depleted regions that is observed in many experiments . while this mechanism is not covered by the present model , it is not at variance with our considerations : in a neutral dislocation arrangement , the formation of dislocation - dense and dislocation - depleted regions might occur without disturbing the burgers vector balance . we note , however , that in our simulations large - scale dislocation patterning is not observed either because it does indeed not occur in single slip , or because the dislocation numbers in our simulations might be too small . in conclusion we discuss the relevance of the processes discussed in the present paper for real - world systems . the relaxation of a random dislocation system has no direct counterpart in real deformation experiments , since it is impossible to ` prepare ' such a random system in the first place . our analysis of the screening of same - sign dislocation systems is , however , of general importance for understanding real dislocation patterns since it demonstrates that wall formation , though a conspicuous feature , can by itself not account for screening . this observation points to the importance of investigating long - range correlations between dislocation positions both within the walls and across different walls , and offers ample scope for future investigations . the investigated processes are of significant importance for discrete dislocation dynamics simulations of plasticity as the slow nature of the relaxation makes it difficult to obtain well - defined and energetically stable initial configurations . our comparison of strain - rate relaxation experiments with the results of miguel et al . demonstrates that the collective behaviour of dislocation systems may depend significantly on initial conditions . the analysis of this dependence is still in its infancy , yet understanding it is indispensable for carrying out dislocation plasticity simulations in a controlled and well - defined manner . financial support of the hungarian scientific research fund ( otka ) under contract no . k 67778 , of the european community s human potential programme under contract nos . mrtn - ct-2003 - 504634 [ sizedepen ] and nmp3-ct-2006 - 017105 [ digimat ] and of nest pathfinder programme trigs under contract nest-2005-path - com-043386 are gratefully acknowledged . we consider a wall of infinite height running along the plane @xmath160 in an isotropic material . edge dislocations of burgers vector @xmath161 are distributed randomly along the wall with average linear density @xmath162 . the geometry corresponds to a plane - strain situation , hence the elastic energy density can be written as @xmath163 . \label{edens}\ ] ] the total energy of the system is obtained by integrating over the system volume , @xmath164 where we have used that , for an infinite system , the second and third terms on the right - hand side do not contribute to the total energy . this can be shown as follows : for plane - strain deformation , the stresses can be written as derivatives of the airy stress function @xmath16 , @xmath165 , @xmath166 and @xmath167 . hence , @xmath168 { \rm d}^2 r = \int_v [ \partial_x\partial_y\chi \partial_x \partial_y \chi - \partial_x^2\chi \partial_y^2 \chi ] { \rm d}^2 r. \label{airy}\ ] ] partially integrating the second term in the integral on the right - hand side with respect to @xmath8 and @xmath115 shows that this integral contributes only surface terms to the total energy . these terms are negligible in the infinite - system limit . the ensemble - averaged stress at any point is given by summing over the stress fields of the individual dislocations in the wall and averaging over the different realizations of the random dislocation positions : @xmath169 where @xmath170 is the @xmath171 component of the stress created at @xmath172 by the @xmath106th dislocation . the elastic energy of the system depends on the averages of products @xmath173 where @xmath174 $ ] . in evaluating these averages we use that the @xmath115 coordinates of the individual dislocations are independent random variables : @xmath175 . \label{indep}\end{aligned}\ ] ] we now make the following observations : * the average stresses @xmath176 and their products @xmath177 depend on the @xmath8 coordinate only . * the average single - dislocation stresses @xmath178 and @xmath179 become zero in the limit @xmath180 , since these stresses are antisymmetric functions of the @xmath115 coordinate . the same is true for the average total stresses @xmath181 and @xmath182 . with these observations and using we can write the system energy as @xmath183 , \label{efinal}\end{aligned}\ ] ] where the second step follows by interchanging the averaging and the integration . @xmath2 is the system size ( tending to infinity ) which , in the absence of any other screening mechanism , delimits the divergence of the dislocation self - energy . for periodic boundary conditions , as used in our simulations , @xmath2 must be understood as the size of the periodic simulation box which in this case defines the screening length for an otherwise uncorrelated dislocation arrangement . @xmath184 is the total number of dislocations in the system . it follows from that the energy per dislocation is equal to the energy of a single unscreened dislocation and , hence , equals the energy in a completely random 2d arrangement . in other words , the arrangement of dislocations of the same sign in a random wall does ( with the possible exception of surface terms that are negligible in the infinite system limit ) not produce any reduction of the total energy . accordingly , the thermodynamic driving force towards forming such a wall is zero . zaiser m , _ statistical modelling of dislocation systems _ , 2001 _ mat . a * 309310 * 30415 groma i , csikor f f and zaiser m , _ spatial correlations and higher - order gradient terms in a continuum description of dislocation dynamics _ , 2003 _ acta mater . _ * 51 * 127181 zaiser m , _ scale invariance in plastic flow of crystalline solids _ , 2006 _ adv . * 55 * 185245 miguel m c , vespignani a , zaiser m and zapperi s , _ dislocation jamming and andrade creep _ , 2002 165501 miguel m c , moretti p , zaiser m and zapperi s , _ statistical dynamics of dislocations in simple models of plastic deformation : phase transitions and related phenomena _ , 2005 _ mat _ a * 400401 * 1918 miguel m c , laurson l and alava m , _ material yielding and irreversible deformation mediated by dislocation motion _ , 2008 _ eur . phys . b * 64 * 44350 wilkens m , _ das spannungsfeld einer anordnung von regellos verteilten versetzungen _ , 1967 _ _ * 15 * 14127 wilkens m , _ das mittlere spannungsquadrat @xmath185 begrenzt regellos verteilter versetzungen in einem zylinderfrmigen krper _ , 1969 _ 11559 zaiser m , miguel m - c and groma i , _ statistical dynamics of dislocation systems : the influence of dislocation dislocation correlations _ , 2001 b * 64 * 224102 zaiser m and seeger a , _ long - range internal stresses , dislocation patterning and work hardening in crystal plasticity _ , 2002 , _ dislocations in solids vol . 11 _ , ed f r n nabarro and m s duesbery ( elsevier ) csikor f f and groma i , _ probability distribution of internal stress in relaxed dislocation systems _ , 2004 b * 70 * 064106 krivoglaz m a 1969 _ theory of x - ray and thermal neutron scattering by real crystals _ ( new york : plenum ) csikor f f , kocsis b , bak b and groma i , _ numerical characterisation of the relaxation of dislocation systems _ , 2005 _ mat . eng . _ a * 400401 * 2147 csikor f f and zaiser m , _ scaling and glassy dynamics in the relaxation of dislocation systems _ , 2006 _ proc . conf . on statistical mechanics of plasticity and related instabilities ( 29 august2 september 2005 bangalore ) _ ed m zaiser _ et al _ ( proceedings of science ) 058 hirth j p and lothe j 1982 _ theory of dislocations _ ( new york : wiley - interscience ) bak b , groma i , gyrgyi g and zimnyi g , _ dislocation patterning : the role of climb in meso - scale simulations _ , 2006 _ comp . * 38 * 228 ovchinnikov a a and zeldovich ya b , _ role of density fluctuations in bimolecular reaction kinetics _ , 1978 _ chem . _ * 28 * 2158 groma i , gyrgyi g and kocsis b , _ debye screening of dislocations _ , 2006 165503 thomson r , koslowski m and lesar r , _ energetics and noise in dislocation patterning _ , 2006 b * 73 * 024104 nabarro f r n , _ complementary models of dislocation patterning _ , 2000 _ phil . mag . _ a * 80 * 75964

we study the relaxation dynamics of systems of straight , parallel crystal dislocations , starting from initially random and uncorrelated positions of the individual dislocations . a scaling model of the relaxation process is constructed by considering the gradual extinction of the initial density fluctuations present in the system . the model is validated by ensemble simulations of the discrete dynamics of dislocations . convincing agreement is found for systems of edge dislocations in single slip irrespective of the net burgers vector of the dislocation system . it is also demonstrated that the model does not work in multiple slip geometries . _ keywords _ : defects ( theory ) , fluctuations ( theory ) , plasticity ( theory )

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Proceed to summarize the following text: during the last two decades there has been much interest in using the electron spin in electronic devices . this research field , often referred to as spintronics , has already made great impact on metal - based information storage systems . there are hopes that a similar success can also be achieved in semiconductor based systems @xcite . manipulating the spins of the electrons via external magnetic fields over nanometer length scales is not considered feasible . another , more attractive , method is to use electric fields to manipulate electron spins via spin - orbit interaction . the spin - orbit interaction arises from the fact that an electron moving in an external electrical field experiences an effective magnetic field in its own reference frame , that in turn couples to its spin via the zeeman effect@xcite . in condensed matter systems , the spin - orbit interaction is found in crystals with asymmetry in the underlying structure @xcite . in bulk this is seen in crystals without an inversion center ( e.g zincblende structures ) and is termed the dresselhaus spin - orbit interaction@xcite . on the other hand the structural asymmetry of the confining potential in heterostructures gives rise to the so called rashba term@xcite . the rashba interaction has practical advantages in that it depends on the electronic environment of the heterostructure which can be modified in sample fabrication and in - situ by gate voltages@xcite . this results in the possibility of varying the spin - orbit interaction on the nanometer scale . interestingly , even structurally symmetric heterostructures can present spin - orbit interaction provided that coupling between subbands of distinct parities is allowed@xcite . the spin - orbit strength can be measured in a variety of different experimental setups@xcite : in a magnetoresistance measurement via shubnikov - de haas oscillations@xcite , weak ( anti- ) localization @xcite , or electron spin resonances in semiconducting nanostructures@xcite or quantum dots@xcite , or optically via spin relaxation @xcite , spin precession @xcite , spin - flip raman scattering @xcite , or radiation - induced magnetoresistance oscillations @xcite . in this paper we propose a method for extracting the strength of the rashba spin - orbit interaction via a charge conductance measurement . this method does not require the use of external magnetic fields or radiation sources . we consider a quantum wire modulated by an external periodic potential , fig . [ fig : schematicpic ] . essentially , the method relies on the fact that the rashba - induced shifts of the band gap positions in energy dramatically alter the charge conductance of the superlattice . via direct diagonalization , we determine the band structure of an infinite parabolically confined quantum wire . the energy bands clearly show band gaps that are renormalized by the rashba interaction . interestingly , the band gaps shift in energy as the strength of the rashba interaction is varied . the location of the band gaps are at the crossing of energy bands from adjacent brillouin zones . these energy bands can be calculated via an analytical approximation scheme@xcite . using non - equilibrium green s functions ( negf ) and the landauer formalism we calculate the charge conductance through a finite region containing both a periodic potential and the rashba interaction . in the conductance we find several dips appearing at different location in energy . moreover , these conductance dips coincide with renormalized band gaps of the superlattice . as with the band gaps the positions of the conductance dips are at the crossing points of the energy bands from next neighbor brillouin zones . for a wide range of rashba spin - orbit interaction strengths some of the conductance dips shift linearly in energy as function of the strength of the rashba coupling . for this range we derive a relation , eq . ( [ eq : linearealpha ] ) , describing the location of linearly shifting dips in energy as a function of rashba interaction strength . the rashba coupling can therefore be extracted by fitting the shift of conductance dips via eq . ( [ eq : linearealpha ] ) . figure 1 shows a schematic of the proposed experimental setup . this paper is organized as follows . in sec . [ sec : infwire ] we calculate the energy bands of the infinite wire superlattice via direct diagonalization . by using an analytical approximation scheme@xcite we then show how the resulting band gaps depend on the rashba spin - orbit interaction . via the negf method and the landauer formalism we introduce in sec . [ sec : finwire ] a numerical scheme to calculate the charge conductance through a finite region . this region is connected to electron reservoirs to the left and right , and contains both a periodic potential and a rashba spin - orbit coupling . in sec . [ sec : result ] we then show that the band picture of the infinite wire superlattice , developed in sec . [ sec : infwire ] , is applicable to the finite length periodic potential . we close sec . [ sec : result ] with a discussion about possible experimental procedures . lastly , in sec . [ sec : pervar ] we show that our results are robust against fluctuations in the strength and width of the periodic potential . we investigate an infinite quasi-1d parabolic wire with a uniform rashba spin - orbit coupling in the presence of a longitudinal modulation described by the potential @xmath0 where @xmath1 is the period of the superlattice . the hamiltonian that describes this system is @xmath2 were @xmath3 is the effective mass , @xmath4 and @xmath5 are the momentum operators in the longitudinal and transverse direction of the wire , @xmath6 is the rashba spin - orbit strength , and @xmath7 is the confinement frequency of the parabolic potential . to find the eigenvalues of the hamiltonian in eq . ( [ eq : originalh ] ) it is convenient to introduce the standard ladder operator @xmath8 of the parabolic confinement and rotate the spin operators so that the @xmath4 part in the rashba interaction term couples to the @xmath9 operator@xcite @xmath10 + v{_{\mbox{\scriptsize p}}}(x)\nonumber\\ & = \widetilde{h}_0+\widetilde{h}_1 , \label{eq : rotatedh}\end{aligned}\ ] ] here @xmath11 is the spin ladder operator , @xmath12 is the rescaled rashba strength and @xmath13 are the eigenvalues of the operator @xmath4 . in eq . ( [ eq : rotatedh ] ) we scale all lengths in oscillator length @xmath14 and all energies in @xmath15 . we separate the hamiltonian @xmath16 into a diagonal part , @xmath17 and a non - diagonal part @xmath18 + \sum_{n\neq0 } c_n e^{in\frac{2\pi}{\lambda}}.\ ] ] the eigenstates of the @xmath19 hamiltonian are represented by the kets @xmath20 . here @xmath21 is the quantum number of the harmonic transverse energy bands , i.e. the eigenvalue of the @xmath22 operator , and @xmath23 is the eigenvalue of the @xmath9 operator with @xmath24 and @xmath25 denoting the spin up and spin down states , respectively . the corresponding eigenenergies are @xmath26 a plot of the @xmath27 energy bands vs @xmath28 in half of the brillouin zone , i.e. , @xmath29 , can be seen in fig . [ fig : h0enbands ] ( a ) . for @xmath30 nm , @xmath31 nm@xmath32 , @xmath33 mevnm ( @xmath34 ) , @xmath35 , @xmath36 , and @xmath37 . ( b ) crossing of the energy bands corresponding to the @xmath38 , @xmath39 , and @xmath40 states . ( c ) crossing of the energy bands corresponding to the @xmath41 , @xmath42 , @xmath43 , and @xmath44 states . ( d ) same as ( c ) but for higher bands , i.e. @xmath45.,scaledwidth=49.0% ] the hamiltonian @xmath46 introduces couplings _ i ) _ between the @xmath20 and the @xmath47 states , due to the rashba interaction , and _ ii ) _ between the @xmath20 and the @xmath48 states , due to the periodic potential . note that the coupling is strongest where the energy bands corresponding to these states cross each other . for some particular rashba strength @xmath49 the energy bands associated with the states @xmath38 and @xmath39 of @xmath19 cross at @xmath50 , where @xmath51 . if we choose the period @xmath1 of the superlattice potential , eq . ( [ eq : periodiceq ] ) , as @xmath52 the @xmath40 energy band will also cross at @xmath53 . this crossing point occurs in energy at @xmath54 see fig . [ fig : h0enbands ] ( b ) . in the following we refer to this particular choice of parameter @xmath49 as the reference spin - orbit coupling strength . similar crossing also occur at @xmath53 with the energy bands associated with the states @xmath55 , @xmath56 , and @xmath57 for energies @xmath58 with the energy band corresponding to the state @xmath59 crossing close by . the crossings for @xmath60 and @xmath61 can be seen , respectively , in fig . [ fig : h0enbands ] ( c ) and ( d ) . we calculate the eigenenergies of the full hamiltonian , @xmath16 , via direct diagonalization . the resulting energy bands are plotted in fig . [ fig : hfullenbands ] ( a ) . the energy bands that cross at @xmath53 and the one crossing close by , see sec . [ sec : h0 ] , corresponds to the states coupled by @xmath46 . a blow up of the resulting energy gaps can be seen in fig . [ fig : hfullenbands ] ( b ) . at @xmath62 the coupling results in a double energy gap , see fig . [ fig : hfullenbands ] ( c ) , and a triple energy gap at the higher energy crossings , see fig . [ fig : hfullenbands ] ( d ) and ( e ) . in fig . [ fig : hfullenbands ] the spin - orbit strength is at the reference value @xmath49 . note that for a non - zero rashba coupling , as in fig . [ fig : hfullenbands ] , the energy gaps have shifted from the bragg plane at @xmath63 . this is because the spin - orbit interaction shifts the wave - number @xmath28 of the electrons in the longitudinal direction and thus renormalizes the locations of interferences . for the same parameters as used in fig . [ fig : h0enbands ] . the energy bands are calculated via direct diagonalization of @xmath16 . ( b ) a blow up focusing on the band gaps of the first three crossings at @xmath53 . ( c ) the band gaps formed at the energy band crossing shown in fig . [ fig : h0enbands ] ( b ) . in ( d ) and ( e ) , the energy band crossings shown in fig . [ fig : h0enbands ] ( c ) and ( d ) , respectively.,scaledwidth=47.0% ] when the strength of the rashba interaction is changed the crossing points @xmath64 of the energy - bands shift and with them the energy gaps . these crossing points can be worked out analytically by using the eigenenergies of the effective rashba hamiltonian of the parabolic wire@xcite , which are @xmath65 and @xmath66 for @xmath67 . for @xmath68 @xmath69 . in eq . ( [ eq : effenup ] ) and ( [ eq : effendown ] ) @xmath70^{1/2},\end{aligned}\ ] ] and we have added the energy constant @xmath71 resulting from the periodic potential and replaced @xmath28 with @xmath72 . note that the energy bands described by eq . ( [ eq : effenup ] ) and eq . ( [ eq : effendown ] ) are derived for wires without a longitudinal periodic potential and therefore do not contain energy gaps that result from the periodic potential , i.e. they are the solution of the @xmath16 , see eq . ( [ eq : rotatedh ] ) , with @xmath73 and @xmath74 for @xmath75 . having determined the crossing points , we insert them into either of the crossing energy bands , eq . ( [ eq : effenup ] ) or eq . ( [ eq : effendown ] ) , to obtain the location of the crossing points in energy as a function of @xmath6 . in the appendix we present the equation for the crossing point between the energy bands @xmath76 and @xmath77 corresponding to states @xmath56 and @xmath57 , see eq . ( [ eq : kcross ] ) . to determine the location in energy of this crossing point as a function of @xmath6 we insert @xmath64 from eq . ( [ eq : kcross ] ) into either @xmath76 or @xmath77 . . we also mark the values @xmath78 mevnm and @xmath79 mevnm onto the @xmath80-axis . these values are used for the conductance results in fig . [ fig : gapvsg].,scaledwidth=45.0% ] now , as the crossing points shift in energy with @xmath6 they can be thought of as trajectories . the trajectories of the crossing points , calculated from eq . ( [ eq : effenup ] ) and ( [ eq : effendown ] ) , can be seen in fig . [ fig : trajectories ] . for higher energies there are further trajectories . we index the trajectories via @xmath21 and the letters `` c '' , `` l '' , and `` r '' . the trajectories that we label with the letter c only shift a little to the left for @xmath81 ( see lowest horizontal line in fig . [ fig : trajectories ] ) . relative to the c labeled trajectories and again for @xmath81 , we see trajectories that make a large shift to the left and right . those trajectories we , respectively , label with the letters `` l '' and `` r '' . for spin - orbit strengths close to the reference strength , @xmath49 , the crossing points follow nonlinear trajectories . as the spin - orbit coupling becomes larger or lower than the reference strength the trajectories quickly become more linear . the choice of the reference rashba spin - orbit strength @xmath49 determines the period of the periodic potential , see eq . ( [ eq : period ] ) . we will show in sec . [ sec : finwire ] that the band gaps appear as dips in the charge conductance through a finite periodic potential . by fitting the measured energy shift of the conductance dips to the trajectories in fig . [ fig : trajectories ] it is possible to extract the value of @xmath6 . the linear parts of the trajectories are best suited for fitting . it would therefore be convenient that the range of @xmath6 extracted via the fitting is contained within the linear region . this can be achieved by choosing a sufficiently low @xmath49-value ( and thus @xmath1 , eq . [ eq : period ] ) . we present below a linearized equation for the @xmath6-value of the crossing point between the energy bands corresponding to the states @xmath56 and @xmath57 as a function of fermi energy . by taylor expanding around some point @xmath82 in the linear region of the trajectories and making a linear approximation we obtain @xmath83 , \label{eq : linearealpha}\ ] ] here @xmath84 and @xmath85 are known functions of the energy band index @xmath21 , see eq . ( [ eq : f ] ) and eq . ( [ eq : g ] ) . the derivation of eq . ( [ eq : linearealpha ] ) is shown in the appendix . in fig . [ fig : trajectories ] we plot as dashed lines linearized trajectories described by eq . ( [ eq : linearealpha ] ) where we have chosen @xmath86 mevnm . in this section we calculate in the linear response the conductance through a finite periodic potential . this is done via the landauer formula together with the negf method . we consider a hardwalled wire of width @xmath87 with a transverse parabolic potential . we divide the wire into a finite central region of length @xmath88 and semi - infinite left and right parts , see fig . [ fig : numschema ] . the central region includes a rashba spin - orbit interaction described by a symmetrized hamiltonian which is turned on smoothly , at both the left and right ends . in the central region we also assume a longitudinal periodic potential that represents the potential due to the fingergates . we describe the central region by the hamiltonian @xmath89 here @xmath90 denotes an anticommutator . the left and right leads contain the same parabolic potential as in the center region but neither a rashba spin - orbit interaction nor a periodic potential in the longitudinal direction , i.e. , they are described by the hamiltonian @xmath91 the total system , @xmath92 , is discretized via the finite difference method on a grid of @xmath93 points with a mesh size @xmath94 . a schematic of the system can be seen in fig . [ fig : numschema ] . , described by the hamiltonian @xmath95 and two semi - infinite wires , described by the hamiltonians @xmath96 and @xmath97 . the total system is discretized on a grid with mesh size @xmath94 . at the left and right edges of the central region the rashba spin - orbit coupling is smoothly turned on.,scaledwidth=47.0% ] here we use the negf method@xcite to calculate the charge conductance . the method requires us to find the retarded green s function of the central region @xmath98 . to do this we have to isolate @xmath98 from the infinite matrix equation describing the retarded green s function of the total system @xmath99 by separating the total green s function into a left , right , and central part we can write eq . ( [ eq : infinitegreen ] ) as @xmath100 \left[\begin{array}{lll } \mathbf{g}{_{\mbox{\scriptsize l}}}{^{\mbox{\scriptsize r } } } & \mathbf{g}{_{\mbox{\scriptsize lc}}}{^{\mbox{\scriptsize r } } } & \mathbf{g}{_{\mbox{\scriptsize lr}}}{^{\mbox{\scriptsize r } } } \\ \mathbf{g}{_{\mbox{\scriptsize cl}}}{^{\mbox{\scriptsize r } } } & \mathbf{g}{_{\mbox{\scriptsize c}}}{^{\mbox{\scriptsize r } } } & \mathbf{g}{_{\mbox{\scriptsize cr}}}{^{\mbox{\scriptsize r } } } \\ \mathbf{g}{_{\mbox{\scriptsize rl}}}{^{\mbox{\scriptsize r } } } & \mathbf{g}{_{\mbox{\scriptsize rc}}}{^{\mbox{\scriptsize r } } } & \mathbf{g}{_{\mbox{\scriptsize r}}}{^{\mbox{\scriptsize r } } } \\ \end{array}\right]\nonumber\\ = & \left[\begin{array}{ccc } \mathbf{i } & 0 & 0 \\ 0 & \mathbf{i } & 0 \\ 0 & 0 & \mathbf{i } \\ \end{array}\right ] . \label{eq : hgmatrixeq}\end{aligned}\ ] ] the matrices @xmath101 , couple together the central region to the left ( @xmath102 l ) and right leads ( @xmath102 r ) . here @xmath103 is the tight - binding hopping parameter that results from the discretization . multiplying out eq . ( [ eq : hgmatrixeq ] ) gives nine matrix equations from which we can isolate a _ finite _ matrix equation for @xmath98 . this is done by treating the contributions from the infinite leads as self - energies@xcite . the matrix equation for @xmath98 is @xmath104 and the self - energy of lead @xmath102l , r is @xmath105 where @xmath106 is the green s function of lead @xmath107 and is determined analytically@xcite . here @xmath108 is the discretized hamiltonian of eq . ( [ eq : originalh ] ) over the central region . all the matrices are @xmath109 matrices . in order to save computational power only the necessary green s function matrix elements are calculated with the recursive green s function method@xcite . here we are interested in low biases and hence focus on the linear response regime . from the green s function the charge conductance at the fermi energy @xmath110 can be calculated , via the fisher - lee relation@xcite , as @xmath111 , \label{eq : conductance}\ ] ] where @xmath112 and @xmath113 . in what follows we use eq . ( [ eq : conductance ] ) to calculate the charge conductance through our system . in our simulations we consider a ga@xmath114in@xmath115as alloy with an effective mass @xmath116 , with @xmath117 being the bare electron mass . the width of the wire is set as @xmath118 nm and its length as @xmath119 m . the oscillator length is @xmath30 nm which corresponds to @xmath120 mev . for the discretization we use @xmath121 points with @xmath122 nm being the distance between nearest neighbors . this corresponds to a tight - binding hopping parameter @xmath123 mev@xmath124 . we choose the reference rashba strength as @xmath78 mev nm ( i.e. @xmath125 ) which results in an energy band crossing point at @xmath126 and a period @xmath127 nm . this corresponds to @xmath12880 fingergates . the strength of the periodic potential is set as @xmath129 mev . in fig . [ fig : gapvsg ] we compare the calculated conductance , fig . [ fig : gapvsg ] ( a ) , through the finite wire to the energy bands of the infinite wire , fig . [ fig : gapvsg ] ( b ) and ( c ) , as a function of fermi energy . results for two values of @xmath6 are presented , @xmath78 mevnm and @xmath79 mevnm . for comparison we also plot in fig . [ fig : gapvsg ] ( a ) the conductance through a non - periodic potential @xmath130 for the same values @xmath6 mentioned above , i.e. the corresponding gapless systems@xcite . each energy gap is associated to its corresponding conductance dip via a labeled arrow . the labeling on the arrows is the same labeling as those of the trajectories in fig . [ fig : trajectories ] . we see that there is a good correspondence between the dips in conductance and the gaps in the energy band for both @xmath49 and @xmath131 . this indicates that the behavior of the _ finite _ periodic potential can be properly described with the band model from the infinite periodic potential . mevnm , and @xmath78 mevnm . note that the conductance curve for @xmath131 has been shifted by 2@xmath132 for clarity . the results for a wire with a non - periodic potential @xmath130 is plotted with black solid lines . in ( b ) and ( c ) the energy bands of the infinite wire superlattice are plotted for the same @xmath6 values as in ( a ) , respectively . note that ( c ) is the same figure as fig . [ fig : hfullenbands ] ( b ) , but rotated clockwise by @xmath133 . the correspondence between the nine pairs of dips and band gaps are indicated by the labeled arrows . , scaledwidth=45.0% ] in fig . [ fig : trajectories ] we can see how the band gaps shift in energy as a function of @xmath6 . we want see if the energy shift of conductance dips correlates with the shift of the band gaps . this is most easily seen by considering the differential conductance @xmath134 in fig . [ fig : standardsurface ] we plot @xmath135 as a function of @xmath136 and @xmath6 . there we see the conductance steps of the parabolic wire as relatively straight vertical trajectories appearing at the tic marks on the horizontal axis . the rest of the trajectories in fig . [ fig : standardsurface ] correspond to dips in the charge conductance . by comparing the trajectories , that the dips in charge conductance form , to the ones of the band gaps in fig . [ fig : trajectories ] we see an excellent match . this firmly confirms that the conductance dips of the finite wire can be described via the band model for the superlattice . we also plot in fig . [ fig : standardsurface ] as light - gray lines the linearized trajectories of crossing points between the energy bands corresponding to the @xmath56 and @xmath57 states , see eq . ( [ eq : linearealpha ] ) . the trajectories are linearized around @xmath6=20 mevnm and are identical to the straight dashed lines in fig . [ fig : trajectories ] . through the finite superlattice as a function of the fermi energy @xmath110 and spin - orbit coupling , @xmath6 . the linearized trajectories of the crosspoints between the energy bands corresponding to the @xmath56 and @xmath59 states are shown with gray solid lines . the trajectories are linearized around @xmath86 mevnm . the rashba spin - orbit values , @xmath78 mevnm and @xmath79 mevnm , corresponding to the curves in fig . [ fig : gapvsg ] are marked on the right border . note that @xmath137 and that the differential conductance @xmath138 is in arbitrary units.,scaledwidth=47.0% ] from fig . [ fig : standardsurface ] we can extract the rashba spin - orbit coupling by fitting the linear energy shift of the conductance dips , via eq . ( [ eq : linearealpha ] ) . the energy shift , i.e. the change in fermi energy , is controlled by the chemical potential of the leads . the backgate that controls the spin - orbit strength introduces an extra shift in the fermi energy ( due to the electrostatic coupling of the backgate to the 2deg ) . this shift can be compensated by changing the chemical potential of the leads by an equal amount . similar methods have been used to probe the energy spectrum of quantum dots using transport methods @xcite . another way to compensate for this shift is to use a combination of back and front gates . this has been experimentally demonstrated to control the rashba spin - orbit interaction strength without introducing charging in the 2deg@xcite . the periodic potential plays a crucial role in in the formation and behavior of the energy gaps . we therefore devote this section to study the effect that variations of parameters in the periodic potential has on the conductance . in what follows we consider variations on the length and strength of the periodic potential ( sec . [ sec : varlenstrength ] ) as well as random fluctuations on the period and strength of each finger gate potential ( sec . [ sec : imprecision ] ) . through the finite periodic potential as a function of fermi energy @xmath110 for ( a ) different values of @xmath139 and ( b ) different number of fingergates . all the conductances were calculated with @xmath140 mevnm . note that the conductances have been separated by @xmath141.,scaledwidth=47.0% ] here we examine the effects of changing the strength of the periodic potential , @xmath139 , and the length @xmath88 . in fig . [ fig : diff ] ( a ) the length of the wire is varied such that it contains 10 , 40 , 80 , or 160 fingergates with a fixed period @xmath142 nm . there we see the dips easily for @xmath143 40 fingergates and they become clearer for more fingergates . there is not much change between the conductance results for 80 and 160 fingergates , i.e. @xmath144 80 fingergates is an adequate number of fingergates . [ fig : diff ] ( b ) shows results where the strength of the periodic potential has been varied between values of 0.2 , 0.5 , 1.0 , and 1.5 @xmath15 . we see that the width of the conductance dips grows with larger @xmath139 . but as @xmath139 increases the total conductance strength @xmath145 deteriorates due to interferences . this makes the detection of the conductance dips difficult . through the finite periodic potential as a function of fermi energy @xmath110 and spin - orbit coupling , @xmath6 . four cases are considered : a system with ( a ) 320 , ( b ) 80 , ( c ) 40 , and ( d ) 10 fingergates . in all the plots we introduce a 5% gaussian error in the length and height of the potential in each period . , scaledwidth=49.0% ] the periodic potential used in the previous section can be considered as being ideal . a more realistic case would be if there were some fluctuations in the potential . to verify the robustness of the results in the previous section we rerun the simulations with a non - ideal periodic potential . this is done by introducing a @xmath146 gaussian error in both the height and length of each potential hill created by the fingergates . figure [ fig : gausserror ] ( b ) shows that extra noise is added to our conductance result ; still the trajectories of the conductance dips are fairly visible . to counter against the noise introduced by fluctuations we could add extra periods to the periodic potential . this helps averaging out the noise , as can be seen in fig . [ fig : gausserror ] ( a ) where we have quadrupled the length of the wire and the number of fingergates . this ( self)averaging is however slow and requires a large number of fingergates . another solution , as we are mainly interested in the slope of the crosspoint trajectories , is actually to reduce the number of fingergates to an optimal number . as can be seen in fig . [ fig : gausserror ] ( c ) we obtain many extra trajectories resulting from the imprecision of the period . but the slope of the trajectories of the conductance dips can be easily fitted for @xmath147 . as expected if we go down to as few as 10 fingergates the effects of the periodic potential is nearly washed out . we have studied a parabolic quantum wire with rashba spin - orbit interaction and a longitudinal periodic potential . we find that the energy gaps resulting from the periodic potential split up and shift from the bragg plane due to the rashba spin - orbit interaction . above a certain spin - orbit strength some of the new energy gaps shift linearly in energy as a function of the rashba spin - orbit strength . we propose that this effect can be used to measure the change in the rashba spin - orbit strength , e. g. resulting from a voltage gate@xcite . the energy gaps result in dips in the charge conductance . the energy shift of these dips can be fitted , by an analytical equation that we derive , eq . ( [ eq : linearealpha ] ) , to extract the strength of the rashba spin - orbit interaction . the advantage of the this method is that it only requires conductance measurement and not any external magnetic or radiation source . in this appendix we calculate the crossing point @xmath64 of the @xmath55 and @xmath59 energy bands . we need to solve the equation @xmath148 for @xmath28 . here we used eq . ( [ eq : effenup ] ) and ( [ eq : effendown ] ) and have defined @xmath149 equation [ eq : encrosseq ] is a nonlinear equation which is hard to solve directly . an easier way is to linearize the @xmath150 and @xmath151 functions around the point @xmath152 . this approximation introduces little errors and allows us to write the crossing point as @xmath153 where @xmath154 and @xmath155 the trajectories of the crossing points across the energy - rashba spin - orbit coupling surface are then @xmath156 or @xmath157 . the crossing points and their trajectories for the other energy bands , namely the ones corresponding to the states @xmath56 and @xmath57 , the @xmath55 and @xmath57 , and the @xmath56 and @xmath59 , can be worked out in the same way . the @xmath28 value of the crossing point between the @xmath76 and @xmath77 bands remains almost constant as a function of @xmath6 . this is because these bands move at similar rate in opposite directions in @xmath28 space as @xmath6 is changed . we take an advantage of this when finding a linear equation around some point @xmath82 in the straight segments of the trajectories . we know that @xmath53 is a crossing point for @xmath158 so we can , instead of using eq . ( [ eq : kcross ] ) , make the approximation that @xmath159 for all @xmath6 . we then linearize @xmath160 by taylor expanding around @xmath82 . solving for @xmath161 in @xmath160 and scaling back to @xmath6 we obtain @xmath162 , \label{eq : applinearealpha}\ ] ] where @xmath163 and @xmath164 in the last steps of the eq . ( [ eq : f ] ) and eq . ( [ eq : g ] ) we have plugged in the values @xmath165 and @xmath166 , which correspond to @xmath78 mevnm and @xmath86 mevnm . note that all lengths are scaled in the oscillator length @xmath14 and all energies in @xmath15 .

in this work we study the effects of a longitudinal periodic potential on a parabolic quantum wire defined in a two - dimensional electron gas with rashba spin - orbit interaction . for an infinite wire superlattice we find , by direct diagonalization , that the energy gaps are shifted away from the usual bragg planes due to the rashba spin - orbit interaction . interestingly , our results show that the location of the band gaps in energy can be controlled via the strength of the rashba spin - orbit interaction . we have also calculated the charge conductance through a periodic potential of a finite length via the non - equilibrium green s function method combined with the landauer formalism . we find dips in the conductance that correspond well to the energy gaps of the infinite wire superlattice . from the infinite wire energy dispersion , we derive an equation relating the location of the conductance dips as a function of the ( gate controllable ) fermi energy to the rashba spin - orbit coupling strength . we propose that the strength of the rashba spin - orbit interaction can be extracted via a charge conductance measurement .

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Proceed to summarize the following text: the serpens cloud is a low mass star - forming cloud in the gould belt . the cloud is known for its high star formation rate ( sfr ) and high surface density of young stellar objects ( ysos ) ( eiroa et al . a 10 deg@xmath3 optical extinction ( a@xmath6 ) map made by cambresy ( 1999 ) originally defined the serpens cloud . more recent studies treat the serpens cloud as two much smaller ( @xmath7 1.0 deg@xmath8 ) regions : serpens main ( centered on r.a . @xmath9 , dec . @xmath10 ( j2000 ) ) and serpens south ( centered on r.a . @xmath11 , dec . @xmath12 ( j2000 ) ) ( enoch et al . 2007 ; harvey et al . 2007a ; gutermuth et al . 2008 ; eiroa et al . 2008 ; bontemps et al . 2010 ) . serpens main is known mainly for its northernmost region , the serpens core , which has the highest yso surface density in the cambresy ( 1999 ) a@xmath6 map ( eiroa et al . serpens south is part of the aquila rift molecular complex . it was first studied in detail by gutermuth et al . ( 2008 ) , and has now been mapped at 70 - 500 @xmath13 m as part of the _ herschel _ gould belt survey ( andre et al . 2010 ; bontemps et al . serpens main is the focus of this study ( see fig . [ fig : mapped_reg ] ) . the total molecular mass of the serpens core is uncertain by at least a factor of 5 . some studies estimate @xmath7250 - 300 m@xmath14 ( mcmullin et al . 2000 ; olmi & testi 2002 ) while others find @xmath71450 m@xmath14 ( white et al . these two results were based on c@xmath15o j=1 - 0 and c@xmath15o j=@xmath2 lines , respectively , so the large discrepancy may be due to the different gas properties traced by each c@xmath15o rotational transition ( eiroa et al . the distance to serpens assumed by these studies may also be too low ( see section 1.2 ) . clearly , the gas mass must be measured with better accuracy to determine the efficiency and history of star formation in the serpens cloud . the best estimate for the distance to serpens main is @xmath16 pc , measured from vlbi trigonometric parallax of the yso , ec 95 ( dzib et al . ec 95 is located at the center of the serpens core and is therefore almost certainly a member , so we adopt the dzib et al . ( 2010 ) 415 pc distance . this is almost twice the previously accepted value of @xmath17 pc ( eiroa et al . 2008 ) , so care must be used in comparing physical properties derived using the lower distance with our results in this paper . we mapped about 1.04 deg@xmath8 of serpens main in the co and @xmath1co j = @xmath2 emission lines . our study complements existing survey data , from the _ spitzer _ c2d legacy ( evans et al . 2003 ; harvey et al . 2006 ; harvey et al . 2007 ) and bolocam 1.1 mm continuum ( enoch et al . 2007 ) surveys . other molecular tracer data exist for serpens main , such as n@xmath18h@xmath19 ( testi et al . 2000 ) , but these data are almost always limited to a sub - region of serpens main ( e.g. , the serpens core ) [ fig : mapped_reg ] shows our mapped region ( red polygon ) , the _ spitzer _ c2d regions with mips ( gray polygon ) and irac ( thin - black polygon ) , and the bolocam 1.1 mm region ( thick black polygon ) overlaid on the cambresy ( 1999 ) @xmath20 map of serpens main . the cambresy ( 1999 ) map suggests @xmath20 @xmath21 10 mag in serpens main , but the more recent @xmath20 map derived from c2d _ spitzer _ data ( enoch et al . 2007 ) shows @xmath20 @xmath22 25 mag . continuum 1.1 mm emission reveals the locations of the coldest and densest dust . n@xmath18h@xmath19 traces the highest density star forming gas ( testi et al . 2000 ) . from the _ spitzer _ c2d data , harvey et al . ( 2007 ) presented a high - confidence set " of 235 ysos associated with serpens main . most ysos have masses @xmath23 m@xmath14 ( eiroa et al . 2008 ) , but there is at least one ( vv serpentis ) with mass @xmath24 m@xmath14 ( ripepi et al . 2007 ; dzib et al . enoch et al . ( 2007 ) identified 35 sub - mm sources in serpens main from the bolocam 1.1 mm survey . we will compare the positions of these ysos and sub - mm sources with our co and @xmath1co data ( see section 5 ) . we divide serpens main into three sub - regions with the following naming scheme : the _ serpens core _ ( eiroa et al . 2008 ) , _ serpens g3-g6 _ ( cohen & kuhi 1979 ) , and _ vv serpentis _ ( chavarria et al . 1988 ) . these sub - regions are labeled _ 1 , 2 , _ and _ 3 _ , respectively , in fig . [ fig : mapped_reg ] . other names previously used for the serpens core include serpens dark , serpens north , and cluster a ( harvey et al . 2007 ; eiroa et al . 2008 ; bontemps et al . 2010 ) ; cluster b for serpens g3-g6 ( harvey et al . 2007 ; enoch et al . 2007 ) ; and cluster c for vv serpentis ( harvey et al . 2007 ; eiroa et al . 2008 ) . this study is a continuation of a molecular cloud mapping project with the arizona radio observatory . previous papers in this series mapped the w51 region in co j = @xmath2 and @xmath1co j = @xmath2 ( bieging , peters , & kang 2010 ) and the w3 region in co j = @xmath2 and j = @xmath25 and @xmath1co j = @xmath2 ( bieging & peters 2011 ) . for further details about the serpens cloud , we refer the reader to : harvey et al . ( 2007 ) ; enoch et al . ( 2007 ) ; gutermuth et al . ( 2008 ) ; eiroa et al . ( 2008 ) ; and bontemps et al . ( 2010 ) . the goal of our study was to map the serpens core , serpens g3-g6 , and vv serpentis regions with high resolution in the j=@xmath2 rotational lines of @xmath0c@xmath26o and @xmath1c@xmath26o ( hereafter co and @xmath1co respectively ) . co emission will be determined principally by cloud temperature and global turbulence . in contrast , @xmath1co will generally be more useful as a measure of the column density . section 2 describes our observations and data reduction techniques . in section 3 , we show our final brightness temperature image cubes for co and @xmath1co j=2 - 1 , and present associated velocity moment maps . section 4 gives the website where our calibrated co and @xmath1co image cubes can be downloaded . in section 5 , we overlay the harvey et al . ( 2007 ) ysos , enoch et al . ( 2007 ) sub - mm sources , and testi et . al ( 2000 ) n@xmath18h@xmath19cores on our co and @xmath1co maps . section 6 is discussion , and section 7 is our summary . the observations were made between 2008 november and 2010 june with the heinrich hertz submillimeter telescope ( hht ) on mt . graham , az , at an elevation of 3200 m. the hht has a 10-m diameter paraboloidal dish and observes in the frequency range from 210 to 500 ghz . using prototype alma band 6 sideband separating mixers ( courtesy of the national radio astronomy observatory ) in a dual polarization receiver on the hht ( lauria et al . 2006 ) , we were able to take co ( 230.5 ghz ) and @xmath1co ( 220.4 ghz ) , j=@xmath2 , spectra simultaneously through the same telescope optics . we used a filterbank spectrometer to record both the co and @xmath1co emission simultaneously . each of the two j=@xmath2 lines was measured with 256 filters of 0.25 mhz bandwidth , giving a total spectral coverage of @xmath27 km s@xmath5 at a resolution of 0.33 km s@xmath5 . co and @xmath1co j = @xmath2 emission in our mapped region of serpens spans a velocity range from @xmath28 to 20 km s@xmath5 ( lsr ) . our spectra extend from @xmath29 to 30 km s@xmath5 ( lsr ) so we detect all gas in the cloud . observations were made by the on - the - fly " ( otf ) method , where the telescope was scanned in a boustrophedonic pattern in right ascension at a scan rate of 10@xmath30/sec , with spectra sampled every 0.1 sec . in subsequent processing , the data were smoothed by 4 samples or 0.4 sec , corresponding to a telescope motion of 4@xmath30 per spectrum . the beam size at the co j=@xmath2 observing frequency is 32@xmath30 ( fwhm ) so this choice of spatial sampling does not cause any significant loss of angular resolution in the scan direction . a raster of lines with 10@xmath30 spacing in declination or @xmath31 of the beam fwhm was observed in this way . the telescope resolution is well - sampled in both coordinates with this choice of otf mapping parameters . the field to be mapped was divided into 39 subfields , each @xmath32 in size , plus a small overlap . each subfield was observed at least once in each of the two co isotopes . total scanning time per subfield was therefore 2.0 hours , plus @xmath730% overhead for telescope slewing and measurement of reference spectra after every other row . subfields were re - observed in whole or part if weather conditions resulted in excessively high noise levels . the telescope pointing accuracy was checked at the start of each observing period by measuring the co j=@xmath2 emission from the red giant star v1111 oph , which has strong co emission lines from its circumstellar molecular envelope . this object is a good pointing calibrator because the stellar emission is compact and centered on the well - known position of the star , which is only about 10@xmath33 from our serpens field . a 5-point cross pattern centered on the star was measured and the integrated line intensities at the 5 positions were fitted to determine the pointing corrections in azimuth and elevation . these corrections were generally @xmath34 . the pointing was checked and corrected about every 2 hours throughout each observing run , so the pointing errors during the otf mapping should be @xmath34 . the intensity scale was determined with the standard method of kutner & ulich ( 1981 ) by comparing an ambient temperature load with on - sky spectra near the target field , to give the spectral line intensity on the @xmath35 scale . the sideband rejections for the mixers were measured each time the receiver was tuned . we observed a standard position , w51d , to place all the otf data on a consistent main - beam brightness temperature scale , in the manner described in bieging et al . ( 2010 ) . all of the otf data were processed in the _ class _ software package of the grenoble astrophysics group . a linear baseline was removed from each spectrum and the data for each spectral channel were convolved to a square grid in r.a . and dec . , with a 10@xmath30 grid spacing in both coordinates . the gridding algorithm uses a circular gaussian weighting function with a fwhm of 0.3@xmath36(beam fwhm ) , to convolve the otf - sampled spectra onto the regular grid points . the gridding thereby increased the effective angular resolution by a factor of @xmath37 . the resulting data cubes were transferred to the _ miriad _ software format ( sault , teuben , & wright 1995 ) for subsequent processing and analysis . to facilitate the comparison of co and @xmath1co lines ( for example , to calculate line ratios at each pixel ) , we used the _ miriad _ task _ regrid _ to resample the data cubes on the velocity axis by 3rd order interpolation . we chose to sample velocities at 0.15 km s@xmath5 intervals from @xmath29 to @xmath38 km s@xmath5 ( lsr ) , so that the original velocity resolution was oversampled by a factor of 2.2 . to match the angular resolutions of the co and @xmath1co j=@xmath2 maps , which differ by @xmath39 because of the difference in line rest frequencies , we convolved the images with gaussians chosen to give identical resolutions for the two isotopologues . the effective resolution of the final maps was 38@xmath30 ( fwhm ) for both the co and @xmath1co images . this additional spatial smoothing also reduced the surface brightness noise level , at only a small cost in angular resolution . we calculated the rms noise in an emission - free velocity range for each map pixel . [ fig : rms_map ] shows the resulting distribution of the noise . the co j=@xmath2 image _ ( left ) _ has relatively uniform noise over the whole field , with a mean value of @xmath40 k ( main beam brightness ) per pixel and per velocity channel , with variations of @xmath41 about this value over most of the map . for the entire @xmath1co image _ ( right ) _ the mean @xmath42 k with comparable levels of variation due to differences in weather ( i.e. , sky noise ) and source elevation when the individual subfields were observed . in fig . [ fig : mean_spectra ] we show the spatially averaged co and @xmath1co spectra for the serpens core ( region 1 ) , serpens g3-g6 ( region 2 ) , and vv serpentis ( region 3 ) in fig . [ fig : mapped_reg ] . the radial velocity of the emission ranges from about @xmath43 to 18 km s@xmath5 for co and from 2 to 13 km s@xmath5 for @xmath1co . the peak brightness temperature for @xmath1co occurs at 8 km s@xmath5 in all three regions , consistent with previous studies ( duarte - cabral et al . 2010 ; graves et al . 2010 ) which concluded that this is the velocity of the bulk of the gas in serpens . we adopt 8 km s@xmath5 as the systemic velocity of serpens ( indicated by the vertical line in fig . [ fig : mean_spectra ] ) . the @xmath1co spectra for all three regions are nearly gaussian in shape . the co profiles , in contrast , are markedly asymmetric and show a dip or inflection at 8 km s@xmath5 , the velocity of the @xmath1co peak emission . this comparison strongly suggests that the co lines are self - absorbed over at least some of the brightest emission regions , possibly as a result of colder gas which is optically thick in co and is located on the near side of a warmer central part of the cloud . [ fig : co_channel ] shows representative velocity slices from our co image cube , averaging over 0.6 km s@xmath5 and stepping by 1.2 km s@xmath5 . at velocities of about 4.8 and 10.8 km s@xmath5 the emission appears elongated and filamentary . [ fig:13co_channel ] shows a similar set of velocity slices from our @xmath1co image cube , averaging over 0.6 km s@xmath5 , at the same velocities as in fig . [ fig : co_channel ] for the brightest co emission ( 4.4 to 10.4 km s@xmath5 ) . outside this velocity range , @xmath1co is mostly below our detection limit . the @xmath1co j=@xmath2 emission , which is generally optically thin , should be a good tracer of molecular column density in the cloud . [ fig : peak_bt ] shows maps of maximum brightness temperature ( @xmath44 ) in co and @xmath1co , over the full range of emission , -11 to 30 km s@xmath5 for co and -5 to 20 km s@xmath5 for @xmath1co . the co @xmath44 distribution is very clumpy with temperatures of 5 to 10 k in serpens g3-g6 and parts of vv serpentis , and temperatures of 15 to 25 k in the serpens core . when co is not self - absorbed or depleted , co @xmath44 approaches the value of the kinetic temperature of the gas ( @xmath45 ) . in contrast , @xmath1co @xmath44 is much more uniform . we present integrated co and @xmath1co brightness temperature ( @xmath46 ) maps in fig . [ fig : integrated_bt ] , displayed in units of k km s@xmath5 . we only consider velocities associated with serpens by summing over -1 to 18 km s@xmath5 for co , and 2 to 13 km s@xmath5 for @xmath1co . we use a 3 rms cutoff to ensure that detected emission is real . @xmath1co is generally optically thin and its spectrum well - behaved , so the @xmath1co integrated @xmath46 map is nearly identical to the @xmath1co @xmath44 map . @xmath1co integrated @xmath46 should trace the highest column density gas , as long as @xmath1co is optically thin . compared to @xmath44 for co , co integrated @xmath46 retains its clumpy structure in the serpens g3-g6 region , but is smoothly distributed otherwise . we show centroid velocity maps ( first - moment ) for co and @xmath1co in fig . [ fig : vel_centroid ] , considering the same range of velocities as we used to make the integrated @xmath46 maps . note that even where co is self - absorbed , the first moment is the mean velocity of the entire line profile and not ( necessarily ) the velocity of the highest peak bracketing the self - absorption . the color palette extends @xmath47 3 km s@xmath5 on either side of 8 km s@xmath5 , the systemic lsr velocity of serpens . red colors are redshifted with respect to the systemic velocity . the centroid velocity of co in the serpens core and @xmath1co in all regions is mostly at the systemic velocity of serpens and spatially uniform . this uniformity contrasts with the centroid velocity of co southward of the serpens core , which is not spatially uniform but appears in filamentary structures with lsr velocities @xmath47 3 km s@xmath5 of the systemic velocity of serpens . this difference in apparent spatial distribution is most likely due to a change in the velocity structure and/or the opacity of the colder absorbing co outside the serpens core . there are two kinematically distinct areas in the serpens core , seen as red and blue components in co centroid velocity separated by about 3 km s@xmath5 . it has been suggested that these are two interacting sub - clouds ( testi et al . 2000 ; eiroa et al . 2008 ; duarte - cabral et al . 2010 ) . [ fig : vel_width ] shows velocity dispersion ( @xmath48 ) ( second - moment ) maps for co and @xmath1co , in units of km s@xmath5 . we apply a 3 rms cutoff @xmath49 k for both isotopes . the velocity range for computing the co @xmath48 map is -1 to + 16 km s@xmath5 ( which avoids the high velocity gas component at r.a . @xmath50 , dec . @xmath51 ) , and 4 to 12 km s@xmath5 for the @xmath1co @xmath48 map . the maximum co @xmath48 is 4.2 km s@xmath5 , and that of @xmath1co is 2.0 km s@xmath5 , while the average co @xmath48 is 2 km s@xmath5 , and that of @xmath1co is 0.8 km s@xmath5 . large @xmath48 reveals extended emission - line wings , which can be caused by large scale gas motions or turbulence due to protostellar outflows . the @xmath1co map is relatively noisy ( i.e. , pixelated ) for small ( @xmath52 km s@xmath5 ) @xmath48 , which occurs away from regions of high column density . unlike @xmath1co @xmath44 , integrated @xmath46 , and large co @xmath48 ( @xmath53 km s@xmath5 ) , large @xmath1co @xmath48 ( @xmath54 km s@xmath5 ) occurs in small ( 0.01 x 0.01 deg@xmath8 ) regions . there is one such region in the serpens core and three to five in serpens g3-g6 . note that we exclude a @xmath55 high velocity ( 18 - 20 km s@xmath5 ) gas component ( r.a . @xmath50 , dec . @xmath51 ) from our moment maps . we assume that this feature is not associated with serpens main . following bieging et al . ( 2011 ) , fig . [ fig : ratio_map ] shows the ratio ( @xmath56 ) of co to @xmath1co line intensities ( @xmath46 ) , averaged over 0.6 km s@xmath5 and spaced 0.6 km s@xmath5 , as a function of velocity . for an assumed @xmath0co/@xmath1co abundance ratio of 50 , the color palette indicates where @xmath1co reaches optical depth @xmath57 for @xmath58 ( gray regions ) , @xmath59 for @xmath60 ( colored regions ) , and co is self - absorbed for @xmath61 ( white regions ) . there is a spatial gradient in optically thick @xmath1co across serpens , appearing in vv serpentis at low velocities and shifting north - easterly until @xmath1co is optically thick in the serpens core at higher velocities . co is self - absorbed and @xmath1co is optically thick in the serpens core . the spatial and velocity structure of the co line ratios imply that a detailed 3-dimensional radiative transfer model of the cloud will be necessary for proper interpretation of these data . the calibrated brightness temperature image cubes of co and @xmath1co j=2 - 1 can be downloaded as fits files from the online version of this paper through the astrophysical journal supplement series . the evolutionary stage of a yso ( after protostar formation but before reaching the main sequence ) can be inferred from its observed sed ( slope of log@xmath64 f@xmath65 vs. log@xmath64 , @xmath66 ) ( lada 1987 ; andre & montmerle 1994 ; greene et al . 1994 ) and/or comparing its sed to theoretical models ( whitney et al . 2003 ; robitaille et al . these evolutionary stages are ( in order of youngest to oldest ) : class i , flat , class ii , and class iii ( lada 1987 ; andre & montmerle 1994 ; greene et al . harvey et al . ( 2007 ) classified their high - confidence set of 235 serpens ysos into the above four stages , 41 class i ( 17 % ) , 25 flat ( 11 % ) , 130 class ii ( 55 % ) , and 39 class iii ( 17 % ) . of the 15 ysos with the coldest seds " ( 70 to 24 @xmath67 flux ratio @xmath68 , harvey et al . 2007 ) , 10 are class i. these 10 ( which we refer to as cold class i ) should be the youngest class i ysos as they are the most embedded or obscured . note that harvey et al . ( 2007 ) state that there are 39 class i and 132 class ii ysos , while the online supplementary table from harvey et al . ( 2007 ) shows 41 class i and 130 class ii . we use the data from the harvey et al . ( 2007 ) online supplementary table . in fig . [ fig : yso_olay ] , we compare our @xmath1co integrated intensity map with locations of the harvey et al . ( 2007 ) ysos : cold " class i ( magenta @xmath69 s ) , all other class i ( magenta @xmath70 s ) , flat ( yellow @xmath70 s ) , class ii ( white @xmath69 s ) , and class iii ( white @xmath70 s ) . the distribution of class i and flat ysos follows regions of large ( @xmath71 k km s@xmath5 ) @xmath1co integrated line intensity . there are three class i ysos ( near r.a . 18@xmath7228@xmath7345@xmath74 , dec . @xmath75 ) that do not coincide with large @xmath1co integrated line intensity ( see section 5.3 ) . in the serpens core , all cold " class i ysos are in locations of peak ( @xmath76 k km s@xmath5 ) @xmath1co integrated line intensity . class ii and iii ysos appear more widely distributed across serpens main . [ fig : serpens_core ] shows the @xmath1co velocity centroid ( 1st moment ) for the serpens core computed over line center ( 6 to 10.5 km s@xmath5 ) in red - green - blue palette , with positions of n@xmath18h@xmath19 j=@xmath77 cores ( cyan @xmath69 s with diameter 1 ) from testi et al . ( 2000 ) and ysos ( symbols as in fig . @xmath78 ) from harvey et al . 2007 . the n@xmath18h@xmath19 cores and class i and flat ysos form a cluster that is strikingly elongated , with a major axis nearly parallel to the boundary separating the red and blue - shifted gas in the serpens core . the four n@xmath18h@xmath19 cores , which should mark the coldest , densest pre - stellar locations in the cloud , also coincide closely with the youngest ysos . class ii and iii ysos are less clustered and more symmetrically distributed about this axis . this distribution raises the question of whether the younger ( class i and flat ) or older ( class ii and iii ) ysos , or both , formed along this boundary . to examine these possibilities , we define two reference positions for serpens core ysos : the unweighted centroid r.a . , dec . of cold class i ysos ( _ cold centroid _ , @xmath79 , dec . @xmath80 ) and that of flat ysos ( _ flat centroid _ , r.a . @xmath81 , dec . @xmath82 ) . fig . [ fig : serpens_core ] shows these two positions as black @xmath83s . the flat centroid lies within a tight grouping of flat ysos @xmath842 southeast of the cold centroid . we note that these two centroid positions lie near the centers of the two sub - clusters " described by duarte - cabral et al . ( 2010 ) , based on their observations of the very optically thin c@xmath85o and c@xmath15o isotopologues . nw sub - cluster " encompasses the position of our _ cold centroid _ , consistent with a very young age for the ysos contained in this sub - cluster . in contrast , duarte - cabral et al . ( 2010 ) find that the se sub - cluster " displays greater kinematic complexity and a wider distribution of yso ages . this se sub - cluster contains our _ flat centroid _ position , consistent with the observed clustering of the flat - sed sources having somewhat greater ages than the cold class i sources . in figures [ fig : yso_hist_sep ] and [ fig : yso_hist_pa ] , we plot histograms of yso projected separation and position angle ( pa ) relative to the cold ( solid black ) and flat ( dashed blue ) centroids , separated by yso class . ( here class i " includes cold class i " ysos . ) the sample contains only ysos within the serpens core . this restriction gives 23 class i , 12 flat , 23 class ii , and 7 class iii objects . the histograms for the youngest classes ( i and flat ) in fig . 13 show that these ysos cluster within 0.5 pc of either centroid position , with small offsets ( @xmath70.2 pc ) as expected from the definitions of the 2 centroids . the distribution of pas for the youngest classes ( fig . 14 ) clearly shows that the clustering is oriented along a nw - se ( @xmath86 ) axis , an orientation which is consistent with the location of the sub - clusters " discussed by duarte - cabral et al . in contrast , the older ysos , classes ii and iii , show a wider distribution relative to either of the centroids ( fig . 13 ) , with the class iii ysos lying almost entirely at separations @xmath87 pc , while the class ii ysos are broadly peaked out to @xmath71 pc . both the class ii and iii objects are uniformly distributed in pa , with no preference for the nw - se axis of the younger yso classes . the spatial distrobutions of the various yso classes suggest that the most recent star formation in the serpens core , i.e. , within the past few@xmath88 years , has occurred almost entirely within a volume of space about 1 pc in length and elongated in a nw - se direction as projected on the sky . the wider distribution of the class iii objects could be a result of their having formed over a more extended volume in an earlier episode of star formation , as suggested by duarte - cabral et al . the class ii ysos , however , show a distribution that is clustered about the reference centroid positions but less tightly than the class i or flat ysos . the progression from a more concentrated to a more dispersed distribution with age could also result from a diffusion of ysos outward away from the serpens core over the past few million years , due to random motions acquired from the internal velocity dispersion of the parent gas clouds . davis et al . ( 1999 ) presented co j=2 - 1 maps of the serpens core region ( @xmath89 ) with 23 resolution , and found evidence in the line wings ( 14 to 22 km s@xmath5 and @xmath90 to 4 km s@xmath5 ) for a burst of outflows " , produced by the cluster of ysos in the core . our data have higher sensitivity but lower angular resolution ( 38 ) than davis et al . ( 1999 ) ; a detailed comparison with their line wing maps shows excellent agreement for both the red- and blue - shifted components , including the broad red - shifted wing associated with the herbig - haro object hh106 ( located @xmath91 west of the core cluster ) . more recently , graves et al . ( 2010 ) have mapped the j=3 - 2 lines of co , @xmath1co , and c@xmath15o over the serpens core with the harp heterodyne receiver array on the jcmt , with effective resolutions of 17 to 20 . their co and @xmath1co integrated intensity maps ( their fig . 2 ) agree well with ours ( fig . 7 ) , allowing for the different transitions and angular resolution . at the position of hh106 ( 18@xmath7229@xmath7318@xmath74 , @xmath92 ) , the co j=2 - 1 line shows the same broad red - shifted wing as their j=3 - 2 line ( their fig . 5 ) , but the lower excitation j=2 - 1 transition has a more pronounced narrow self - absorption feature at 8 km s@xmath5 , the systemic velocity of the cloud . our line ratio channel maps ( fig . 7 ) show that near the systemic velocity , much of the serpens core cloud is optically thick in @xmath1co j=2 - 1 , and the co line is likely to be self - absorbed . graves et al . ( 2010 ) also find from the j=3 - 2 line ratios that the @xmath1co line is optically thick with @xmath93 having values up to @xmath77 toward the densest submm continuum cores . duarte - cabral et al . ( 2010 ) presented maps of the c@xmath85o and c@xmath15o j=2 - 1 and 1 - 0 transitions made with the iram 30 m telescope , covering only the central [email protected] of the serpens core . these optically thin transitions should be less distorted by opacity effects than the @xmath1co or co lines , revealing more accurately the distribution and kinematics of the highest column density regions . duarte - cabral et al . ( 2010 ) found that the sub - clusters have different kinematic properties . the nw center has a single , nearly gaussian line profile , while the se center has 2 blended components . they interpret the profiles as a superposition of 2 clouds with different velocities and argue that these are colliding in the area containing the se sub - cluster . in a second paper , duarte - cabral et al . ( 2011 ) show that a smoothed particle hydrodynamics ( sph ) 3d simulation of two colliding gas cylinders , one rotated @xmath7 45@xmath33 relative to the other , can reproduce many of the observational properties of the serpens core . especially the onset of star formation in the zone where the clouds collide . their sph simulation includes gas hydrodynamics , self - gravity , turbulence , and sink particles . given the large differences in optical depths of the c@xmath85o and c@xmath15o lines compared with the co and @xmath1co transitions we present here , it is difficult to make direct comparisons . we note however , that duarte - cabral et al . ( 2010 ) find a velocity gradient in the c@xmath15o j=1 - 0 map with velocity increasing from the se to nw sub - clusters , that is consistent with that in our @xmath1co velocity centroid map ( fig . our data show a gradient over a much larger region ( @xmath710 ) than that covered by duarte - cabral et al . ( 2010 , 2011 ) . their sph simulation may therefore underestimate the physical size of the colliding clouds . in fig . [ fig : bolocam_olay ] , we overlay contours of 1.1 mm continuum emission ( enoch et al . 2007 ) on our @xmath1co integrated @xmath46 map ( convolved to 90 resolution to match that of the published bolocam map ) . contour levels are at 5 , 10 , 20 , 40 , 80 , and 160 times the rms noise @xmath94 mjy beam@xmath5 . the 1.1 mm emission coincides with high @xmath1co integrated @xmath46 , except for a filamentary feature in serpens g3-g6 centered at r.a . @xmath95 , dec . @xmath75 called the starless cores region " ( enoch et al . i ysos and two class ii ysos are located in the starless cores region ( see fig . [ fig : yso_olay ] ) . we detect only a faint ( @xmath96 k km s@xmath5 ) local maximum in @xmath1co integrated @xmath46 at r.a . @xmath97 , dec . @xmath98 . submm maps of the serpens core have been presented by davis et al . ( 1999 ) , made with the scuba bolometer array at wavelengths of 850 and 450 . these images have considerably better resolution ( 14 and 8 respectively ) but like the bolocam 1.1 mm map , also suffer from spatial filtering so that mainly the bright compact sources are detected . four of these ( smm 1 , 5 , 9 , and 10 ) lie within the nw sub - cluster and lie on the secondary maximum of the integrated @xmath1co intensity ( fig . 7 , right ) @xmath73 nw of the brightest @xmath1co peak . the other 6 submm compact sources are associated with the brightest peak in fig . 7 ( right ) , and with the se sub - cluster of duarte - cabral et al . this positional agreement implies that , as expected , the current most active star formation coincides with the region of greatest gas column density . serpens main has yet to be extensively studied using data from the _ herschel _ gould belt survey , which spans far - ir to sub - mm wavelengths . such data could reveal even younger prestellar cores ( class 0 ) in serpens main . for example , goicoechea et al . ( 2012 ) presented a submm spectrum for a single serpens core class 0 yso . hundreds of class 0 ysos have been identified in the nearby aquila rift molecular complex using _ herschel _ data ( andre et al . 2010 ; bontemps et al . 2010 ) . the spatial distribution of co , as seen in the peak and integrated brightness temperature maps ( figs . 6 & 7 ) as well as the individual channel maps ( fig . 4 ) gives the impression of a highly disturbed gas cloud . there are shell - like structures and holes especially in the serpens core region . elsewhere the co has a flocculent appearance with many small clumps ( @xmath70.1 ) all having approximately the same peak brightness temperature . overall , there is a gradient of increasing peak and integrated brightness temperatures from south to north over the 2 declination extent of our maps . although co emission is detected in every pixel , there are large contrasts in some regions . the serpens core , at the northern end of the cloud , stands out sharply against a lower @xmath99 background , as a result of the high concentration of ysos which energize the gas by radiation heating of the associated dust , and by injecting kinetic energy via numerous bipolar outflows . the class i and flat ysos in the serpens core form a cluster that is nearly parallel to the intersection of the red and blue - shifted gas components and is strikingly linear - shaped ( see fig . [ fig : serpens_core ] ) , similar to the corresponding distribution of sub - mm cores ( enoch et al . 2007 ; duarte - cabral et al . 2010 ) . the velocity gradient might result from a collective action of outflows in the center of the serpens core , and the linear - shaped cluster might indicate a preferred orientation for proto - stellar jets . duarte - cabral et al . ( 2011 ) show that a cloud - cloud collision also explains the velocity gradient and linear - shaped cluster . however , their sph simulation should be treated with caution until the result is independently verified using an adaptive mesh refinement ( amr ) code , such as athena or flash , to calculate accurately the effects of shocks in the collision . from the observed @xmath1co velocity dispersion ( fig . [ fig : vel_width ] ) and present yso distribution ( fig . [ fig : yso_hist_sep ] ) , we find that class i , flat , and class ii ysos in the serpens core had sufficient drift velocities and lifetimes to reach their present locations , even if they all formed in the same limited volume . most class i and flat ysos are within @xmath840.5 projected pc of both the flat and cold centroids , while most class ii ysos are within @xmath841.0 projected pc ( see fig . [ fig : yso_hist_sep ] ) . assuming a mean inclination angle of 45@xmath33 to the line of sight , class i and flat , and class ii ysos would need to drift @xmath840.7 pc and 1.4 pc , respectively . class ii ysos have lifetimes @xmath84 2 myr , which sets class i and flat yso lifetimes at @xmath840.5 and 1.0 myr , respectively ( evans et al . assuming drift velocity ( @xmath100 ) is constant since yso formation , class i , flat , and class ii ysos would need minimum @xmath100 @xmath842.0 , 1.0 , and 1.0 km s@xmath5 , respectively . because co traces bulk gas motion , we can assume that @xmath100 @xmath84 @xmath101 ( i.e. , protostars inherit the velocity dispersion of accreted gas ) , where @xmath101 is the spatial average of @xmath102 . we calculate @xmath101 over r.a . @xmath103 @xmath104 and dec . @xmath106 , the smallest region containing the yso sample from figs . @xmath101 = 2.0 km s@xmath5 and rms @xmath107 km s@xmath5 , giving @xmath100 @xmath108 km s@xmath5 . @xmath100 exceeds or equals what is necessary for nearly all class i , flat , and class ii ysos to have formed in a small volume at either the flat or cold centroid positions . since we do not detect co and @xmath1co in the starless cores " region , co is most likely depleted there . average co @xmath44 from fig . [ fig : peak_bt ] is @xmath8410 - 20 k , so @xmath45@xmath8410 - 20 k ( assuming no co self - absorption or depletion ) . at 10 - 20 k co becomes depleted for number densities @xmath109 @xmath110 ( goldsmith 2001 ) . the lowest ( inferred ) gas number density that bolocam can detect is @xmath84@xmath111 @xmath110 ( enoch et al . co is unlikely to be depleted in the serpens core and serpens g3-g6 regions because the temperature of the surrounding gas is too high , so co will not freeze out onto dust grains , despite sufficiently high number densities . there are in fact five ysos in the starless cores " region ( i.e. , not truly starless ) , but heating may be insufficient ( co and @xmath1co peak t@xmath112 is about 3 and 1.5 k , respectively ) to prevent co from freezing out onto gains . we mapped the j=@xmath2 rotational lines of co and @xmath1co over 3900 square arcmin of the serpens core , serpens g3-g6 , and vv serpentis regions of the serpens main cloud ( 1.04 deg@xmath8 in total ) with high spatial ( 38 ) and spectral ( 0.3 km s@xmath5 ) resolution . our final data are calibrated @xmath46 image cubes for co and @xmath1co ( available online in fits format ) . the _ spitzer _ c2d legacy survey ( evans et al . 2003 ; harvey et al . 2007 ) and the bolocam 1.1 mm continuum survey ( enoch et al . 2007 ) are the only other studies that have mapped the entirety of the serpens main molecular cloud . these latter two surveys were limited to measuring dust temperature and column density ; our co and @xmath1co data measure gas kinematics as well as physical properties . at the systemic velocity of serpens main ( 8 km s@xmath5 ) , co is self - absorbed and @xmath1co is optically thick in the serpens core . the gas traced by co in serpens g3-g6 and vv serpentis appears in filamentary structures having lsr velocities between 6 and 8 km s@xmath5 . the spatial and velocity structure of the co line ratios implies that a detailed 3-dimensional radiative transfer model of the cloud would be necessary for a complete interpretation of our spectral data . it is likely that the observed class i , flat , and class ii ysos in the serpens core formed in a @xmath71 pc volume between the flat and/or cold centroid positions . the distributions of projected separations and pas suggest a co - spatial formation site ( figs . [ fig : yso_hist_sep]-[fig : yso_hist_pa ] ) . our measured @xmath1co velocity dispersion of @xmath84@xmath113 km s@xmath5 implies that the serpens core ysos could have formed within a small volume and then diffused away to their observed spatial distributions on a timescale of @xmath114 yr . the blue- and red - shifted regions in the @xmath1co velocity first moment ( fig . [ fig : serpens_core ] ) could be the result of a collective action of outflows , such as a preferred protostellar jet orientation . an alternative explanation is a cloud - cloud collision between the blue - shifted gas and foreground gas along the line of sight ( duarte - cabral et al . 2010 ; duarte - cabral et al . 2011 ) . the starless cores " region is likely to be the site of further star formation in serpens . the detection of 1.1 mm cold dust emission but no co or @xmath1co emission in that region , suggests that co is largely depleted due to high density and low termperatures in this part of the serpens cloud . in a future paper we will employ a grid of statistical equilibrium / radiative transfer models for co line emission , incorporating co excitation and photodissociation , to derive the distribution of total hydrogen ( h@xmath18 and hi ) column densities and masses of the serpens core , serpens g3-g6 , and vv serpentis regions . this analysis should resolve the disagreement between previous mass estimates noted by eiroa et al . this material is based upon work supported by the national science foundation graduate research fellowship under grant no . dge 1106400 . this work was supported in part by national science foundation grant ast-0708131 to the university of arizona , and by the nasa space grant program . we thank the anonymous referee for suggestions that strengthened the discussion and conclusion sections in this paper . we thank steve stahler , chris mckee , yancy shirley , neil evans , and kevin hardegree - ullman for helpful discussions . we also thank dr . a. r. kerr of the national radio astronomy observatory for providing the prototype alma band 6 mixers used in this work . we are grateful to butler burton and greg schwarz for a prompt submission process and storing our data online with the astrophysical journal supplemental series . the heinrich hertz submillimeter telescope is operated by the arizona radio observatory , which is part of steward observatory at the university of arizona . andre , a. , & montmerle t. 1994 , apj , 420 , 837 andre , ph . , menshchikov , a. , bontemps , s. , et al . 2010 , a&a , 518 , l102 bieging , j. h. , peters , w. l. , & kang , m. 2010 , apjs , 191 , 232 bieging , j. h. , & peters , w. l. 2011 , apjs , 196 , 18 bontemps , s. , andre , ph . , konyves , v. , et al . 2010 , a&a , 518 , l85 cambresy , l. 1999 , a&a , 345 , 965 chavarria , k. , lara , e. de , finkenzeller , u. , mendoza , e. e. , & ocegueda , j. 1988 , a&a , 197 , 151 cohen , m. , & kuhi , l. v. 1979 , apjs , 41 , 743 davis , c.j . , matthews , h.e . , ray , t.p . , dent , w.r.f , & richer , j.s . 1999 mnras , 309 , 141 duarte - cabral , a. , fuller , g. a. , peretto , n. , et al . 2010 , a&a , 519 , a27 duarte - cabral , a. , dobbs , c. l. , peretto , n. , & fuller , g. a. 2011 , a&a , 528 , a50 dzib , a. , loinard , l. , mioduszewski , a. j. , et al . 2010 , apj , 718 , 610 eiroa , c. , djupvik , a. a. , & casali , m. m. 2008 , in _ handbook of star forming regions ii : the southern sky _ , ed . b. reipurth , ( asp monograph publications ) , 693 enoch , m. l. , glenn , j. , evans , n. j. , et al . 2007 , 666 , 982 evans , n. j. , allen , l. e. , blake , g. a. , et al . 2003 , pasp , 115 , 965 evans , n. j. , dunham , m. m. , jorgensen , j. k. , et al . 2009 , apj , 181 , 321 goicoechea , j. r. , 2012 , a&a , 548 , 17 pp goldsmith , p. f. 2001 , apj , 557 , 736 graves , s. f. , richer , j. s. , buckle , j. v. , et al . 2010 , mnras , 409 , 1412 greene , t. p. , wilking , b. a. , andre , p. , young , e. t. , & lada , c. t. 1994 , apj , 434 , 614 gutermuth , r. a. , bourke , t. l. , allen , l. e. , et al . 2008 , apj , 673 , l151 harvey , p. m. , chapman , n. , lai , s. , et al . 2006 , apj , 644 , 307 harvey , p. m. , merin , b. , huard , t. l. , et al . 2007 , apj , 663 , 1149 kutner , m. l. , & ulich , b. l. 1981 , apj , 250 , 341 lada , c. j. 1987 , in _ star forming regions _ , eds . m. peimbert & j. jugaku ( dordrecht : reidel ) , 1 lauria , e.f . , kerr , a.r . , reiland , g. , et al . 2006 , alma memo 553 , ( nrao : charlottesville , va ) mcmullin , j. p. , mundy , l. g. , blake , g. a. , et al . 2000 , apj , 536 , 845 olmi , l. , & testi , l. 2002 , a&a , 392 , 1053 ripepi , v. , bernabei , s. , marconi , m. , et al . 2007 , a&a , 462 , 1023 sault , r. j. , teuben , p. j. , & wright , m. c. h. 1995 , in _ astronomical data analysis software and systems iv _ r. shaw , h. e. payne , & j. j. e. hayes ( asp conf . 77 ) , p. 433 testi , l. , sargent , a.i . , olmi , l , & onello , j.s . 2000 , apj , 540 , l53 white , g.j . , casali , m.m . , & eiroa , c. 1995 , a&a , 298 , 594

we mapped @xmath0co and @xmath1co j = @xmath2 emission over 1.04 deg@xmath3 of the serpens molecular cloud with @xmath4 spatial and 0.3 km s@xmath5 spectral resolution using the arizona radio observatory heinrich hertz submillimeter telescope . our maps resolve kinematic properties for the entire serpens cloud . we also compare our velocity moment maps with known positions of young stellar objects ( ysos ) and 1.1 mm continuum emission . we find that @xmath0co is self - absorbed and @xmath1co is optically thick in the serpens core . outside of the serpens core , gas appears in filamentary structures having lsr velocities which are blue - shifted by up to 2 km s@xmath5 relative to the 8 km s@xmath5 systemic velocity of the serpens cloud . we show that the known class i , flat , and class ii ysos in the serpens core most likely formed at the same spatial location and have since drifted apart . the spatial and velocity structure of the @xmath0co line ratios implies that a detailed 3-dimensional radiative transfer model of the cloud will be necessary for full interpretation of our spectral data . the starless cores " region of the cloud is likely to be the next site of star formation in serpens .

You are an expert at summarizing long articles. Proceed to summarize the following text: a massive star builds up onion - like layers of different chemical elements synthesized by hydrostatic nuclear burning processes during its lifetime . at the end of its evolution , the innermost fe core collapses into a neutron star , which triggers a core - collapse supernova ( sn ) explosion . the detailed process of the explosion is complicated and poorly understood , but a consensus from theoretical studies suggests that the explosion should be asymmetric and turbulent , especially near the core ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * and references therein ) . multi - dimensional numerical simulations have shown that the explosion also leads to extensive mixing and inversion among the stratified layers by hydrodynamic instabilities ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? a distinct method to explore the explosion dynamics of sne , therefore , would be to investigate the detailed chemical and kinematic properties of sn ejecta material in nearby young galactic supernova remnants ( snrs ) where the imprints of explosion remain . cassiopeia a ( hereafter cas a ) , at the age of @xmath3 years @xcite , is one of the best studied young galactic snrs . its sn explosion was classified as type iib based on the optical spectra of light echoes @xcite , which implies that the progenitor was probably a star of @xmath4@xmath5 that had lost a significant portion , but not all , of its h - rich envelope before the explosion . over the past several decades , cas a has been extensively studied in almost all wavebands from radio to gamma - rays . the complex spatial distribution of _ shocked _ sn ejecta , such as the x - ray - emitting fe - rich ejecta plumes beyond the si - rich material and o / s - rich optical knots outlying the bright ejecta shell ( e.g. , * ? ? ? * ; * ? ? ? * ) , indicates an explosion resulting in an inversion of the chemical layers . furthermore , the inhomogeneous distribution of _ unshocked _ sn ejecta radiating si and ti emission lines @xcite implies that the explosion was turbulent near the progenitor core . in the visible waveband , many bright optical knots are presented in and around cas a. they have been classified into two major groups based on their proper motions and line - of - sight velocities : ( 1 ) fast - moving knots ( fmks ) and ( 2 ) quasi - stationary flocculi ( qsfs ) . the fmks show large proper motions and high radial velocities , corresponding to expansion velocities of up to @xmath6 ( e.g. , * ? ? ? * ; * ? ? ? they have spectral features strongly enhanced in o and other heavy elements ( e.g. , s and ar ) that are mostly synthesized from the nuclear burning process in a deep stellar layer , while showing no detectable h or he emission lines ( e.g. , * ? ? ? * ; * ? ? ? based on their high expansion velocities and chemical composition , they have been regarded as dense material ejected from the disrupted layer of the progenitor after the sn explosion . the dynamical and chemical properties of the qsfs are very different from those of the fmks . the qsfs have considerably slower velocities ( @xmath7 ) , and so their typical expansion age is @xmath8 years @xcite , which is much larger than the age of the remnant . they are bright in [ ] and h@xmath9 emission , with a handful of other h and he emission lines , in optical spectra ( e.g. , * ? ? ? * ; * ? ? ? considering these properties , qsfs are believed to be dense circumstellar medium ( csm ) blown out from the progenitor prior to the sn explosion . in addition , some intermediate optical knots , that is , the so called fast - moving flocculi ( fmfs ) or nitrogen knots ( nks ) , were reported by @xcite . while their spectra , which show strong [ ] 6548 , 6583 accompanied by weak h@xmath9 without any lines of o and s , are analogous to those of qsfs @xcite , their proper motions are larger than @xmath10 , corresponding to @xmath11 @xcite . most of them have been found outside of the bright main ejecta shell . therefore , these outlying n - rich knots have been interpreted as being fragments of the progenitor s photosphere expelled by the sn blast wave at the time of explosion . the dynamical and chemical properties of these different types of optical knots have provided important clues to unveiling the snr s origin and evolution . for example , the expansion center and age were determined from the proper motions of the fmks ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) , and the three - dimensional ( 3d ) structure of the ejecta knots reconstructed from spectral mapping observations has shown that the sn ejecta is expanding spherically but is systematically receding at a speed of @xmath12 at a distance of 3.4 kpc @xcite . the dense , slow - moving qsfs indicate that the progenitor had undergone significant and inhomogeneous mass loss during the red supergiant phase @xcite . although this velocity - based classification is an efficient way to classify the knots as sn ejecta and csm , which have distinctive expansion velocities , i.e. , a few @xmath13 vs. a few @xmath14 , it encounters limitations when characterizing the sn ejecta material from the different nucleosynthetic layers . according to previous numerical simulations for core - collapse sn explosions , the radial velocity profiles of heavy elements in sn ejecta are almost identical @xcite . several models in @xcite predict the velocity separation of up to @xmath15 between different heavy elements . their velocity profiles , however , are very broad ( a few @xmath13 ) , so that their distributions largely overlap . these numerical simulations may imply that sn material from different nucleosynthetic layers is barely distinguishable in velocity space . in order to comprehend the explosion dynamics , therefore , a more systematic classification of sn ejecta based on their chemical composition is needed . in this paper , we report the results of broad near - infrared ( nir ) spectroscopic observations toward the main ejecta shell of cas a , focusing on classification of the emission knots based on their _ spectrochemical _ properties . the nir study of cas a has been relatively limited in the literature , although there are many bright forbidden lines of various elements in the nir waveband , some of which may arise from the deep nucleosynthetic layers . as far as we are aware , the only nir spectroscopic study covering the entire _ jhk _ bandpass was conducted by @xcite , who obtained low - resolution ( @xmath16 ) spectra of five fmks and three qsfs that were previously known . they showed that the spectra of the fmks are dominated by [ ] 1.029 , 1.032 , 1.034 , 1.037 ( hereafter [ ] 1.03 multiplets ) and [ ] 1.188 , as well as two high - ionization si lines , [ ] 1.963 and [ ] 1.430 , while those of the qsfs show strong he i 1.083 accompanied by h emission lines . a dozen bright [ ] lines are also detected in both fmks and qsfs . our spectra confirm these features . here we present the spectrochemical classification of knots and discuss their characteristics . the organization of the paper is as follows . in section [ sec - obs ] , we outline our spectroscopic observations and data reduction procedures . an explanation of how we identified individual knots from two - dimensional ( 2d ) dispersed images and derived their spectral properties is given in section [ sec - ide ] . in sections [ sec - pca ] and [ sec - dis ] , we carry out a classification of the knots using principal component analysis ( pca ) and discuss the origin of knots in different classes . finally , the paper is summarized in section [ sec - sum ] . we carried out nir spectroscopic observations of cas a using triplespec mounted on the palomar 5 m hale telescope . triplespec is a cross - dispersed nir spectrograph that provides simultaneous wavelength coverage from 0.94 to 2.46 at a spectral resolving power of @xmath17 @xcite . the spectrograph uses two adjacent quadrants , i.e. , @xmath18 pixels , of a rockwell scientific hawaii - ii array . the slit width and length are 1 and 30 , respectively . on 2008 june 29 and august 8 , we obtained spectra at eight slit positions along the main ejecta shell ( figure [ fig - slit ] ) . the orientations of slits 1 and 4 were set perpendicular to the shell , while those of the others were largely parallel to the shell . the nearby a0v star hd223386 was observed as a spectroscopic standard right before and/or after the target observations at similar airmasses . in addition to the target spectra , we also obtained spectra of sky background by dithering along the slit or taking spectra of nearby sky , depending on the complexity of the target fields . the total on - source exposure time at each slit position ranges from 300 to 1800 s. the detailed parameters of the spectroscopic observations are provided in table [ tbl - log ] . we developed an idl - based data reduction pipeline to reduce the obtained triplespec data . the pipeline first performed subtraction of a dark frame and flat fielding , followed by subtraction of sky background emission . for the latter , we used sky background emission obtained in a dithered frame or in a frame from nearby empty sky , depending on the complexity of the source emission in a given slit . because of the non - uniform dispersion by the three cross - dispersing prisms in triplespec @xcite , some orders of the triplespec spectra are severely curved . by carefully tracing the spectra of standard stars and airglow emission lines , we obtained a 2d wavelength solution for each order that can correct this effect in a satisfactory manner . we conducted fifth - order order polynomial fits to the wavelengths of the oh airglow emission lines @xcite in the triplespec spectra to obtain the wavelength solutions at 0.5 @xmath19 uncertainty for each order . heliocentric velocity correction was also performed when calculating the velocity of the emission line detected in the triplespec spectra . we used an a0v - type standard star ( hd223386 ) in the photometric calibration and confirmed that the fluxes of the [ ] 1.644 emission line are consistent with those from the narrow - band imaging observations of the same areas ( see section [ sec - obs - img ] ) . in 2005 august 28 and 2008 august 11 , we performed nir imaging observations for the remnant using the wide - field infrared camera ( wirc ; * ? ? ? * ) attached to the palomar 5 m telescope . the camera consists of a single @xmath20 rockwell hawaii - ii hgcdte nir detector with a pixel scale of @xmath21 pixel@xmath22 , which provides a field of view of @xmath23 . we used an [ ] narrow - band filter that has a mean wavelength of 1.644 and a bandwidth of 252 . while the single exposure time per frame in 2005 and 2008 is 60 s and 200 s , the multiple dithering observations yield total integration times of 1200 s and 5400 s , respectively . we also obtained h - continuum narrow - band images ( mean wavelength of 1.570 and bandwidth of 236 ) in order to subtract the bright stars in the [ ] narrow - band images . the average seeing throughout our observations was @xmath24 fwhm . we followed standard procedures for the reduction of nir imaging data . first , the dark and sky background were subtracted from each individual dithered frame . then , all of the frames were divided by the normalized flat image . the astrometric solution was derived on the basis of unsaturated stars around the remnant in the two micron all sky survey ( 2mass ) point - source catalog ( psc ; * ? ? ? we coadded all dithered frames which were astrometrically aligned . in terms of photometric calibration , h_-band magnitude in the 2mass system was used by assuming that the [ ] narrow - band magnitude is the same as the _ h_-band magnitude . the uncertainty in the zero - point magnitude we derived is less than 0.1 mag , corresponding to the 10% of the flux . figure [ fig-2dspec ] shows 2d dispersed images of strong emission lines detected in our triplespec spectra : red for [ ] 1.644 , green for he i 1.083 , and blue for [ ] 1.188 + [ ] 0.953 . the emission features are complex , often with multiple peaks , and so the identification of individual ` knots ' by visual inspection is not straightforward . we used the idl routine clumpfind @xcite , which was developed for the identification of clumps in molecular clouds . the routine identifies ` clumps ' by searching for local maxima above some intensity threshold and following them down them to low - intensity levels . for a given 2d dispersed image , a `` mask '' locating individual knots was generated from [ ] 1.644 emission features or , if they are weak , [ ] 0.953 or [ ] 1.188 emission features , and shifted along the wavelength to find other emission lines associated with the knots . the details of this knot identification process are given in the supplement material of @xcite . in total , we identified 63 knots of distinctive kinematical and spectral properties in the 2d dispersed images . figure [ fig - clumps ] shows their locations and table [ tbl - knot ] lists their positions , sizes , and radial velocities . we extracted one - dimensional ( 1d ) spectra of individual knots using their mask files ( figure [ fig - clumps ] ) . we identified 46 emission lines in total and derived their parameters by performing single gaussian fits to the detected lines . table [ tbl - flx ] lists the derived line widths and fluxes . [ ] 1.644 line is detected in all 63 knots and their fluxes are also listed in table [ tbl - knot ] . among the 46 emission lines , 43 lines are previously detected lines in snrs @xcite , whereas three lines of [ ] at 2.145 , 2.218 , and 2.242 are detected for the first time in snrs . the [ ] lines originate from transitions between levels in @xmath25 g and @xmath25h terms with high excitation energies ( @xmath26 k ) and have been detected in a few objects , such as sgra@xmath27/irs16 complex ( * ? ? ? * and references therein ) , classical novae @xcite , h ii regions @xcite , and planetary nebulae @xcite . among the previously reported lines , on the other hand , the o i 1.1286 , 1.1287 , [ ] 1.1881 , and [ ] 1.430 lines are not detected in our spectra . @xcite reported detection of the o i lines in the fmks in cas a , but we were unable to confirm the detection with our spectra . they also reported detection of the highly ionized [ ] 1.430 line , but again we were unable to confirm the detection . the [ ] 1.1881 line was included in the list of identified lines in three qsfs in cas a and the kepler snr by @xcite , but we consider that this was a misidentification of the [ ] 1.1883 line . the expected flux of the [ ] 1.1881 line in typical conditions ( e.g. , @xmath28 k , @xmath29 @xmath30 ) in snrs is almost negligible , i.e. , its flux relative to the [ ] 1.257 line is @xmath31 , whereas the [ ] 1.1883 line could be as strong as @xmath32 of the [ ] 1.257 line for cosmic abundance , or even higher if fe atoms are locked into dust grains @xcite . in order to systematically characterize the spectral properties of the 63 knots , which have 46 emission lines in total , we applied the pca method . pca measures the variances among the original input variables ( i.e. , brightness of the emission lines in this study ) and then sets new orthogonal axes called principal components ( pcs ) along the largest variances . therefore , the largest variance ( or information ) is contained in the first pc ( pc1 ) , the second most in pc2 , and so forth . if there are significant correlations among the original input variables , then the majority of the information is confined within the first few pcs , which makes it possible to categorize the objects into a few groups based on the first few pcs . before performing the pca , we apply an extinction correction using the line ratio of the [ ] 1.257 and 1.644 emission . these two lines originate from the same upper level ( @xmath33 ) and therefore their intrinsic flux ratio ( @xmath34_{\rm int}$ ] ) depends on the einstein a - coefficients ( @xmath35 ) and their wavelengths , i.e. , @xmath34_{\rm int}=(a_{ki,1.257}/1.257)/(a_{ki,1.644}/1.644)$ ] . the a - coefficients , however , are considerably uncertain so the line ratio ranges from 0.98 to 1.49 ( * ? ? ? * ; * ? ? ? * and references therein ) in literature . we adopt a line ratio of 1.36 , which is the value suggested by @xcite and @xcite . ( we found that the uncertainty of the theoretical line ratio does not affect our classification of the knots because they are well grouped in pc spaces , as we will show in section [ sec - pca - result - pc ] . the criteria of the groups , however , may change depending on the intrinsic line ratio we adopt . in section [ sec - pca - result - pc ] , we will describe this in more detail . ) then , by comparing the observed line flux ratio to the intrinsic ratio , we obtain the extinction of the knots and deredden the observed fluxes of all of the lines . we use the general interstellar extinction curve derived from a carbonaceous - silicate grain model with a milky way size distribution for @xmath36 @xcite . the derived extinctions are listed in table [ tbl - knot ] . in our previous work @xcite , we showed that the extinction toward the west is systematically larger than that toward the east , which is consistent with the previous optical / x - ray extinction estimates ( see figure 1 in * ? ? ? we further showed that the extinctions of red - shifted knots are systematically higher than those of the blue - shifted knots , implying the presence of a large amount of sn dust inside and around the main ejecta shell ( see * ? ? ? * for details ) . the lines from the same upper level in the dereddened spectral data do not provide independent information any more , so that the number of attributes in the pca are now reduced from 46 to 23 . in order to prevent a few bright lines dominating the pca , the line intensities are standardized by subtracting the mean and dividing by the standard deviation . since we use the mean - subtracted data , the zero pcs represent the location of mean brightness , not the location of the zero fluxes of the lines ( hereafter convergent point ) , which is important for our classification ( see section [ sec - pca - result - pc ] ) . in order to get the pc coefficients of a knot indicating the convergent point when all the emission lines of a knot get close to zero flux , we add one artificial knot into the data set that has emission lines with zero flux . table [ tbl - pca ] contains the relative and cumulative fraction of variances contributed by the 10 most significant pcs . the first three pcs account for the majority , i.e. , @xmath3785% , of the spectral information ; thus , we use them in our classification of the knots . figure [ fig - atr ] shows the projection of the 23 attributes on the combination plane of the three most significant pcs . ( this type of plots is known as an _ h - plot _ ; see @xcite and @xcite . ) while figure [ fig - atr](a ) and [ fig - atr](b ) show 2d projections on the planes of ( pc1 vs. pc2 ) and ( pc1 vs. pc3 ) , figure [ fig - atr](c ) shows 3d projections for the three pcs together . the lengths of the vectors in the plots are proportional to the fractional contributions by the spectral line to the given pc , and their quadratic sum is equal to unity . in figure [ fig - atr](d ) , we visualize the 3d projections using the coordinate of ( longitude vs. latitude ) representing the two projection angles on the surface of a sphere . we see in figure [ fig - atr ] that the attributes can be largely divided into three groups , each of which is composed of several strongly correlated spectral lines . ( note that the cosine of the angle between the vectors on the plots measures the linear correlation between the emission lines ; see @xcite for example . ) the first group ( hereafter ` he group ' ) is composed of h i , he i , and [ ] lines . these lines are almost perfectly correlated with each other . the lengths of their vectors are close to unity in the pc1-pc3 plane , which means that these spectral lines are properly accounted for by pc1 and pc3 . the second group ( ` s group ' ) is composed of [ ] , [ ] , and ionized s emission lines . these lines are also strongly correlated with each other and mostly contributed by pc2 and pc3 in the direction orthogonal to the he group lines . the last group ( ` fe group ' ) is composed of ionized fe emission lines , i.e. , [ ] and [ ] lines . these lines are rather loosely correlated but are still generally well separated from the lines in the other two groups . the [ ] 2.046 and 2.224 lines in particular appear somewhat distinct from the other [ ] lines . this might be caused by the higher excitation energies of the two lines than those of the other lines , i.e. , @xmath38 k vs. @xmath39 k. we plot the pc coefficients of the 63 knots on the pc planes in figure [ fig - obj ] . the central positions of the planes , where ( pc1 , pc2 , pc3 ) = ( 0 , 0 , 0 ) , represent the spectrum made by averaging all of the spectra of all 63 knots . in the lower panels of the figure , which are the enlarged plots of the central areas of the pc planes , we draw dashed lines in order to group the knots ( see below ) originating from ( pc1 , pc2 , pc3 ) = ( -0.10 , 0.23 , 0.10 ) . this convergent point is the location of the virtual knot with zero flux ( section [ sec - pca - method ] ) , and the radial distance from the convergent point is proportional to the brightness of the emission line . the distributions of the pc coefficients in figure [ fig - obj ] are very similar to those in figure [ fig - atr ] . there appear to be three groups of knots in figure [ fig - obj ] that have pc coefficients similar to those of the three groups in figure [ fig - atr ] , i.e. , the he , s , and fe groups . we therefore group the knots in figure [ fig - obj ] as he - rich , s - rich , and fe - rich knots using the dashed lines . as a result , we identify 7 he - rich knots , 45 s - rich knots , and 11 fe - rich knots . figure [ fig-1dspec ] shows sample 1d spectra of the three groups . as expected , the he - rich knots have strong lines of he i 1.083 compared to [ ] and some of them also show several emission lines of h i , [ ] , [ ] , and [ ] as well . the h i and [ ] lines are detected only in the he - rich knots . the s - rich knots also have bright [ ] lines , but they have much stronger lines of [ ] 0.953 , [ ] 1.03 multiplets , [ ] 1.188 , and [ ] 1.963 . the low - ionized emission lines of refractory elements , i.e. , [ ] and [ ] , are also detected in several s - rich knots . although some s - rich knots show the he i 1.083 line , their intensities are significantly less than those of the he - rich knots . the fe - rich knots have strong lines of [ ] and some of them have a weak he i 1.083 line as well . the brightest fe - rich knot , knot 10 in slit 5 , also emits [ ] lines in the _ k _ band . as in the he - rich knots , no lines of c , si , and s are detected in the fe - rich knots . the three knot groups are easily distinguishable when the flux ratios of [ ] 1.03 multiplets , he i 1.083 , and [ ] 1.188 are compared ( figure [ fig - fluxratio ] ) . the he - rich knots are well separated from the other groups in f(he i-1.083)/f([]-1.644 ) , i.e. , the ratio is greater than two for the he - rich knots but lower than two for the s - rich and fe - rich knots . the f([]-1.03)/f([]-1.644 ) and f([]-1.188)/f([]-1.644 ) of the he - rich knots are mostly smaller than a few times 1.0 and 0.1 , respectively . although the s - rich and fe - rich knots are not clearly distinguished in the f([]-1.03)/f([]-1.644 ) and f([]-1.188)/f([]-1.644 ) comparisons , we see that these ratios are higher in the s - rich knots than the fe - rich knots , i.e. , f([]-1.03)/f([]-1.644 ) @xmath40 and f([]-1.188)/f([]-1.644 ) @xmath41 for the s - rich knots and vice versa . it is worth noting that the flux ratios of the knots in figure [ fig - fluxratio ] and the criteria mentioned above are based on the assumption that the intrinsic flux ratio of [ ] 1.257 and 1.644 is 1.36 ( section [ sec - pca - method ] ) . if we adopt different line ratios , e.g. , 0.98 to 1.49 ( * ? ? ? * ; * ? ? ? * and references therein ) , then the criteria of those three flux ratios will be f(he i-1.083)/f([]-1.644 ) = 1.02.4 , f([]-1.03)/f([]-1.644 ) = 2.36.0 , and f([]-1.188)/f([]-1.644 ) = 0.190.34 . figure [ fig - hist ] compares the distributions of the knot sizes , radial velocities , and line widths of the three knot groups . the angular sizes of most of the knots are in the range 27(or 0.030.1 pc at a distance of 3.4 kpc ) and there is no significant difference among the three groups in their sizes , although some of the s - rich and fe - rich knots are as large as 10 . on the other hand , there are significant differences in their radial velocities and line widths . the radial speeds of he - rich knots are @xmath42 , while those of s - rich knots range from @xmath43 to @xmath44 with a median of @xmath45 . the radial velocities of fe - rich knots range from @xmath46 to @xmath47 with a median of @xmath48 . in line width , the he - rich knots have widths of 510 , while the s- and fe - rich knots have widths of 1035 . ( note that our spectral resolution at 1.64 is @xmath49 . ) figure [ fig - hist ] also compares the distribution of the [ ] 1.644 line fluxes among the three knot groups . while the s - rich and fe - rich knots have a similar distribution with an increased number of knots that have faint [ ] emission , the pattern is absent in the he - rich knots . there is no apparent correlation among the four physical parameters of the knots . one of the physical parameters of the knots that can be straightforwardly obtained is electron density using [ ] lines originating from levels with similar excitation energies , because their ratios are mainly determined by electron densities ( e.g. , * ? ? ? [ ] 1.644 and 1.677 are such lines , and figure [ fig - ne ] compares their expected ratios as a function of density for the assumed temperatures of 5000 , 10,000 , and 20,000 k ( left panel ) with observed values as a function of the [ ] 1.644 flux ( right panel ) . we see that the ratio is quite insensitive to temperature and can be used to estimate electron density in the range @xmath50@xmath51 . the electron density of the he - rich knots is a few @xmath52 , while the s - rich knots show electron densities over a broad range of @xmath53 to @xmath51 . the fe - rich knots have somewhat lower ( @xmath54@xmath52 ) electron densities compared to the other two groups . there appears to be no correlation between the electron densities and [ ] 1.644 line fluxes . in this section , we discuss the origin of the knots using their spectral characteristics described in the previous section . the he - rich and s - rich knots have quite distinct spectral properties ; the he - rich knots have high f(he i-1.083)/f([]-1.644 ) and low f([]-1.03)/f([]-1.644 ) , f([]-1.188)/f([]-1.644 ) , while the s - rich knots have low f(he i-1.083)/f([]-1.644 ) and high f([]-1.03)/f([]-1.644 ) , f([]-1.188)/f([]-1.644 ) . their kinematic properties are also quite different ; the he - rich knots have low ( @xmath42 ) line - of - sight speeds , while the s - rich knots have high ( up to @xmath55 ) line - of - sight speeds . these spectral and kinematical properties suggest that the he - rich knots are dense , slow - moving csm swept up by the sn blast wave , while the s - rich knots are fast - moving sn ejecta that have been shocked . the same conclusion was reached by @xcite , who performed an abundance analysis using [ ] 1.188 and [ ] 1.257 lines . they showed that the relative abundance of p ( a major product of the stellar ne - burning layer ) to fe ( in number ) for the he - rich knots is similar to the solar abundance , whereas that of the s - rich knots is 10100 times higher than the solar abundance . the characteristics of the two types of knots match well with those of qsfs and fmks known from previous optical studies ( see section [ sec - int ] for a summary of their properties ) . similar to the optical qsfs , the he - rich knots have bright he i lines together with h i and [ ] lines ; all the he - rich knots have a he i 1.083 line , 6 out of the 7 show h i pa@xmath56 , br@xmath57 lines , and the three brightest ones also have [ ] 1.040 , 1.041 lines . like the optical fmks dominated by lines of ionized heavy elements o , s , ar , the s - rich knots show strong s lines plus [ ] and [ ] lines ; all the s - rich knots have [ ] 1.03 multiplets , and 39 out of 45 s - rich knots also have [ ] 1.188 and/or [ ] 1.963 lines . indeed , the overall spectra of he - rich and s - rich knots are similar to the nir spectra of qsfs and fmks obtained by @xcite , respectively . furthermore , the radial velocities of the two nir knot groups are well consistent with those of the optical groups . for example , the radial velocity of the he - rich knots is @xmath58 , while the generally accepted radial velocity of qsfs is @xmath59 @xcite . in addition , the median line - of - sight velocity of s - rich knots is @xmath60 , while the systematic velocity of the fmks is @xmath61 @xcite . rich emission lines of si , p , and s in the s - rich knots and fmks imply that they are the sn ejecta originated from the ne- and o - burning layers . the two bright emission lines , [ ] 1.188 and [ ] 1.644 , have comparable excitation energies and critical densities , and so their line ratios are strongly dependent on their abundance ratio , @xmath62(p / fe ) @xcite . as seen in figure [ fig - fluxratio ] , there is a large scatter in this line ratio for s - rich knots , which implies that the abundance ratio @xmath62(p / fe ) varies almost two orders of magnitude for these knots . we also found that 13 out of 45 s - rich knots have clear but relatively weak emission lines of he i and/or [ ] . the detection of the he , c , and fe lines in the s - rich ejecta , which are either lighter or heavier elements than the o - burning materials , might infer microscopic mixing during the sn explosion . in many s - rich knots , a highly ionized si line , [ ] 1.963 , is detected , while in a few s - rich knots , a [ ] 1.645 line is also detected . the detection of si in very different ionization stages indicates the broad range of temperatures in the s - rich knots . in section [ sec - pca - result ] , we found that the fe - rich knots exhibit intermediate characteristics between he - rich and s - rich knots ; they emit strong [ ] lines without any si , p , and s lines , but have high line - of - sight speeds of up to @xmath63 a few knots also emit an he i 1.083 line . the high velocities , however , suggest that they are not dense qsfs represented by the he - rich knots . their line widths are also considerably broader than those of he - rich knots , i.e. , 1035 vs. 510 ( figure [ fig - hist ] ) . on the other hand , the missing si , p , and s lines indicate that the abundances of these ne- and o - burning elements are very low in these fe - rich knots . we can consider two possible explanations regarding the origin of the fe - rich knots : ( 1 ) swept - up csm around contact discontinuity ( cd ) or ( 2 ) shocked sn ejecta enriched with fe elements that had been synthesized in explosive si burning . the ambient medium that the cas a sn blast wave is propagating into is believed to be csm with an @xmath64 density distribution ( e.g. , * ? ? ? * ) . in such a case , 1d similarity solutions show that the shocked csm accumulates at the cd with infinite density asymptotically @xcite . we thus expect `` dense '' csm expanding at a speed comparable to the shocked sn material . in the real situation , however , this interacting region between the shocked sn ejecta and the shocked csm is hydrodynamically unstable and distorted , with the density of the shocked csm limited to @xmath65 times the density at the ambient shock @xcite . the temperature of the shocked csm near the cd , therefore , may be lower than the typical temperature ( @xmath66 kev ) of the shocked csm @xcite , but probably no more than a factor of 10 , and all of the fe in the shocked csm will be in high ionization stages . furthermore , h i lines are not detected in all fe - rich knots with an upper limit of f(h i - pa@xmath56)/f([]-1.257 ) @xmath67 . note that the observed ratio of these line intensities ranges between 0.05 and 10 for snrs while it is @xmath68 for orion , which should be the typical ratios for shocked and photoionized gases of cosmic abundance , respectively @xcite . therefore , h must be depleted in fe - rich knots . the non - detection of he and n lines might also indicate that the abundances of these ` circumstellar ' elements are very low in fe - rich knots , although this needs to be confirmed from other waveband observations . ( see the next paragraph for an explanation of the faint he lines detected in some fe - rich knots . ) we may therefore conclude that the fe - rich knots are not likely the shocked csm . this leaves the second possibility , i.e. , the fe - rich knots are fe - enriched sn ejecta . the high velocities and large velocity widths are consistent with sn ejecta being swept up by the reverse shock . the low [ ] and [ ] fluxes compared to the [ ] flux , however , implies that the abundances of p and s , which are ne- and o - burning materials , are very low , which is in sharp contrast to the s - rich knots . these characteristics strongly suggest that _ the fe - rich knots are most likely `` pure '' fe ejecta synthesized in the deepest stellar interior . _ the weak he i 1.083 line detected in some fe - rich knots could be due to an @xmath9-rich freeze - out process during the explosive si burning ; just after the explosion , complete si burning with an @xmath9-rich freeze - out occurs under high temperatures and low density conditions in the stellar deep layer , and many alpha particles are `` frozen out '' without participating in further nucleosynthetic processes @xcite . similar dense , fe - predominant ejecta , likely from the @xmath9-rich freeze - out process , have been detected in another young core - collapse snr g11.2 - 0.3 @xcite . pure fe ejecta have been detected in x - rays ( see below ) but not in optical or nir wavebands . this is surprising considering the extensive optical / nir studies carried out for cas a since its discovery . figure [ fig - knotpos ] partly gives an answer . in the right panel of figure [ fig - knotpos ] , red is an [ ] 1.644 narrow - band image while green and blue are _ hubble space telescope _ ( _ hst _ ) acs / wfc f850lp and f775w images which are dominated by ionized s and o lines , respectively @xcite . previous optical observations had been mostly toward the northern ejecta shell bright in ionized o , s , and ar lines ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) or toward fmks outside the main shell ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? figure [ fig - knotpos ] , however , shows that the southwestern ( sw ) main ejecta shell , which is bright in the [ ] line but faint in the ionized o and s lines , is the region where fe - rich ejecta can be found . indeed , our result shown in the left panel of figure [ fig - knotpos ] confirms this ; 9 out of 11 fe - rich knots are located in slits 46 . ( note that the compact red knots in the interior and beyond the sw shell are mostly qsfs , and they are bright in [ ] 6548 , 6583 and h@xmath9 images too @xcite . ) the slit positions are determined from an [ ] 1.644 image , so that some of them were placed toward the red portions of the main ejecta shell and , by decomposing the emission into individual velocity components , we were able to identify fe - rich ejecta . it is worth noting that @xcite also noted the bright [ ] emission in the sw shell in their [ ] 1.644 image of cas a. meanwhile , we can see that the [ ] 17.9 emission is much brighter than the emission from o - burning elements such as ar and s in the sw shell in the _ spitzer _ mid - infrared maps of ionic lines @xcite . our result suggests that this [ ] emission - predominant area , i.e. , the red area of the sw ejecta shell in figure [ fig - knotpos ] , is probably mainly composed of fe ejecta . it is not easy to identify the counterpart of fe - rich knots in optical images because several velocity components are usually superposed along the line of sight toward the main ejecta shell . the brightest fe - rich knot ( knot 10 in slit 5 ; hereafter k10 ) , however , is somewhat isolated and we can identify its counterpart . in figure [ fig - k10 ] , the upper two images are [ ] 1.644 images at different epochs and they show that k10 is a clump of @xmath69 elongated along the slit . the two [ ] images clearly show that the clump is moving fast tangentially . the proper motion is measured @xmath70 yr@xmath22 , implying a tangential velocity of @xmath71 . in the lower f775w and f850lp images , we see diffuse faint emission at the position of k10 ( see the red contour ) . its brightness distribution is different , with two small ( @xmath72 ) bright spots ( s1 and s2 in the figure ) in the lower part of the clump . one of these bright spots , s1 , is coincident with an s - rich knot ( knot 9 in slit 5 ) , which is spatially coincident with k10 but has a line - of - sight velocity ( @xmath47 ) that is very different from that of k10 ( @xmath73 ) . the other bright spot , s2 , must also be due to an s - rich knot not detected in our spectroscopy . so excluding these two bright knots , k10 appears faint in f775w and f850lp images . we suspect that most of the diffuse emission in the f775w and f850lp images is due to optical [ ] lines . this can be confirmed by optical spectroscopy . recently , there have been optical spectral mapping observations of cas a @xcite and , in principle , a similar analysis can be done to detect fe - rich knots , although the optical [ ] lines , e.g. , [ ] 7155 and 8617 lines , will be much fainter than the [ ] 1.644 line because of large interstellar extinction ( @xmath7410 mag ) toward cas a ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . the distribution and amount of fe ejecta in cas a have been a subject of controversy . previous x - ray studies detected hot and diffuse `` pure '' fe ejecta with mass @xmath75 that might have formed by @xmath9-rich freeze - out during the complete si burning @xcite . these x - ray - emitting shocked fe ejecta are distributed mainly in the southeastern and northern regions of the remnant ( figure [ fig - fedist ] ) . on the other hand , the hard x - ray emission from the radioactive decay of @xmath2ti has been detected in the interior of the main ejecta shell ( figure [ fig - fedist ] ; * ? ? ? * ) . since @xmath2ti is essentially synthesized in complete si burning with @xmath9-rich freeze - out in the innermost region ( e.g. , * ? ? ? * ) , @xmath2ti traces `` pure '' @xmath76ni or its stable nuclei @xmath76fe . the majority of the observed @xmath2ti is inside the reverse shock and therefore from unshocked fe ejecta . the inferred mass of the unshocked fe ejecta is @xmath75 @xcite . such unshocked fe ejecta , however , have not yet been detected , presumably because they are cool ( @xmath77 k ; e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . instead , the x - ray - emitting fe ejecta located just outside the @xmath2ti emission was attributed to the corresponding shocked ejecta ( figure [ fig - fedist ] ; * ? ? ? however , the missing x - ray - emitting fe ejecta toward the south and northeast directions from the explosion center have been puzzling . we note that the nir [ ] -bright , red portion of the sw shell appears to be in contact with the interior @xmath2ti - emitting region ( figure [ fig - fedist ] ) . this might be the case for the small red portion near slit 2 in the northeastern shell , too . therefore , if these [ ] -bright regions are composed of shocked , dense fe ejecta , as implied from our spectroscopic result , it explains why we do not see x - ray - emitting diffuse fe ejecta toward these directions ; the unshocked `` pure '' fe ejecta traced by the radioactive @xmath2ti emission in the interior of the main shell is composed of both dense and diffuse ejecta , and when these are swept up by a reverse shock , we observe either nir [ ] emission or x - ray emission depending on their densities . we do not see nir emission associated with the central bright @xmath2ti emission but this could be because the shock is face - on , as suggested by the large red - shifted central velocity ( 11003000 @xmath78 ) of the @xmath2ti line @xcite , or because the reverse shock has not yet reached the dense , unshocked fe ejecta . recent multi - dimensional simulations show that such global asymmetry in fe ( or @xmath76ni ) density can arise from the low - mode convection of the innermost region just after the core bounce ( e.g. , * ? ? ? future nir spectral mapping observations revealing the 3d distribution of the dense fe ejecta will be helpful for understanding the sn explosion dynamics of the innermost region . we have carried out nir spectroscopic observations toward the main ejecta shell of the young snr cas a. in total , 63 individual knots were identified from eight slit positions by using a clump - finding algorithm . each of these knots has distinct kinematical and spectral properties . within the _ jhk _ spectral range ( 0.942.46 ) , we found 46 emission line features including a dozen bright [ ] lines , forbidden lines of other metallic species , and h and he lines . we employed the pca method to classify the knots based on their relative line fluxes into three distinctive groups : he - rich knots of pre - supernova circumstellar wind material , plus s - rich and fe - rich knots of sn ejecta material . the he - rich and s - rich knots correspond to qsfs and fmks studied in the visible waveband , while fe - rich knots , showing in general only [ ] emission lines , are likely ` pure ' dense fe ejecta from the innermost layer of the progenitor . we summarize our main results as follows . the pca showed that the nir spectral lines can be grouped into three groups : ( 1 ) group 1 , composed of h i and he i lines together with [ ] lines , ( 2 ) group 2 , composed of forbidden lines of si , p , and s , and ( 3 ) group 3 , composed of forbidden fe lines . the lines in the first two pcs are strongly correlated with each other , while the correlation is rather weak among the forbidden fe lines in group 3 . these three spectral groups of the emission lines are almost independent in 3d pc space ( figure [ fig - atr ] ) . the distribution of the knots in the pc planes matches well with the above spectral groups , and we classified the knots into three groups : ( 1 ) he - rich , ( 2 ) s - rich , and ( 3 ) fe - rich knots . the knots belonging to these three groups are well separated from each other in f([]-1.03)/f([]-1.644 ) vs. f(he i-1.083)/f([]-1.644 ) plane ( figure [ fig - fluxratio ] ) , so that one may use these line ratios to classify the knots in cas a. it would be interesting to determine whether this classification methodology applies for other core - collapse snrs . \3 . the he - rich knots show bright emission lines of he i 1.083 and [ ] together with [ ] and h i lines . their line - of - sight speeds are small ( @xmath42 ) . from these chemical and kinematical characteristics , we conclude that the he - rich knots are dense csm swept up by the sn blast wave . these knots correspond to the previously known qsfs . the s - rich knots show strong forbidden lines of s together with [ ] and [ ] , and their line - of - sight speeds reach a few 1000 @xmath78 . these chemical and kinematical properties indicate that the s - rich knots are dense sn ejecta material mostly originating from the o - burning layers and swept up by a reverse shock . these knots correspond to the fmks detected in previous optical studies . the fe - rich knots only show strong [ ] and [ ] lines , and no or weak he i 1.083 lines . like the s - rich knots , they have large line - of - sight speeds ( up to @xmath63 ) and broad line widths ( 1035 ) , but they do not show the lines from si , p , and s. some fe - rich knots show he i 1.083 but their fluxes compared to the [ ] lines are much weaker than those of the he - rich knots . these spectroscopic properties suggest that the fe - rich knots are most likely `` pure '' dense fe ejecta from the innermost layer of the sn . the comparison of [ ] 1.644 images with the _ hst _ acs / wfc f850lp and f775w and _ nustar _ @xmath2ti images reveals that these fe ejecta are mainly distributed in the sw main ejecta shell , just outside the unshocked @xmath2ti in the interior . this supports that there could be a large amount of unshocked `` pure '' fe ejecta associated with @xmath2ti . together with the diffuse , x - ray - emitting `` pure '' fe ejecta detected by _ chandra _ , our result implies that the fe ejecta synthesized in the innermost region develop large - scale non - uniformity during the sn explosion and are expelled asymmetrically . this seems to be consistent with the low - mode , convection - driven sn explosion model . we wish to thank the anonymous referee for the very useful comments and suggestions which helped us improve the quality of the paper . we thank brian grefenstette and fiona harrison for providing the _ nustar _ data . we also want to thank john raymond and sung - chul yoon for helpful discussions . the interactive 3d figures were made by using asymptote which is a descriptive vector graphics language . this research was supported by basic science research program through the national research foundation of korea(nrf ) funded by the ministry of science , ict and future planning ( 2014r1a2a2a01002811 ) . cccrccr 2008 jun 29 & 1 & 23:23:28.54 + 58:50:32.2 & 0.0 & both & 50.50 & 300 @xmath79 4 + & 8 & 23:23:18.13 + 58:50:14.4 & 13.4 & ab & 51.00 & 300 @xmath79 5 + & 2 & 23:23:37.60 + 58:49:59.7 & 288.7 & ab & 50.25 & 300 @xmath79 4 + & st & 23:48:53.97 + 59:58:44.3 & 0.0 & ab & 30.00 & 30 @xmath79 6 + 2008 aug 08 & 3 & 23:23:41.11 + 58:48:45.2 & 36.1 & ab & 45.25 & 300 @xmath79 6 + & 4 & 23:23:24.77 + 58:47:12.2 & 12.0 & ab & 46.25 & 300 @xmath79 6 + & 5 & 23:23:18.50 + 58:47:25.1 & 307.3 & os & 30.00 & 300 @xmath79 1 + & 6 & 23:23:18.33 + 58:47:23.5 & 307.3 & os & 30.00 & 300 @xmath79 1 + & 7 & 23:23:13.57 + 58:47:49.2 & 5.2 & ab & 45.25 & 300 @xmath79 6 + & st & 23:48:53.97 + 59:58:44.3 & 0.0 & ab & 30.00 & 30 @xmath79 8 + cc|c|rcrrr 1 & 1 & 23:23:28.54 + 58:50:47.6 & 8.75 & s & 8.0 ( 0.5 ) & @xmath80 ( 2 ) & 589 ( 8) + 1 & 2 & 23:23:28.54 + 58:50:34.1 & 3.25 & s & 10.6 ( 0.7 ) & @xmath81 ( 2 ) & 165 ( 4 ) + 1 & 3 & 23:23:28.54 + 58:50:33.8 & 5.00 & he & 5.8 ( 0.1 ) & @xmath82 ( 2 ) & 793 ( 5 ) + 1 & 4 & 23:23:28.54 + 58:50:32.3 & 5.50 & s & 3.3 ( 2.0 ) & @xmath83 ( 17 ) & 96 ( 13 ) + 1 & 5 & 23:23:28.54 + 58:50:31.1 & 5.75 & s & 4.9 ( 0.2 ) & @xmath84 ( 2 ) & 383 ( 3 ) + 1 & 6 & 23:23:28.54 + 58:50:27.6 & 3.00 & s & 3.3 ( 0.5 ) & @xmath85 ( 3 ) & 132 ( 4 ) + 1 & 7 & 23:23:28.54 + 58:50:26.6 & 5.75 & fe & 9.8 ( 0.4 ) & @xmath86 ( 2 ) & 561 ( 10 ) + 1 & 8 & 23:23:28.54 + 58:50:25.8 & 3.00 & s & 3.4 ( 0.4 ) & @xmath87 ( 2 ) & 198 ( 5 ) + 2 & 1 & 23:23:35.19 + 58:50:06.0 & 4.50 & fe & 5.5 ( 0.9 ) & @xmath88 ( 6 ) & 204 ( 10 ) + 2 & 2 & 23:23:35.10 + 58:50:06.3 & 5.75 & s & 9.3 ( 3.6 ) & @xmath89 ( 14 ) & 75 ( 10 ) + 2 & 3 & 23:23:35.83 + 58:50:04.3 & 8.00 & s & 4.2 ( 1.7 ) & @xmath90 ( 20 ) & 140 ( 12 ) + 2 & 4 & 23:23:38.21 + 58:49:58.1 & 9.25 & s & 7.0 ( 0.5 ) & @xmath91 ( 3 ) & 480 ( 10 ) + 2 & 5 & 23:23:38.21 + 58:49:58.1 & 6.50 & s & 11.0 ( 1.0 ) & @xmath92 ( 3 ) & 339 ( 12 ) + 2 & 6 & 23:23:38.70 + 58:49:56.8 & 3.75 & s & 9.2 ( 1.7 ) & @xmath93 ( 2 ) & 164 ( 7 ) + 3 & 1 & 23:23:41.64 + 58:48:50.9 & 6.75 & s & 8.2 ( 0.3 ) & @xmath94 ( 3 ) & 460 ( 6 ) + 3 & 2 & 23:23:41.28 + 58:48:47.0 & 5.00 & s & 7.6 ( 0.3 ) & @xmath95 ( 3 ) & 310 ( 4 ) + 3 & 3 & 23:23:40.92 + 58:48:43.2 & 7.25 & s & 7.3 ( 0.1 ) & @xmath96 ( 2 ) & 972 ( 4 ) + 3 & 4 & 23:23:40.52 + 58:48:38.9 & 6.50 & s & 7.2 ( 0.3 ) & @xmath97 ( 4 ) & 434 ( 8) + 3 & 5 & 23:23:40.16 + 58:48:35.1 & 5.75 & s & 6.2 ( 1.2 ) & @xmath98 ( 7 ) & 121 ( 7 ) + 4 & 1 & 23:23:24.90 + 58:47:16.8 & 5.25 & fe & 15.4 ( 1.4 ) & @xmath99 ( 5 ) & 240 ( 6 ) + 4 & 2 & 23:23:24.82 + 58:47:13.9 & 3.75 & s & 10.6 ( 0.6 ) & @xmath100 ( 4 ) & 289 ( 4 ) + 4 & 3 & 23:23:24.77 + 58:47:12.2 & 5.75 & fe & 7.5 ( 0.4 ) & @xmath101 ( 3 ) & 298 ( 5 ) + 4 & 4 & 23:23:24.73 + 58:47:10.7 & 6.25 & s & 9.9 ( 0.3 ) & @xmath102 ( 3 ) & 606 ( 5 ) + 4 & 5 & 23:23:24.73 + 58:47:10.7 & 4.00 & fe & 6.6 ( 0.4 ) & @xmath103 ( 4 ) & 236 ( 4 ) + 4 & 6 & 23:23:24.72 + 58:47:10.2 & 5.75 & s & 10.3 ( 0.3 ) & @xmath104 ( 3 ) & 577 ( 6 ) + 5 & 1 & 23:23:17.21 + 58:47:32.7 & 5.25 & s & 11.5 ( 0.8 ) & @xmath105 ( 3 ) & 313 ( 7 ) + 5 & 2 & 23:23:17.28 + 58:47:32.3 & 5.25 & s & 8.8 ( 0.5 ) & @xmath106 ( 3 ) & 561 ( 10 ) + 5 & 3 & 23:23:17.34 + 58:47:32.0 & 4.50 & fe & 8.7 ( 0.9 ) & @xmath107 ( 4 ) & 286 ( 8) + 5 & 4a & 23:23:17.26 + 58:47:32.4 & 6.25 & s & 8.3 ( 0.3 ) & @xmath108 ( 3 ) & 537 ( 7 ) + 5 & 4b & 23:23:17.34 + 58:47:32.0 & 6.25 & he & 8.3 ( 0.1 ) & @xmath108 ( 3 ) & 2149 ( 7 ) + 5 & 5 & 23:23:17.62 + 58:47:30.3 & 5.25 & s & 9.3 ( 0.5 ) & @xmath109 ( 3 ) & 456 ( 6 ) + 5 & 6 & 23:23:17.88 + 58:47:28.8 & 4.75 & s & 8.8 ( 0.8 ) & @xmath110 ( 4 ) & 273 ( 8) + 5 & 7 & 23:23:18.00 + 58:47:28.0 & 3.75 & s & 7.2 ( 0.6 ) & @xmath111 ( 3 ) & 215 ( 5 ) + 5 & 8 & 23:23:18.72 + 58:47:23.8 & 5.00 & s & 10.8 ( 0.5 ) & @xmath112 ( 3 ) & 492 ( 9 ) + 5 & 9 & 23:23:19.41 + 58:47:19.7 & 1.75 & s & 11.9 ( 1.8 ) & @xmath113 ( 8) & 99 ( 5 ) + 5 & 10 & 23:23:19.49 + 58:47:19.3 & 9.75 & fe & 7.9 ( 0.0 ) & @xmath114 ( 3 ) & 4959 ( 8) + 6 & 1 & 23:23:17.02 + 58:47:31.3 & 3.50 & s & 6.6 ( 0.9 ) & @xmath115 ( 6 ) & 245 ( 8) + 6 & 2 & 23:23:17.07 + 58:47:31.0 & 3.25 & s & 9.0 ( 0.7 ) & @xmath116 ( 4 ) & 340 ( 7 ) + 6 & 3 & 23:23:16.94 + 58:47:31.7 & 4.25 & s & 10.2 ( 0.3 ) & @xmath117 ( 3 ) & 667 ( 7 ) + 6 & 4a & 23:23:17.22 + 58:47:30.0 & 2.75 & he & 10.0 ( 2.1 ) & @xmath118 ( 6 ) & 58 ( 6 ) + 6 & 4b & 23:23:17.30 + 58:47:29.6 & 2.75 & s & 10.4 ( 3.6 ) & @xmath118 ( 6 ) & 34 ( 6 ) + 6 & 5 & 23:23:17.22 + 58:47:30.0 & 3.00 & s & 7.4 ( 0.8 ) & @xmath119 ( 5 ) & 272 ( 6 ) + 6 & 6 & 23:23:17.53 + 58:47:28.2 & 6.25 & s & 10.4 ( 0.4 ) & @xmath120 ( 3 ) & 714 ( 8) + 6 & 7 & 23:23:17.99 + 58:47:25.5 & 4.50 & s & 7.8 ( 0.3 ) & @xmath121 ( 3 ) & 623 ( 8) + 6 & 8 & 23:23:18.47 + 58:47:22.6 & 2.75 & s & 9.3 ( 0.7 ) & @xmath122 ( 4 ) & 171 ( 5 ) + 6 & 9 & 23:23:18.75 + 58:47:21.0 & 2.75 & fe & 8.3 ( 0.5 ) & @xmath123 ( 4 ) & 258 ( 5 ) + 6 & 10 & 23:23:19.04 + 58:47:19.3 & 3.50 & fe & 7.3 ( 0.4 ) & @xmath124 ( 3 ) & 283 ( 5 ) + 6 & 11 & 23:23:19.29 + 58:47:17.8 & 6.50 & fe & 6.9 ( 0.2 ) & @xmath125 ( 3 ) & 676 ( 7 ) + 6 & 12 & 23:23:19.52 + 58:47:16.4 & 4.25 & fe & 9.9 ( 1.7 ) & @xmath126 ( 6 ) & 92 ( 6 ) + 7 & 1 & 23:23:13.66 + 58:47:56.4 & 3.25 & s & 10.2 ( 0.7 ) & @xmath127 ( 3 ) & 161 ( 3 ) + 7 & 2 & 23:23:13.62 + 58:47:53.2 & 5.50 & he & 8.4 ( 0.2 ) & @xmath128 ( 3 ) & 424 ( 6 ) + 7 & 3 & 23:23:13.51 + 58:47:44.2 & 4.00 & he & 8.5 ( 0.2 ) & @xmath129 ( 2 ) & 647 ( 9 ) + 7 & 4 & 23:23:13.49 + 58:47:42.0 & 2.75 & he & 8.5 ( 0.3 ) & @xmath130 ( 3 ) & 480 ( 10 ) + 7 & 5 & 23:23:13.46 + 58:47:39.7 & 3.25 & he & 9.1 ( 0.2 ) & @xmath131 ( 3 ) & 516 ( 7 ) + 8 & 1 & 23:23:18.56 + 58:50:28.1 & 3.25 & s & 8.1 ( 0.9 ) & @xmath132 ( 3 ) & 180 ( 7 ) + 8 & 2 & 23:23:18.43 + 58:50:24.0 & 6.00 & s & 7.6 ( 0.5 ) & @xmath133 ( 3 ) & 369 ( 9 ) + 8 & 3 & 23:23:18.22 + 58:50:17.1 & 6.25 & s & 7.3 ( 0.4 ) & @xmath134 ( 2 ) & 443 ( 9 ) + 8 & 4 & 23:23:18.07 + 58:50:12.3 & 5.25 & s & 9.2 ( 0.4 ) & @xmath135 ( 3 ) & 364 ( 6 ) + 8 & 5 & 23:23:17.93 + 58:50:07.7 & 4.50 & s & 10.8 ( 1.6 ) & @xmath136 ( 6 ) & 170 ( 8) + 8 & 6 & 23:23:17.88 + 58:50:06.2 & 4.00 & s & 13.6 ( 1.7 ) & @xmath137 ( 3 ) & 139 ( 5 ) + 8 & 7 & 23:23:17.76 + 58:50:02.3 & 4.75 & s & 9.4 ( 0.7 ) & @xmath138 ( 4 ) & 378 ( 10 ) + 8 & 8 & 23:23:17.76 + 58:50:02.1 & 4.00 & s & 6.8 ( 0.8 ) & @xmath139 ( 5 ) & 239 ( 12 ) + 8 & 9 & 23:23:17.67 + 58:49:59.1 & 4.00 & s & 8.1 ( 0.6 ) & @xmath140 ( 3 ) & 294 ( 6 ) + cc|lr|rr|r 1 & 1 & [ s iii ] @xmath141 - @xmath142 & 0.95311 & 7.2 ( 0.2 ) & 9233 ( 261 ) & + 1 & 1 & [ c i ] @xmath143 - @xmath142 & 0.98241 & @xmath144 ( @xmath144 ) & @xmath144 ( 15 ) & + 1 & 1 & [ c i ] @xmath141 - @xmath142 & 0.98503 & @xmath144 ( @xmath144 ) & @xmath144 ( 15 ) & + 1 & 1 & [ s ii ] @xmath145 - @xmath146 & 1.02867 & 8.4 ( 0.1 ) & 2317 ( 18 ) & line - fix + 1 & 1 & [ s ii ] @xmath147 - @xmath146 & 1.03205 & 8.4 ( - ) & 3182 ( - ) & line - fix + 1 & 1 & [ s ii ] @xmath145 - @xmath148 & 1.03364 & 8.4 ( - ) & 2185 ( 13 ) & line - fix + 1 & 1 & [ s ii ] @xmath147 - @xmath148 & 1.03705 & 8.4 ( - ) & 1058 ( - ) & line - fix + 1 & 1 & [ n i ] @xmath147 - @xmath149 & 1.03979 & @xmath144 ( @xmath144 ) & @xmath144 ( 12 ) & + 1 & 1 & [ n i ] @xmath145 - @xmath149 & 1.04074 & @xmath144 ( @xmath144 ) & @xmath144 ( 14 ) & + 1 & 1 & [ s i ] @xmath141 - @xmath142 & 1.08212 & 9.9 ( 0.5 ) & 207 ( 15 ) & + 1 & 1 & he i @xmath150 - @xmath151 & 1.08302 & @xmath144 ( @xmath144 ) & @xmath144 ( 11 ) & + 1 & 1 & h i pa@xmath57 & 1.09381 & @xmath144 ( @xmath144 ) & @xmath144 ( 9 ) & + 1 & 1 & [ s i ] @xmath143 - @xmath142 & 1.13059 & @xmath144 ( @xmath144 ) & @xmath144 ( 13 ) & + 1 & 1 & [ p ii ] @xmath143 - @xmath142 & 1.14682 & 7.8 ( 0.7 ) & 293 ( 48 ) & oh - cont + 1 & 1 & [ p ii ] @xmath141 - @xmath142 & 1.18828 & 9.8 ( 0.1 ) & 736 ( 15 ) & + 1 & 1 & [ fe ii ] @xmath152 - @xmath153 & 1.24854 & @xmath144 ( @xmath144 ) & @xmath144 ( 7 ) & + 1 & 1 & [ fe ii ] @xmath154 - @xmath155 & 1.25214 & @xmath144 ( @xmath144 ) & @xmath144 ( 8 ) & + 1 & 1 & [ fe ii ] @xmath156 - @xmath33 & 1.25668 & 10.2 ( 0.3 ) & 388 ( 15 ) & + 1 & 1 & [ fe ii ] @xmath157 - @xmath155 & 1.27035 & @xmath144 ( @xmath144 ) & @xmath144 ( 18 ) & + 1 & 1 & [ fe ii ] @xmath154 - @xmath158 & 1.27878 & 4.0 ( 0.6 ) & 30 ( 6 ) & + 1 & 1 & h i pa@xmath56 & 1.28181 & @xmath144 ( @xmath144 ) & @xmath144 ( 12 ) & + 1 & 1 & [ fe ii ] @xmath159 - @xmath153 & 1.29427 & 9.3 ( 1.0 ) & 133 ( 22 ) & oh - cont + 1 & 1 & [ fe ii ] @xmath157 - @xmath158 & 1.29777 & @xmath144 ( @xmath144 ) & @xmath144 ( 12 ) & + 1 & 1 & [ fe ii ] @xmath152 - @xmath33 & 1.32055 & 17.5 ( 2.6 ) & 92 ( 18 ) & + 1 & 1 & [ fe ii ] @xmath154 - @xmath153 & 1.32778 & 12.3 ( 2.1 ) & 67 ( 15 ) & + 1 & 1 & [ fe ii ] @xmath160 - @xmath153 & 1.53347 & 8.7 ( 0.9 ) & 95 ( 12 ) & oh - cont + 1 & 1 & [ fe ii ] @xmath161 - @xmath158 & 1.59947 & 9.8 ( 1.1 ) & 76 ( 11 ) & oh - cont + 1 & 1 & [ si i ] @xmath143 - @xmath142 & 1.60683 & @xmath144 ( @xmath144 ) & @xmath144 ( 5 ) & + 1 & 1 & [ fe ii ] @xmath160 - @xmath33 & 1.64355 & 13.7 ( 0.1 ) & 589 ( 8 ) & + 1 & 1 & [ si i ] @xmath141 - @xmath142 & 1.64545 & @xmath144 ( @xmath144 ) & @xmath144 ( 7 ) & + 1 & 1 & [ fe ii ] @xmath162 - @xmath155 & 1.66377 & 14.0 ( 1.5 ) & 73 ( 11 ) & oh - cont + 1 & 1 & [ fe ii ] @xmath161 - @xmath153 & 1.67688 & 15.4 ( 0.5 ) & 146 ( 8 ) & + 1 & 1 & [ fe ii ] @xmath162 - @xmath158 & 1.71113 & @xmath144 ( @xmath144 ) & @xmath144 ( 9 ) & + 1 & 1 & [ fe ii ] @xmath163 - @xmath155 & 1.74494 & @xmath144 ( @xmath144 ) & @xmath144 ( 5 ) & + 1 & 1 & [ fe ii ] @xmath163 - @xmath158 & 1.79710 & @xmath144 ( @xmath144 ) & @xmath144 ( 62 ) & + 1 & 1 & [ fe ii ] @xmath162 - @xmath153 & 1.80002 & @xmath144 ( @xmath144 ) & @xmath144 ( 240 ) & + 1 & 1 & [ fe ii ] @xmath161 - @xmath33 & 1.80939 & @xmath144 ( @xmath144 ) & @xmath144 ( 689 ) & + 1 & 1 & [ si vi ] @xmath146 - @xmath148 & 1.96287 & 16.4 ( 0.6 ) & 501 ( 27 ) & + 1 & 1 & [ fe ii ] @xmath164 - @xmath165 & 2.04601 & @xmath144 ( @xmath144 ) & @xmath144 ( 20 ) & + 1 & 1 & he i @xmath166 - @xmath167 & 2.05813 & @xmath144 ( @xmath144 ) & @xmath144 ( 12 ) & + 1 & 1 & [ fe ii ] @xmath168 - @xmath165 & 2.13277 & @xmath144 ( @xmath144 ) & @xmath144 ( 9 ) & + 1 & 1 & [ fe iii ] @xmath169 - @xmath170 & 2.14511 & @xmath144 ( @xmath144 ) & @xmath144 ( 10 ) & + 1 & 1 & h i br@xmath57 & 2.16553 & @xmath144 ( @xmath144 ) & @xmath144 ( 11 ) & + 1 & 1 & [ fe iii ] @xmath171 - @xmath172 & 2.21779 & @xmath144 ( @xmath144 ) & @xmath144 ( 13 ) & + 1 & 1 & [ fe ii ] @xmath173 - @xmath174 & 2.22379 & @xmath144 ( @xmath144 ) & @xmath144 ( 10 ) & + 1 & 1 & [ fe iii ] @xmath169 - @xmath175 & 2.24209 & @xmath144 ( @xmath144 ) & @xmath144 ( 13 ) & + ccrr 1 & 9.8 & 42.7 & 42.7 + 2 & 5.1 & 22.2 & 64.9 + 3 & 4.7 & 20.4 & 85.3 + 4 & 1.1 & 4.7 & 90.0 + 5 & 1.0 & 4.2 & 94.2 + 6 & 0.4 & 1.8 & 96.0 + 7 & 0.3 & 1.3 & 97.4 + 8 & 0.2 & 0.7 & 98.1 + 9 & 0.1 & 0.5 & 98.6 + 10 & 0.1 & 0.4 & 98.9 +

we report the results of broadband ( 0.952.46 ) near - infrared spectroscopic observations of the cassiopeia a supernova remnant . using a clump - finding algorithm in two - dimensional dispersed images , we identify 63 ` knots ' from eight slit positions and derive their spectroscopic properties . all of the knots emit [ ] lines together with other ionic forbidden lines of heavy elements , and some of them also emit h and he lines . we identify 46 emission line features in total from the 63 knots and measure their fluxes and radial velocities . the results of our analyses of the emission line features based on principal component analysis show that the knots can be classified into three groups : ( 1 ) he - rich , ( 2 ) s - rich , and ( 3 ) fe - rich knots . the he - rich knots have relatively small , @xmath0 , line - of - sight speeds and radiate strong he i and [ ] lines resembling closely optical quasi - stationary flocculi of circumstellar medium , while the s - rich knots show strong lines from o - burning material with large radial velocities up to @xmath1 indicating that they are supernova ejecta material known as fast - moving knots . the fe - rich knots also have large radial velocities but show no lines from o - burning material . we discuss the origin of the fe - rich knots and conclude that _ they are most likely `` pure '' fe ejecta synthesized in the innermost region during the supernova explosion . _ the comparison of [ ] images with other waveband images shows that these dense fe ejecta are mainly distributed along the southwestern shell just outside the unshocked @xmath2ti in the interior , supporting the presence of unshocked fe associated with @xmath2ti .

You are an expert at summarizing long articles. Proceed to summarize the following text: classification of networked data is a quite attractive field with applications in computer vision , bioinformatics , spam detection and text categorization . in recent years networked data have become widespread due to the increasing importance of social networks and other web - related applications . this growing interest is pushing researchers to find scalable algorithms for important practical applications of these problems . + in this paper we focus our attention on a task called _ node classification _ , often studied in the semi - supervised setting @xcite . recently , different teams studied the problem from a theoretic point of view with interesting results . for example @xcite developed on - line fast predictors for weighted and unweighted graphs and herbster et al . developed different versions of the perceptron algorithm to classify the nodes of a graph ( @xcite ) . @xcite introduced a game - theoretic framework for node classification . we adopt the same approach and , in particular , we obtain a scalable algorithm by finding a nash equilibrium on a special instance of their game . the main difference between our algorithm and theirs is the high scalability achieved by our approach . this is really important in practice , since it makes possible to use our algorithm on large scale problems . given a weighted graph @xmath0 , a labeling of @xmath1 is an assignment @xmath2 where @xmath3 . + we expect our graph to respect a notion of regularity where adjacent nodes often have the same label : this notion of regularity is called _ homophily_. most machine learning algorithms for node classification ( @xcite ) adopt this bias and exploit it to improve their performances . + the learner is given the graph @xmath1 , but just a subset of @xmath4 , that we call training set . the learner s goal is to predict the remaining labels minimizing the number of mistakes . @xcite introduce also an irregularity measure of the graph @xmath1 , for the labeling @xmath4 , defined as the ratio between the sum of the weights of the edges between nodes with different labels and the sum of all the weights . intuitively , we can view the weight of an edge as a similarity measure between two nodes , we expect highly similar nodes to have the same label and edges between nodes with different labels being `` light '' . based on this intuition , we may assign labels to non - training nodes so to minimize some function of the induced weighted cut . in the binary classification case , algorithms based on min - cut have been proposed in the past ( for example @xcite ) . generalizing this approach to the multiclass case , naturally takes us to the _ multi - way cut _ ( or multi - terminal cut see @xcite ) problem . given a graph and a list of terminal nodes , find a set of edges such that , once removed , each terminal belongs to a different component . the goal is to minimize the sum of the weights of the removed edges . + unfortunately , the multi - way cut problem is max snp - hard when the number of terminals is bigger than two ( @xcite ) . furthermore , efficient algorithms to find the multi - way cut on special instances of the problem are known , but , for example , it is not clear if it is possible to reduce a node classification problem on a tree to a multi - way cut on a tree . in this section we describe the game introduced by @xcite that , in a certain sense , aims at distributing over the nodes the cost of approximating the multi - way cut . this is done by expressing the labels assignment as a nash equilibrium . we have to keep in mind that , since this game is non - cooperative , each player maximizes its own payoff disregarding what it can do to maximize the sum of utilities of all the players ( the so - called social welfare ) . the value of the multi - way cut is strongly related to the value of the social welfare of the game , but in the general case a nash equilibrium does not give any guarantee about the collective result . + in the graph transduction game ( later called gtg ) , the graph topology is known in advance and we consider each node as a player . each possible label of the nodes is a pure strategy of the players . since we are working in a batch setting , we will have a train / test split that induces two different kind of players : * * determined players*(@xmath5 ) those are nodes with a known label ( train set ) , so in our game they will be players with a fixed strategy ( they do not change their strategy since we can not change the labels given as training set ) * * undetermined players*(@xmath6 ) those that do not have a fixed strategy and can choose whatever strategy they prefer ( we have to predict their labels ) the game is defined as @xmath7 , where @xmath8 is the set of players , @xmath9 is the joint strategy space ( the cartesian product of all strategy sets @xmath10 ) , and @xmath11 is the combined payoff function which assigns a real valued payoff @xmath12 to each pure strategy profile @xmath13 and player @xmath14 . a mixed strategy of player @xmath14 is a probability distribution @xmath15 over the set of the pure strategies of @xmath16 . each pure strategy @xmath17 corresponds to a mixed strategy where all the strategies but the @xmath17-th one have probability equals to zero . we define the utility function of the player @xmath16 as @xmath18 where @xmath19 is the probability of @xmath20 . we assume the payoff associated to each player is additively separable ( this will be clear in the following lines ) . this makes gtg a member of a subclass of the multi - player games called polymatrix games . for a pure strategy profile @xmath21 , the payoff function of every player @xmath14 is : @xmath22 where @xmath23 means that @xmath16 and @xmath24 are neighbors , this can be written in matrix form as @xmath25 where @xmath26 is the partial payoff matrix between @xmath16 and @xmath24 , defined as @xmath27 , where @xmath28 is the identity matrix of size @xmath29 and @xmath30 represent the element of @xmath31 at row @xmath15 and column @xmath4 . the utility function of each player @xmath32 can be re - written as follows : [ cols= " > , < " , ] the results of our experiments , shown in table [ t : multi ] , are not conclusive , but we can observe some interesting trends : * it is not really clear which one between gtg - ess and labprop is the most accurate algorithm , but anyway @xmath33 is always competitive with them . * @xmath33 is always much better than wmv . as expected wmv works better on `` not too sparse '' graphs such ghgraph , but even in this case it is outperformed by @xmath33 . * gtg - ess and labprop s time complexity did not permit us to run them in a reasonable amount of time with our computational resources . we introduced a novel scalable algorithm for multiclass node classification in arbitrary weighted graphs . our algorithm is motivated within a game theoretic framework , where test labels are expressed as the nash equilibrium of a certain game . in practice , mucca works well even on binary problems against competitors like label propagation and shazoo that have been specifically designed for the binary setting . several questions remain open . for example , committees of mucca predictors work well but we do not know whether there are better ways to aggregate their predictions . also , given their common game - theoretic background , it would be interesting to explore possible connections between committees of mucca predictors and gtg - ess .

we introduce a scalable algorithm , mucca for multiclass node classification in weighted graphs . unlike previously proposed methods for the same task , mucca works in time linear in the number of nodes . our approach is based on a game - theoretic formulation of the problem in which the test labels are expressed as a nash equilibrium of a certain game . however , in order to achieve scalability , we find the equilibrium on a spanning tree of the original graph . experiments on real - world data reveal that mucca is much faster than its competitors while achieving a similar predictive performance .

You are an expert at summarizing long articles. Proceed to summarize the following text: working out a quantitative description of the properties of dense strongly interacting matter produced in ultrarelativistic heavy ion collisions presents one of the most fascinating problems in high energy physics . the simplest ( albeit not unique ) way of putting the experimental data from rhic @xcite and lhc @xcite into a coherent framework is to describe the essential physics of these collisions as a hydrodynamical expansion of primordial quark - gluon matter that , after a short transient period , reaches sufficient level of local equilibration allowing the usage of hydrodynamics . the features of the experimentally observed energy flow , in particular the presence of a strong elliptic flow , suggest early equilibration of the initially produced matter and small shear viscosity of the expanding fluid , see e.g. the discussion in @xcite and @xcite devoted to rhic and lhc results respectively . can stylized features of primordial quark - gluon matter , in particular its anomalously low viscosity , be described within a weakly coupled theory , i.e. as a plasma composed of quasiparticles with the quantum numbers of quarks and gluons ? to address this question let us recall that extensive experimental studies of `` ordinary '' electromagnetic plasma has demonstrated , see e.g. @xcite , that it is practically never observed in the state of textbook thermal equilibrium . realistic description of the properties of experimentally observed qed plasma is possible only through taking into account the presence , in addition to thermal excitations , of randomly excited fields . the resulting state was termed _ turbulent plasma_. collective properties of turbulent plasmas are markedly different from those of the ordinary equilibrium plasmas . in particular , they are characterized by anomalously low shear viscosity and conductivity , dominant effects of coherent nonlinear structures on transport properties . thus it is natural to consider turbulent qcd plasma as a natural candidate for describing the primordial quark - gluon matter in the weak coupling regime . calculation of shear viscosity of turbulent qgp performed in @xcite has indeed demonstrated that its shear viscosity is anomalously small . in the present paper we focus on studying the leading turbulent contributions to polarization properties of turbulent relativistic plasma . for simplicity we restrict our consideration to the abelian case . the non - abelian generalization is briefly described in section [ conc ] . a weakly turbulent plasma is described as perturbation of an equilibrated system of ( quasi-)particles by weak turbulent fields @xmath0 . in the collisionless vlasov approximation , the plasma properties are defined by the following system of equations ( @xmath1 is a regular non - turbulent field ) : @xmath2f(p , x , q)=0 \nonumber\\ & & \partial^{\mu}\left ( f^r_{\mu \nu } + f^t_{\mu \nu } \right)= j_{\nu}(x ) = e \sum_{q , s}\int dp\ , p_{\nu}\ , q\ , f(p , x , q ) . \label{kinetic+maxw}\end{aligned}\ ] ] the stochastic ensemble of turbulent fields is assumed to be gaussian and characterized by the following correlators : @xmath3 in the present study we use the following parametrization of the two - point correlator @xmath4 @xcite : @xmath5\ ] ] turbulent polarization arises as a ( linear ) response to a regular perturbation that depends on turbulent fields . it is fully characterized by the polarization tensor @xmath6 defined as a variational derivative of the averaged induced current @xmath7 over the regular gauge potential @xmath8 : @xmath9 let us rewrite the kinetic equation in ( [ kinetic+maxw ] ) in the following condensed form @xmath10 where @xmath11 is a distribution function characterizing the original non - turbulent plasma and introduce the following systematic expansion in the turbulent and regular fields : @xmath12 where powers of @xmath13 count those of @xmath14 and powers of @xmath15 count those of @xmath16 . turbulent polarization is described by contributions of the first order in the regular and the second in the turbulent fields . the lowest nontrivial contribution to the induced current ( [ incur ] ) is thus given by @xmath17 . we have @xmath18 where @xmath19 generic expression for the polarization tensor taking into account turbulent effects can be written as @xmath20 where @xmath21 . both longitudinal and transverse components can be presented as a sum of hard thermal loops ( htl ) contributions and the gradient expansion in the turbulent scale @xmath22 : @xmath23 \label{gradexp } \nonumber\end{aligned}\ ] ] @xmath24 and the standard htl contribution @xmath25 , \nonumber \\ & & \pi_{t}^{\mathrm htl } ( \omega,{\left| \mathbf { k } \right|})= m^2_d \dfrac{x^2}{2 } \left[1+\dfrac{1}{2 x } \ ; ( 1-x^2 ) \ ; l(x ) \right ] \nonumber \\ & & l(x ) \equiv \ln\left|\dfrac{1+x}{1-x}\right|-\imath\pi\theta(1-x ) ; \;\;\ ; m^2_d = e^2 t^2/3 . \label{htl}\end{aligned}\ ] ] the computation of turbulent polarization was carried out to second order in the gradient expansion @xcite . in what follows we restrict ourselves to discussing the leading contribution to the imaginary part of the polarization function corresponding to the turbulent modification of landau damping in ( [ htl ] ) : @xmath26 the functions @xmath27 and @xmath28 are shown in fig . [ pct1 ] . $ ] ( solid lines ) and @xmath29 $ ] ( dashed lines ) . left : transverse response ; right : longitudinal response.,title="fig:",scaledwidth=45.0% ] $ ] ( solid lines ) and @xmath29 $ ] ( dashed lines ) . left : transverse response ; right : longitudinal response.,title="fig:",scaledwidth=45.0% ] the conclusions following from fig . [ pct1 ] can be formulated as follows : 1 . * timelike domain . * from fig . [ pct1 ] we see that the sign of the imaginary part of the turbulent contribution to the polarization operator in the timelike domain @xmath30 is negative and corresponds to turbulent damping of timelike collective excitations . this refers to both transverse and longitudinal modes . as the htl contribution in this domain is absent , this turbulent damping is a universal phenomenon present for all @xmath31 such that @xmath32 and all values of the parameters involved ( @xmath22 , @xmath33 , @xmath34 ) . the turbulent damping leads to an attenuation of the propagation of collective excitations at some characteristic distance . * spacelike domain . * the situation in the spacelike domain @xmath35 is more diverse . in contrast with the timelike domain the gradient expansion for the imaginary part of the polarization tensor starts from the negative htl contribution corresponding to landau damping . as seen from fig . [ pct1 ] the imaginary parts of turbulent contributions to the longitudinal polarization tensor are negative and are thus amplifying the landau damping . the most interesting contributions come from the turbulent contributions to the transverse polarization tensor . we see that the electric contribution @xmath36 $ ] in the spacelike domain is positive at all @xmath37 while the magnetic contribution @xmath38 $ ] is negative for @xmath39 and positive for @xmath40 . this means that the turbulent plasma becomes unstable for sufficiently strong turbulent fields . let us briefly discuss some relevant points : * 1 . * the above presented results are obtained in the framework of a perturbative expansion based on two crucial assumptions . first , one assumes slow temporal evolution of the distribution function due to particle interaction with turbulent fields thus neglecting the corresponding @xmath41 contributions . second , changes in the distribution function are treated as small . this , in turn , means that turbulent fields should be small enough . in this sense the reliable results refer to small modifications of landau damping but , as the onset of turbulent instability takes place for parametrically large fields , this result should be considered as a qualitative indication . the observed instability can be termed `` secondary '' because the turbulent fields themselves result from some `` first level '' instabilities . the origin of the effect is in turbulent stochastic inhomogeneity and thus similar to stochastic transition radiation , which vanishes in the limit @xmath42 ) @xcite . the non - abelian generalization of the above - described results for the imaginary part of the polarization tensor leads to identical expressions , the only difference being in trivial color factors , just as in the htl case . i. arsene , et al . ( brahms ) , _ nucl . phys . _ * a757 * ( 2005 ) , 1 + b.b . back , et al . ( phobos ) , _ nucl . * a757 * ( 2005 ) , 28 + j. adams , et al . ( star ) , _ nucl . phys . _ * a757 * ( 2005 ) , 102 + k. adcox , et al . ( phenix ) , _ nucl . phys . _ * a757 * ( 2005 ) , 184 v.v . tamoykin , _ astrophysics and space science _ ( 1972 ) , 120 + m.r . kirakosyan , a.v . leonidov , `` stochastic jet quenching in high energy nuclear collisions '' , arxiv:0810.5442 [ hep - ph ] + m.r . kirakosyan , a.v . leonidov , `` energy loss in stochastic abelian medium '' , proc . quarks 2008 , zagorsk , russia , arxiv:0809.2179 [ hep - ph ]

polarization properties of turbulent stochastically inhomogeneous ultrarelativistic qed plasma are studied . it is shown that the sign of nonlinear turbulent landau damping corresponds to an instability of the spacelike modes and , for sufficiently large turbulent fields , to an actual instability of a system .

You are an expert at summarizing long articles. Proceed to summarize the following text: given an input sequence , _ segmentation _ is the problem of identifying and assigning tags to its subsequences . many natural language processing ( nlp ) tasks can be cast into the segmentation problem , like named entity recognition @xcite , opinion extraction @xcite , and chinese word segmentation @xcite . properly representing _ segment _ is critical for good segmentation performance . widely used sequence labeling models like conditional random fields @xcite represent the contextual information of the segment boundary as a proxy to entire segment and achieve segmentation by labeling input units ( e.g. words or characters ) with boundary tags . compared with sequence labeling model , models that directly represent segment are attractive because they are not bounded by local tag dependencies and can effectively adopt segment - level information . semi - markov crf ( or semi - crf ) @xcite is one of the models that directly represent the entire segment . in semi - crf , the conditional probability of a semi - markov chain on the input sequence is explicitly modeled , whose each state corresponds to a subsequence of input units , which makes semi - crf a natural choice for segmentation problem . however , to achieve good segmentation performance , conventional semi - crf models require carefully hand - crafted features to represent the segment . recent years witness a trend of applying neural network models to nlp tasks . the key strengths of neural approaches in nlp are their ability for modeling the compositionality of language and learning distributed representation from large - scale unlabeled data . representing a segment with neural network is appealing in semi - crf because various neural network structures @xcite have been proposed to compose sequential inputs of a segment and the well - studied word embedding methods @xcite make it possible to learn entire segment representation from unlabeled data . in this paper , we combine neural network with semi - crf and make a thorough study on the problem of representing a segment in neural semi - crf . @xcite proposed a segmental recurrent neural network ( srnn ) which represents a segment by composing input units with rnn . we study alternative network structures besides the srnn . we also study segment - level representation using _ segment embedding _ which encodes the entire segment explicitly . we conduct extensive experiments on two typical nlp segmentation tasks : named entity recognition ( ner ) and chinese word segmentation ( cws ) . experimental results show that our concatenation alternative achieves comparable performance with the original srnn but runs 1.7 times faster and our neural semi - crf greatly benefits from the segment embeddings . in the ner experiments , our neural semi - crf model with segment embeddings achieves an improvement of 0.7 f - score over the baseline and the result is competitive with state - of - the - art systems . in the cws experiments , our model achieves more than 2.0 f - score improvements on average . on the pku and msr datasets , state - of - the - art f - scores of 95.67% and 97.58% are achieved respectively . we release our code at https://github.com/expresults/segrep-for-nn-semicrf . figure [ fig : ne - and - cws ] shows examples of named entity recognition and chinese word segmentation . for the input word sequence in the ner example , its segments ( _ `` michael jordan'':per , `` is'':none , `` a'':none , `` professor'':none , `` at'':none , `` berkeley'':org _ ) reveal that `` michaels jordan '' is a person name and `` berkeley '' is an organization . in the cws example , the subsequences ( utf8gkai``/pudong '' , `` /development '' , `` /and '' , `` /construction '' ) of the input character sequence are recognized as words . both ner and cws take an input sequence and partition it into disjoint subsequences . formally , for an input sequence @xmath0 of length @xmath1 , let @xmath2 denote its subsequence @xmath3 . segment _ of @xmath4 is defined as @xmath5 which means the subsequence @xmath6 is associated with label @xmath7 . a _ segmentation _ of @xmath4 is a _ segment _ sequence @xmath8 , where @xmath9 and @xmath10 . given an input sequence @xmath4 , the _ segmentation _ problem can be defined as the problem of finding @xmath4 s most probable _ segment _ sequence @xmath11 . 0.241 0.241 0.241 0.241 semi - markov crf ( or semi - crf , figure [ fig : std ] ) @xcite models the conditional probability of @xmath11 on @xmath4 as @xmath12 where @xmath13 is the feature function , @xmath14 is the weight vector and @xmath15 is the normalize factor of all possible _ segmentations _ @xmath16 over @xmath4 . by restricting the scope of feature function within a segment and ignoring label transition between segments ( 0-order semi - crf ) , @xmath13 can be decomposed as @xmath17 where @xmath18 maps segment @xmath19 into its representation . such decomposition allows using efficient dynamic programming algorithm for inference . to find the best segmentation in semi - crf , let @xmath20 denote the best segmentation ends with @xmath21^th^ input and @xmath20 is recursively calculated as @xmath22 where @xmath23 is the maximum length manually defined and @xmath24 is the transition weight for @xmath25 in which @xmath26 . previous semi - crf works @xcite parameterize @xmath27 as a sparse vector , each dimension of which represents the value of corresponding feature function . generally , these feature functions fall into two types : 1 ) the _ crf style features _ which represent input unit - level information such as `` the specific words at location @xmath28 '' 2 ) the _ semi - crf style features _ which represent segment - level information such as `` the length of the segment '' . @xcite proposed the segmental recurrent neural network model ( srnn , see figure [ fig : rnn ] ) which combines the semi - crf and the neural network model . in srnn , @xmath29 is parameterized as a bidirectional lstm ( bi - lstm ) . for a segment @xmath9 , each input unit @xmath30 in subsequence @xmath31 is encoded as _ embedding _ and fed into the bi - lstm . the rectified linear combination of the final hidden layers from bi - lstm is used as @xmath29 . @xcite pioneers in representing a segment in neural semi - crf . bi - lstm can be regarded as `` neuralized '' _ crf style features _ which model the input unit - level compositionality . however , in the srnn work , only the bi - lstm was employed without considering other input unit - level composition functions . what is more , the _ semi - crf styled _ segment - level information as an important representation was not studied . in the following sections , we first study alternative input unit - level composition functions ( [ sec : alt - inp - rep ] ) . then , we study the problem of representing a segment at segment - level ( [ sec : seg - rep ] ) . besides recurrent neural network ( rnn ) and its variants , another widely used neural network architecture for composing and representing variable - length input is the convolutional neural network ( cnn ) @xcite . in cnn , one or more filter functions are employed to convert a fix - width segment in sequence into one vector . with filter function `` sliding '' over the input sequence , contextual information is encoded . finally , a pooling function is used to merge the vectors into one . in this paper , we use a filter function of width 2 and max - pooling function to compose input units of a segment . following srnn , we name our cnn segment representation as scnn ( see figure [ fig : cnn ] ) . however , one problem of using cnn to compose input units into segment representation lies in the fact that the max - pooling function is insensitive to input position . two different segments sharing the same vocabulary can be treated without difference . in a cws example , utf8gkai`` '' ( racket for sell ) and `` '' ( ball audition ) will be encoded into the same vector in scnn if the vector of utf8gkai`` '' that produced by filter function is always preserved by max - pooling . concatenation is also widely used in neural network models to represent fixed - length input . although not designed to handle variable - length input , we see that in the inference of semi - crf , a maximum length @xmath23 is adopted , which make it possible to use padding technique to transform the variable - length representation problem into fixed - length of @xmath23 . meanwhile , concatenation preserves the positions of inputs because they are directly mapped into the certain positions in the resulting vector . in this paper , we study an alternative concatenation function to compose input units into segment representation , namely the sconcate model ( see figure [ fig : concate ] ) . compared with srnn , sconcate requires less computation when representing one segment , thus can speed up the inference . for segmentation problems , a segment is generally considered more informative and less ambiguous than an individual input . incorporating segment - level features usually lead performance improvement in previous semi - crf work . segment representations in section [ sec : alt - inp - rep ] only model the composition of input units . it can be expected that the segment embedding which encodes an entire subsequence as a vector can be an effective way for representing a segment . in this paper , we treat the segment embedding as a lookup - based representation , which retrieves the embedding table with the surface string of entire segment . with the entire segment properly embed , it is straightforward to combine the segment embedding with the composed vector from the input so that multi - level information of a segment is used in our model ( see figure [ fig : with - seg ] ) . however , how to obtain such embeddings is not a trivial problem . a natural solution for obtaining the segment embeddings can be collecting all the `` correct '' segments from training data into a lexicon and learning their embeddings as model parameters . however , the in - lexicon segment is a strong clue for a subsequence being a correct segment , which makes our model vulnerable to overfitting . unsupervised pre - training has been proved an effective technique for improving the robustness of neural network model @xcite . to mitigate the overfitting problem , we initialize our segment embeddings with the pre - trained one . word embedding gains a lot of research interest in recent years @xcite and is mainly carried on english texts which are naturally segmented . different from the word embedding works , our segment embedding requires large - scale segmented data , which can not be directly obtained . following @xcite which utilize automatically segmented data to enhance their model , we obtain the auto - segmented data with our neural semi - crf baselines ( srnn , scnn , and sconcate ) and use the auto - segmented data to learn our segment embeddings . another line of research shows that machine learning algorithms can be boosted by ensembling _ heterogeneous _ models . our neural semi - crf model can take knowledge from heterogeneous models by using the segment embeddings learned on the data segmented by the heterogeneous models . in this paper , we also obtain the auto - segmented data from a conventional crf model which utilizes hand - crafted sparse features . once obtaining the auto - segmented data , we learn the segment embeddings in the same with word embeddings . a problem that arises is the fine - tuning of segment embeddings . fine - tuning can learn a task - specific segment embeddings for the segments that occur in the training data , but it breaks their relations with the un - tuned out - of - vocabulary segments . figure [ fig : wo - ft ] illustrates this problem . since oov segments can affect the testing performance , we also try learning our model without fine - tuning the segment embeddings . in this section , we describe the detailed architecture for our neural semi - crf model . following @xcite , we use a bi - lstm to represent the input sequence . to obtain the input unit representation , we use the technique in @xcite and separately use two parts of input unit embeddings : the pre - trained embeddings @xmath32 without fine - tuning and fine - tuned embeddings @xmath33 . for the @xmath28th input , @xmath34 and @xmath35 are merged together through linear combination and form the input unit representation @xmath36 + b^\mathcal{i})\ ] ] where the notation of @xmath37 $ ] equals to @xmath38 s linear combination @xmath39 and @xmath40 is the bias . after obtaining the representation for each input unit , a sequence @xmath41 is fed to a bi - lstm . the hidden layer of forward lstm @xmath42 and backward lstm @xmath43 are combined as @xmath44+b^\mathcal{h})\ ] ] and used as the @xmath28^th^ input unit s final representation . given a segment @xmath9 , a generic function scomp@xmath45 stands for the segment representation that composes the input unit representations @xmath46 . in this work , scomp is instantiated with three different functions : srnn , scnn and sconcate . besides composing input units , we also employ the segment embeddings as segment - level representation . embedding of the segment @xmath9 is denoted as a generic function semb@xmath47 which converts the subsequence @xmath48 into its embedding through a lookup table . at last , the representation of segment @xmath19 is calculated as @xmath49+b^\mathcal{s})\ ] ] where @xmath50 is the embedding for the label of a segment . .hyper - parameter settings [ cols= " > , < " , ] table [ tbl : cws - stoa ] shows the comparison with the state - of - the - art cws systems . the first block of table [ tbl : cws - stoa ] shows the neural cws models and second block shows the non - neural models . our neural semi - crf model with multi - level segment representation achieves the state - of - the - art performance on pku and msr data . on ctb6 data , our model s performance is also close to @xcite which uses semi - supervised features extracted auto - segmented unlabeled data . according to @xcite , significant improvements can be achieved by replacing character embeddings with character - bigram embeddings . however we did nt employ this trick considering the unification of our model . semi - crf has been successfully used in many nlp tasks like information extraction @xcite , opinion extraction @xcite and chinese word segmentation @xcite . its combination with neural network is relatively less studied . to the best of our knowledge , our work is the first one that achieves state - of - the - art performance with neural semi - crf model . domain specific knowledge like capitalization has been proved effective in named entity recognition @xcite . segment - level abstraction like whether the segment matches a lexicon entry also leads performance improvement @xcite . to keep the simplicity of our model , we did nt employ such features in our ner experiments . but our model can easily take these features and it is hopeful the ner performance can be further improved . utilizing auto - segmented data to enhance chinese word segmentation has been studied in @xcite . however , only statistics features counted on the auto - segmented data was introduced to help to determine segment boundary and the entire segment was not considered in their work . our model explicitly uses the entire segment . in this paper , we systematically study the problem of representing a segment in neural semi - crf model . we propose a concatenation alternative for representing segment by composing input units which is equally accurate but runs faster than srnn . we also propose an effective way of incorporating segment embeddings as segment - level representation and it significantly improves the performance . experiments on named entity recognition and chinese word segmentation show that the neural semi - crf benefits from rich segment representation and achieves state - of - the - art performance . this work was supported by the national key basic research program of china via grant 2014cb340503 and the national natural science foundation of china ( nsfc ) via grant 61133012 and 61370164 . chris dyer , miguel ballesteros , wang ling , austin matthews , and noah a. smith . transition - based dependency parsing with stack long short - term memory . in _ acl-2015 _ , pages 334343 , beijing , china , july 2015 . acl . wenbin jiang , meng sun , yajuan l , yating yang , and qun liu . discriminative learning with natural annotations : word segmentation as a case study . in _ acl-2013 _ , pages 761769 , sofia , bulgaria , august 2013 . john d. lafferty , andrew mccallum , and fernando c. n. pereira . conditional random fields : probabilistic models for segmenting and labeling sequence data . in _ icml 01 _ , pages 282289 , san francisco , ca , usa , 2001 . daisuke okanohara , yusuke miyao , yoshimasa tsuruoka , and junichi tsujii . improving the scalability of semi - markov conditional random fields for named entity recognition . in _ acl-2006 _ , pages 465472 , sydney , australia , july 2006 . acl . xu sun , yaozhong zhang , takuya matsuzaki , yoshimasa tsuruoka , and junichi tsujii . a discriminative latent variable chinese segmenter with hybrid word / character information . in naacl-2009 _ , pages 5664 , boulder , colorado , june 2009 . yiou wang , junichi kazama , yoshimasa tsuruoka , wenliang chen , yujie zhang , and kentaro torisawa . improving chinese word segmentation and pos tagging with semi - supervised methods using large auto - analyzed data . in _ ijcnlp-2011 _ , pages 309317 , chiang mai , thailand , november 2011 .

many natural language processing ( nlp ) tasks can be generalized into segmentation problem . in this paper , we combine semi - crf with neural network to solve nlp segmentation tasks . our model represents a segment both by composing the input units and embedding the entire segment . we thoroughly study different composition functions and different segment embeddings . we conduct extensive experiments on two typical segmentation tasks : named entity recognition ( ner ) and chinese word segmentation ( cws ) . experimental results show that our neural semi - crf model benefits from representing the entire segment and achieves the state - of - the - art performance on cws benchmark dataset and competitive results on the conll03 dataset .

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Proceed to summarize the following text: we consider a piece of cortex @xmath5 ( the _ neural field _ ) , which is a regular compact subset when representing locations on the cortex , or periodic domains such as the torus of dimension 1 @xmath6 in the case of the representation of the visual field , in which neurons code for a specific orientation in the visual stimulus : in that model , @xmath5 is considered to be the feature space @xcite . ] of @xmath7 for some @xmath8 , and the density of neurons on @xmath5 is given by a probability measure @xmath9 assumed to be absolutely continuous with respect to lebesgue s measure @xmath10 on @xmath5 , with strictly positive and bounded density @xmath11 $ ] . on @xmath5 , we consider a spatially extended network composed of @xmath12 neurons at random locations @xmath13 drawn independently with law @xmath14 in a probability space @xmath15 , and we will denote by @xmath16 the expectation with respect to this probability space . a given neuron @xmath17 projects local connections in its neighborhood @xmath18 , and long - range connections over the whole neural field . we will consider here that the local microcircuit connectivity consists of a fully connected graph with @xmath1 nearest - neighbors . the synaptic weights corresponding to these connections are assumed equal to @xmath19 where @xmath20 ( it is generally positive since local interactions in the cortex tend to be excitatory ) . a central example is the case @xmath21 with @xmath22 . with zero probability , it may occur for a given neuron @xmath23 that its local microcircuit @xmath18 is not well defined . this occurs if there exists @xmath24 such that the number of neurons at distance strictly smaller than @xmath25 of neuron @xmath23 , denoted @xmath26 , is strictly smaller than @xmath0 and the number of neurons at a distance smaller or equal to @xmath25 is strictly larger than @xmath0 , meaning in particular that there exists several neurons at distance precisely @xmath25 . this event has a null probability , @xmath18 is defined as the union of all neurons at distance strictly smaller than @xmath25 , completed by @xmath27 neurons randomly chosen among those at distance exactly @xmath25 of neuron @xmath23 . the neurons also send non - local connections which are specific ( i.e. depend on the type of neurons , indexed here by the spatial location ) , which are much sparser than the local microcircuit . we will consider that the macro - connections are random variables @xmath28 drawn in @xmath15 and frozen during the evolution of the network , with law : @xmath29 where @xmath30 is a bernoulli random variable with parameter @xmath31 @xmath32 the coefficient @xmath33 governs the connectivity weight between neurons at location @xmath34 and @xmath35 . for instance , in the visual cortex , if the neurons of the cortical column at location @xmath34 codes for the collinear ( resp , orthogonal ) orientation as neurons in the column at @xmath35 , @xmath33 is positive ( negative ) . these coefficients are assumed to be smooth ( see assumption [ assump : spacecontinuity ] ) and bounded , and we denote : @xmath36 the scaling coefficient @xmath37 corresponds to the total incoming connections from the microcircuit related to neuron @xmath38 . the parameter @xmath31 accounts for the connectivity level of the macrocircuit . in particular , if populations are not connected , we will set @xmath39 . in that sense , the function @xmath31 does not account for all absent links in the network , but rather for the sparsity of the macro - circuit . motivated by the fact that the macro - circuit is very sparse and that micro - circuits form non - trivial patches of connectivity , we will assume that , when @xmath40 , @xmath41 the hypothesis on the connectivity ensure the following facts , desirable for a modeling at the neural field scale ( see fig . [ fig : neurons ] ) : * the local micro - circuit shrinks to a single point in the limit @xmath40 ( see lemma [ lem : sizemicro ] ) , and * the macro - circuit is sparse at the level of single cells ( @xmath42 ) , but non - sparse at the level of cortical columns ( @xmath43 ) . note that in all our developments , one only needs the assumption that @xmath44 as @xmath45 . this is of course a consequence of our current assumption . a schematic topology usually considered could be the 2-dimensional regular lattice @xmath46 approximating the unit square @xmath47 ^ 2 $ ] with @xmath48 points . in this model , typical micro - circuit size could be chosen to be @xmath49 with @xmath22 , and @xmath31 of order @xmath50 with @xmath51 . our model takes into account the fact that in reality , neurons are not regularly placed on the cortex , and therefore such a regular lattice case is extremely unlikely to arise ( this architecture has probability zero ) . moreover , in contrast with this more artificial example , the probability distribution of the location of one given neuron do not depend on the network size . in our setting , @xmath14 accounts for the density of neurons on the cortex , and as the network size is increased , new neurons are added on the neural field at locations independent of that of other neurons , with the same probability @xmath14 , so that neuron locations sample the asymptotic cell density . these elements describe the random topology of the network . prior to the evolution , a number of neurons @xmath12 and a configuration @xmath52 is drawn in the probability space @xmath15 . the configuration of the network provides : * the locations of the neurons @xmath53 i.i.d . with law @xmath14 * the connectivity weights , in particular the values of the i.i.d . bernoulli variables @xmath54 of parameter @xmath31 . let us start by analyzing the topology of the micro - circuit . at the macroscopic scale , we expect local micro - circuits to shrink to a single point in space , which would precisely correspond to the scale at which imaging techniques record the activity of the brain ( a pixel in the image ) . the micro - circuit connects a neuron to its @xmath0 nearest neighbors . we made the assumptions that @xmath0 tends to infinity as @xmath40 while keeping @xmath1 . this property ensures that for a fixed neuron @xmath55 and for any @xmath56 , the distance and @xmath57 , @xmath58 the euclidean norm of @xmath59 , regardless of the space involved and the dimension @xmath60 considered . ] @xmath61 is , with overwhelming probability , upperbounded by a constant multiplied by @xmath62 . in the regular lattice case , this property is trivial . in our random setting , we introduce the maximal distance between two neurons in the microcircuit associated to neuron @xmath23 is noted : @xmath63 this quantity has a law that is independent of the specific neuron @xmath23 chosen . [ lem : sizemicro ] the microcircuit shrinks to a single point in space as @xmath12 to infinity . more precisely , for any @xmath17 , the maximal distance @xmath64 between two neurons in the microcircuit associated to neuron @xmath23 decreases towards @xmath65 , in the sense that there exists @xmath66 such that the maximal distance between two points in a microcircuit satisfies the inequality : @xmath67 \leq c \left(\left(\frac{v(n)}{n}\right)^{\frac 1 d } + \frac{1}{v(n)}\right).\ ] ] we have assumed that the locations @xmath34 are iid with law @xmath14 absolutely continuous with respect to lebesgue s measure with density lowerbounded by some positive quantity @xmath68 . let us fix a neuron @xmath23 at location @xmath34 which is almost surely in the interior of @xmath5 . we are interested in the distances between different neurons within the microcircuit around @xmath23 , and will therefore consider the distribution of relative locations of neurons belonging to @xmath18 conditionally to the location @xmath34 of neuron @xmath23 . we will denote @xmath69 the expectation under this conditioning . it is clear that the set of random variables @xmath70 are identically distributed . moreover , these are independent conditionally on the value of @xmath34 . we will show that , for any neuron @xmath71 , the distance @xmath72 tends to zero as @xmath12 increases with probability one . to this end , we use the characterization of the maximal distance @xmath73 as the minimal radius @xmath74 such that the ball centered at @xmath34 with diameter @xmath74 contains @xmath0 points : @xmath75 we will show that there exists a quantity @xmath76 tending to zero such that @xmath77 with large probability , i.e. @xmath78 . to this end , we start by noting that conditionally on @xmath34 , the random variables @xmath79 are independent , identically distributed . moreover , for @xmath34 fixed , there exists @xmath80 such that for any @xmath81 , @xmath82 . therefore , for @xmath81 , the random variables @xmath83 are such that : @xmath84}=\int_{b(r_i,\alpha(n ) ) } d\lambda(r)\in [ \lambda_{min},\lambda_{max}]\times \gamma(n)\\ \text{var}_i(z_j^n ) = { \mathcal{e}}_i{[z_j^n ] } - { \mathcal{e}}_i{[z_j^n]}^2 \in [ \lambda_{\min } - \lambda_{\max } \gamma(n),\lambda_{\max}-\lambda_{\min } \gamma(n ) ] \times \gamma(n ) \end{cases}\ ] ] where @xmath85\times c = [ ac , bc]$ ] and @xmath86 with @xmath87 the volume of the unit ball in @xmath7 . the radius @xmath76 is chosen such that : @xmath88 with @xmath89 . this assumption implies that @xmath90\geq \eta^d \frac{v(n)}{n}\ ] ] we have : @xmath91}= \frac 1 { v(n)}\sum_{j=1}^n ( z_j^n-{\mathcal{e}}_{i}{[z_j^n ] } ) .\ ] ] which tends to zero in probability , since ( we recall that the variance is of order @xmath92 for @xmath12 large ) @xmath93}\right)\right)^2\right ] = \frac n { v^2(n ) } \text{var}_i(z_j^n ) = o \left(\frac 1 { v(n)}\right ) . \end{aligned}\ ] ] the quantity @xmath94 is therefore lowerbounded , with overwhelming probability , by @xmath95 which is , under our assumption on @xmath96 , is greater than @xmath97 for @xmath12 large enough : with overwhelming probability , for large @xmath12 , the microcircuit is fully included in the ball of radius @xmath76 . we have assumed that the set @xmath5 is bounded . let us denote by @xmath98 its diameter ( i.e. the maximal distance between two points in @xmath5 ) . we have : @xmath99 & = { \mathcal{e}}_i[d^n_m(i ) \mathbbm{1}_{d^n_m(i)\leq 2\alpha(n ) } ] + { \mathcal{e}}_i[d^n_m(i ) \mathbbm{1}_{d^n_m(i ) > 2\alpha(n)}]\\ & \leq 2\alpha(n ) + d(\gamma){\mathcal{p}}_i\left[m_n(\alpha(n))<v(n ) \right]\\ & \leq 2\alpha(n ) + d(\gamma){\mathcal{p}}_i\left [ \left \vert \frac{m_n(\alpha(n))}{v(n)}-\frac{n}{v(n ) } { \mathcal{e}}_i{[z_j^n ] } \right\vert > \eta^d-1 \right]\\ & \leq 2\alpha(n ) + d(\gamma)\frac{{\mathcal{e}}_i { \left[\left(\frac 1 { v(n)}\sum_{j=1}^n \left(z_j^n-{\mathcal{e}}_i{[z_j^n]}\right)\right)^2\right]}}{(\eta^d-1)^2 } \\ & \leq c \left(\left(\frac{v(n)}{n})\right)^{\frac 1 d } + \frac{1}{v(n)}\right ) \end{aligned}\ ] ] which ends the proof . let us now introduce the dynamics of the neurons activity . the state of each neuron @xmath23 in the network is described by a @xmath100-dimensional variable @xmath101 , typically corresponding to the membrane potential of the neuron and possibly additional variables such as those related to ionic concentrations and gated channels . these variables have a stochastic dynamics . in order to deal with these stochastic evolutions , we introduce a new complete probability space @xmath102 endowed with a filtration @xmath103 satisfying the usual conditions , and we denote by @xmath104 the expectation with respect to this probability space . note that this space is distinct from the configuration space @xmath15 . once a configuration @xmath52 is fixed for a @xmath12-neurons network , it is frozen and each neuron will have a random evolution following the equations : @xmath105 where @xmath106 governs the intrinsic dynamics of each cell , @xmath107 is a sequence of independent @xmath108 brownian motions of dimension @xmath109 modeling the external noise , @xmath110 a bounded and measurable function of @xmath111 modeling the level of noise at each space location , and @xmath112 the interaction function . the map @xmath113 is the interaction delay between neurons located at @xmath74 and those at @xmath114 which is assumed to be of the form : @xmath115 where @xmath116 is the synaptic transmission time and @xmath117 the transport time ( @xmath118 is the transmission speed assumed constant ) . since @xmath5 is bounded , all delays are bounded by a finite quantity @xmath119 ( in our notations , @xmath120 ) . in what follows , we will use the shorthand notation @xmath121 . the parameters of the system are assumed to satisfy the following assumptions : 1 . [ assump : loclipschspace ] @xmath122 is @xmath123-lipschitz - continuous with respect to all three variables , 2 . [ assump : loclipschbspace ] @xmath124 is @xmath125-lipschitz - continuous with respect to both variables and bounded . we denote @xmath126 3 . [ assump : lineargrowth ] the drift satisfies uniformly in space ( @xmath74 ) and time ( @xmath127 ) , the inequality : @xmath128 4 . [ assump : spacecontinuity ] the drift , delay , diffusion and connectivity functions are regular with respect to the space variables @xmath129 ( we will assume for instance that these are all @xmath130-lipschitz continuous ) . let us first state the following proposition ensuring well - posedness of the network system under the assumptions of the section : [ pro : existenceuniquenessnetwork ] let @xmath131}$ ] a square integrable process with values in @xmath132 . under the current assumptions , for any configuration @xmath52 of the network , there exists a unique strong solution to the network equations with initial condition @xmath133 . this solution is square integrable and defined for all times . the proof of this proposition is classical . it is a direct application of the general theory of sdes in infinite dimensions ( * ? ? ? * chapter 7 ) , and elementary proof in our particular case of delayed stochastic differential equations can be found in ( * ? ? ? * theorem 5.2.2 ) : for any fixed configuration , we have a regular @xmath12-dimensional sde with delays satisfying a monotone growth condition [ assump : lineargrowth ] ensuring a.s . boundedness for all times of the solution . the proof of this property is essentially based on the same arguments as those of the proof of theorem [ thm : existenceuniquenessspace ] , and the interested reader is invited to follow the steps of that demonstration . it is important to note that the bound one obtains on the expectation of the squared process depends on the configuration of the network . indeed , the macroscopic interaction term involves the sum of a random number of terms @xmath134 rescaled by @xmath135 . the quantity @xmath136 can take large values ( up to @xmath12 ) with positive ( but small ) probability , and therefore the scaling coefficient is not enough to properly control such cases . the bound obtained by classical methods will therefore diverge in @xmath12 , and this will be a deep question for our aim to prove convergence results as @xmath45 . in the present manuscript , we will be able to handle these terms properly in that limit by using fine estimates related to @xmath137 , see lemma [ lem : sumchi ] . we are interested in the limit , as @xmath45 , of the behavior of the neurons . since we are dealing with diffusions in random environment , there are at least two notions of convergence : _ quenched _ convergence results valid for almost all configuration @xmath138 , and _ annealed _ results valid for the law of the network averaged across all possible configurations . here , we will show averaged convergence results as well as quenched properties along subsequences . similarly to what was observed in @xcite , the limit of such spatially extended mean - field models will be stochastic processes indexed by the space variable , which , as a function of space , are not measurable with respect to the borel algebra @xmath139 . as noted in @xcite , this is not a mathematical artifact of the approach , since neurons accumulating on the neural field are driven by independent brownian motions , and therefore no regularity is to be expected in the limit . however , even if trajectories are highly irregular , this will not be the case of the law of these solutions . in order to handle this irregularity , we will use the _ spatially chaotic _ brownian motion on @xmath5 , a two - parameter process @xmath140 such that for any fixed @xmath111 , the process @xmath141 is a @xmath109-dimensional standard brownian motion , and for @xmath142 in @xmath5 , the processes @xmath143 and @xmath144 are independent of _ spatially chaotic _ if the processes @xmath145 and @xmath146 are independent for any @xmath142 . ] . this process is relatively singular seen as a spatio - temporal process : in particular , it is not measurable with respect to @xmath147 . the spatially chaotic brownian motion is distinct from other more usual spatio - temporal processes . in particular , its covariance is @xmath148}=(t\wedge t ' ) \delta_{r = r'}$ ] : the covariance is not measurable with respect to @xmath139 . in contrast , the more classical space - time brownian motion ( the process corresponding to space - time white noise differential terms ) on the positive line ( @xmath149 ) has a covariance @xmath150 : it is continuous with respect to space . it is also distinct from wiener processes on hilbert spaces ( * ? ? ? * chapter 4.1 . ) ( a.k.a . cylindrical brownian motions ) which have a covariance defined through a trace - class operator on the hilbert space , and may be decomposed as the sum of standard brownian motions on a basis of that hilbert space ( i.e. , there is a countable number of brownian motions involved ) . the chaotic brownian motion , due to his high singularity as a space - time process , is more suitably seen as an infinite collection of wiener processes . we will show that the network equations satisfies the propagation of chaos property in the limit where @xmath12 goes to infinity , and that the state of the network converges towards a very particular mckean - vlasov equation involving a spatially chaotic brownian motion . the propagation of chaos property ( boltzmann s molecular chaos hypothesis , or stozahlansatz ) states that , provided that the initial conditions of all neurons are independent , the law of any finite set of neurons converge to a product of laws ( loosely speaking , are asymptotically independent ) for all times ( see @xcite ) . in our network , this property means that the heuristic notion of boltzmann s stozahlansatz applies in that the dependence relationship between the state of , say 2 , fixed neurons in the network , dilute away in the thermodynamic limit , and these end up being independent in the large @xmath12 limit . in detail , for almost all configuration of the network , the asymptotic law of neurons located at @xmath74 in the support of @xmath14 will be measurable with respect to @xmath151 and converge towards the stochastic neural field mean - field equation with delays : @xmath152\,dt\\ + \int_{\gamma } j(r , r'){\mathbbm{e}}_{\bar{z}}[b(\bar{x}_t(r),\bar{z}_{t-\tau(r , r')}(r ' ) ) ] \ , d\lambda(r ' ) \,dt \end{gathered}\ ] ] where @xmath153 is a spatially chaotic brownian and the process @xmath154 is independent and has the same law as @xmath155 . in other words , we will show that the law of the solution @xmath156 , noted @xmath157 , is measurable with respect to @xmath147 , and that the mean - field equation can be expressed as the integro - differential mckean - vlasov equation : @xmath158 let us eventually give the fokker - planck equation on the possible density @xmath159 of @xmath160 with respect to lebesgue s measure : @xmath161 . \end{gathered}\ ] ] the mean - field equations are similar to those found in the setting of @xcite but present an additional term related to the presence of a micro - circuit , showing the local averaging effects arising in our setting . interestingly , this shows a kind of universal behavior across all possible choices of parameters @xmath0 and @xmath31 , i.e. across possible local statistics of the topology . the limit equations are very complex : similarly to what was discussed in @xcite , they resemble mckean - vlasov equations but involve delays , spatially chaotic brownian motions and an ` integral over spatial locations ' ( in a sense that will be made clearer in the sequel ) . this is hence a very unusual stochastic equation we need to thoroughly study in order to ensure that these make sense and are well - posed . the existence and uniqueness of solutions to this mean - field equation are addressed in section [ sec : existenceuniquenessspace ] , and the proof of the propagation of chaos and convergence of the network equations towards the solution of the mean - field equation is performed in section [ sec : propachaspace ] . the mean - field equation involves two unusual terms : a stochastic integral involving spatially chaotic brownian motions and an integrated mckean - vlasov mean - field term with delays . the spatially chaotic brownian motion was introduced in @xcite . in order to handle these equations , we start by introducing and discussing the functional spaces in which we are working , and the notion of solutions to these singular equations . first of all , in order to make sense of the mean - field equation , we need to show that the lebesgue s integral over @xmath5 term is well defined . this integral involves the expectation of the process , so even though the solution is not measurable with respect to space , its expectation may be depending on the regularity of its law with respect to space . in this view , we consider the set @xmath162 of spatially chaotic processes @xmath163 defined for times @xmath164 $ ] that have the following continuity property : there exists a _ coupled process _ @xmath165 indexed by @xmath111 , such that for any fixed @xmath111 , @xmath165 has the same law as @xmath156 , and moreover , there exists a constant @xmath66 such that for any @xmath166 : @xmath167}\left\vert \hat{x}_t(r)-\hat{x}_t(r')\right\vert \right]}\leq c ( \vert r - r'\vert + \sqrt{\vert r - r'\vert}).\ ] ] note that in this case , for any lipschitz - continuous function @xmath168 , the map @xmath169}$ ] is continuous ( hlder @xmath170 ) . in particular , it is measurable . one can then define the lebesgue s integral of it over @xmath5 . such processes are called _ chaotic processes with regular law_. we note that in the absence of space - dependent delays , the process is more regular ( lipschitz - continuous ) . moreover , stochastic processes @xmath171 are said square integrable for all @xmath111 if : @xmath172}<\infty.\ ] ] we further consider the subset @xmath173 composed of processes @xmath163 that satisfy the following regularity in time : there exists @xmath66 such that : @xmath174}\leq c ( \sqrt{\vert t - t'\vert } + \vert t - t'\vert).\ ] ] eventually , for a process @xmath175 , we define the squared norm : @xmath176}\vert{z_s({r})}\vert^2 \right]}\,d\lambda(r)\ ] ] and the @xmath177 norm : @xmath178}\vert{z_s({r})}\vert \right]}\,d\lambda(r).\ ] ] these clearly define norms on random variables indexed by @xmath111 , when identifying processes that are @xmath179-a.s . we denote @xmath180 the set of of random variables in @xmath162 such that @xmath181 . note that this norm depends on @xmath14 the distribution over @xmath5 of neurons . it is of course possible to define a norm using lebesgue s measure on @xmath5 , which would be in that case independent of the choice of @xmath14 . the two obtained measures will be of course equivalent since here we assumed that @xmath14 was equivalent to lebesgue s measure . let us start by giving a simple yet informative example of such process . let @xmath143 be a spatially chaotic brownian motion , and consider @xmath182 , r\in\gamma}$ ] a @xmath183-progressively measurable real - valued process indexed by @xmath184 that belongs to @xmath185 and which is independent of the collection of brownian motions @xmath186 . we denote by @xmath187 the coupled process corresponding to the regularity condition . we assume that for any @xmath188 we have @xmath189 } < c<\infty , \text { and}\\ { \mathbbm{e}\left [ \vert \hat{\delta}_t(r)-\hat{\delta}_t(r')\vert^2 \right]}\leq c^2 ( \vert r - r'\vert + \sqrt{\vert r - r'\vert})^2 \end{cases}\ ] ] since for any fixed @xmath111 , the process @xmath143 is a standard brownian motion , the process @xmath190 defined by the stochastic integral : @xmath191 is well defined . it is spatially chaotic since for @xmath142 the brownian motions @xmath143 and @xmath144 and the processes @xmath192 and @xmath193 are independent . moreover , they have a regular law in the sense of our definition . indeed , let @xmath194 be a standard brownian motion independent of @xmath187 . the process @xmath195 has the same law as @xmath190 , and moreover , @xmath196}\leq \left(\int_0^t { \mathbbm{e}\left [ \vert \hat{\delta}_s(r)-\hat{\delta}_s(r')\vert^2 \right]}\,ds\right)^{1/2}\leq \sqrt{t } c ( \vert r - r'\vert+\sqrt{\vert r - r'\vert}).\ ] ] the process @xmath190 therefore belongs to @xmath162 . moreover , it is a square integrable martingale with quadratic variation @xmath197}\,ds$ ] . this implies that for any @xmath198 in @xmath199 $ ] : @xmath200 } = { \mathbbm{e}\left [ \vert\int_t^{t ' } \delta_s(r ) dw_s(r)\vert \right]}\leq \left(\int_{t}^{t ' } { \mathbbm{e}\left [ \vert\delta_s(r)\vert^2 \right]}\,ds\right)^{1/2}\leq c \sqrt{t'-t}.\ ] ] the process therefore belongs to @xmath201 . note that this example illustrates an important fact . the process @xmath190 involves two processes , @xmath192 and @xmath143 , and in order to build up the coupled @xmath202 , we used the fact that we were able to find two processes @xmath187 and @xmath203 such that the pairs @xmath204 and @xmath205 had the same law ( here , the two components are independent ) . this fact will be also prominent in the definition of the solutions to the mean - field equation . [ def : solution ] a _ strong solution _ to the mean - field equation on the probability space @xmath206 , with respect to the chaotic brownian motion @xmath186 and with an initial condition @xmath207 is a spatially chaotic process @xmath208 , i.e. with continuous sample paths and regular law , such that : 1 . there exists a coupling @xmath209 , r\in\gamma)$ ] , such that for any fixed @xmath184 @xmath210 and moreover : @xmath211}\left\vert \hat{\zeta}^0_t(r)-\hat{\zeta}^0_t(r')\right\vert \right]}\leq c ( \vert r - r'\vert + \sqrt{\vert r - r'\vert}),\\ { \mathbbm{e}\left [ \sup_{t\in [ -\tau,0]}\left\vert \hat{x}_t(r)-\hat{x}_t(r')\right\vert \right]}\leq c ( \vert r - r'\vert + \sqrt{\vert r - r'\vert } ) . \end{cases}\ ] ] this regularity ensures that for any lipschitz - continuous map @xmath212 , the map @xmath213}$ ] is continuous . in particular , it is measurable and hence the integral @xmath214}d\lambda(r)$ ] can be computed in the usual ( lebesgue s ) sense . 2 . for any @xmath184 , @xmath215)$ ] is a strong solution , in the usual sense ( see ( * ? ? ? * defintion 5.2.1 ) ) , i.e. it is adapted to the filtration @xmath216 , almost surely equal to @xmath217 for @xmath218 $ ] , and the equality : @xmath219 \,ds\\ \quad+ \int_0^t \int_{\gamma } j(r , r'){\mathbbm{e}}_{z } [ b(x_s(r ) , z_{s-\tau(r , r ' ) } ( r ' ) ) ] d\lambda(r')\,ds , \qquad t>0\\ \zeta^0_t(r ) , \qquad t\in [ -\tau , 0]\\ ( z_t)\operatorname{\stackrel{\mathcal{l}}{=}}(x_t ) \text { independent of $ ( x_t)$ and $ ( w_t(\cdot))$ } \end{cases}\ ] ] holds almost surely . [ thm : existenceuniquenessspace ] let @xmath220,\ ; r\in\gamma ) \in \mathcal{z}_{0}^2 $ ] a square - integrable process with a regular law . the mean - field equation with initial condition @xmath221 has a unique strong solution on @xmath222 $ ] for any @xmath223 . the solution belongs to @xmath224 . this theorem is proved through a usual fixed point argument for a map @xmath225 acting on stochastic processes @xmath226 in @xmath180 defined by : @xmath227 \,ds\\ \quad+ \int_0^t \int_{\gamma } j(r , r'){\mathbbm{e}}_{z } [ b(x_s(r ) , z_{s-\tau(r , r ' ) } ( r ' ) ) ] d\lambda(r')\,ds , \qquad t>0\\ \zeta^0_t(r ) \qquad , \qquad t\in [ -\tau , 0]\\ ( z_t)\operatorname{\stackrel{\mathcal{l}}{=}}(x_t ) \text { independent of $ ( x_t)$ and $ ( w_t(\cdot))$ } \end{cases } \end{aligned}\ ] ] we aim at building a sequence of processes by iterating the map @xmath225 starting from a given initial process , and showing that this constitutes a cauchy sequence , converging to the unique fixed point of the map , i.e. the unique solution of the mean - field equations . this classical scheme appears relatively complex to handle in our present case . indeed , the construction of the sequence is not trivial , as we need to be able to integrate the expectation of a function of the processes , hence we need this expectation to be measurable with respect to space . second is the fact that we aim at showing existence and uniqueness in a relatively strong sense ( condition ( ii ) of the definition ) valid for any @xmath111 ( and not @xmath14-almost surely as would be the case under the norm ) . let us start by showing that we can iterate the map @xmath225 . to this end , we analyze the processes @xmath228 , image of processes @xmath229 under the map @xmath225 . it is easy to see that @xmath230 is spatially chaotic . let us start by showing that @xmath230 is square integrable ( in what follows , @xmath231 denotes a constant independent of time , that may vary from line to line ) . we note that : @xmath232 + \int_{\gamma } j(r , r ' ) { \mathbbm{e}}_z[b(x , z_{s-\tau(r , r')})]d\lambda(r')\ ] ] is lipschitz - continuous with respect to @xmath233 ( with constant @xmath234 ) , and hence @xmath235 . standard inequalities allow showing that the value @xmath236 } \vert y_s(r)\vert^2 \right]}$ ] satisfies the relationship : @xmath237 } \vert \zeta^0_s(r)\vert^2 \right ] } + t \,c\int_0^t ( 1+n_s^x(r))\,ds + 4\,t \vert\sigma(r)\vert^2 \bigg ) \ ] ] which is finite under the assumption that @xmath156 and @xmath221 are square integrable . note that this property readily implies , by application of gronwall s lemma , that any possible solution is square integrable . the regularity in time is then a direct consequence of this inequality and of the fact that the lipschitz continuity of @xmath238 implies that @xmath239 . indeed , for @xmath198 , we have @xmath240 } & \leq \int_{t}^{t ' } { \mathbbm{e}\left [ \vert \psi(r , s , x_s(r))\vert \right ] } ds + { \mathbbm{e}\left [ \vert \sigma(r ) ( w_{t'}(r)-w_t(r))\vert \right]}\\ & \leq c ( 1+n_t^x(r)^{1/2 } ) ( t'-t ) + \vert \sigma(r)\vert \sqrt{t'-t}. \end{aligned}\ ] ] it therefore remains to show that @xmath230 is regular in law . let @xmath203 be a standard brownian motion , and assume that @xmath241 is a coupling of @xmath242 in the sense that they are equal in law for any fixed @xmath74 , and that both @xmath165 and @xmath243 have the regularity property ( @xmath226 is a process satisfying the assumptions of definition [ def : solution ] ) . we define @xmath244 as : @xmath245 \,ds\\ + \int_0^t \int_{\gamma } j(r , u){\mathbbm{e}}_{z } [ b(\hat{x}_s(r ) , z_{s-\tau(r , u ) } ( u ) ) ] d\lambda(u)\,ds \end{gathered}\ ] ] it is clear that this process has the same law as @xmath230 since @xmath246 , and this obviously also holds for the processes @xmath247 and @xmath248 . let us denote @xmath249 } \vert \hat{x}_s(r)-\hat{x}_s(r')\vert \right]}$ ] . we have : @xmath250 } \int_0^s \bigg\{k_f(\vert r - r'\vert + \vert \hat{x}_u(r)-\hat{x}_u(r')\vert ) + \vert \bar{j}\vert l \big(\vert \hat{x}_u(r)-\hat{x}_u(r')\vert \\ & + d_{u-\tau_s } \big)+ k_{\gamma } ( 1+\vert b\vert_{\infty})\vert r - r'\vert + \vert j\vert_{\infty } l \vert \hat{x}_u(r)-\hat{x}_u(r')\vert + \int_{\gamma } \mathbbm{e}[\vert \hat{x}_{u-\tau_s(r , v)}(v)-\hat{x}_{u-\tau_s(r',v)}(v)]d\lambda(v)\bigg\}\,du\bigg]\\ & \leq \big(c+t ( k_{\gamma}(1+\vert b\vert_{\infty})+k_f)\big ) \vert r - r'\vert + c\big(1+t\sqrt{k_{\gamma}}\vert j\vert_{\infty } l \big)\sqrt{\vert r - r'\vert } + \int_0^t \big(k_f + 2\vert\bar{j}\vert l + \vert j\vert_{\infty}\big ) d_s^x\,ds \end{aligned}\ ] ] and we conclude , using the assumption that @xmath251 , on the regularity of the law of the process @xmath244 . in particular , let us emphasize the fact that for any @xmath252 a @xmath97-lipschitz - continuous function , @xmath253 } \vert { \mathbbm{e}\left [ \varphi(x_t(r))-\varphi(x_t(r ' ) ) \right ] } \vert\leq c \big(\vert r - r'\vert + \sqrt{\vert r - r'\vert}\big),\ ] ] which implies that the expectation @xmath254 $ ] is measurable with respect to the borel algebra @xmath139 in @xmath114 , allowing to make sense of the integral over the space variable @xmath114 . let us eventually remark that , again , by gronwall s lemma , any possible solution has a coupled process satisfying the regularity condition . these properties ensure that we can make sense of the spatial integral term in the definition of @xmath225 for iterates of that function . a sequence of processes can therefore be defined by iterating the map . we fix @xmath226 a process in @xmath255 satisfying the coupling assumptions above ( related to definition [ def : solution ] ) , and build the sequence @xmath256 by induction through the recursion relationship @xmath257 . we show that these processes constitute a cauchy sequence for @xmath258 . this will not be enough for our purposes : we are interested in proving existence and uniqueness of solutions for all @xmath74 . equipped with the estimates on the distance , we will come back to the sequence of processes at single locations , show that these also constitute a cauchy sequence in the space of stochastic processes in @xmath259 ( which is complete ) and conclude . again , one needs to be careful in the definition of the above recursion and build recursively a sequence of processes @xmath260 independent of the collection of processes @xmath261 and having the same law as follows : * @xmath262 is independent of @xmath133 and has the same law as @xmath133 * for @xmath263 , @xmath264 is independent of the sequence of processes @xmath265 and is such that the collection of processes @xmath266 has the same joint law as @xmath265 , i.e. @xmath267 is chosen such as its conditional law given @xmath268 is the same as that of @xmath256 given @xmath269 . once all these ingredients have been introduced , it is easy to show that @xmath270 satisfies a recursion relationship , by decomposing this difference into the sum of elementary terms : @xmath271\\ & \quad + \int_0^t \int_{\gamma } j(r , r')\big\{\big ( { \mathbbm{e}}_{z } [ b(x_s^{k}(r ) , z^{k}_{s-\tau(r , r ' ) } ( r ' ) ) ] \\ & \qquad \qquad \qquad-{\mathbbm{e}}_{z } [ b(x^{k-1}_s(r ) , z^{k-1}_{s-\tau(r , r ' ) } ( r ' ) ) ] \big)\big\}d\lambda(r')\,ds \\ & = : a_t(r ) + b_t(r)+ c_t(r ) \end{aligned}\ ] ] and checking that the following inequalities apply : @xmath272 through the use of cauchy - schwarz inequality , @xmath273 by standard mckean - vlasov arguments , and @xmath274 } \bigg \vert \int_{\gamma}j(r , r')\int_0^s \big({\mathbbm{e}}_{z } [ b(x_u^{k}(r ) , z_{u-\tau(r , r')}^{k } ( r ' ) ) - b(x_u^{k-1}(r ) , z_{u-\tau(r , r')}^{k-1 } ( r ' ) ) ] \big ) { du\,}d\lambda(r ' ) \bigg\vert d\lambda(r)\bigg]\\ & \leq \vert j\vert_{\infty } \;\int_{\gamma^2 } \int_0^t{\mathbbm{e}}\bigg [ { \mathbbm{e}}_{z } [ \big \vert b(x_u^{k}(r ) , z_{u-\tau(r , r')}^{k } ( r ' ) ) - b(x_u^{k-1}(r ) , z_{u-\tau(r , r')}^{k-1 } ( r ' ) ) ] \big\vert\big ) du\bigg]d\lambda(r)d\lambda(r ' ) \quad ( cs)\\ & \leq 2 \ , l\vert j\vert_{\infty } \int_{\gamma^2 } \int_0^t { \mathbbm{e}\left [ \vert x_s^{k}(r)-x_s^{k-1}(r)\vert \right]}\,ds d\lambda(r)d\lambda(r')\quad \ref{assump : loclipschbspace}\\ & \leq 2 \;l\vert j\vert_{\infty } \int_0^t \vert{x^{k}-x^{k-1}}\vert_s^1\,ds \end{aligned}\ ] ] these inequalities imply : @xmath275 with @xmath276 . let us now denote for @xmath277 the norm @xmath278 } \vert z_s(r ) \vert \right]}$ ] consider @xmath279 . similar developments yield to the inequality : @xmath280 where @xmath281 and @xmath282 correspond to the constants of the penultimate equation . we denote @xmath283 and @xmath284 . by recursion and using equation , we obtain : @xmath285 this implies that the processes @xmath286 for fixed @xmath74 constitute a cauchy sequence in the space of stochastic processes . from this relationship , routine methods allow proving existence and uniqueness of fixed point for @xmath225 ( see e.g. @xcite ) , and that this fixed point is adapted and almost surely continuous . proving uniqueness of the solution using equation is then classical . we therefore proved that there exists a unique solution to the mean - field equation , which moreover is regular in space in the sense defined above . of course , as stated , the solutions are discontinuous at all points @xmath111 . the proposition nevertheless ensures a form of regularity in law , which will be central in the sequel to prove averaging effects in the microcircuit . now that we have introduced suitable spaces in which the mean - field equations are well - defined , and proved that the equation was well - posed , we are in a position to demonstrate the main result of the manuscript , namely the convergence in law of the solutions of the network equations towards the equations , and the fact that the propagation of chaos property occurs . we consider that the network equations have chaotic initial conditions with law continuous in space @xmath287 . in detail , the initial conditions of the @xmath12 neurons in the network are considered independent processes @xmath288 , e)$ ] ( the space of square integrable processes from @xmath289 $ ] on @xmath259 ) with law equal to @xmath290 . our convergence result raises several difficulties compared to more standard models : * first is the fact that at the micro - circuit scale , there will be a local averaging principle ( yielding the convergence towards a local term @xmath291 $ ] . this property is not classical : indeed , in the network equation , the microcircuit interaction term is @xmath292 , and therefore involve the state of neurons located at different places on @xmath5 and different delays . the convergence will be handled using ( i ) the fact that in the limit considered , the neurons belong to the microcircuit collapse at a single space location , and ( ii ) regularity properties of the law of the solution as a function of space and time . this convergence will be the subject of lemma [ lem : convmicro ] . * second , the macro - circuit interaction term involve delocalized terms across the neural field . the sum will be shown to converge to a non - local averaged term involving an integral over space . this will be proved through the use of lemma [ lem : convmacro ] and [ lem : sumchi ] . moreover , we will prove our convergence through a non - classical coupling method that we now describe . let us now fix a configuration @xmath52 of the network . the neuron labeled @xmath23 in the network is driven by the @xmath109-dimensional brownian motion @xmath293 , and has the initial condition @xmath294 . we aim at defining a spatially chaotic brownian motion @xmath295 on @xmath296 such that the standard brownian motion @xmath297 is equal to @xmath298 , and proceed as follows . let @xmath299 , r\in\gamma}$ ] be a @xmath300-dimensional spatially chaotic brownian motion independent of the processes @xmath301 . the process @xmath295 defined by the coupling : @xmath302 is clearly a spatially chaotic brownian motion , and will be used to construct a particular solution of the mean - field equations . in order to completely define a solution of the mean - field equations , we need to specify an initial condition , and aim at coupling it to the initial condition of neuron @xmath23 . to this end , we define a spatially chaotic process @xmath303 equal in law to @xmath304 and independent of @xmath305 , and define a coupled process @xmath306 as : @xmath307 here again , it is clear that this process is spatially chaotic , i.e. that for any @xmath142 , the processes @xmath308 and @xmath309 are independent , and that @xmath308 has the law of @xmath310 . now that these processes have been constructed , we are in a position to define the process @xmath311 as the unique solution of the mean - field equation , driven by the spatially chaotic brownian motion @xmath312 and with the spatially chaotic initial condition @xmath313 : @xmath314d\lambda(r')\,dt } \\ & \displaystyle{\quad + \bar{j}{\mathbbm{e}}_{z}[b(x_t(r),z_{t-\tau_s}(r))]\,dt+ \sigma(r)\ , dw^i_t(r ) \qquad \text{for } t\geq 0}\\ \\ \bar{x}^i_t(r ) & = \zeta^{i,0}_t ( r ) \qquad \text{for } t\in [ -\tau , 0]\\ \\ ( z_t)&\operatorname{\stackrel{\mathcal{l}}{=}}(\bar{x}^i_t ) \in { \mathcal{m}}\quad \text { independent of $ ( \bar{x}^i_t)$ and $ ( w^{i}_t(\cdot))$}. \end{array } \right .\ ] ] the same procedure applied for all @xmath315 allows building a collection of independent stochastic processes @xmath316 . these are clearly independent of the configurations of the finite - size network . let us denote by @xmath317 the probability distribution of @xmath318 solution of the mean - field equation . as previously , the process @xmath319 generically denotes a process belonging to @xmath162 and distributed as @xmath109 . we start by analyzing the local averaging property on the micro - circuit . this is the subject of the following lemma . [ lem : convmicro ] there exists a positive constant @xmath281 such that , for any @xmath12 sufficiently large , averaged across all configurations @xmath52 : @xmath320\right \vert \right]}\right\ } \leq k_1 \sqrt{\left({\frac{v(n)}{n}}\right)^{\frac{1}{d}}+\frac{1}{{v(n)}}}\ ] ] conditioned on @xmath34 , the set @xmath321 is a collection of independent identically distributed random variables . the map @xmath322 is lipschitz continuous . therefore , the regularity properties proved in theorem [ thm : existenceuniquenessspace ] ensure that we have ( in what follows , @xmath281 denotes a constant , independent of @xmath12 , that may change from line to line ) : @xmath323-{\mathbbm{e}}_{{z}}[b(\bar{x}^i_t(r),{z}_{t-\tau_{s}}(r_i ) ] \vert \leq k_1(\sqrt{\vert \tau_{ij}-\tau_s \vert } + \vert r_j - r_i\vert+ \sqrt{\vert r_j - r_i\vert})=k_1(\sqrt{d_{ij}}+d_{ij}).\ ] ] for almost any configuration @xmath52 and any @xmath71 , we have seen that for @xmath12 sufficiently large , by application of proposition [ lem : sizemicro ] , the distances @xmath72 are small ( of order @xmath62 ) . moreover , we have : @xmath324 \\ & \quad = \frac{1}{v(n ) } \sum_{j\in { \mathcal{v}}(i)}\left ( b(\bar{x}^i_t(r_i),\bar{x}^j_{t-\tau(r_i , r_j)}(r_j ) ) - { \mathcal{e}}[{\mathbbm{e}}_{z}[b(\bar{x}^i_t(r_i),z_{t-\tau_{ij}}(r_j))]]\right)\\ & \qquad + \frac{1}{v(n ) } \sum_{j\in { \mathcal{v}}(i)}{\mathcal{e}}[{\mathbbm{e}}_{z}[b(\bar{x}^i_t(r_i),z_{t-\tau_{ij}}(r_j ) ) ] ] - { \mathbbm{e}}_{z}[b(\bar{x}^i_t(r_i),{z}_{t-\tau_s}(r_i ) ) ] \end{aligned}\ ] ] for any measurable function @xmath325 , the quantity @xmath326 $ ] is precisely the average of the random variable @xmath327 . therefore , a quadratic control argument ( see e.g. ( * ? ? ? * theorem 1.4 . ) ) allows to show that the first term is of order @xmath328 , in the sense that : @xmath329\right)]]\leq \frac{k_1}{\sqrt{v(n)}}.\ ] ] this argument consists in showing that the expectation of the square of the sum is of order @xmath330 , which is performed by showing that ( i ) the terms of the sum are centered ( i.e. that the expectation term introduced which was chosen to this purpose is precisely the expectation with respect to @xmath331 of @xmath332 for @xmath331 equal in law to @xmath333 which all have the same law ) and ( ii ) using cauchy - schwarz inequality to bound the term by the square root of the expectation of the squared sum , developing the square and showing that the number of null terms is bounded by some constant multiplied by @xmath0 . this argument is not developed here as it will be the core of the proof of lemma [ lem : convmacro ] . the second term is handled by using the control given by equation and the result of proposition [ lem : sizemicro ] , ensuring that @xmath334 - { \mathbbm{e}}_{z}[b(\bar{x}^i_t(r_i),{z}_{t-\tau_s}(r_i))]\big]\big]\\\leq k_1\bigg(\sqrt{\left({\frac{v(n)}{n}}\right)^{\frac{1}{d}}+\frac 1 { v(n ) } } + \left({\frac{v(n)}{n}}\right)^{\frac{1}{d}}+\frac 1 { v(n)}\bigg ) . \end{gathered}\ ] ] put together , the two last estimates yield the desired result . [ lem : convmacro ] the coupled macroscopic interaction term converges towards a non - local mean - field term with speed @xmath335 , in the sense that there exists a constant @xmath336 independent of @xmath12 such that : @xmath337d\lambda(r)\right \vert\big]\big]\leq \frac{k_2}{\sqrt{n\beta(n)}}.\ ] ] conditioned on the location @xmath34 of neuron @xmath23 , the collection of @xmath338-random variables @xmath339 are independent and identically distributed . the sum @xmath340 is therefore , conditionally on @xmath341 and @xmath34 , the sum of independent and identically distributed processes , with finite mean and variance ( since @xmath124 is a bounded function ) . the expectation of each term in the sum , conditionally on @xmath342 and @xmath34 , is equal to : @xmath343 d\lambda(r).\ ] ] let us denote by @xmath344 the term under consideration is simply the empirical average @xmath345 , and conditionally on @xmath341 and @xmath34 , the terms are independent , identically distributed , centered @xmath346-random variables with second moment : @xmath347\leq \frac{1}{\beta(n)}m_2\ ] ] where @xmath348 is a finite constant independent of @xmath12 , @xmath34 and @xmath341 . let us denote by @xmath349 $ ] the expectation on @xmath338 conditioned on @xmath34 and @xmath342 . we have : @xmath350 & \leq \sqrt{\hat{{\mathbbm{e}}}_i\big[\big(\frac 1 n \sum_{j=1}^n \phi_{ij}\big)^2\big]}\\ & \leq \frac 1 n \sqrt{\sum_{j=1}^n \hat{{\mathbbm{e}}}_i\big[\phi_{ij}^2\big ] } \leq \sqrt{\frac{m_2}{n\beta(n)}}. \end{aligned}\ ] ] thanks to the fact that @xmath348 is independent of @xmath342 , we conclude that : @xmath351 \big ] = { \mathbbm{e}}\big[\hat{{\mathbbm{e}}}_i\big[\vert \frac 1 n \sum_{j=1}^n \phi_{ij}\vert\big]\big\vert r_i\big]\leq \sqrt{\frac{m_2}{n\beta(n)}}\ ] ] now that we have analyzed the local and macroscopic interaction terms defined with the coupled processes , we are in a position to demonstrate our main result , namely the full multiscale convergence result . [ thm : propagationchaosspace ] let @xmath352 a fixed neuron in the network . under the assumptions [ assump : loclipschspace]-[assump : spacecontinuity ] , for almost all configuration @xmath52 of the neuron locations @xmath353 and connectivity links @xmath354 , the process @xmath355 solution of the network equations converges in law towards the process @xmath356 solution of the mean - field equations with initial condition @xmath304 and moreover , the speed of convergence is given by : @xmath357\right ) = o\left(\sqrt{\left(\frac{v(n)}{n}\right)^{\frac 1 { d}}+\frac 1 { { v(n)}}}+\frac 1 { \sqrt{n\beta(n)}}\right)\ ] ] the notation @xmath358 denotes the expectation on @xmath15 the distribution of network configurations @xmath52 , i.e. space locations @xmath359 and connectivity links @xmath360 . the expectation @xmath361 $ ] is therefore the global expectation , i.e. on @xmath362 . the result shows that the expectation tends to zero . this implies quenched convergence ( i.e. for almost all configuration @xmath52 ) along subsequences . in detail , the speed of converge @xmath363 announced in the theorem ( on the righthand side of equation ) allows to define subsequences ( i.e. a sequence of network size @xmath364 ) for which we have almost sure convergence . these sequences are such that borel - cantelli lemma can be applied , namely subsequences extracted through a strictly increasing application @xmath365 such that @xmath366 is summable . we prepare for the proof by demonstrating the following fine estimate that will be used to control configurations with more links than the expected value : [ lem : sumchi ] under our assumptions , for any @xmath17 and @xmath367 , we have for @xmath12 sufficiently large : @xmath368 where @xmath369 . in order to demonstrate the result , we make use of chernoff - hoeffding theorem @xcite controlling the deviations from the mean of bernoulli random variables with fixed mean @xmath60 , is such that , for any @xmath370 , @xmath371\leq \left(\left(\frac p { p+\varepsilon}\right)^{p+\varepsilon}\left(\frac { 1-p } { 1-p-\varepsilon}\right)^{1-p-\varepsilon}\right)^m.\ ] ] ] . let us denote by @xmath372 . this is a binomial variable of parameters @xmath373 . chernoff - hoeffding theorem ensures that : @xmath374 & \leq \left(\left(\frac{\beta(n)}{\gamma\beta(n)}\right)^{\gamma \beta(n)}\left(\frac{1-\beta(n)}{1-\gamma\beta(n)}\right)^{1-\gamma \beta(n)}\right)^n\\ & \leq \exp\left(-\gamma \log(\gamma ) n \beta(n ) + n\big(1-\gamma\beta(n)\big)\big(\log\big(1-\beta(n)\big)-\log\big(1-\gamma\beta(n)\big)\big)\right ) . \end{aligned}\ ] ] using a taylor expansion of the logarithmic terms for large @xmath12 ( using the fact that @xmath31 tends to zero at infinity ) , it is easy to obtain @xmath375 \leq \exp\left ( ( -\gamma\log(\gamma)+\gamma-1 ) n\beta(n ) + o(n\beta(n)^2)\right)\ ] ] note that for @xmath367 , the quantity @xmath376 is strictly positive . we therefore have , for @xmath12 sufficiently large , that the probability is bounded by : @xmath375 \leq \exp\left ( -\frac 1 2 ( \gamma\log(\gamma)-\gamma+1 ) n\beta(n))\right).\ ] ] this allows to conclude the lemma as follows . it is clear by definition that @xmath377 . therefore , @xmath378 \end{aligned}\ ] ] yielding the desired result . we are now in a position to perform the proof of theorem [ thm : propagationchaosspace ] . the proof is based on evaluating the distance @xmath379 $ ] , and breaking it into a few elementary , easily controllable terms . a substantial difference with usual mean - field proofs is that network equations correspond to processes taking values in @xmath259 in which the interaction term is sum over a finite number of neurons in the network equation , while the mean - field equation is a spatially extended equation with an effective interaction term involving an integral over @xmath5 . this will be handled using the result of lemma [ lem : convmacro ] . we introduce in the distance coupled interaction terms that were controlled in lemmas [ lem : convmicro ] and [ lem : convmacro ] and obtain the following elementary decomposition ( each line of the righthand side corresponds to one term of the decomposition , @xmath380 ) : @xmath381\big)\ , ds\\ & \quad + \frac{1}{n\beta(n ) } \sum_{j=1}^{n } \int_0^t j(r_i , r_j)\chi_{ij } \big(b(x^{i,{\mathcal{a}}_n}_s , x^{j,{\mathcal{a}}_n}_{s-\tau_{ij}})-b(\bar{x}^{i}_s(r_{i}),\bar{x}^{j}_{s-\tau_{ij}}(r_{j } ) ) \big)\,ds\\ & \quad + \int_0^t \big(\frac{1}{n\beta(n ) } \sum_{j=1}^{n } j(r_i , r_j)\chi_{ij } b(\bar{x}^{i}_s(r_{i}),\bar{x}^{j}_{s-\tau_{ij}}(r_{j } ) ) -\int_{\gamma } j(r_i , r ' ) { \mathbbm{e}}_z[b(\bar{x}^i_s(r_{i}),z_{s-\tau(r_{i},r')}(r'))]d\lambda(r')\big)\,ds\\ & \nonumber \qquad = : a^i_t(n)+b^i_t(n)+c^i_t(n)+d^i_t(n)+e^i_t(n ) \end{aligned}\ ] ] it is easy to show , using assumptions [ assump : loclipschspace ] and [ assump : loclipschbspace ] , that the terms @xmath382 and @xmath383 satisfy the inequalities : @xmath384 & \leq k_f\,\int_0^{t } { \mathbbm{e}}\big[\sup_{-\tau\leq u\leq s } \vert x_u^{i,{\mathcal{a}}_n}-\bar{x}_u^i(r_{i})\vert \big]\ , ds\\ \max_{i=1\cdots n}\mathcal{e}\big[{\mathbbm{e}}\big[\sup_{-\tau\leq s\leq t } \vert b_s^i(n ) \vert\big]\big ] & \leq \frac{v(n)+1}{v(n ) } l\ , \int_0^{t } \max_{k=1\cdots n}{\mathcal{e}}\big[{\mathbbm{e}}\big[\sup_{-\tau\leq u\leq s}\vert x^{k,{\mathcal{a}}_n}_u-\bar{x}^k_u(r_{i } ) \vert\big]\big ] \ , ds \end{aligned}\ ] ] the term @xmath385 requires to be handled with care , because of the sparsity of the macrocircuit . indeed , this term involves the sum of @xmath12 random variables and is rescaled by @xmath386 . as we assumed @xmath42 in order to account for the sparsity in the macrocircuit , most terms in the sum are equal to zero such that @xmath387 . we have : @xmath388\leq \vert j\vert_{\infty } \frac 1 { n\beta(n ) } \sum_{j } \chi_{ij } \int_0^t { \mathbbm{e}}\big[\sup_{0 \leq u \leq s } \vert b(x^{i,{\mathcal{a}}_n}_u , x^{j,{\mathcal{a}}_n}_{u-\tau_{ik}})-b(\bar{x}^i_u,\bar{x}^j_{u-\tau_{ik } } ) \vert \big]\,ds\ ] ] this expression shows how critical the singular sparse coupling is to our estimates . indeed , the random variable @xmath389 almost surely tends to @xmath97 as @xmath12 goes to infinity , but it can reach very large values ( up to @xmath390 which diverges as @xmath12 goes to infinity ) . configurations @xmath52 for which the sum is large are increasingly improbable , but for these configurations , the deterministic scaling @xmath391 is not fast enough to overcome the divergence of the input term . there is therefore a competition between the probability of having configurations with large values of @xmath392 and the divergence of the solutions . however , in the present case , this control will be possible using the estimate of the probability that the number of links exceeds @xmath393 using the result of lemma [ lem : sumchi ] . indeed , fixing @xmath367 , and distinguishing whether @xmath394 or not , we obtain : @xmath395\big ] & \leq 2 \gamma \ ; l \ ; \vert j \vert_{\infty } \int_0^t \max_{k=1\cdots n}{\mathcal{e}}\big [ { \mathbbm{e}}\big[\sup_{-\tau \leq u \leq s } \vert x^{k,{\mathcal{a}}_n}_{u}-\bar{x}^k_{u } \vert \big]\big]\,ds\\ & \qquad + 2 \vert b\vert_{\infty}\vert j\vert_{\infty } { \mathcal{e}}\left(\frac 1 { n\beta(n ) } \sum_{j } \chi_{ij } { \mathbbm{1}_{\mathcal{d}_{\gamma}}}\right ) \end{aligned}\ ] ] where @xmath369 as defined in lemma [ lem : sumchi ] . by application of this lemma , and using the fact that the second term of the upper bound is negligible compared to @xmath396 , we conclude that : @xmath397\big ] & \leq 2 \gamma \ ; l \ ; \vert j \vert_{\infty } \int_0^t \max_{k=1\cdots n}{\mathcal{e}}\big [ { \mathbbm{e}}\big[\sup_{-\tau \leq u \leq s } \vert x^{k,{\mathcal{a}}_n}_{u}-\bar{x}^k_{u } \vert \big]\,ds + \frac{k_c}{\sqrt{n\beta{(n)}}}. \end{aligned}\ ] ] we are left with controlling the terms @xmath398 and @xmath399 . these consist of sums only involving the coupled processes , and were analyzed in the previous sections . by direct application of the results of lemmas [ lem : convmicro ] and [ lem : convmacro ] , we have : @xmath400\big ] } & \leq \displaystyle{k_1\sqrt{\left(\frac{v(n)}{n}\right)^{\frac{1}{d}}+\frac 1 { { v(n)}}}}\\ \displaystyle{\max_{k=1\cdots n}{\mathcal{e}}\big [ { \mathbbm{e}}\big[\sup_{-\tau\leq s \leq t } \vert e^k_t \vert \big]\big ] } & \leq \displaystyle{\frac{k_2}{\sqrt{n\beta{(n ) } } } } \end{cases}\ ] ] all together , we hence have , for some constants @xmath401 and @xmath402 independent of @xmath12 @xmath403\right),\ ] ] the inequality : @xmath404 which proves the theorem by application of gronwall s lemma . [ cor : propachaosspace ] let @xmath405 and fix @xmath406 neurons @xmath407 . under the assumptions of theorem [ thm : propagationchaosspace ] , the process @xmath408 converges in law towards @xmath409 . we have : @xmath410\right ) \\ & \quad \leq l \max_{k=1\cdots n}\mathcal{e}\left({\mathbbm{e}}\left [ \sup_{-\tau\leq t \leq t } \left\vert x^{k,{\mathcal{a}}_n}_t-\bar{x}^{k}_t\right\vert^2 \right]\right)\\ \end{aligned}\ ] ] which tends to zero as @xmath12 goes to infinity , hence the law of @xmath408 converges towards that of @xmath411 which is equal by definition to @xmath412 . the dynamics of neuronal networks in the brain lead us to analyze a class of spatially extended networks which display multiscale connectivity patterns that are singular in at least two aspects : * the network display local dense connectivity patterns in which neurons are connected to their @xmath0-nearest neighbors , where @xmath1 . * the macro - circuit was also singular , in the sense that the probability of two neurons @xmath23 and @xmath38 to be connect tends towards zero . this is very far from usual mean - field models that consider full connectivity patterns , or partial connectivity patterns proportional to the network size @xcite . in these cases , the convergence is substantially slower , and the rescaling actually required thorough controls on the number of incoming connections to each neurons . the introduction of local microcircuits with negligible size was suggested in @xcite , in the context of the fluctuations induced by the microcircuit . our scaling , motivated by characterizing the macroscopic activity at the scale of the neural field @xmath5 , lead us to consider local microcircuits with spatial extension of order @xmath413 , which tends to zero in the limit @xmath45 . at this scale , the fluctuations related to the microcircuit vanish , which allowed identifying the large @xmath12 limit process . however , at the scale of one neuron ( or considering , similarly to @xcite , a field of size @xmath12 , i.e. typical distances between neurons of order @xmath97 ) , the microcircuits may induce more complex phenomena in which fluctuations become prominent . this interesting problem remains largely open and can not be addressed with the techniques presented in the manuscript . the developments presented in this article also go way beyond what was done in the domain of mean - field analysis of large spatially extended systems . in that domain , probably the two most relevant contributions to date are @xcite and @xcite . in @xcite , a relatively sketchy model of neural field was proposed , in which the system was fully connected and neurons gathered at discrete space location that eventually filled the neural field . the model presented here is considerably more relevant from the biological viewpoint , and necessitated to deeply modify the proofs proposed in that manuscript . in particular , the connectivity patterns are now randomized , and the proof is now made independent of results arising in finite - populations networks . moreover , the two main contributions of the article , namely the singular coupling , was absent of the above cited manuscript . such coupling was discussed in @xcite , where the authors consider the case of network with nearest - neighbors topology ( only a local micro - circuit ) in which neurons connect to a non - trivial proportion of neurons @xmath3 . there is a substantial difficulty in considering only very local micro - circuits connectivity and sparse macro - circuits . here , we solved this problem and framed it in a more general setting with multiscale coupling . the proof presented in the present manuscript is relatively general . in particular , it can be extended to models with non locally lipchitz continuous dynamics ( as is the case of the classical fitzhugh - nagumo model @xcite ) as was presented in @xcite , or to networks with multiple layers . the results enjoy a relatively broad universality . indeed , we observe that the limit obtained is independent of the choice of the size of the micro - circuit and sparsity of the macro - circuit ( as long as proper scaling is considered ) . this property shows that the limit is universal : for any choice of function @xmath0 and @xmath31 , the macroscopic limit of our networks are identical . an interesting question is then what would be an optimal choice of functions @xmath0 and @xmath31 so that the convergence is the fastest . the speed of convergence towards the mean - field equation is hence governed by three quantities : * the term @xmath414 controls the regularity of the law solution of the mean - field equation with respect to space . the larger @xmath0 , the wider the micro - circuit , and therefore the slowest the local convergence . * the term @xmath415 controls the speed of averaging at the micro - circuit scale , which decreases with the size of the micro - circuit @xmath0 . * the term @xmath396 controls the speed of the averaging at the macro - circuit scale . this term is of course the smallest when @xmath31 is large . in the biological system under consideration , there is nevertheless an energetic cost to increasing the connectivity level . the two first term corresponding to the micro - circuit convergence properties can give an information on the order of the optimal micro - circuit size . minima are obtained when @xmath0 is of order @xmath416 , e.g. @xmath417 in dimension @xmath97 . other choices may be analyze to optimize other criteria such that information capacity vs energetic considerations , anatomical constraints , size of clusters sharing resources , . eventually , this result has also implications in neuroscience modeling . in this domain , authors widely use the so - called wilson - cowan neural field model ( see @xcite for a review ) . this model is given by non - local differential equations of type : @xmath418 where @xmath419 represents the mean firing - rate of neurons and @xmath420 corresponds to a sigmoidal function . this type of equations is similar to those obtained in the analysis of fully connected neural fields , as shown in @xcite , when considering a discrete wilson - cowan type of dynamics for the underlying network , i.e. a case where @xmath421 and @xmath422 for @xmath420 a smooth sigmoidal function . in this case , we showed @xcite that the solutions were attracted by gaussian spatially chaotic processes with mean @xmath423 and standard deviation @xmath424 satisfying the integro - differential equations : @xmath425 where @xmath426 . these are compatible with the neural field equations . however , these actually appear to overlook the complex connectivity pattern , and in particular neglect the additional local averaging term that we found here using rigorous probabilistic methods . taking into account local microcircuitry would actually yield an additional term in the equation on @xmath423 : @xmath427 the study of these new equations will , with no doubt , present substantial different dynamics , are offer a new neural field model well worth analyzing in order to understand the qualitative role of local microcircuits on the dynamics .

the cortex is a very large network characterized by a complex connectivity including at least two scales : a microscopic scale at which the interconnections are non - specific and very dense , while macroscopic connectivity patterns connecting different regions of the brain at larger scale are extremely sparse . this motivates to analyze the behavior of networks with multiscale coupling , in which a neuron is connected to its @xmath0 nearest - neighbors where @xmath1 , and in which the probability of macroscopic connection between two neurons vanishes . these are called singular multi - scale connectivity patterns . we introduce a class of such networks and derive their continuum limit . we show convergence in law and propagation of chaos in the thermodynamic limit . the limit equation obtained is an intricate non - local mckean - vlasov equation with delays which is universal with respect to the type of micro - circuits and macro - circuits involved . ' '' '' ' '' '' the purpose of this paper is to provide a general convergence and propagation of chaos result for large , spatially extended networks of coupled diffusions with multi - scale disordered connectivity . such networks arise in the analysis of neuronal networks of the brain . indeed , the brain cortical tissue is a large , spatially extended network whose dynamics is the result of a complex interplay of different cells , in particular neurons , electrical cells with stochastic behaviors . in the cortex , neurons interact depending on their anatomical locations and on the feature they code for . the neuronal tissue of the brain constitute spatially - extended structures presenting complex structures with local , dense and non - specific interactions ( microcircuits ) and long - distance lateral connectivity that are function - specific . in other words , a given cell in the cortex sends its projections at ( i ) a local scale : the neurons connect extensively to anatomically close cells ( the _ microcircuits _ ) , forming a dense local network , and ( ii ) superimposed to this local architecture , a very sparse functional architecture arises , in which long - range connections are made with other cells that are anatomically more remote but that respond to the same stimulus ( the functional _ macrocircuit _ ) . this canonical architecture was first evidenced by electrophysiological recordings in the 70 s @xcite and made more precise as experimental techniques developed ( see @xcite for striking representations of this architecture in the striate cortex ) . the primary visual cortex of certain mammals is a paradigmatic well documented cortical area in which this architecture was evidenced . in such cortical areas , neurons organize into columns of small spatial extension containing a large number of cells ( on the order of tens of thousands cells ) responding preferentially to specific orientations in visual stimuli @xcite , constituting local microcircuits that distribute across the cortex in a continuous map , each cell connecting densely with its nearest neighbors and sparsely with remote cells coding for the same stimulus @xcite . these spatially extended networks are called _ neural fields_. such organizations and structures are deemed to subtend processing of complex sensory or cortical information and support brain functions @xcite . in particular , the activity of these neuronal assemblies produce a mesoscopic , spatially extended signal , which is precisely at the spatial resolution of the most prominent imaging techniques ( eeg , meg , mri ) . these recordings are good indicators of brain activity : they are a central diagnostic tool used by physicians to assert function or disfunction . in these spatially extended systems , the presence of delays in the communication of cells , chiefly due to the transport of information through axons and to the typical time the synaptic machinery needs to transmit it , is essential to the dynamics . these transmission delays will chiefly affect the long connections of the macrocircuit , which are orders of magnitude longer than those of the microcircuit . the mathematical and computational analysis of the dynamics of neural fields relies almost exclusively on the use of heuristic models since the seminal work of wilson , cowan and amari @xcite . these propose to describe the mesoscopic cortical activity through a deterministic , scalar variable whose dynamics is given by integro - differential equations . this model was widely studied analytically and numerically , and successfully accounted for hallucination patterns , binocular rivalry and synchronization @xcite . justifying these models starting from biologically realistic settings has since then been a great endeavor @xcite . this problem was undertaken recently using probabilistic methods . the first contribution @xcite introduced an approximation of the underlying connectivity of the neural network involved , considering a fully connected architecture ( each neuron was connected to all the others ) and neurons in the same column were considered to be precisely at the same spatial location . they showed propagation of chaos and convergence to some intricate mckean - vlasov equation . more recently , an heterogeneous macrocircuit model was analyzed in @xcite . in that paper , the authors considered a network with heterogeneous and non - global connectivity : neurons were connected with their @xmath2-nearest neighbors , where @xmath3 with @xmath4 , or with power - law synaptic weights , and obtained a limit theorem for the behavior of the empirical density . in both cases , the connectivity was considered at a single scale , and did not reproduce the actual type of connectivity pattern observed in the brain . in the present manuscript , we come back to these models with a more plausible architecture including local microcircuit together with non - local macroscopic sparse connectivity . using statistical methods and in particular an extension of the coupling method @xcite , we will demonstrate the propagation of chaos property , and convergence towards a complex nonlinear markov equation similar to the classical _ mckean - vlasov _ equations , but with a non - local integral over space locations and delays . interestingly , this object presents substantial differences with the usual mckean - vlasov limits : beyond the presence of delays , the neural field limit regime is at a mesoscopic scale where averaging effects locally to occur , but is fine enough to resolve brain s structure and its activity , resulting in the presence of an integral term over space . the solution , seen as a function of space , is everywhere discontinuous , which makes the limiting object highly singular . the present work is distinct of that of @xcite in that we consider local connectivity patterns in which neurons connect to a negligible portion of the neurons . this includes non - trivial issues , and necessitate to thoroughly control the regularity of the law of the solution as a function of space . on the other hand , beyond the presence of random locations of individual neurons and the presence of a dense microcircuit , the sparse macro - circuit generalizes non - trivially the work done in @xcite . indeed , at the macro - circuit scale , the probability of connecting two fixed neurons tends to zero . we therefore need to deal with a non - globally connected network , and address the problem by using fine estimates on the interaction terms and chernoff - hoeffding theorem @xcite . the speed of convergence towards the mean - field equations is quantified and involves three terms , one governing the local averaging effects arising from the micro - circuits , one arising from the regularity properties of the solutions , and one corresponding to the speed of convergence of the macro - circuit interaction term towards a continuous limit . in the neural field regime , the limit equations are very singular , in particular trajectories are not measurable with respect to the space . these limits are very hard to analyze at this level of generality . however , in the type of models usually considered in the study of neural fields , namely the firing - rate model , it was shown in @xcite that the behavior can be rigorously and exactly reduced to a system of deterministic integro - differential equations that are compatible with the usual wilson and cowan system in the zero noise limit . noise intervenes in these equations a nonlinear fashion , fundamentally shaping in the macroscopic dynamics . the paper is organized as follows . we start in section [ sec : model ] by introducing precisely our model and proving a few simple results on the network equations and on the topology of the micro - circuit . this being shown , we will turn in section [ sec : existenceuniquenessspace ] to the analysis of the network equations , and will in particular make sense of the intricate non - local mckean - vlasov equation , show well - posedness and some regularity estimates on the law of the mean - field equations . section [ sec : propachaspace ] will be devoted to the demonstration of the convergence of the network equations towards the mean - field equations .

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Proceed to summarize the following text: gallager in @xcite showed that for @xmath5 and @xmath6 , there exist @xmath7 regular low - density parity - check ( ldpc ) codes for which the bit error probability tends to zero asymptotically whenever we operate below the threshold . richardson and urbanke in @xcite derived the capacity of ldpc codes for various message passing algorithms and described density evolution , a deterministic algorithm to compute thresholds . zyablov and pinsker @xcite analyzed ldpc codes under a simpler decoding algorithm known as the bit flipping algorithm and showed that almost all the codes in the regular ensemble with @xmath8 can correct a constant fraction of worst case errors . sipser and spielman in @xcite used expander graph arguments to analyze bit flipping algorithm . burshtein and miller in @xcite applied expander based arguments to show that message passing algorithms can also correct a fixed fraction of worst case errors when the degree of each variable node is at least five . @xcite showed that linear programming decoder @xcite is also capable of correcting a fraction of errors . recently , burshtein in @xcite showed that regular codes with variable nodes of degree four are capable of correcting a linear number of errors under bit flipping algorithm . he also showed tremendous improvement in the fraction of correctable errors when the variable node degree is at least five . in this paper , we consider the error correction capability of the ensemble @xmath9 of @xmath10 regular ldpc codes as defined in @xcite when decoded using the gallager a algorithm . we analyze decoding failures using the notion of trapping sets and prove that a code with girth @xmath11 can not correct all @xmath12 or fewer errors . using this result , we prove that for any @xmath13 , for sufficiently large block length @xmath14 , no code in the @xmath15 ensemble can correct @xmath16 fraction of errors . this result settles the problem of error correction capability of column - weight - three codes . the rest of the paper is organized as follows . in section [ section2 ] we establish the notation and describe the gallager a algorithm . we then characterize the failures of the gallager a decoder with the help of fixed points . we also introduce the notions of trapping sets , failure sets and critical number . in section [ section3 ] we investigate the relation between error correction capacity and girth of the code . we extend the results to bit flipping algorithm in section [ section4 ] and conclude in section [ section5 ] . ldpc codes @xcite are a class of linear block codes which can be defined by sparse bipartite graphs @xcite . let @xmath17 be a bipartite graph with two sets of nodes : @xmath14 variable nodes and @xmath18 check nodes . the check nodes ( variable nodes ) connected to a variable node ( check node ) are referred to as its neighbors . the degree of a node is the number of its neighbors . this graph defines a linear block code @xmath19 of length @xmath14 and dimension at least @xmath20 in the following way : the @xmath14 variable nodes are associated to the @xmath14 coordinates of codewords . a vector @xmath21 is a codeword if and only if for each check node , the sum of its neighbors is zero . such a graphical representation of an ldpc code is called the tanner graph @xcite of the code . the adjacency matrix of @xmath17 gives @xmath22 , a parity check matrix of @xmath23 . an @xmath7 regular ldpc code has a tanner graph with @xmath14 variable nodes each of degree @xmath24 ( column weight ) and @xmath25 check nodes each of degree @xmath26 ( row weight ) . this code has length @xmath14 and rate @xmath27 @xcite . in the rest of the paper we consider codes in the @xmath10 , @xmath28 , regular ldpc code ensemble . note that the column weight and row weight are also referred to as left degree and right degree in literature . it should also be noted that the tanner graph is not uniquely defined by the code and when we say the tanner graph of an ldpc code , we only mean one possible graphical representation . the girth @xmath29 is the length of the shortest cycle in @xmath17 . in this paper , @xmath30 represents a variable node , @xmath31 represents an even degree check node and @xmath32 represents an odd degree check node . gallager in @xcite proposed two simple binary message passing algorithms for decoding over the binary symmetric channel ( bsc ) ; gallager a and gallager b. see @xcite for a detailed description of gallager b algorithm . for column - weight - three codes , which are the main focus of this paper , these two algorithms are the same . every round of message passing ( iteration ) starts with sending messages from variable nodes ( first half of the iteration ) and ends by sending messages from check nodes to variable nodes ( second half of the iteration ) . let @xmath33 , a binary @xmath14-tuple be the input to the decoder . let @xmath34 denote the message passed by a variable node @xmath35 to its neighboring check node @xmath36 in @xmath37 iteration and @xmath38 denote the message passed by a check node @xmath36 to its neighboring variable node @xmath35 . additionally , let @xmath39 denote the set of all messages from @xmath35 , @xmath40 denote the set of all messages from @xmath35 except to @xmath36 , @xmath41 denote the set of all messages to @xmath36 . @xmath42 and @xmath43 are defined similarly . the gallager a algorithm can be defined as follows . @xmath44 at the end of each iteration , an estimate of each variable node is made based on the incoming messages and possibly the received value . the decoded word at the end of @xmath37 iteration is denoted as @xmath45 . the decoder is run until a valid codeword is found or a maximum number of iterations @xmath46 is reached , whichever is earlier . the output of the decoder is either a codeword or @xmath47 . _ a note on the decision rule : _ different rules to estimate a variable node after each iteration are possible and it is likely that changing the rule after certain iterations may be beneficial . however , the analysis of various scenarios is beyond the scope of this paper . for column - weight - three codes only two rules are possible . * decision rule a : if all incoming messages to a variable node from neighboring checks are equal , set the variable node to that value ; else set it to received value * decision rule b : set the value of a variable node to the majority of the incoming messages ; majority always exists since the column - weight is three we adopt decision rule a throughout this paper . we now characterize failures of the gallager a decoder using fixed points and trapping sets @xcite . consider an ldpc code of length @xmath14 and let @xmath48 be the binary vector which is the input to the gallager a decoder . let @xmath49 be the support of @xmath48 . the support of @xmath48 is defined as the set of all positions @xmath50 where @xmath51 . without loss of generality , we assume that the all zero codeword is sent over bsc and that the input to the decoder is the error vector . hence , throughout this paper a message of @xmath52 is alternatively referred to as an incorrect message , a received value of @xmath52 is referred to as an initial error . @xcite a decoder failure is said to have occurred if the output of the decoder is not equal to the transmitted codeword . @xmath48 is called a _ fixed point _ if @xmath53 that is , the message passed from variable nodes to check nodes along the edges are the same in every iteration . since the outgoing messages from variable nodes are same in every iteration , it follows that the incoming messages from check nodes to variable nodes are also same in every iteration and so is the estimate of a variable after each iteration . in fact , the estimate after each iteration coincides with the received value . it is clear from above definition that if the input to the decoder is a fixed point , then the output of the decoder is the same fixed point . @xcite let @xmath48 be a fixed point . then @xmath49 is known as a trapping set . a @xmath54 trapping set @xmath55 is a set of @xmath56 variable nodes whose induced subgraph has @xmath57 odd degree checks . [ thm1 ] let @xmath19 be a code in the ensemble of @xmath10 regular ldpc codes . let @xmath55 be a set consisting of @xmath56 variable nodes with induced subgraph @xmath58 . let the checks in @xmath58 be partitioned into two disjoint subsets ; @xmath59 consisting of checks with odd degree and @xmath60 consisting of checks with even degree . let @xmath61 and @xmath62 . @xmath55 is a trapping set iff : ( a ) every variable node in @xmath58 is connected to at least two checks in @xmath60 and ( b ) no two checks of @xmath59 are connected to a variable node outside @xmath58 . see appendix [ appendix1 ] we note that theorem [ thm1 ] is a consequence of fact 3 in @xcite . we also remark that theorem [ thm1 ] can be extended to higher column weight codes but in this paper we restrict our attention to column - weight - three codes . if the variable nodes corresponding to a trapping set are in error , then a decoder failure occurs . however , not all variable nodes corresponding to trapping set need to be in error for a decoder failure to occur . @xcite the minimal number of variable nodes that have to be initially in error for the decoder to end up in the trapping set @xmath55 will be referred to as _ critical number _ for that trapping set . a set of variable nodes which if in error lead to a decoding failure is known as a _ failure set_. _ remarks _ 1 . to `` end up '' in a trapping set @xmath55 means that , after a possible finite number of iterations , the decoder will be in error , on at least one variable node from @xmath63 at every iteration @xcite . the notion of a failure set is more fundamental than a trapping set . however , from the definition , we can not derive necessary and sufficient conditions for a set of variable nodes to form a failure set . 3 . a trapping set is a failure set . subsets of trapping sets can be failure sets . more specifically , for a trapping set of size @xmath56 , there exists at least one subset of size equal to the critical number which is a failure set . 4 . the critical number of a trapping set is not fixed . it depends on the outside connections of checks in @xmath60 . however , the maximum value of critical number of a @xmath54 trapping set is @xmath56 . fig.[sixcycle ] shows a subgraph induced by a set of three variable nodes @xmath64 . if no two odd degree check nodes from @xmath65 are connected to a variable outside the subgraph , then by theorem [ thm1 ] , fig.[sixcycle ] is a @xmath66 trapping set . on the other hand , if two odd degree checks , say @xmath67 and @xmath68 , are connected to another variable node , say @xmath69 , the subgraph resembles fig . [ 42trappingset ] . assuming no other connections , fig.[42trappingset ] is a @xmath70 trapping set . we make the following observations : 1 . the three variable nodes in a @xmath66 trapping set form a six cycle . however , not all six cycles are @xmath66 trapping sets . apart from the subgraph induced by variable nodes , the outside connections should be known to determine whether a given subgraph is a trapping set or not . the @xmath70 trapping set in fig.[42trappingset ] illustrates this point . the critical number of a @xmath66 trapping set is three . there exist @xmath70 trapping sets with critical number three and it is highly unlikely that a @xmath70 trapping set does not contain a failure set of size three . however , we can show by a counterexample that this is indeed possible . a @xmath54 trapping set is not unique i.e. , two trapping sets with same @xmath56 and @xmath57 can have different underlying topological structures ( induced subgraphs ) . so , when we talk of a trapping set , we refer to a specific topological structure . in this paper , the induced subgraph is assumed to be known from the context . 4 . to avoid a trapping set in a code , it is sufficient to avoid topological structures isomorphic to the subgraph induced by the trapping set . for example to avoid @xmath66 trapping sets of fig.[sixcycle ] , it is sufficient to avoid six cycles . it is possible that a code has six cycles but no @xmath66 trapping sets . in this case all six cycles are part of @xmath70 or other trapping sets . [ sixcycle ] [ 42trappingset ] burshtein and miller in @xcite applied expander based arguments to message passing algorithms . they analyzed ensembles of irregular graphs and showed that if the degree of each variable node is at least five , then message passing algorithms can correct a fraction of errors . codes with column weight three and four can not achieve the expansion required for these arguments . recently , burshtein in @xcite developed a new technique to investigate the error correction capability of regular ldpc codes and showed that at sufficiently large block lengths , almost all codes with column weight four are also capable of correcting a fraction of errors under bit flipping algorithm . for column - weight - three codes he notes that such a result can not be proved . this is because a non negligible fraction of codes have parallel edges in their tanner graphs and such codes can not correct a single worst case error . in this paper we prove a stronger result by showing that for any given @xmath71 , at sufficiently large block lengths @xmath14 , no code in the @xmath15 ensemble can correct all @xmath4 or fewer errors under gallager a algorithm and show that this holds for the bit flipping algorithm also . @xcite [ lemma1 ] a code whose tanner graph has parallel edges can not correct a single worst case error . see @xcite . the proof is for bit flipping algorithm , but also applies to gallager a algorithm . [ lemma2 ] let @xmath23 be an @xmath72 regular ldpc code with girth @xmath73 . then @xmath23 has at least one failure set of size two or three . see appendix [ appendix2 ] . [ lemma3 ] let @xmath23 be an @xmath72 regular ldpc code with girth @xmath74 . then @xmath23 has least one failure set of size three or four . since @xmath74 , there is at least one six cycle . without loss of generality , we assume that @xmath64 together with the three even degree checks @xmath75 and the three odd degree checks @xmath65 form a six cycle as in fig.[sixcycle ] . if no two checks from @xmath65 are connected to a variable node , then @xmath64 is a @xmath66 trapping set and hence a failure set of size three . on the contrary , assume that @xmath64 do not form a @xmath66 trapping set . then there exists @xmath69 which is connected to at least two checks from @xmath65 . if @xmath69 is connected to all the three checks , @xmath76 is a codeword of weight four and it is easy to see that @xmath64 is a failure set . now assume that @xmath69 is connected to only two checks from @xmath65 . without loss of generality , let the two checks be @xmath67 and @xmath68 . let the third check connected to @xmath69 be @xmath77 as shown in fig.[42trappingset ] . if @xmath78 and @xmath77 are not connected to a common variable node then @xmath76 is a @xmath70 trapping set and hence a failure set of size four . if @xmath78 and @xmath77 are connected to say @xmath79 , we have two possibilities : ( a ) the third check is @xmath80 and ( b ) the third check of @xmath79 is @xmath81 ( the third check can not be @xmath82 or @xmath83 as this would introduce a four cycle ) . we claim that in both cases @xmath76 is a failure set . the two cases are discussed below . case ( a ) : let @xmath84 . @xmath85 the messages in the second half of first iteration are , @xmath86 similar equations hold for @xmath87 . for @xmath78 we have @xmath88 similar equations hold for @xmath77 . at the end of first iteration , we note that @xmath89 and @xmath90 receive all incorrect messages , @xmath91 and @xmath79 receive two incorrect messages and all other variable nodes receive at most one incorrect message . we therefore have @xmath92 and @xmath93 . the messages sent by variable nodes in the second iteration are , @xmath94 the messages passed in second half of second iteration are same as in second half of first iteration , except that @xmath95 . at the end of second iteration , we note that @xmath89 and @xmath90 receive all incorrect messages , @xmath91 and @xmath79 receive two incorrect messages and all other variable nodes receive at most one incorrect message . the situation is same as at the end of first iteration . the algorithm runs for _ m _ iterations and the decoder outputs @xmath96 which implies that @xmath76 is a failure set . case ( b ) : the proof is along the same lines as for case ( a ) . the messages for first iteration are the same . the messages in the first half of second iteration are , @xmath97 the messages passed in second half of second iteration are same as in second half of first iteration , except that @xmath98 and @xmath99 . at the end of second iteration , @xmath100 and @xmath79 receive two incorrect messages and all other variable nodes receive at most one incorrect message and hence @xmath101 . the messages passed in first half of third iteration ( and therefore subsequent iterations ) are same as the messages passed in first half of second iteration . the algorithm runs for _ m _ iterations and the decoder outputs @xmath96 which implies that @xmath76 is a failure set . let @xmath23 be an @xmath72 regular ldpc code with girth @xmath102 . then @xmath23 has at least one failure set of size four or five . see appendix [ appendix2 ] . _ remark : _ it might be possible that lemmas [ lemma2][lemma4 ] can be made stronger by further analysis , i.e. , it might be possible to show that a code with girth four has a failure set of size two , a code with girth six has failure set of size three and a code with girth eight has a failure set of size four . however , these weaker lemmas are sufficient to establish the main theorem . [ lemma5 ] let @xmath23 be an @xmath72 regular ldpc code with girth @xmath11 . then the set of variable nodes @xmath103 involved in the shortest cycle is a trapping set of size @xmath12 . since @xmath23 has girth @xmath29 , there is at least one cycle of length @xmath29 . without loss of generality , assume that @xmath103 form a cycle of minimum length as shown in fig.[mincycle ] . let the even degree checks be @xmath104 and the odd degree checks be @xmath105 . note that each variable node is connected to two checks from @xmath60 and one check from @xmath59 and @xmath106 is connected to @xmath107 . we claim that no two checks from @xmath59 can be connected to a common variable node . the proof is by contradiction . assume @xmath108 and @xmath109 ( @xmath110 are connected to a variable node @xmath111 . then @xmath112 form a cycle of length @xmath113 and @xmath114 form a cycle of length @xmath115 . since @xmath11 , @xmath116 this implies that there is a cycle of length less than @xmath29 , which is a contradiction as the girth of the graph is @xmath29 . by theorem [ thm1 ] , @xmath103 is a trapping set . [ corollary1 ] for a code to correct all @xmath0 or fewer errors , it is necessary to avoid all cycles up to length @xmath1 . we now state and prove the main theorem . [ thm2 ] consider the standard @xmath10 regular ldpc code ensemble . let @xmath71 . let @xmath117 be the smallest integer satisfying @xmath118 then , for @xmath119 , no code in the @xmath120@xmath10 ensemble can correct all @xmath4 or fewer errors . first observe that for any @xmath121 , we have @xmath122 from [ theorem c.1 @xcite ] and [ lemma c.1 @xcite ] , we have the girth @xmath29 of any code in @xmath15 is bounded by @xmath123 for @xmath121 , equations ( [ girtheq1 ] ) and ( [ girtheq2 ] ) imply that for any code in the @xmath15 ensemble , the girth is bounded by @xmath124 the result now follows from corollary [ corollary1 ] . the bit flipping algorithm does not belong to the class of message passing algorithms . however , the definitions from section [ section2 ] and the results from section [ section3 ] can be generalized to the parallel bit flipping algorithm @xcite . without loss of generality we assume that the all zero codeword is sent . we begin with a few definitions . @xcite a variable node is said to be corrupt if it is different from its original sent value . in our case , a variable node is corrupt if it is @xmath52 . a check node is said to be satisfied if it is connected to even number of corrupt variables and unsatisfied otherwise . let @xmath33 be the input to the parallel bit flipping decoder . @xmath125 is a trapping set for bit flipping algorithm if the set of corrupt variables after every iteration is @xmath125 . [ thm3 ] let @xmath126 be a set of variable nodes satisfying the conditions of theorem [ thm1 ] . then @xmath126 is a trapping set for the bit flipping algorithm . let @xmath127 . then @xmath126 is the set of corrupt variable nodes . observe that a variable flips if it is connected to at least two unsatisfied checks . since no variable is connected to two unsatisfied checks , the set of corrupt variable nodes is unchanged and by definition @xmath126 is a trapping set . we note that theorem [ thm3 ] is also a consequence of fact 3 from @xcite . a trapping set for gallager a is also a trapping set for bit flipping algorithm . it can be shown that lemmas [ lemma1][lemma5 ] and theorem [ thm2 ] also hold for the bit flipping algorithm . in this paper we have investigated the error correction capability of column - weight - three codes under gallager a and extended the results to bit flipping algorithm . future work includes investigation of sufficient conditions to correct a given number of errors for column - weight - three as well as higher column weight codes . [ [ appendix1 ] ] _ proof of theorem [ thm1 ] : _ let @xmath33 be the input to the decoder with @xmath128 . then , @xmath129 let a check node @xmath130 . then , @xmath131 let a check node @xmath132 . then , @xmath133 for any other check node @xmath36 , @xmath134 . by the conditions of the theorem , at the end of first iteration , any @xmath135 receives at least two @xmath52 s and any @xmath136 receives at most one @xmath52 so , we have @xmath137 by definition , @xmath126 is a trapping set . to see that the conditions stated are necessary observe that for a variable node to send the same messages as in the first iteration , it should receive at least two messages which coincide with the received value . @xmath32 [ [ appendix2 ] ] _ proof of lemma [ lemma2 ] : _ let @xmath138 be the variable nodes that form a four cycle with even degree checks @xmath139 and odd degree checks @xmath140 . if @xmath83 and @xmath78 are not connected to a common variable node , then @xmath141 is a @xmath142 trapping set and hence a failure set of size two . now assume that @xmath83 and @xmath78 are connected to a common variable node @xmath90 . then , @xmath64 is a @xmath143 trapping set and therefore a failure set of size three . @xmath32 _ proof of lemma [ lemma4 ] : _ let @xmath144 be the variable nodes that form an eight cycle ( see fig.[44trappingset ] ) . if no two checks from @xmath145 are connected to a common variable node , then @xmath146 is a @xmath147 trapping set and hence a failure set of size four . on the other hand , if @xmath146 is not a trapping set , then there must be at least one variable node which is connected to two checks from @xmath145 . assume that @xmath67 and @xmath77 are connected to @xmath79 and the third check of @xmath79 is @xmath148 ( see fig.[53trappingset ] ) . we claim that @xmath149 is a failure set . let @xmath150 and @xmath151 be as defined in theorem [ thm1 ] . [ 44trappingset ] [ 53trappingset ] * case 1 : * no two checks from @xmath152 are connected to a common variable node . then @xmath153 is a @xmath154 trapping set and hence a failure set of size five . * case 2 : * all the three checks in @xmath151 are connected to a common variable node , say @xmath155 . then @xmath156 is a codeword of weight six and it is easy to see that @xmath153 is a failure set . * case 3 : * there are variable nodes connected to two checks from @xmath151 . there can be at most two such variable nodes ( if there are three such variable nodes , they will form a cycle of length less than or equal to six violating the condition that the graph has girth eight ) . note that if @xmath157 , the decoder has a chance of correcting only if a check node in @xmath150 receives an incorrect message from a variable node outside @xmath153 in some @xmath37 iteration . we now prove that this is not possible . indeed in the first iteration @xmath158 by similar arguments as in the proof for theorem [ thm1 ] , it can be seen that the only check nodes which send incorrect messages to variable nodes outside @xmath153 are @xmath159 and @xmath148 . there are now two subcases . * subcase 1 : * there is one variable node connected to two checks from @xmath151 . let @xmath155 be connected to @xmath68 and @xmath80 . it can be seen that the third check connected to @xmath155 can not belong to @xmath150 as this would violate the girth condition . so , let the third check be @xmath160 . in the first half of second iteration , we have @xmath161 the only check nodes which send incorrect messages to variable nodes outside @xmath153 , are @xmath162 and @xmath160 . the variable node @xmath155 is connected to @xmath68 and @xmath80 . if @xmath148 and @xmath160 are not connected to any common variable node , we are done . on the other hand , let @xmath148 and @xmath160 be connected to a variable node , say @xmath163 . the third check of @xmath163 can not be in @xmath150 . proceeding as in the case of proof for lemma [ lemma3 ] , we can prove that @xmath153 is a failure set by observing that there can not be a variable node outside @xmath153 which sends an incorrect message to a check in @xmath150 . * subcase 2 : * there are two variable nodes connected to two checks from @xmath151 . let @xmath68 and @xmath80 be connected to @xmath155 and @xmath68 and @xmath148 connected to @xmath163 . proceeding as above , we can conclude that @xmath153 is a failure set . this work is funded by nsf under grant ccf-0634969 , itr-0325979 and insic - ehdr program . the authors would like to thank anantharaman krishnan for illustrations . a. shokrollahi , `` an introduction to low - density parity - check codes , '' in _ theoretical aspects of computer science : advanced lectures_.1em plus 0.5em minus 0.4emnew york , ny , usa : springer - verlag new york , inc . , 2002 , pp . 175197 . s. k. chilappagari , s. sankaranarayanan , and b. vasic , `` error floors of ldpc codes on the binary symmetric channel , '' in _ international conference on communications _ , vol . 3 , june 11 - 15 2006 , pp . 10891094 .

in this paper , we investigate the error correction capability of column - weight - three ldpc codes when decoded using the gallager a algorithm . we prove that the necessary condition for a code to correct @xmath0 errors is to avoid cycles of length up to @xmath1 in its tanner graph . as a consequence of this result , we show that given any @xmath2 such that @xmath3 , no code in the ensemble of column - weight - three codes can correct all @xmath4 or fewer errors . we extend these results to the bit flipping algorithm . * index terms * low - density parity - check codes , gallager a algorithm , trapping sets , error correction capability

You are an expert at summarizing long articles. Proceed to summarize the following text: the dynamics and roughening of moving interfaces in disordered media has been a subject of great interest in non - equilibrium statistical physics since the 1980s . relevant examples of physically and technologically important processes include thin film deposition @xcite , fluid invasion in porous media @xcite , and wetting and propagation of contact lines between phase boundaries @xcite . the understanding of the underlying physics involved in interface roughening is crucial to the control and optimization of these processes . significant progress in the theoretical study of interface dynamics has been made and a number of theories have been developed @xcite , which in some selected cases agree well with the experimental findings @xcite . most of the theoretical understanding in this field is based on modeling interface roughening with a local stochastic equation of motion for the single - valued height variable of the interface . however , there are several cases of interest where such an approach can not be justified _ e.g. _ due to conservation laws in the bulk . this is especially true for processes such as fluid invasion in porous media , which is often experimentally studied by hele - shaw cells @xcite or imbibition of paper @xcite . it has been shown that in such cases spatially local theories can not provide a complete description of the underlying dynamics . for describing the diffusive invasion dynamics in such systems , a phase - field model explicitly including the local liquid bulk mass conservation law has been developed and applied to the dynamics of 1d imbibition fronts in paper @xcite . this was achieved by a generalized cahn - hilliard equation with suitable boundary conditions , which couple the system to the reservoir . numerical results for roughening from the model are in good agreement with relevant experiments @xcite . one of the great advantages of the phase - field approach is that it s possible to analytically derive equations of motion for the phase boundaries in the so - called sharp interface limit @xcite . most recently , we have developed a systematic formalism to derive such equations for the 2d meniscus and 1d contact line dynamics of fluids in capillary rise @xcite . the equations are derived from the 3d bulk phase - field formulation , using variational approach as applied to relevant rayleigh dissipation and free energy functionals . through successive projections , equations of motion for the 2d meniscus and 1d contact line can be derived . the leading terms of such equations ( for small amplitude , long wavelength fluctuations ) can be shown to agree with results obtained from the sharp interface equations in the appropriate limits @xcite . in addition to the need for non - local models to account for mass conservation , in hele - shaw and imbibition type of problems the inherent quenched disorder should be properly taken care of . unlike thermal disorder , which is relatively easy to handle , quenched disorder depends on the height of the 1d interface @xmath0 as @xmath1 . this makes its influence on interface roughening highly nontrivial , often leading to anomalous scaling @xcite . currently , for such cases good agreement between theory , simulations and experiments has not been achieved . even on the experimental side some results such as the quantitative values of the scaling exponents , are not consistent and difficult to interpret . very recently , soriano _ et al . _ @xcite conducted an experimental study of forced fluid invasion in a specially designed hele - shaw cell . the quenched disorder pattern in hele - shaw cell is realized by creating large number of copper islands that randomly occupy the sites of a square grid on a fiberglass substrate fixed on the bottom hele - shaw cell . three different disorder patterns were used . two of them are obtained by random selection of the sites of a square lattice . the third kind of disorder is formed by parallel tracks , continuous in the interface growth direction and randomly distributed along the lateral direction . it was found that for forced flow the temporal growth exponent @xmath2 which is nearly independent of experimental parameters and disorder patterns . however , the spatial roughness exponent @xmath3 was found to be sensitive to experimental parameters and disorder patterns . anomalous scaling with @xmath4 , and a local roughness exponent @xmath5 was found in disorder pattern with parallel tracks along the growth direction . it was also demonstrated that such anomalous scaling is a consequence of different local velocities on the tracks and the coupling in the motion between neighboring tracks . on the theoretical side , for non - local hele - shaw and paper imbibition problems there are two different approaches within the phase - field models to include additive quenched disorder . et al . _ @xcite put the quenched disorder inside the chemical potential , the gradient of which is the driving force for interface motion . on the other hand , hernandez - machado _ et al . _ @xcite accounted for the effect of fluctuation of hele - shaw gap thickness as a mobility with quenched disorder in the phase - field model , while keeping the chemical potential free of noise . these two approaches lead to quantitatively different roughening properties . when considering the problem of capillary rise in a typical hele - shaw cell set - up between two rough walls more microscopically , the location of the surface of such corrugated walls in the cartesian coordinate system is a spatially fluctuating quantity , which indicates the presence of quenched disorder . an experimental realization is given in @xcite . to treat this problem faithfully , in solving the phase - field equation such a fluctuating wall surface should be treated as a physical boundary without phenomenologically adding quenched noise to the equation of motion as done previously . consequently it is evident that a rigorous analytic treatment of such a problem is overwhelmingly difficult . however , in this paper we demonstrate that with proper mathematical formulation of the problem , it is possible - albeit with some approximations - to analytically derive equations of motion for the meniscus and contact line dynamics . most importantly , these equations incorporate the wall disorder in a natural way . to achieve this , we utilize an explicit curvilinear coordinate transformation of the phase - field equation in order to apply projection methods to unravel the relevant physics in the limit of small disorder . to some extent , this kind of curvilinear coordinate transformation is similar to the boundary - fitted coordinate system frequently used in computational fluid dynamics ( cfd ) @xcite . the outline of this paper is as follows : in chapter [ 2d ] we will consider the phenomenological 2d phase field model of capillary rise with quenched disorder in the mobility , similar to that in refs . we will adapt the systematic projection method of kawasaki and ohta @xcite to obtain a linearized interface equation ( lie ) that describes small fluctuations of an interface , whose deterministic part reduces to the previous result @xcite in a special limit . to treat the problem rigorously , in chapter [ 3d ] we will consider the full 3d phase field model with corrugated walls as the source of quenched disorder . the transformation to curvilinear coordinates , as discussed above , is introduced to obtain linearized , effective bulk disorder from the original curvilinear boundary condition . following this we develop and apply a general projection scheme @xcite to obtain the effective equations of motion for small fluctuations of the 2d meniscus , and ultimately for the 1d contact line between the meniscus and the wall . again , the deterministic parts of these equations reduce to previously known limits in special cases . however , we demonstrate that the forms of the quenched noise terms derived here are different from the previous works . a 2d phase - field model explicitly including the local conservation of bulk mass was introduced to study capillary rise by dub _ the bulk disorder in their model was included in the effective chemical potential . recently a similar model , where the disorder was considered through a stochastic mobility coefficient , was studied by hernandez - machado _ et al . in particular , they assumed a one - sided mobility coefficient , which vanishes on one side of the interface . from this model they derived an equation of motion for small interface fluctuations . in this section , we will use the systematic projection method introduced by kawasaki and ohta @xcite , to derive the corresponding linearized interface equation ( lie ) describing small fluctuations in a sharp interface in a similar model . in our model we assume the mobility to be independent of the phase , as in the previous works @xcite , but spatially stochastic , as in @xcite . this corresponds to considering the invading fluid and the porous medium , but not the receding fluid . this picture is valid when the receding fluid has low density and viscosity . in practice this would mean a gas , such as air , being displaced by a liquid , such as water or oil . the model allows a systematic projection of the effective noise term at the interface . the phase field model describes capillary rise at a coarse - grained level with a phase field @xmath6 that obtains the value @xmath7 in the phase of the displaced fluid , and @xmath8 in the phase of the displacing fluid . the phase field thus describes the effective component densities , and thus must be locally conserved . an energy cost for an interface is included to obtain the free energy as @xmath9=\frac{1}{2}[\nabla\phi(\mathbf{x , t})]^2 + v(\phi(\mathbf{x , t})),\ ] ] where @xmath10 has two minima at @xmath8 and @xmath7 . the details of @xmath10 are not relevant in the sharp interface limit , except to define the surface tension , so we can choose the standard ginzburg - landau form @xmath11 , where one of the phases can be set metastable by nonzero coefficient @xmath12 . the equation of motion for the conserved phase field is given by the continuity equation @xmath13 and fick s law @xmath14 , where @xmath15 , and @xmath16 is the mobility that we choose to be a position dependent stochastic variable here . the resulting equation of motion for the phase field is then given by @xmath17 = m\nabla\cdot ( 1+\xi(x))\nabla\left[v'(\phi)-\nabla^2\phi\right],\ ] ] where the variable @xmath18 is now the dimensionless , quenched noise . the sharp interface limit of this model without the noise is well known , and discussed e.g. in ref . the geometry of the problem is that of a half - plane , where a reservoir of the displacing fluid is located at the @xmath19-axis . the boundary condition of the chemical potential at the half - plane boundary can be connected to the physical effect that is driving the capillary rise . in this paper we will consider spontaneous imbibition , where the rise is driven by a chemical potential difference in the medium , which favors the displacing fluid @xcite . this means that the two minima of @xmath10 are at different heights . in our notation the chemical potential difference is @xmath20 , and we consider chemically homogenous medium , where @xmath21 . spontaneous imbibition corresponds to a dirichlet boundary condition at the reservoir @xcite . forced flow , where flow is caused by an imposed mass flux into the system from the reservoir , can be modeled with the neumann boundary condition , where @xmath22 is the flux @xcite . an analysis along the lines presented in this paper can also be conducted for the case of forced imbibition . a recent review of phase field modeling of imbibition is given in refs . @xcite . using the green s function @xmath23 for the 2d laplacian , equation can be inverted using gauss s theorem . this leads to the integro - differential form @xmath24 notation here has been shortened by omitting the function arguments , and using unprimed and primed functions for functions of unprimed and primed coordinates , respectively . the green s functions always take both primed and unprimed coordinates as argument . also the coordinate invariant form is used , with integration measure given by @xmath25 . the boundary term @xmath26 vanishes in the case of spontaneous imbibition , or dirichlet boundary condition in half - plane geometry . using the standard 1d kink solution method for projection to sharp interface @xcite in normal coordinates gives eq . as @xmath27\\ + \int du\partial_u\phi_0\int ds'du'\sqrt{\det(g')}g\nabla'^2\xi'(\kappa'\partial_{u'}\phi_0'+\alpha ) , \end{split}\ ] ] where the normal coordinates @xmath28 are distances along and perpendicular to the interface , respectively , @xmath29 is the local curvature of the interface , @xmath30 is the surface tension of the phase field model , and finally @xmath31 is the 1d kink solution @xmath32 . in the ginzburg - landau form of @xmath10 this would be given by @xmath33 , with the appropriate choice of dimensionless units . we have assumed a disorder correlation length that is larger than the interface width , which leads to the constant surface tension obtained . with two further approximations is approximated with its form for the 1d disorder free system : @xmath34 , where @xmath35 is the normal velocity . this approximation is similar to the 1d kink solution in the standard procedure , but its range of extent is not only near the interface , but between the reservoir and interface . another assumption we made in the derivation is neglecting boundary terms of form @xmath36 . this is justified when the width of the kernel @xmath37 along @xmath19-direction , which is of order @xmath38 , is much larger than the correlation length of the disorder @xmath18 . ] the standard procedure @xcite can be followed . transforming the equation to cartesian coordinates is made somewhat more tedious by the necessity to transform derivatives w.r.t . @xmath39 and @xmath40 , but standard differential geometry methods can be applied . after the sharp interface limit , i.e. @xmath41 , the transformation to cartesian coordinates , and linearization in small fluctuations of the interface @xmath42 and the noise @xmath18 , which also eliminates cross terms proportional to @xmath43 , we get the lie as @xmath44\partial_t\big[h_0(t)+h(x',t)\big]=\\ -\sigma\partial_x^2h(x , t)-\alpha+\frac{\partial_th_0(t)}{m}\xi(x , h_0(t ) ) , \end{split}\ ] ] where the disorder term @xmath45 is given by @xmath46 note that the linearization has been carried out in full here . this means that the disorder term does not include any dependence on the interface fluctuations . this eliminates the non - linearity of the quenched noise , which is one of its characteristic properties , but we believe it is not crucial in the regime where the linearization is appropriate . in other words , our results show non - trivial features that arise in the effective noise at the interface level with this type of multiplicative bulk disorder , even in the linear regime of weak disorder . the green s function for the dirichlet boundary condition in half - plane geometry is explicitly given by @xmath47 using this , the fourier space representation of the interface equation becomes @xmath48 where @xmath49 , @xmath50 is the fourier transform of the chemical potential difference ( @xmath51 , if @xmath52 , when @xmath53 ) , and the disorder in fourier space is given by @xmath54 in the case of columnar disorder , which does nt depend on @xmath55 , the interface equation simplifies to @xmath56 it is noteworthy that the limit @xmath57 the interface equation is @xmath58 , readily giving the correct washburn law @xcite , if we associate @xmath59 , and @xmath60 . our method of analysis can be applied to the case of forced flow by simply changing the boundary condition of the phase field model at the reservoir , and applying the corresponding green s function @xcite . the dispersion relation above is the main result in this section . it involves two length scales : a crossover length scale @xmath61 @xcite , and the distance from the reservoir @xmath62 . the deterministic part of the dispersion relation is plotted in fig . [ fig : dispersion ] , at the two limits of these length scales : the limit @xmath63 brings out the `` deep '' limit , @xmath64 , behavior . the limit @xmath65 shows the `` shallow '' limit , @xmath66 , behavior . a plot from the intermediate regime with @xmath67 is also shown . ( vertical dashed line ) and @xmath62 ( vertical dotted line ) . the upper figure focuses on the `` deep '' regime , with @xmath63 , in the middle figure these length scales are the same , and the lowest figure focuses on the `` shallow '' regime , with @xmath65.,width=302,height=377 ] the deterministic part of the dispersion relation here is identical to that previously obtained by dub _ @xcite for the case of disorder in the chemical potential . in the `` deep '' limit where @xmath68 , our result also reproduces that of hernandez - machado _ et al . _ @xcite for the one - sided mobility case . using different methods the same result has also been obtained for the hele - shaw setup by paune and casademunt @xcite , and ganesan and brenner @xcite . our disorder term in the lie is similar to those obtained in refs . @xcite in the sense that in all cases the effective noise is linearly proportional to the velocity of the interface propagation . however , the quantitative forms of the noise terms are different when using different methods . how these differences influence the kinetic roughening of interfaces would need to be determined by extensive numerical comparison between the different results , which at this point has not been conducted . as a linear @xmath69 proportionality in the fourier space representation of the effective noise term is linked to the @xmath55-derivative of the green s function of the laplacian in real space representation , it appears to us that the linear @xmath69 is present in the noise terms of refs . @xcite and @xcite , but not in ref . the linear @xmath69 dependence is in general characteristic of effective interface noise caused by conserved bulk disorder @xcite . however , the @xmath70 dependence ( @xmath71 ) dimensionally cancels the integral over the kernel in the effective noise , eq .. this is explicit in the case of columnar disorder in eq . , but is equally valid with the noise in the non - columnar case . thus the multiplicative bulk disorder in the mobility leads to a different type of effective noise than the chemical potential disorder , which is considered in @xcite . dimensionally this can be seen from the definition of the model , eq . , where the noise term is in front of the gradient of the chemical potential . the fact that the columnar disorder leads to effective noise , which is local in fourier space , is in accordance with the conclusions of experiments of soriano _ et al . _ @xcite , and with the numerical results from the one - sided model @xcite . this would indicate that the phase dependence of the mobility is not crucially important when considering the invasion of a viscous fluid into a fluid with negligible viscosity , and when the interface is consequently always stable . when the direction of the invasion is reversed , as studied recently with the one - sided model in @xcite , the situation is naturally quite different . while the stochastic mobility case of the previous section is heuristically appealing , a more faithful treatment of the wall disorder should start from the microscopic roughness of the walls . to this end , in this chapter we will study a 3d version of the same phase - field model as the 2d model , but where the mobility is constant and the disorder is explicitly included as fluctuations of the wall position . thus the geometry of the model is that of a hele - shaw cell : the 3d volume between two walls that are planar on average , but fluctuate . we will show here that by proper mathematical formulation this model can indeed be analyzed by a generalized projection formalism @xcite . the basic idea is to perform a mathematical transformation from the basic cartesian to a local curvilinear coordinate system as defined by the wall itself . to this end , we consider the one - wall setup as shown in fig . 2 . the one - wall setup neglects the meniscus - mediated interaction between the two contact lines at the two walls . the one - wall approximation also neglects finite gap spacing , i.e. the distance between the two walls in the hele - shaw cell , which fluctuates as result of the wall fluctuations . this induces additional disorder effects when the wall fluctuations are comparabale to the gap spacing , but it remains to be studied if the gap effect can be separated from the contact line interaction , which would be represented by two coupled equations of motion for the two contact lines . in the present work , we only consider to the one - wall approximation , or the limit of large gap spacing . disorder at the wall surface is taken into account by describing local corrugations in the wall position around @xmath72 by a ( small ) function @xmath73 . the explicit coordinate transformation to the local , curvilinear wall coordinate system is defined by @xmath74 which corresponds to @xmath75 when @xmath73 . this means that in the new coordinate system the wall is located back at @xmath75 . given the proper green s function , @xmath76 the phase field equation can be inverted in any geometry and coordinate system as @xmath77 where @xmath78 is the corresponding surface term , and @xmath79 is the volume element for the coordinate system . the green s function appropriate for the above mentioned coordinate system is considered in some detail in appendix [ curvy ] . here we compute the correction to the original cartesian green s function to linear order in @xmath80 . the final result we obtain , after neglecting some surface contributions that are discussed in more detail in appendix [ curvy ] , is what one would expect by simply plugging the above definitions into the cartesian green s function and linearizing in @xmath81 : @xmath82 here @xmath83 is the green s function for the laplacian in 3d cartesian coordinates as given in eq . . here we again only consider spontaneous capillary rise , where the boundary conditions for the phase field model are zero chemical potential at the reservoir and zero flux at the walls . thus the surface integral term in eq . is identically zero . the projection and linearization of the integral equation follows the standard projection operation theory @xcite , which we already used in the previous chapter for the 2d model . the generalization for the present case is straightforward . after projection the integral equation is expressed in terms of the 2d meniscus variable @xmath84 and has the following form : @xmath85 when linearizing the above equation , it must be done simultaneously in the meniscus fluctuations , _ i.e. _ @xmath86 , and in the wall fluctuations using the linearized green s function of eq .. this results in the linearized green s function evaluated at the meniscus : @xmath87 substituting this into the meniscus equation gives @xmath88 where the left hand side equation is to the zeroth order , and the right hand side is to the first order in @xmath89 or @xmath90 . these terms are defined in the same fashion as those in cartesian coordinate system @xcite . the terms @xmath91 and @xmath92 arise from the fluctuating wall . they are given by @xmath93 the zeroth order equation would give the washburn law , if we used the green s function for the geometry between two walls and assumed a constant curvature for the meniscus . we will assume an average profile @xmath94 , which can be considered to obey washburn s law even though we have only a single vertical wall in the system . since @xmath62 is not needed for determining the form of the evolution equation for the fluctuating part @xmath42 of eq . at that single wall , the precise time - dependence of @xmath62 is not crucial for the analysis to be presented below . a local equation of motion for the meniscus fluctuations can be obtained by fourier - cosine transformation following @xcite . the above terms become @xmath95&= & \frac{1}{2}\dot{h}_0h(\vec{k},t ) ; \\ { \mathcal{f}}_{x / k_x}{\mathcal{f}}^{cos}_{y / k_y}[i_{c}]&= & \frac{1}{2}e^{-2kh_0}\dot{h}_0h(\vec{k},t ) ; \\ { \mathcal{f}}_{x / k_x}{\mathcal{f}}^{cos}_{y / k_y}[i_{d}]&= & \frac{1}{2k}\dot{h}(\vec{k},t)\left(1-e^{-2kh_0}\right ) ; \\ i_{\textrm e}&=&0 ; \\ { \mathcal{f}}_{x / k_x}{\mathcal{f}}^{cos}_{y / k_y}[i_{\textrm f}]&=&\frac{\dot{h}_0(t)}{2k } \left(1-e^{-2kh_0}\right)\delta h(k_x , h_0(t)),\end{aligned}\ ] ] where @xmath96 . we then have the meniscus equation of motion using the above in the fourier transform of eq . : @xmath97 the deterministic part of the above meniscus equation is identical to the deterministic part of the lie derived from the 2d phase - field model , apart from the dimensionality . this is by construction , since the same method was used for the same theory in different dimensions by applying the corresponding green s functions . a similar analysis can also be performed for the case where the disorder at the walls consists of chemical impurities ( _ i.e. _ spatially fluctuating surface tension ) instead of spatial roughness @xcite . in this case the deterministic part of the meniscus equation is by construction identical to that of the above . however , there is no effective noise at the meniscus level , since the effect of the disorder comes in from the contact line that serves as a boundary condition for the meniscus . to proceed to the level of the 1d contact line we consider the generalized variational approach @xcite . formally , one can write the 3d phase field model in terms of variations of a rayleigh dissipation functional , and a free energy functional . then , using approximations that express higher dimensional entities in term of the relevant lower dimensional ones , we obtain a chain of projection equations as @xmath98 } { \delta \dot { \phi}(x , y , z;t ) } & = & - \frac { \delta f_{\textrm { 3d}}[\phi ] } { \delta \phi(x , y , z;t ) } \\ \label{2d_functional_eom } \rightarrow \frac { \delta r_{\textrm { 2d } } [ \dot h ] } { \delta \dot h ( x , y;t ) } & = & - \frac { \delta f_{\textrm { 2d}}[h ] } { \delta h(x , y;t ) } \\ \rightarrow \frac { \delta r_{\textrm { 1d } } [ \dot c ] } { \delta \dot c ( x;t ) } & = & - \frac { \delta f_{\textrm { 1d}}[c ] } { \delta c(x;t)}\ , , \label{ray}\end{aligned}\ ] ] where @xmath99 refers to the rayleigh dissipation functional , and @xmath100 refers to the free energy functional in @xmath101 dimensional space . here the relevant 3d , 2d and 1d objects are the phase field , the meniscus profile and the contact line profile , respectively . the variable @xmath102 denotes the fluctuating contact line profile , and @xmath103 for the one - wall case . the corresponding expansion for the contact line is @xmath104 . for small fluctuations @xmath42 and @xmath105 , consistency requires that @xmath106 . the projection from 3d to 2d is made possible by the 1d kink approximation in the direction normal to the interface , as demonstrated in the preceding section . the corresponding approximation we have used to make the 2d to 1d projection possible is the quasi - stationary ( qs ) approximation @xmath107 ; @xmath108 , which corresponds to the minimum of energy constrained by the contact line profile . the meniscus can then be expressed in terms of the contact line as @xmath109 the explicit forms for the rayleigh dissipation and free energy functionals that reproduce the meniscus equation when plugged into eq . are @xmath110&=&\int_{-\infty}^{\infty}~dx_1dx_2\int_0^{\infty}~dy_1dy_2\int~dt_1 \dot{h}(x_1,y_1,t_1);\nonumber\\ & & \times \tilde{g}_{3d}(x_1,y_1,h(x_1,y_1,t_1);x_2,y_2,h(x_2,y_2,t))\dot{h}(x_2,y_2,t_1 ) \label{2d_rayleigh}\\\hspace{2 cm } f_{2d}[h]&=&\sigma_b\int_{-\infty}^{\infty}~dx_1\int_0^{\infty}~dy_1\int~dt_1~\sqrt{1+|\nabla h(x_1,y_1,t_1)|^2}.\label{2d_energy}\end{aligned}\ ] ] the effective 1d functionals can be obtained from the above by inserting the quasi - stationary approximation @xmath111 into and . in order to obtain the 1d equation of motion to linear order in small fluctuations one needs to expand the functionals to second order in both @xmath112 and @xmath113 , and then take the variation with respect to the contact line as shown in eq .. neglecting the zeroth order equation for the reasons mentioned earlier , the general fourier space equation of motion we obtain for the first order fluctuations is @xmath114=-\sigma_b |k_x| c(k_x , t),\ ] ] where the lhs is the variation of the rayleigh dissipation functional , and the rhs is the variation of the free energy . the rhs is recognized as the deterministic restoring force acting on the contact line @xcite . the shorthand notations stand for @xmath115 not all of these integrals are solvable in closed form , but can be approximated to a good degree of accuracy by the following expressions : @xmath116&=&\frac{\dot{c}_0}{2|k_x|}c(k_x , t ) ; \label{eq_3d_corrug_first_cleom_term}\\ { \mathcal{f}}_{x / k_x}[i_{3}]&=&\frac{2\dot{c}_0}{\pi \frac{e^{-2|k_x|c_0s}}{s^3\sqrt{s^2 - 1}}\nonumber\\ & & \approx \frac{1.14\cdot 4}{3\pi { \mathcal{f}}_{x / k_x}[i_{4}]&=&\frac{2\dot{c}(k_x , t)}{k_x^2\pi}\int_{1}^{\infty}ds \frac{1- e^{-2|k_x|c_0s}}{s^4\sqrt{s^2 - 1}}\nonumber\\ & & \approx \frac{4\dot{c}(k_x , t)}{3\pi k_x^2 } \left(1-e^{-2.28|k_x|c_0}\right);\\ { \mathcal{f}}_{x / k_x}[i_{5}]&=&0;\\ { \mathcal{f}}_{x / k_x}[i_{6}]&=&\frac{2}{\pi |k_x|}\dot{c}_0\delta h(k_x , c_0 ) \int_1^\infty ds \frac{1- e^{-2|k_x|c_0s}}{s^2\sqrt{s^2 - 1}}\nonumber\\ & & \approx\frac{2}{\pi |k_x|}\dot{c}_0\delta h(k_x , c_0 ) \left(1- e^{-2.56|k_x|c_0}\right).\end{aligned}\ ] ] we note that the corrections to the free energy functional in the curvilinear coordinates are of third order in @xmath81 and @xmath42 . this can be seen by coordinate transforming the area element , which in cartesian coordinates is @xmath117 . finally , after approximating @xmath118 and @xmath119 , the equation of motion for the contact line fluctuations becomes @xmath120 note that all of the approximations above are for dimensionless quantities , with errors depending on the physical parameter @xmath121 . the relative errors in the approximated functional forms are under 3% , when compared against numerical integration of the respective integrals for different values of @xmath122 . an exception to this are relative errors of @xmath123 $ ] and @xmath124 $ ] when @xmath125 , as both @xmath126 and @xmath127 vanish at the limit , causing the relative error behave badly . however , at machine precision away from @xmath128 then these errors are no more than 15% , and more importantly the error of the complete dispersion relation stays within the 3% error margin . this is due to the fact that the dispersion remains finite , as one can see from figure [ fig : dispersion ] . apart from simple numerical factors , the contact line equation above has the same functional form as the results derived in the previous sections . in particular , @xmath129 in the `` deep '' limit @xmath130 , which thus agrees with the previous works discussed earlier @xcite . this form of dispersion relation is always obtained by our method for interfaces in model b. this has to do with the quasi - stationary approximation , which essentially assumes that meniscus fluctuations dampen quickly in the direction perpendicular to the contact line , in order to obtain temporally local equations . how this leads to the coupling of the meniscus and contact line dynamics is discussed in more detail in another publication @xcite . the effective noise term we obtain from the 3d model shares the property of linear dependence on the velocity of the propagation with the 2d mobility noise , and with the previous analyses @xcite . in the case of surface tension impurities at the wall @xcite the effective contact line noise is proportional to @xmath131 , whereas in the fluctuating wall case in eq . the @xmath132 dependence is linear . this is analogous to the 2d mobility disorder in the sense that the effective noise is different from that obtained for conserved disorder . the extra factor of @xmath133 as compared to the 2d model ( eq . ) comes from the fact that the disorder in the 3d model comes from the walls , whereas in the 2d model the disorder is in the bulk . we note that the more complicated properties of the noise in the form of non - locality in fourier space are lost by our approximations . note that the noise is still non - local in real space , as is apparent from its real space representation @xmath134 in eq . . in addition to fourier space non - localities , our one - wall approach does nt explicitly include the gap spacing , which provides an additional physical length scale . in spite of this , our results are in accordance with those of ref . @xcite , where the gap fluctuations were considered as the only source of disorder in context of darcy s law . in this paper , we have studied the effective interface dynamics of the three - phase contact line in a hele - shaw experiment by deriving the meniscus and contact line equations of motion from higher dimensional bulk phase - field theories by projection methods . the projection methods take into account the non - local dynamics of the system caused by local mass conservation , and can be systematically applied starting from a full 3d description . we have here considered two particular models , namely an _ ad hoc _ model where the disorder is in the effective mobility in 2d @xcite , and a more microscopic model where wall corrugation in 3d is explicitly treated with a curvilinear coordinate transformation . in both cases , we have focused on the limit of small disorder by linearizing in disorder strength and in the fluctuations caused by the disorder . by construction this linearization , performed in real space , causes the fourier space representations of the equations of motion to be local . the upside of this is that the effective dynamics are written in a concise manner , and the physical predictions are easily interpreted and the equations we obtain are readily affable to numerical analysis . the obvious downside is that the procedure involves a number of approximations , the validity of which is not certain _ a priori_. in particular , the quasi - stationary approximation of eq . , which ultimately enables our contact line analysis , requires a critical assessment . a more rigorous approach would in fact consider the contact line as the boundary condition to the meniscus equation of motion . however , explicitly solving the meniscus profile as a function of the contact line leads to an equation that is , among other things , non - local in time . thus we are forced to simplify the model by using the qs approximation , the validity of which we can consider both from a physical perspective , or more rigorously by considering the limits of the meniscus equation of motion . physically , the qs approximation comes from the minimum of meniscus energy constrained by the boundary condition of the contact line . this is expected to define the meniscus profile when the meniscus moves slowly , and thus it s called the quasi stationary approximation . mathematically the meniscus equation reduces to the diffusion equation when @xmath135 , and @xmath136 . in this limit the meniscus level disorder @xmath137 acts as a source term @xmath138 this leads to an additional disorder term in the qs approximation , which then leads to a plethora of new first order disorder terms in @xmath139 and @xmath140 . however , all these new disorder terms arising from @xmath139 are proportional to @xmath141 , and thus not relevant in the qs limit . additionally , the two new disorder terms created in @xmath140 due to @xmath137 cancel each other out exactly . thus , we expect our results with the simplified version of the qs approximation to hold in this particular limit . in addition to the detailed derivations and new projection formalism presented here , our purpose has been to quantitatively compare two different approaches to modeling rough wall hele - shaw experiments , namely that based on the 2d phase field model with a stochastic mobility , eq . , and the 3d phase field model with a fluctuating geometry . the projection method we use for both cases produces the linear response of the meniscus and contact line to small fluctuations . for both cases , the @xmath142 dependence of the meniscus and contact line deterministic lies is the same . in particular , in the special case of the `` deep '' limit where @xmath143 , the asymptotic forms of our general dispersion relations are in agreement with previous works on the hele - shaw problem by paune and casademunt @xcite , ganesan and brenner @xcite and hernandez - machado _ et al . the main advantage of our method is the way it incorporates the noise into the projection , and thus allows us to study the effective noise caused to the contact line level by bulk or wall disorder . the main result of this analysis is that in both cases the effective noise is linearly proportional to the velocity of the interface . while this result qualitatively agrees with the other works cited above , quantitative differences remain in the form of the noise terms . the relevance of these differences to the actual kinetic roughening of the interfaces remains a challenging numerical problem . we would like to thank k. elder for inspirational and pleasant discussions , as well as for sharing his expertise and results on the sharp interface analysis . this work has been supported in part by the academy of finland through its center of excellence grant ( comp ) . s.m . has been supported by a personal grant from the academy of finland . in this appendix we will consider in more detail the green s function of the laplacian in the fluctuating coordinate system @xmath144 which is schematically depicted in figure [ schem ] . the generalization to the two - wall setup is straightforward , but very tedious including two independent disorder functions . in particular , we will consider the correction to the green s function to linear order in small wall fluctuations @xmath80 . first , the metric tensor of the above coordinate system can be obtained by transformation of the cartesian metric tensor as @xmath145=\sum_i \frac{\partial x^{i}}{\partial x^{i ' } } \frac{\partial x^{i}}{\partial x^{j'}}= \begin{bmatrix } 1+(\partial_x\delta h)^2 & \partial_x\delta h & \partial_z\delta h\partial_x\delta h\\ \partial_x\delta h & 1 & \partial_z\delta h\\ \partial_z\delta h\partial_x\delta h & \partial_z\delta h & 1+(\partial_z\delta h)^2 \end{bmatrix}.\ ] ] the coordinate transformation from cartesian coordinates is merely a shift , and thus the integration measure should not change , meaning that the jacobian in the coordinate transformation of an integral should be unity . this is indeed so , since @xmath146)\equiv 1.\ ] ] in the case of cartesian coordinates we can obtain the green s function , which we denote @xmath147 , using the image charge method with the dirichlet boundary condition at @xmath148 and the neumann boundary condition at @xmath149 : @xmath150 @xmath151.\ ] ] we work in the limit of small fluctuations , so we write the laplacian in the curvilinear coordinates as the cartesian laplacian plus a correction , @xmath152 . note that we unconventionally denote @xmath153 for any coordinates @xmath154 $ ] . the correction @xmath155 is explicitly shown below : @xmath156 \frac { \partial^2}{\partial x \partial y } + 2 \left [ \frac { \partial \delta h(x , z ) } { \partial z } \right ] \frac { \partial^2}{\partial z \partial y } + [ \frac { \partial^2 \delta h(x , z ) } { \partial x^2 } + \frac { \partial^2 \delta h(x , z ) } { \partial z^2 } ] \frac { \partial } { \partial y}. \label{correctionoperator}\end{aligned}\ ] ] in order to use eq . for the curvilinear coordinates , we need the green s function , which has the property of @xmath157 \tilde { g}_{\textrm { 3d } } ( \boldsymbol { r}',\boldsymbol { r_1 } ' ) = - \delta ( \boldsymbol { r } ' - \boldsymbol { r_1}')$ ] . since the laplacian in the curvilinear coordinates can be expressed as the cartesian laplacian plus a correction , we can find the inverse of the curvilinear laplacian , or @xmath158 , to first order in @xmath81 as @xmath159 the above operator notation can be written in full form as : @xmath160 substituting @xmath147 into the above , we can work out the correction to the green s function . after using a similar argument to neglect surface integrals of type @xmath36 as we did with eq . , we find @xmath161 at this point the primes can just be dropped , since @xmath162 . this result is hardly surprising , since a simple substitution of @xmath163 to @xmath164 yields identical results to linear order . the neglected surface integrals include a reservoir term and a wall term . the reservoir term can be readily seen to be small when the meniscus is further away from the reservoir than the disorder correlation length . additionally , the reservoir boundary correction is zero when considering columnar _ i.e. _ @xmath165-independent disorder . the wall term is more problematic , since it involves the boundary correction due to fluctuation in the direction of the wall normal . we have to observe the meniscus further away from the wall than the disorder correlation length in order to neglect this boundary correction . the absence of boundary disorder corrections is highly desirable if we are to keep our formalism tractable , so we have neglected the boundary corrections to the green s function .

we consider the influence of quenched noise upon interface dynamics in 2d and 3d capillary rise with rough walls by using phase - field approach , where the local conservation of mass in the bulk is explicitly included . in the 2d case the disorder is assumed to be in the effective mobility coefficient , while in the 3d case we explicitly consider the influence of locally fluctuating geometry along a solid wall using a generalized curvilinear coordinate transformation . to obtain the equations of motion for meniscus and contact lines , we develop a systematic projection formalism which allows inclusion of disorder . using this formalism , we derive linearized equations of motion for the meniscus and contact line variables , which become local in the fourier space representation . these dispersion relations contain effective noise that is linearly proportional to the velocity . the deterministic parts of our dispersion relations agree with results obtained from other similar studies in the proper limits . however , the forms of the noise terms derived here are quantitatively different from the other studies .

You are an expert at summarizing long articles. Proceed to summarize the following text: the experimental data on kaons and other mesons containing strange quarks are both abundant and rather precise [ ] . these particles are expected to play an important role in meson vacuum phenomenology and at non - zero temperatures , most notably in the restoration of the chiral symmetry that is spontaneously [ ] and explicitly [ ] broken in vacuum . their structure - at least in the scalar sector - is ambiguous , as in the case of the non - strange mesons [ ] . + open questions regarding the strange mesons can be addressed , for example , within the linear sigma model [ ] , as has been done , e.g. , in ref . [ ] . however , our approach is different in comparison to ref . [ ] in that our calculations are based on a globally chirally invariant @xmath1 lagrangian that contains the scalar and pseudoscalar but also vector and axial - vector degrees of freedom . in this paper , we briefly outline our calculations and present some first results . the paper is organized as follows : in sec . 2 we present the model lagrangian and its implications and in sec . 3 we summarize our results . the lagrangian of the @xmath1 model with global chiral invariance has an analogous form as the corresponding linear sigma model lagrangian for @xmath2 [ ] : @xmath3-m_{0}^{2 } \mathrm{tr}(\phi^{\dagger}\phi)-\lambda_{1}[\mathrm{tr}(\phi^{\dagger}% \phi)]^{2 } -\lambda_{2}\mathrm{tr}(\phi^{\dagger}\phi)^{2}\nonumber\\ & -\frac{1}{4}\mathrm{tr}[(l^{\mu\nu})^{2}+(r^{\mu\nu})^{2}]+\frac{m_{1}^{2}% } { 2 } \mathrm{tr}[(l^{\mu})^{2}+(r^{\mu})^{2}]+\mathrm{tr}[h(\phi+\phi ^{\dagger})]\nonumber\\ & + c(\det\phi+\det\phi^{\dagger})-2ig_{2}(\mathrm{tr}\{l_{\mu\nu}[l^{\mu } , l^{\nu}]\ } + \mathrm{tr}\{r_{\mu\nu}[r^{\mu},r^{\nu}]\})\nonumber\\ & -2g_{3}\left [ \mathrm{tr}\left ( \left\ { \partial_{\mu}l_{\nu}+\partial_{\nu}l_{\mu } \right\ } \{l^{\mu},l^{\nu}\}\right ) + \mathrm{tr } \left ( \left\ { \partial_{\mu}% r_{\nu } + \partial_{\nu}r_{\mu } \right\ } \{r^{\mu},r^{\nu}\}\right ) \right ] \nonumber\\ & + \frac{h_{1}}{2}\mathrm{tr}(\phi^{\dagger}\phi)\mathrm{tr}[(l^{\mu})^{2 } + ( r^{\mu})^{2}]+h_{2}\mathrm{tr}[(\phi r^{\mu})^{2}+(l^{\mu}\phi)^{2 } ] + 2h_{3}\mathrm{tr}(\phi r_{\mu}\phi^{\dagger}l^{\mu } ) . \label{lagrangian}%\end{aligned}\ ] ] where @xmath4{ccc}% \frac{(\sigma_{n}+a_{0}^{0})+i(\eta_{n}+\pi^{0})}{\sqrt{2 } } & a_{0}^{+}% + i\pi^{+ } & k_{s}^{+}+ik^{+}\\ a_{0}^{-}+i\pi^{- } & \frac{(\sigma_{n}-a_{0}^{0})+i(\eta_{n}-\pi^{0})}% { \sqrt{2 } } & k_{s}^{0}+ik^{0}\\ k_{s}^{-}+ik^{- } & { \bar{k}_{s}^{0}}+i{\bar{k}^{0 } } & \sigma_{s}+i\eta_{s}% \end{array } \right ) \label{phi}\ ] ] is a matrix containing the scalar and pseudoscalar degrees of freedom and @xmath5{ccc}% \frac{\omega_{n}^{\mu}+\rho^{\mu0}}{\sqrt{2}}+\frac{f_{1n}^{\mu}+a_{1}^{\mu0}% } { \sqrt{2 } } & \rho^{\mu+}+a_{1}^{\mu+ } & k_{\star}^{\mu+}+k_{1}^{\mu+}\\ \rho^{\mu-}+a_{1}^{\mu- } & \frac{\omega_{n}^{\mu}-\rho^{\mu0}}{\sqrt{2}}% + \frac{f_{1n}^{\mu}-a_{1}^{\mu0}}{\sqrt{2 } } & k_{\star}^{\mu0}+k_{1}^{\mu0}\\ k_{\star}^{\mu-}+k_{1}^{\mu- } & { \bar{k}}_{\star}^{\mu0}+{\bar{k}}_{1}^{\mu0 } & \omega_{s}^{\mu}+f_{1s}^{\mu}% \end{array } \right ) , \nonumber \\ r^{\mu } : = \frac{1}{\sqrt{2}}\left ( \begin{array } [ c]{ccc}% \frac{\omega_{n}^{\mu}+\rho^{\mu0}}{\sqrt{2}}-\frac{f_{1n}^{\mu}+a_{1}^{\mu0}% } { \sqrt{2 } } & \rho^{\mu+}-a_{1}^{\mu+ } & k_{\star}^{\mu+}-k_{1}^{\mu+}\\ \rho^{\mu-}-a_{1}^{\mu- } & \frac{\omega_{n}^{\mu}-\rho^{\mu0}}{\sqrt{2}}% -\frac{f_{1n}^{\mu}-a_{1}^{\mu0}}{\sqrt{2 } } & k_{\star}^{\mu0}-k_{1}^{\mu0}\\ k_{\star}^{\mu-}-k_{1}^{\mu- } & { \bar{k}}_{\star}^{\mu0}-{\bar{k}}_{1}^{\mu0 } & \omega_{s}^{\mu}-f_{1s}^{\mu}% \end{array } \right ) \label{lr}\end{aligned}\ ] ] are , respectively , the left - handed and right - handed matrices containing the vector and axial - vector degrees of freedom . furthermore , @xmath6 is the covariant derivative ; @xmath7 and @xmath8 are , respectively , the left - handed and right - handed field strength tensors ; the term tr@xmath9 $ ] [ @xmath10 , @xmath11 , @xmath12 explicitly breaks chiral symmetry due to nonzero quark masses , and the term @xmath13 describes the @xmath14 anomaly [ ] . + as in ref . [ ] , in the non - strange sector , we assign the fields @xmath15 and @xmath16 to the pion and the @xmath17 counterpart of the @xmath18 meson , @xmath19 . the fields @xmath20 and @xmath21 represent the @xmath22 and @xmath23 vector mesons , respectively , and the fields @xmath24 and @xmath25 represent the @xmath26 and @xmath27 mesons , respectively . in the strange sector , we assign the @xmath28 fields to the kaons ; the @xmath29 field is the strange contribution to the physical @xmath18 and @xmath30 fields ; the @xmath31 , @xmath32 , @xmath33 , and @xmath34 fields correspond to the @xmath35 , @xmath36 , @xmath37 , and @xmath38 mesons , respectively . in accordance with ref . [ ] , we assign the scalar kaon @xmath39 to the physical @xmath40 state . the assignment of the non - strange scalar states depends on the results for their masses ( see next paragraph ) . + in order to implement spontaneous symmetry breaking , we shift the @xmath41 and @xmath42 fields by their respective vacuum expectation values @xmath43 and @xmath44 , resulting in @xmath45-@xmath46 and @xmath47-@xmath48 mixing [ ] as well as in @xmath29-@xmath32 , @xmath39-@xmath33 , and @xmath28-@xmath34 mixing . these are removed as described in ref . we observe consequently the mixing between the @xmath41 ( @xmath49 , with @xmath50 standing for non - strange quarks ) and @xmath42 ( @xmath51 ) states , yielding two scalar states that are respectively predominantly of @xmath52 or @xmath53 nature [ note that the lagrangian ( [ lagrangian ] ) also yields @xmath45 - @xmath29 mixing ] . in this paper , it suffices to say that our preliminary results , in the limit where the large-@xmath54 suppressed parameter @xmath55 is set to zero , show the existence of a predominantly @xmath52 state with a mass between 1.0 and 1.3 gev and of a predominantly @xmath53 state with mass between 1.3 and 1.6 gev , depending on the choice of parameters . our work concerning these ( and other ) issues is still ongoing and will allow us to determine the parameters in the model more exactly so that the masses of the two mentioned scalars will become more precisely determined [ ] . note that our @xmath1 model possesses no free parameters due to the preciseness of the experimental data [ ] . we have presented an @xmath1 linear sigma model with ( axial-)vector degrees of freedom and global chiral invariance . our preliminary results suggest that the notion of the scalar states above 1 gev as quarkonia may hold also when the @xmath2 lagrangian of ref . [ ] is extended to three flavors . clearly , the parameters in our model still need to be calculated precisely and , regarding the scalar sector , one needs to include the tetraquark and glueball fields in order to study their mixing with the scalar quarkonia . 0 j. goldstone , nuovo cim . * 19 * , 154 ( 1961 ) ; j. goldstone , a. salam and s. weinberg , phys . rev . * 127 * , 965 ( 1962 ) . g. t hooft , phys . rept . * 142 * , 357 ( 1986 ) . d. parganlija , f. giacosa and d. h. rischke , arxiv:1003.4934 [ hep - ph ] .

we outline the extension of the globally chirally invariant @xmath0 linear sigma model with vector and axial - vector degrees of freedom to @xmath1 . we present preliminary results concerning the scalar meson masses .

You are an expert at summarizing long articles. Proceed to summarize the following text: the paradigm of inflation provides nowadays a simple and accurate description of many of the aspects of the universe we observe . it is able to naturally explain several questions like the particle horizon or the flatness problems of early cosmology , among others @xcite . remarkably , this paradigm also explains the origin of the large scale structure . it requires , however , the quantum principles for this mechanism to work @xcite . then , it is one of the most favorable situations where quantum gravity phenomena can potentially be detected . regrettably , the traditional paradigms based on quantum field theories on classical spacetimes are not valid close to the big bang singularity . they simply assume suitable initial conditions at the onset of inflation or during the slow roll phase , where the only relevant quantum phenomena are those coming from small perturbations of matter and geometry . in this manuscript we are interested in the extension of cosmological perturbation theory to those regimes where classical general relativity is not valid anymore . among the different candidates for a quantum theory of gravity , loop quantum gravity ( lqg ) is one of the most developed approaches @xcite . loop quantum cosmology ( lqc ) , the application of lqg techniques to cosmological scenarios , has demonstrated to be a trustworthy formalism @xcite , where the classical singularity is replaced by a quantum bounce , as well as it preserves upon evolution the semiclassicality of quantum states @xcite . this formalism provides an extension of the inflationary scenarios to the planck era , opening the possibility of potentially studying high energy physics and quantum gravity phenomena . in addition , if inhomogeneities are included , even if one follows the formalism of cosmological perturbation theory , the traditional assumptions adopted there must be revisited if one wants to extend this paradigm of the early universe to the deep planck regime . for instance , the classical spacetime approximation might not be valid anymore for non - semiclassical states or close to the high curvature regime of the background geometry . in addition , the criteria that pick out an initial state for the inhomogeneities must also be revised in these new quantum gravity scenarios , since these models admit an extension into the high curvature regime , where general relativity breaks down , but the geometry is still regular ( though not necessarily classical ) . among the different proposals to study cosmological models that include small inhomogeneities in the framework of lqc so far , there is a well known proposal that adopts the so - called _ dressed metric _ approach @xcite . there , the equations of motion of the perturbations evolving on a quantum geometry are influenced by ( not all but some of ) the fluctuations of the background quantum state . this proposal is inspired by the _ hybrid quantization _ @xcite , since it also assumes that the main contributions of quantum geometry are incorporated in the background degrees of freedom , while the inhomogeneities are described by means of a standard representation ( for instance a fock quantization ) . this is a regime between full quantum gravity and quantum field theories on curved spacetimes . the hybrid quantization was originally applied to gowdy cosmologies , providing a consistent and successful quantization free of singularities . afterwards , this quantization was extended to other models , like gowdy cosmologies coupled to matter @xcite . it was also adopted for the study of cosmological inflationary models with small scalar inhomogeneities , where it was provided a full quantization @xcite . however , the dynamics has not been fully understood yet . in ref . @xcite the quantum dynamics was partially solved , assuming a born oppenheimer ansatz for the solutions to the homogeneous scalar constraint , allowing one to recover a dressed metric regime ( under certain approximations ) from a more general formalism . finally , in ref . @xcite , a generally covariant formalism for scalar perturbations has been developed in order to strengthen the predictions of this hybrid approach . there , after a canonical transformation in the phase space ( of the perturbed theory ) , the authors are able to explicitly identify the true physical degrees of freedom ( mukhanov sasaki variables ) without carrying out any gauge fixing . they adopt a born oppenheimer ansatz , derive the dressed metric regime within this fully covariant formalism and provide the effective equations of motion . the genuine ( loop ) quantum dynamics of the background spacetime is a key issue for the dressed metric approach , since all its richness lays in the quantum fluctuations . it is very well known for a free massless scalar field @xcite , but , whenever a potential is added , the dynamics becomes sufficiently intricate that only an effective description has been considered so far @xcite . nevertheless , one can restrict the study to those states for which the dressed metric and the effective dynamics are in agreement ( assuming that they exist ) . within this approximation , the only relevant magnitude to be provided is the value of the scalar field at the bounce ( once its mass has been fixed ) for the homogeneous model . for the inhomogeneities , nevertheless , one specifies the initial data by choosing a suitable vacuum state ( usually ) at that time . however , this is the most important conceptual question to be understood . as we mentioned before , this is a qualitatively different situation with respect to the traditional treatments in which suitable initial data is provided at the onset of inflation or during the slow - roll regime . here , genuine quantum geometry contributions are negligible and then ignored . presently , there is no well understanding of what is their relevance in these new scenarios . in previous studies @xcite in loop quantum cosmology , there have been proposed some candidates , the so - called adiabatic states @xcite , mainly focusing in the fourth order ones . on the one hand , the physical predictions obtained from these adiabatic initial conditions for the perturbations at the bounce seem to be in good agreement with observations whenever the scales for which one obtains large corrections are beyond the observable ones in the cosmic microwave background ( cmb ) . in those situations the main quantum corrections only affect the amplitude of the large scale temperature anisotropies of the cmb , i.e. the range of modes that are either not observable or just entering today the horizon . on the other hand , considering adiabatic states of high order ( fourth order or more ) , a ultraviolet renormalization scheme is available for the stress - energy tensor in order to compute the physical energy density of the perturbations @xcite . nonetheless , it is not completely clear that such family of states ( or equivalently initial conditions ) is the physically preferred choice , at least for those values of the scalar field at the bounce that do not give primordial power spectra where the scales with strong corrections are at the limit of the range of observability or beyond ( usually those that do not produce large number of e - foldings ) . the purpose of this manuscript is to study different primordial power spectra for scalar perturbations that can be obtained from the hybrid quantized model commented above @xcite , and qualitatively confront the obtained physical predictions with other approaches in loop quantum cosmology , with particular attention to the dressed metric approach , and with observations provided by planck mission @xcite . we will assume that the states of the background are highly peaked on the effective geometry , so that it is in good agreement with the dressed metric regime of the hybrid approach . this consideration allows us to determine the dynamics of the background and the perturbations by means of a set of differential equations that incorporate quantum geometry corrections . for convenience , we will consider initial data at the bounce , since the background dynamics is fully determined by one parameter : the value of the homogeneous mode of the scalar field . we will consider as initial state of the perturbation the adiabatic states studied in refs . @xcite and @xcite of 0th , 2nd and 4th order . for them , we found that the primordial power spectrum at the end of inflation agrees with observations if there is sufficient e - foldings , but none of the adiabatic states considered in this manuscript provides a suppression of the power spectrum , but an enhancement , for the observable large scale modes . in addition , in order to face the question of the vacuum state of the perturbations at the bounce , i.e. their initial conditions , we have investigated an alternative prescription based on a genuine algorithm that computes numerically an initial vacuum state in such a way that the time variation of the amplitude of the primordial power spectrum of the mukhanov sasaki variable from the bounce to the end of the kinematically dominated epoch is relieved . this method provides a vacuum state at the bounce that produces a primordial power spectrum at the end of inflation that is in good agreement with the current observations of planck mission . remarkably , it is suppressed at large scales , as it seems to be favored by current observations . furthermore , there is an important contrast with respect to the predictions of the adiabatic states since it does not present the highly oscillating region and averaged enhancement at large scales typical of those states in this scenario . this paper is organized as follows . in section [ sec : eff - dyn ] we specify the ( semi)classical setting and its dynamics by means of a set of effective equations of motion within the hybrid quantization approach . the initial value problem is studied in section [ sec : init - state ] , where we consider several choices of initial vacuum state for scalar perturbations . there we provide a new constructive method to select a suitable initial vacuum state . we also consider adiabatic states to different order . in section [ sec : ps - atps ] we compute the primordial power spectrum at the end of inflation for these vacuum states . we also provide several examples of the power spectrum of temperature anisotropies for the new vacuum state and compare with observations . we conclude and discuss our results in section [ sec : dis - conc ] . for the sake of completeness , we include an appendix where the previous constructive method is applied to a particular quantum field theory in a de sitter space . the system we will study here is a flat friedmann robertson walker spacetime with @xmath0 topology . it is endowed with a spacetime metric characterized by a homogeneous lapse @xmath1 and a scale factor @xmath2 multiplying the auxiliary three - metric @xmath3 of the three - torus . the angular coordinates in this spatial manifold are @xmath4 such that @xmath5 , where @xmath6 is the period in each orthonormal direction ( for simplicity the three periods have been chosen to be equal ) . the auxiliary three - metric @xmath3 induces a laplace - beltrami operator whose eigenfunctions @xmath7 and eigenvalues @xmath8 are well known ( see e.g. ref . @xcite ) . here @xmath9 is any tuple whose first component is a strictly positive integer . we will adopt a description of this homogeneous geometry in terms of lqg variables . there , one starts with an su(2)-connection and a densitized triad as basic canonical variables . however , the connection itself is not well defined in the full quantum theory but holonomies of the connection . besides , in the context of lqc , we will adhere to the improved dynamics scheme @xcite . this choice is motivated by the fact that such cosmologies bounce whenever the energy density achieves a universal critical value @xmath10 for semiclassical states , where @xmath11 is the planck energy density . in this situation , the classical homogeneous canonical variables of the geometry are the volume @xmath12 and the canonically conjugated variable @xmath13 ( which is proportional to the hubble parameter ) . these variables satisfy the classical algebra @xmath14 , where @xmath15 is the newton constant and @xmath16 the immirzi parameter @xcite . we will couple to this model a massive scalar field @xmath17 fulfilling , with its canonical momentum , @xmath18 . following the analysis of ref . @xcite , we will introduce scalar perturbations around the classical homogeneous variables . this was done in a closely related scenario in a seminal paper by halliwell and hawking @xcite . the perturbative expansion of the einstein - hilbert action of general relativity coupled to a massive scalar field is truncated to second order . the resulting action , in the canonical formalism , can be written in terms of a hamiltonian that is a linear combination of constraints : the homogeneous mode of the scalar constraint that contains quadratic contributions of the perturbations , one hamiltonian and one diffeomorphism constraints , both ( local and ) linear in the perturbations . these two last constraints generate the so - called gauge transformations of the perturbations . the common strategy followed in the theory of cosmological perturbations is to work with gauge invariant potentials @xcite . in this sense , in ref . @xcite it was proposed to implement the gauge invariant potentials at the action itself by means of the hamilton - jacobi theory . however , the explicit dependence of the hamiltonian in terms of the non gauge - invariant variables was ignored as well as the background variables were not properly transformed . these last steps were completed in ref . @xcite . following it , we consider a canonical transformation involving the whole phase space , in order to separate the gauge invariant mukhanov sasaki potential @xmath19 , and its conjugated momentum @xmath20 , with respect to the remaining gauge variables , that will be defined as @xmath21 , for @xmath22 . the details can be found in ref . the total hamiltonian , to second order in the perturbations , will be of the form @xmath23 where @xmath24 and @xmath25 are the fourier modes of the inhomogeneous lagrange multipliers associated with the constraints linear in the perturbations of the final action . we will then adopt a hybrid quantization approach , combining a loop quantization for the homogeneous connection , and a standard representation for the scalar field and the inhomogeneities @xcite . since the model involves first class constraints , we will adopt the dirac quantization approach . the physical states must be annihilated by the operators corresponding to the constraints . in this case , one can easily realize that any physical state must be independent of @xmath26 , for @xmath22 . the remaining condition to be imposed is that physical states must be annihilated by the homogeneous mode of the hamiltonian constraint , i.e. by @xmath27 . since we do not know yet how to solve this quantum constraint , we will compute approximate solutions by adopting first a born oppenheimer ansatz for them , and then additional approximations ( see refs . @xcite for further details ) that are expected to be fulfilled for instance by semiclassical states with small dispersions . in addition , among them we choose those ones where the effective equations of motion of the expectation values of the basic observables agree with the effective dynamics of loop quantum cosmology ( though we do not know yet if this family of states really exists , it seems natural to assume it does ) . this effective dynamics has been studied in a similar scenario @xcite corresponding to a flat frw spacetime filled with a massless scalar field . the effective equations of motion provided there approximate very well the exact quantum evolution of the expectation values of quantum operators for semiclassical states . following this motivation , the effective hamiltonian constraint that we will employ in this manuscript is obtained from the quantum hamiltonian constraint @xmath28 by replacing expectation values of products of operators by products of expectation values of such operators . under these assumptions , the dynamics is generated by the effective hamiltonian constraint @xmath29 with the unperturbed hamiltonian constraint @xmath30 in the last expression , @xmath31 denotes the mass of the massive scalar field and @xmath32 the minimum non - zero eigenvalue of the area operator in lqg . besides , @xmath33 is the quadratic contribution of each of the modes @xmath34 with @xmath35 where @xmath36 and @xmath37 the above expressions are easily obtained from refs . @xcite by using the previously mentioned effective dynamics prescription . it is worth commenting that , for convenience , one can replace @xmath38 by @xmath39 in . this is in agreement with our approximate effective equations of motion and the perturbative scheme we adopt . we have also neglected corrections of the inverse of the volume operator @xmath40~\simeq~1/v+{\cal o}(v^{-2})$].the effective equations of motion for the phase space variables are obtained by computing their poisson brackets with the effective hamiltonian constraint @xmath41 given in the previous expressions . for the background variables the equations of motion are given by [ eq : hom - eqs2 ] @xmath42 where @xmath43 , for any phase space variable @xmath44 . for the sake of completeness we have included backreaction contributions , but for all practical purposes we will neglect them in the following . as a consequence , the effective equations of motion for the background will be equivalent to the ones used in ref . ( 2.9 ) , ( 2.11 ) and ( 4.2). the time evolution of the perturbations , on the other hand , is dictated by an infinite number of first order differential equations : @xmath45 which do not mix different modes and ( given our perturbative truncation ) are linear in the inhomogeneities . in the following , we will only consider conformal time @xmath46 , which involves @xmath47 . in addition , and for the sake of simplicity , we will set @xmath48 but with @xmath49 taking continuous values . this is in agreement with the approximation @xmath50 . it is enough for all practical purposes to consider @xmath6 much bigger than the hubble radius during the whole evolution . in standard inflationary cosmology it is common to give initial data at the onset of inflation for both the background and the perturbations . one can also consider initial data close to the classical singularity , but keeping in mind that einstein s theory breaks down there and any physical prediction can not be fully trusted . nevertheless , in that regime , one expects that quantum gravity effects will become relevant , and the emergent preinflationary scenario can potentially change the traditional picture . this is indeed the case of our model . in the present scenario the classical singularity is replaced by a quantum bounce whenever the energy density of the system reaches a critical value @xmath51 of the order of the planck energy density . in lqc usually one considers initial data when the energy density reaches @xmath51 . this is so because there the hubble parameter vanishes , the value of the scale factor is chosen equal to the unit ( its concrete value is in fact irrelevant for open flat topology ) , and the momentum conjugated to the field is determined by the scalar constraint ( i.e. the friedmann equation ) . therefore , any solution to the effective equations of motion is completely determined by the value of the homogeneous mode of the scalar field at the bounce , i.e. @xmath52 . in conclusion , concerning initial conditions of the background variables , we consider @xmath53 as the only free dynamical parameter . in addition , one can also allow for different values for the mass of the scalar field @xmath31 , and then consider it as an additional ( homogeneous ) free parameter . however , the specification of initial data for the perturbations is not fully understood yet , though there are several natural candidates as initial state of the inhomogeneous sector . this freedom in the choice of initial vacua is tantamount to the fact that quantum field theories in general curved spacetimes do not possess a sufficient number of symmetries allowing to choose a unique vacuum state @xcite . the best one can do , by now , is to assume additional physical or mathematical criteria that select a given candidate among all possible choices . but let us first remind that the selection of a fock vacuum state is equivalent to select a complete set of creation and annihilation variables . in turn , this is equivalent to define a complete set of `` positive frequency '' complex solutions @xmath54 to the equation of motion that satisfy the normalization relation @xmath55 in the last equation the prime stands for derivation with respect to conformal time whereas the asterisk stands for complex conjugation . since the equations of motion for the mukhanov sasaki variables are linear , the different choices of sets of `` positive frequency '' solutions can be translated to different choices of initial conditions @xmath56 at the considered initial time @xmath57 . also the normalization relation is preserved upon evolution and therefore it is only necessary that such condition is satisfied initially . general initial conditions ( up to a irrelevant global phase ) can be written as @xmath58 where @xmath59 is a non - negative function of the mode , @xmath49 , whereas @xmath60 is an arbitrary real function . one can restrict the ultraviolet behavior of those functions using well motivated physical and mathematical conditions . for instance , for compact spatial slices one can impose unitary evolution ( in addition to invariance under spatial symmetries ) to the fock quantization @xcite . extending this criterion to open spatial sections , for instance by choosing a vacuum with a finite particle production per spatial volume upon evolution , one can see that the previous functions must behave like @xmath61 and @xmath62 for large @xmath49 , with @xmath63 . other well known and widely used criteria to select suitable vacuum states , as the hadamard condition @xcite or the adiabatic states @xcite , give a similar ultraviolet restriction @xcite . we will restrict our study to those initial conditions satisfying the unitary evolution requirement . nonetheless , since this requirement only constraints the ultraviolet behavior , there still exists an infinite freedom . we then need to further restrict the possible candidates . one of the most popular ways to constraint or select a set of `` positive frequency '' solutions ( or initial conditions ) is by means of the so - called adiabatic states . in order to define a complete set of complex solutions for them one considers the following form @xmath64 if this expression is introduced in the equations of motion one obtains for the function @xmath65 the following equation @xmath66 here , @xmath67 stands for the corresponding time - dependent mass term which in our case is given by @xmath68 adiabatic solutions are then obtained by choosing conveniently functions @xmath69 that provide approximations to the exact solution to equation that converge to them at least as @xmath70 in the limit of infinite large @xmath49 , with @xmath71 being the _ order _ of the adiabatic solution where the different approximate solutions , i.e. the adiabatic orders , are determined in the limit @xmath72 . nonetheless , it is well known that this is essentially equivalent to consider the limit of infinitely large @xmath49 . ] . it is important to remark that those adiabatic solutions are suitable approximated solutions in the limit of large @xmath49 but it is also well known that they do not provide in general a good approximation for small @xmath49 . they were originally introduced with the purpose of adopting analytical tools in several situations , and later on it was shown they are also useful for other purposes like renormalization of the stress - energy tensor in cosmological scenarios . however , we are not interested in analytical approximations to the exact solutions but instead in using these adiabatic states to obtain suitable initial conditions for our equations of motion , as it was done in ref . @xcite for the dressed metric approach , in order to study the physical predictions of the hybrid quantization approach , in particular , as well as the robustness of loop quantum cosmology , in general . actually , from equation one can define initial conditions associated to an adiabatic solution ( absorbing an irrelevant global phase ) by choosing @xmath73 and @xmath74 , with both expressions in the right hand sides evaluated at @xmath75 . now , we are going to introduce two different procedures that we will use in this work to obtain specific adiabatic initial conditions of different orders . the first procedure mimics the one explained in the reference @xcite . one obtains an adiabatic solution of order @xmath76 , @xmath77 , by plugging in the right hand side of the equation the @xmath78 order solution @xmath69 , starting with @xmath79 ( the solution naturally associated to a free massless scalar field ) . the other procedure that we use to select specific adiabatic initial conditions for each order is to consider the above mentioned solutions and perform an asymptotic expansion in the limit of large @xmath49 and then truncating it at the considered order . this method is equivalent to the one used in ref . the functions obtained by means of this last procedure will be denoted as @xmath80 . in the following we will consider these two specific adiabatic initial conditions until 4th order . to show some instances of the functions considered here , obviously both cases lead to the same initial conditions for the 0th order , @xmath81 , whereas for the 2nd order we have @xmath82 and @xmath83 . it is worth noting that it is not guaranteed that these two procedures give meaningful initial conditions ( and therefore solutions ) since they can lead to functions @xmath69 that are negative or even complex for some values of @xmath49 . in addition to the above mentioned potential problems of the two procedures to obtain initial conditions associated to adiabatic states it is easy to realize that , in general , both of them give different sets of `` positive frequency '' solutions when considering different initial times . this may not be seen as a big drawback but it is important to stress that , for instance , in de sitter spacetimes the privileged bunch davies vacuum is selected by one specific set of solutions irrespectively to the time selected to define the corresponding initial conditions . in summary , these adiabatic conditions allow us to reduce the possible initial states for the inhomogeneities , but from the definition of adiabatic states , it is clear that there still exist infinitely many choices of adiabatic states of any order . recently , it has been proposed a new criterion to select a unique set of initial conditions in the context of adiabatic vacua @xcite . the criterion selects those initial conditions for which the expectation value of the adiabatically regularized stress - energy tensor at the initial time vanishes . besides the previous prescriptions for the choice of initial data , in this work we will also propose a criterion to select a set of initial conditions , but based on different arguments . it focuses on the behavior of the relevant quantity in our computations : the primordial power spectrum of the comoving curvature perturbation @xmath84 . the primordial power spectrum @xmath85 is defined from its 2-point function ( which is in one - to - one relation with a concrete vacuum state ) as @xcite @xmath86 where @xmath87 is the conformal time at which the quantities in the expression above are evaluated ( usually at the end of inflation ) . it can be written , in terms of the `` positive frequency '' solutions of the gauge invariant mukhanov sasaki potential , as @xmath88 where @xmath89 is given in terms of the hubble parameter defined as @xmath90 . our criterion is based on selecting the initial conditions that minimize the temporal variation of @xmath91 in a selected time interval . they can be obtained as follows . for each mode , we define the quantity @xmath92 that clearly depends on the specific solution @xmath93 , and therefore on the initial conditions , and also in the considered temporal integration limits . then , the initial conditions we are looking for are the ones that minimize , i.e. , we pick out the specific values for the real variables @xmath59 and @xmath94 that minimize @xmath95 . the idea behind this constructive criterion is to select solutions for which the quantity @xmath96 does not oscillate rapidly _ and _ with large amplitudes . this choice is justified by the fact that other natural vacuum states in quantum field theories , like the poincar - invariant vacuum for time - independent scenarios or the bunch davies vacuum for de sitter spacetimes , are absent of such fast ( and large ) time - dependent oscillations we show a particular example where we recover the bunch - davies state out of this algorithm . ] . they are reflected in the corresponding primordial power spectra , showing strong oscillations in @xmath49 . we understand that this behavior should be weakened if one is selecting a more suitable vacuum state ( at least approximately ) . the primordial power spectrum obtained by means of this method is non - highly oscillating and provides the lowest power when averaging in smalls bins of @xmath49 for @xmath97 ( among the states considered in our study ) . it is worth comenting that , as we show in app . [ app : bd - vac ] , this criterion allows us to identify the bunch - davies vacuum state of a massless scalar field on a de sitter spacetime . therefore , we will regard it as a good choice of initial data . in this work we will take the initial time @xmath98 as the time of the bounce @xmath99 , which is the one in which we are giving the initial conditions . furthermore , we set @xmath100 as the time in which @xmath101 vanishes for the first time from the bounce , which can be considered as the beginning of the inflationary phase ( do not confuse with the slow roll regime ) . in the following ( clearly abusing of the language ) we will refer to the initial conditions obtained in this way as the `` non - oscillating '' initial conditions . nonetheless , it is important to note that , since the time interval considered is finite , this criterion is not expected to give such desirable non - oscillating solutions ( in @xmath96 ) for @xmath102 . in figure [ fig : dk - ck ] we compare the functions @xmath59 and @xmath60 for the different adiabatic initial conditions considered here as well as the `` non - oscillating '' ones . as we can see , in the case of @xmath59 , all choices give the same behavior for @xmath103 , but they strongly differ otherwise . for the functions @xmath60 we see that the `` non - oscillating '' initial conditions and the adiabatic vacuum ones differ in the shown range of @xmath49 , although all of them go quickly to zero for large values of @xmath49 . and left graph : comparison of the @xmath59 function . right graph : comparison of the @xmath60 functions . here the dashed lines correspond to negative values of @xmath60 . note that for the 0th order adiabatic initial conditions @xmath105 . , title="fig:",scaledwidth=49.0% ] and @xmath104 . left graph : comparison of the @xmath59 function . right graph : comparison of the @xmath60 functions . here the dashed lines correspond to negative values of @xmath60 . note that for the 0th order adiabatic initial conditions @xmath105 . , title="fig:",scaledwidth=49.0% ] the `` non - oscillating '' initial conditions , as well as the considered adiabatic initial data , are almost independent of the values chosen for @xmath53 and @xmath31 in the ranges that are physically interesting . this is easy to understand taking into account that the actual values of @xmath53 and @xmath31 are almost irrelevant in the dynamics of the superinflationary phase after the bounce and the kinematically dominated phase . in order to test potential predictions of our proposal regarding the planck era that can be observed in the cosmic microwave background , we study the primordial power spectrum of the comoving curvature perturbations at the end of inflation see equation . this magnitude is indirectly related with observations through the angular spectrum of temperature anisotropies of the cmb . for any pair of homogeneous initial conditions , i.e. , the mass of the field and value of the field at the bounce , we first compute the primordial power spectrum for the comoving curvature for the `` non - oscillating '' initial conditions , @xmath106 , which is going to be considered as the reference one . such primordial power spectrum is obtained from the set of complex solutions @xmath107 , that in the following we are going to denote simply as @xmath108 . any other set of complex solutions is related with the non - oscillatory one by means of @xmath109 with @xmath110 . these mode - dependent complex coefficients @xmath111 and @xmath112 are completely determined by the particular initial conditions @xmath113 under consideration and the `` non - oscillatory '' ones @xmath114 . actually , @xmath115 , \quad \beta_{k}=-i\left[-u^{\prime}_{k,0}v_{k,0}+u_{k,0}v^{\prime}_{k,0}\right].\ ] ] then , one can write the squared absolute value of the new set of complex solutions , and hence the primordial power spectrum , in terms of the reference one as @xmath116)|u_{k}|^{2},\ ] ] where @xmath117 $ ] denotes the real part . since we are considering as the reference set of solutions the `` non - oscillatory '' ones , and taking into account the behavior of the initial conditions considered in the previous section , it is easy to realize that the term producing time oscillations for @xmath96 is the one containing the real part of @xmath118 . therefore , for every set of initial conditions we will consider as well the primordial power spectrum obtained by removing that oscillatory part @xmath119 it is worth to comment that for @xmath120 , where @xmath121 denotes the conformal time from the bounce until the end of inflation . for the instance with @xmath122 and @xmath123 , the beginning of inflation ( defined as the moment in which @xmath101 equals to zero for the first time ) occurs at @xmath124 , the slow - roll regime starts at @xmath125 and the inflationary era ends at @xmath126 . for other values of @xmath53 and @xmath31 , in the ranges explored , these times take similar values . also we have selected the units to perform numerical simulations such that @xmath127 = 1 . ] , @xmath128 will give a good approximation of the averaged power spectrum @xmath129 with respect to small bins , whose size in @xmath49 ( or even in @xmath130 ) must be selected in order average the oscillations ( which are expected anyway to be unobservable in the present observations of the cmb ) . in figure [ fig : ppss ] , we compare the primordial power spectra obtained for the initial conditions considered in the previous section , i.e. , the `` non - oscillatory '' ones , the free massless initial conditions and the two sets of 2nd and 4th adiabatic initial conditions . also we compare those primordial power spectra with the one obtained with the first order slow roll formula @xmath131 where @xmath132 is the hubble parameter , @xmath133 is the first slow roll parameter in the hubble flow functions , the subscript @xmath134 means that the quantity is evaluated when the mode crosses the hubble horizon @xmath135 and the dot stands for the derivative with respect to the proper time . and @xmath104.,title="fig:",scaledwidth=49.0% ] and @xmath104.,title="fig:",scaledwidth=49.0% ] + and @xmath104.,title="fig:",scaledwidth=49.0% ] and @xmath104.,title="fig:",scaledwidth=49.0% ] + and @xmath104.,title="fig:",scaledwidth=49.0% ] and @xmath104.,title="fig:",scaledwidth=49.0% ] as we can see in the first graph of figure [ fig : ppss ] the power spectra obtained from the non - oscillatory initial conditions and the slow - roll formula are almost equivalent for modes that cross the hubble horizon when the slow - roll approximation is valid and @xmath136 is less than @xmath137 . ] . nonetheless , we obtain slightly more power in such region for the `` non - oscillatory '' initial conditions because the slow - roll formula is indeed an approximation that is not able ( at the considered order ) to achieve the required precision . on the other hand , for modes that exit the hubble horizon before the slow - roll approximation starts to be valid we obtain that the non - oscillating spectrum predicts less power than the slow - roll formula and shows two regions with different behaviors : ( i ) small oscillations for @xmath138 and ( ii ) strong power suppression for @xmath139 . in the rest of the graphs in figure [ fig : ppss ] we show the primordial power spectrum obtained for the adiabatic initial conditions . one can see that for all of them one obtains a highly oscillatory primordial power spectrum . we can distinguish three regions with different behavior . for large @xmath49 the amplitude of the oscillations tends to vanish and therefore it gets a behavior similar to the `` non - oscillatory '' spectrum and the slow roll formula . for small @xmath49 , one gets a suppression of power for @xmath140 , @xmath141 , @xmath142 and @xmath143 and a large constant power for @xmath144 in the limit @xmath145 . in the intermediate region all the spectra show high oscillations with an averaged enhanced power . it is important to note that these large enhancements are not compatible with current observations unless such region correspond to scales that are not currently observable in the cmb . with the aim at comparing the obtained primordial power spectra with the ones preferred by the observations and statistical analysis of planck mission one has to perform first a _ scale matching_. in fact , we have set arbitrarily @xmath146 whereas observationally the scale factor is usually set as the unit nowadays ( @xmath147 ) . then , in order to obtain a correspondence of the comoving scales @xmath49 with the ( physical ) observational ones we ( naively ) match them using the observational power amplitude @xmath148 at some pivot mode @xmath149 . we will use here the pivot scale used by the planck mission @xcite @xmath150 and the amplitude obtained for the best fit . ] of the tt+lowp data given by @xmath151 . therefore , we will define our pivot comoving scale @xmath149 as the one such that @xmath152 and such that it exits the hubble horizon in ( or closer to ) the slow - roll region . the value of the pivot scale @xmath149 obtained in this way depends both on the mass of the scalar field @xmath31 and the value of its homogeneous mode at the bounce @xmath53 . once such scale matching is done we compare the primordial power spectrum of the `` non - oscillatory '' solutions that we suggest in this manuscript with three parametrized primordial power spectra studied by the planck collaboration . the first one is the simple power - law @xmath153 which is parametrized by the power amplitude @xmath148 at the pivot scale and the spectral index @xmath154 . the planck collaboration obtains from the tt+lowp data that @xmath155 . this simple form of the primordial power spectrum is the statistically preferred by planck , not because it provides the best fit to the observational data , but because it yields a good fit with a remarkably small number of parameters . actually , a better fit to the planck data is obtained when one allows the primordial power spectrum to deviate from the simple power - law form , either considering a running of the spectral index or different functional forms . such improvement of the fitting is mainly due to the possibility of suppressing the primordial power spectrum at large scales . in order to obtain a better fit there , the planck collaboration considers two forms for the primordial power spectra that include two additional parameters . the first one consists in a simple power - law spectrum multiplied by an exponential cut - off : @xmath156\right\}.\ ] ] this power spectrum is typical in scenarios in which slow roll is preceded by a stage of kinetic energy domination @xcite . the second one is a broken - power - law ( bpl ) potential of the form @xmath157 with @xmath158 to ensure continuity at @xmath159 . the best fit to the tt+lowp planck data for the first model is given by @xmath160 , @xmath161 and @xmath162 . for the second model the best fit is obtained with @xmath163 , @xmath164 and @xmath165 . both of them give a slightly better fitting for the observational data , although not good enough to be statistically preferred over the simple power - law spectrum @xcite with two parameters less . in figure [ fig : ppsskp ] we compare the primordial power spectra for the `` non - oscillatory '' initial vacuum for different values of the homogeneous initial conditions with planck best fit of the previous parameterized power spectra . . left graph : fixed @xmath166 . the vertical dashed line corresponds to the largest scale observable in the cmb.,title="fig:",scaledwidth=49.0% ] . left graph : fixed @xmath166 . the vertical dashed line corresponds to the largest scale observable in the cmb.,title="fig:",scaledwidth=49.0% ] + as we see our proposal has a large scale power suppression that resembles the one given by the broken - power - law best fit of planck for scales observed in the cosmic microwave background . finally , we can see the qualitative consequences of the `` non - oscillatory '' initial conditions at the bounce in the power spectrum of temperature anisotropies . we have carried out the computation employing the code @xcite and the cosmological parameters provided by the planck collaboration @xcite for the best fits of both the simple power - law and cut - off models @xcite . in fig . [ fig : anist ] we plot the planck collaboration observational data for the temperature correlations , the planck best fit and the predictions of the primordial power spectra obtained for the `` non - oscillating '' initial conditions . we choose a particular value of the mass of the scalar field equal to @xmath167 and several choices of its homogeneous mode at the bounce . we observe that our `` non - oscillating '' initial conditions can explain the power suppression for small angular momenta ( large cosmological scales ) of the anisotropies of the cmb . this suppression is stronger in those cases where there is not enough inflation , but for a sufficiently high number of e - foldings we still recover the simple power - law primordial power spectrum statistically preferred by planck . for the left graph in fig . [ fig : anist ] we have used the best - fit cosmological parameters provided by the planck collaboration whereas for the right graph we use the cosmological parameters for the best fit of cut - off models @xcite . we observe that the peaks of the baryonic resonances are quite sensible to the two sets of cosmological parameters provided by planck collaboration , mainly because the values of @xmath52 and @xmath31 must be accurately selected according to the change of the matching scale at the pivot mode , but in both cases they still share the same qualitative properties . indeed we have not carried out a rigorous statistical analysis in order to determine the best fit parameters for the primordial power spectrum predicted by the `` non - oscillating '' initial conditions . it will be a matter of future research . it is also interesting to notice that it seems that our `` non - oscillating '' vacuum produces a strong suppression of the primordial power spectrum that is not able to explain the anomalies around @xmath168 , without strongly suppressing the power spectrum of temperature anisotropies at larger scales . although we have not carried out a detailed analysis , one possible explanation is that the small oscillations that precede the strong suppression of the primordial power spectrum of the `` non - oscillating '' vacuum be responsible of those anomalies if the amplitude of these oscillations is big enough . however , it is not clear to us by now which physical process could be behind it . in this manuscript we have computed the primordial power spectrum of the comoving curvature perturbation in the framework of hybrid loop quantum cosmology . since the genuine quantum dynamics has not been fully solved yet , we have considered the effective equations coming from such hybrid quantization ( simply replacing operators by expectation values ) . additionally , we have neglected the backreaction of the perturbations to the background dynamics . it simplifies the dynamics and allows us to compare the results of the hybrid and the dressed metric approaches . with these assumptions we have computed the primordial power spectrum obtained for different choices of initial vacuum states at the time of the bounce for the mukhanov sasaki variables . more specifically , we have considered two different procedures to obtain specific adiabatic - like initial conditions of arbitrary order and computed the primordial power spectra for 0th , 2nd and 4th orders . they are in good agreement with the ones obtained within the dressed metric approach @xcite . therefore , it is remarkable that the predictions of loop quantum cosmology seem to be robust , since these two formalisms , constructed following different strategies , provide qualitatively similar predictions under the same physical conditions . note that , although we have neglected backreaction contributions and consider the lqc effective dynamics , the effective equations of motion of the perturbation are different in the hybrid and the dressed metric approaches mainly due to the way in which polymeric corrections are included . the consequence is that the quantitative final results will be different , as well as the adiabatic - like initial data in the two approaches will not agree since they involve time - dependent functions that do not coincide when quantum gravity corrections are important ( i.e. , at the bounce ) . the primordial power spectra for adiabatic vacuum states in the hybrid and the dressed metric approaches have in common three distinct behaviors at different scales : ( i ) smooth slow - roll - like behavior ( with small oscillations ) for @xmath169 , ( ii ) large oscillations with an averaged power enhancement for @xmath170 and ( iii ) strong power suppression for @xmath139 , except for the expanded adiabatic vacuum of order four @xmath171 for which the power remains large and constant , at least in the hybrid approach . it is worth to mention that this kind of primordial power spectra , though they can be in good agreement with observations if the most important corrections correspond to scales that are not currently observable in the cmb , are not able to successfully explain the suppression of the temperature anisotropy power spectrum at large scales , without further considerations @xcite . let us also comment that , at first glance , neither our results nor the ones of the dressed metric proposal are in agreement with the ones obtained within the _ deformed algebra _ approach , that leads to highly oscillatory ( with large amplitude ) primordial power spectra even at small scales @xcite . in addition to adiabatic - like initial conditions for the perturbations at the bounce , we have also provided a new criterion to select a suitable initial vacuum state . such criterion picks out the initial conditions for each mode in such a way that it minimizes the time variation of the amplitude of the mukhanov sasaki variable from the bounce to the beginning of inflation . we have shown that such `` non - oscillating '' initial conditions yield a primordial power spectrum where the large oscillations with the averaged enhanced power are not present , obtaining instead for that range of scales a behavior compatible with the one obtained from the slow - roll formula . remarkably , it presents a strong power suppression for large scales . consequently , it may provide a better fitting to the current observations than the spectrum obtained from a quadratic potential considering only the slow - roll regime or from the usual simple power - law primordial power spectrum . nonetheless , although a rigorous statistical analysis is necessary , it seems that the obtained strong power suppression is not enough to explain the observed anomaly around the multipole @xmath172 in the temperature anisotropy angular spectrum . one appealing possibility is to consider initial conditions for the mukhanov sasaki variables that slightly deviate from the ones obtained with the considered criterion . such initial conditions would lead to a primordial power spectrum with slightly larger oscillations around @xmath173 that might explain the above mentioned anomaly . those slightly differently initial conditions might be obtained by different physical processes . one possibility is , for instance , by minimizing the time integrated ( in conformal time ) value of the energy density for each mode from the bounce to the beginning of inflation . on the other hand , since we expect that the set of complex solutions selected by `` non - oscillating '' initial conditions give approximately similar physical results to the ones obtained by giving minkowski - like initial conditions in the kinematically dominated period , another possible way to obtain a primordial power spectrum with larger ( but still small ) oscillations around @xmath173 is defining minkowski - like initial conditions around the end of the superinflationary era after the bounce . indeed , it is not clear to us that the quantum dynamics of the universe is fully determined by our approach but , instead , there is a decoupling scale right after the bounce where the hybrid quantization ( and so the dressed metric approach ) is valid and where it is natural to give minkowski - like initial conditions . finally , another possibility is to break the hypothesis of isotropy before inflation @xcite , being the anisotropies the main source of oscillations at scales of the order of @xmath173 . the assumption of isotropy of the universe together with the truncation of the perturbations to second order in the action allow us to focus our attention on scalar perturbations , since they decouple dynamically from the vector and tensor modes . in these circumstances , the vector inhomogeneities are non - dynamical degrees of freedom . however , the tensor modes can not be neglected if one wants a complete physical picture of the system ( under the previous hypotheses ) . although the hybrid quantization approach is still incomplete at this respect , our preliminary calculations suggest that it is possible to incorporate tensor modes in this formalism without further considerations . the effective equations of motion are well defined at the planck regime and have a well behaved ultraviolet limit . our purpose in the future is to compute the tensor primordial spectra for several initial adiabatic vacuum states ( as well as for the `` non - oscillatory '' one ) within this hybrid quantization approach . although it is soon to draw any conclusion within this formalism , the analyses by planck collaboration suggest that a quadratic potential is statistically disfavored since it predicts a tensor - to - scalar ratio slightly higher than the upper bound @xmath174 ( 95%cl ) . however , whether this is true in the hybrid formalism and the possible physical phenomena that could deal with this question is something that we will discuss in a forthcoming publication . our results are mainly based on the selection of the `` non - oscillating '' vacuum state , which has been obtained by following a particular algorithm whose main purpose is to minimize the oscillations of the primordial power spectrum . but additional considerations can be further investigated . for instance , the quantity inside the integral in eq . is the absolute value of the derivative of @xmath96 with respect to conformal time . although the algorithm works very well eliminating most of the oscillations in this physical quantity ( let us recall that it is related with the 2-point function ) , one could instead consider minimizing these type of oscillations in other physical quantities like either the total , kinetic or potential time - dependent energy of each mode . besides , it would be interesting to apply this algorithm to different cosmological settings . in particular , there are some situations where this method can be tested since the natural vacuum state is already known . this is the case , for instance , of time - independent scenarios or de sitter spacetimes . if we consider arbitrary initial conditions , like in eq . , it would be very interesting to see if this algorithm is able to reconstruct the privileged initial conditions of the poincar - invariant vacuum or bunch - davies vacuum , respectively . a preliminary study for a test , massive scalar field in a minkowski spacetime shows that the algorithm is able to find in a good approximation the initial conditions for the poincar - invariant vacuum state . all these aspects will be addressed in future publications . in summary , the study carried out in this manuscript provides novel ideas about the extension of the traditional inflationary paradigm of cosmological perturbations theory to the planck era . when the big bang singularity is avoided , and the evolution of the universe can be extended far in the past with respect to the onset of inflation , it is natural to ask again these two important questions : i ) what is the natural initial state of the universe in the past , for instance , at the high curvature regime ? and ii ) how predictive are these new scenarios with respect to the present observations ? we show here that loop quantum cosmology and the hybrid quantization approach of this particular model suggest a possible answer to these questions . besides , the usually ignored freedom about the choice of initial state of the perturbations has been considered in this manuscript . we have provided a new criterion that can seed light on the understanding about the existence of privileged vacuum states that has not been considered before in the way we do here , at least to the knowledge of the authors . we also strongly believe that our criterion is not restricted to scenarios in genuine quantum cosmology but they can also be adopted in many other models in cosmology without further considerations . the authors are greatly thankful to i. agull , l. castell gomar , m. martn benito , g. a. mena marugn and t. pawowski for enlightening conversations and suggestions reflected in the manuscript . we also thank j. torrado for his clarifications about the code . this work was supported in part by pedeciba , and the grants micinn / mineco fis2011 - 30145-c03 - 02 and fis2014 - 54800-c2 - 2-p from spain . d. m - db is supported by the project conicyt / fondecyt / postdoctorado/3140409 from chile . j. o. acknowledges support by the grant nsf - phy-1305000 ( usa ) . in this appendix we will show that the criterion of minimizing ( mode by mode ) the time variation of the power spectrum , introduced in the section [ sec : init - state ] , allows us to pick out the bunch - davies state when considering a test scalar field in a de sitter spacetime . we will restrict the study to a massless scalar field in the cosmological chart of this spacetime , as it was done in ref . we will first briefly summarize what is the bunch davies vacuum and how it can be selected by using the symmetries of the de sitter spacetime and the hadamard condition . further details can be found in refs . the action of the test massless scalar field , @xmath17 , is given by @xmath175 where @xmath176 denotes the determinant of the metric @xmath177 that in conformal time takes the form @xmath178 here , @xmath179 and @xmath180 with @xmath132 the constant hubble parameter ( in cosmic time ) . it is worth commenting that , due to homogeneity and isotropy of this spacetime , the metric is invariant under rotations and translations . in addition , it is invariant under the dilatations @xmath181 . these transformations leave @xmath132 unaltered . the consequence is that the previous action in eq . will be also invariant under this set of transformations . let us now consider the redefinition @xmath182 and decompose the field @xmath183 in fourier modes of the form of @xmath184 . then , the partial differential field equation can be written in terms of a set of infinitely many ordinary differential equations given by @xmath185 with @xmath186 . one can easily see that these equations are invariant under the set of transformations considered above , in agreement with the action in eq . . the solutions to these equations are known , @xmath187 where @xmath188 and @xmath189 are complex constants , that in principle may depend on the mode @xmath190 . let us remind that the choice of a particular vacuum for the fock quantization is tantamount to select a complete set of solutions to the equations of motion satisfying the normalization condition given in eq . . the normalization condition imposes @xmath191 and , taking into account the irrelevance of a complex global phase , the freedom on selecting a vacuum state is given by two real parameters per mode . nonetheless , if one imposes invariance of the vacuum under rotations then the complex constants can only depend on @xmath49 . in addition if one requires invariance under dilations then @xmath188 and @xmath192 must be k - independent , therefore @xmath193 and @xmath194 . the bunch davies vacuum is obtained by taking @xmath195 ( and therefore @xmath196 ) and it is the unique vacuum which is invariant under the afore mentioned symmetries and its 2-point function has the hadamard form @xcite . in order to elaborate more about this point , let us consider symmetry invariant vacuum states that we will denote as @xmath197 and obtain the 2-point function , for simplicity , evaluated at the same time @xmath46 . it is given by @xmath198 let us comment that the 2-point functions in this 2-parameter family are invariant under spatial translations and rotations , and dilations of the space and time . the 2-point functions of the field @xmath17 are easily related with the previous ones by @xmath199 one can compute explicitly these 2-point functions @xcite . if we expand the quantum field @xmath200 as @xmath201 such that @xmath202 , the expectation values in eq . can be computed and yield @xmath203 where @xmath204 stands for the imaginary part and the functions @xmath205 are defined as @xmath206 such that , @xmath207 here , we have introduced the function @xmath208 then , we immediately see that @xmath209 and @xmath210 oscillate as functions of @xmath46 . this property will be essential in our criterion for the choice of vacuum state . the previous integrals can be computed ( see ref . @xcite ) , and one can obtain explicitly @xmath211 . however , we will not give here the explicit result . nonetheless , one can check in ref . @xcite that among these 2-point functions , there is only one that is hadamard @xcite . precisely , it is the only 2-point function where the contributions of @xmath209 and @xmath210 to the power spectrum disappear since it corresponds to @xmath212 and @xmath213 . as we mentioned before , this choice corresponds to the so - called bunch - davies vacuum state . as we have shown any vacuum state is obtained from a complete set of complex solutions @xmath54 . therefore , given an initial time @xmath57 , the initial data @xmath113 that select a set of solutions for the bunch davies vacuum are @xmath214 taking into account that the addition of a global phase to the solutions still defines the same vacuum state , we will consider instead initial data @xmath215 of the form given in eq . with @xmath216 where the superscripts are referred to the bunch - davies vacuum . it is straightforward to see that with these initial data one gets the equivalent set of solutions @xmath217 , where @xmath218 . now , we will show the consequences of our criterion when it is applied to this particular quantum field theory . we will only assume invariance under translations and rotations in what follows . let us then start by choosing a set of arbitrary initial data @xmath219 accordingly , for a given initial time @xmath220 , of the form of eq . . one can easily see that the resulting set of solutions @xmath108 can be written as in eq . . this time , the coefficients , in principle , can depend on @xmath49 . indeed , if @xmath221 and @xmath222 , we get we will now concentrate only in the fourier transform of the two point function evaluated at the same time @xmath46 . it is essentially given by eq . as @xmath224 with @xmath225\frac{1}{2k}\left(1+\frac{1}{k^2\eta^2}\right),\ ] ] and@xmath226 - 2 \arctan\left[-\frac{1}{k \eta_0}\right].\ ] ] these are the basic ingredients we need in order to proceed with our method . we should minimize the integral in eq . with respect to @xmath227 and @xmath94 . numerically , it is possible to minimize this function for any choice of @xmath228 and @xmath229 . of course , for those modes that have not oscillated several times in the interval @xmath230 $ ] , this algorithm will not yield an accurate value of the minimum . this is why we must consider @xmath230 $ ] to be sufficiently extensive , covering a large portion of the spacetime . since it is computationally expensive , we will consider instead the minimization of the integral @xmath231 which yields the same result , at least for conformal time , but with this last expression easier to handle analytically than eq . . the integral can be computed explicitly . nevertheless , we will not show here the result since the expression is considerably long . indeed , it is enough to consider , for any fixed value of @xmath100 , the behavior of the integral when @xmath232 . it is possible to see that it diverges as @xmath233 for any @xmath49 if @xmath234 . here we have used @xmath235 . nevertheless , if @xmath236 for all @xmath49 , the integral always converges , since it behaves as @xmath237.\ ] ] we conclude that , for @xmath232 ( when all the modes are inside the hubble radius initially ) , there is only one minimum for this integral , and therefore , only one vacuum state , correspondingly . moreover , this vacuum state coincides with the already mentioned privileged de sitter vacuum that is invariant under spatial translations and rotations , dilations of space and time , and such that its 2-point function fulfills the hadamard condition ( i.e. the bunch - davies state ) . m. martn - benito , d. martn - de blas and g. a. mena marugn , _ approximation methods in loop quantum cosmology : from gowdy cosmologies to inhomogeneous models in friedmann walker geometries _ , class . quantum grav . * 31 * , 075022 ( 2014 ) . l. castell gomar , j. cortez , d. martn - de blas , g. a. mena marugn , j. m. velhinho , _ uniqueness of the fock quantization of scalar fields in spatially flat cosmological spacetimes _ , jcap 1211 , 001 ( 2012 ) . jeronimo cortez , daniel martin - de blas , guillermo a. mena marugan , jose m. velhinho , _ massless scalar field in de sitter spacetime : unitary quantum time evolution _ , class . quantum grav . 30 ( 2013 ) 075015 . b. bolliet , j. grain , c. stahl , l. linsefors , a. barrau , _ comparison of primordial tensor power spectra from the deformed algebra and dressed metric approaches in loop quantum cosmology _ , d * 91 * , 084035 ( 2015 ) .

we provide the power spectrum of small scalar perturbations propagating in an inflationary scenario within loop quantum cosmology . we consider the hybrid quantization approach applied to a friedmann robertson walker spacetime with flat spatial sections coupled to a massive scalar field . we study the quantum dynamics of scalar perturbations on an effective background within this hybrid approach . we consider in our study adiabatic states of different orders . for them , we find that the hybrid quantization is in good agreement with the predictions of the dressed metric approach . we also propose an initial vacuum state for the perturbations , and compute the primordial and the anisotropy power spectrum in order to qualitatively compare with the current observations of planck mission . we find that our vacuum state is in good agreement with them , showing a suppression of the power spectrum for large scale anisotropies . we compare with other choices already studied in the literature .

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Proceed to summarize the following text: because they are composed of old , passively evolving populations , elliptical galaxies offer great promise for tracing the evolution of the universe . whether elliptical galaxies formed through hierarchical merging or monolithic collapse , one of the major goals in extragalactic studies is the determination of the `` redshift of formation , '' @xmath5 , that marks the age where most of the stars in early - type galaxies formed . recent studies of galaxy clusters out to @xmath6 indicate that most of the star formation had to be completed at high redshift ( @xmath7 ) , followed by quiescent evolution thereafter ( stanford , eisenhardt , & dickinson 1998 ; kodama et al . 1998 ) . the uv upturn is a sharp rise in the spectra of e and s0 galaxies shortward of restframe 2500 . it provides a sensitive tracer of age for the oldest stars in these galaxies , and can potentially constrain @xmath5 . traditionally characterized by the @xmath8 color , the uv upturn in local galaxies originates in a population of hot horizontal branch ( hb ) stars and their uv - bright progeny ( see brown et al . 2000 and references therein ) . as first noted by greggio & renzini ( 1990 ) , the uv upturn should evolve rapidly with age , through the evolution of hb morphology and the main sequence turnoff mass . although all models of elliptical galaxy evolution predict a rapid evolution in the uv upturn ( e.g. , tantalo et al . 1996 ) , the timing for the uv upturn onset depends strongly upon model parameters . we have been undertaking a series of observations to trace the evolution of the uv upturn as a function of redshift . faint object camera ( foc ) observations of abell 370 ( @xmath4 ) provided the first detection of far - uv emission from quiescent elliptical galaxies above @xmath9 ( brown et al . the very strong uv emission found at @xmath4 suggests no evolution + @xmath10based on observations with the nasa / esa hubble space telescope obtained at the space telescope science institute , which is operated by aura , inc . , under nasa contract nas 5 - 26555 . @xmath11 noao research associate . + in the uv upturn between our own epoch and one 4 gyr earlier , a finding inconsistent with some models of galaxy evolution , and apparently consistent only for high values of @xmath5 . here , we describe observations that trace the uv upturn to higher redshift . we used the far - uv camera on the space telescope imaging spectrograph ( stis ) to measure the uv emission from four giant elliptical galaxies in the cluster cl0016 + 16 at @xmath12 . ground - based spectroscopy confirms their passive evolution and cluster membership ( dressler & gunn 1992 ) ; morphological classification comes from wide field planetary camera 2 ( wfpc2 ) imaging in the f555w and f814w bands ( smail et al . we assume the currently popular cosmology of @xmath13 , @xmath14 , and @xmath15 km s@xmath16 mpc@xmath16 , although our results are more sensitive to @xmath5 than to the assumed cosmology . using the stis far - uv camera on 19 aug and 21 aug 1999 , we observed a @xmath17 field centered at ra(j2000)=@xmath18 and dec(j2000)=@xmath19 in the cluster cl0016 + 16 . the total exposure was 27892 sec ; 10 frames were taken in two visits , with dithering by 6 pixels to smooth out small - scale detector variations . we used the crystal quartz filter ( f25qtz ) for a bandpass that spans 14502000 , thus reducing the sky background from geocoronal and lyman-@xmath20 to negligible levels at little cost in galaxy light , as the redshift ( @xmath0 ) puts the lyman limit at the short wavelength cutoff . because the long wavelength cutoff of this bandpass is due to detector sensitivity , red leak is also negligible ; we avoid the problematic red leak and red grating scatter that hampered earlier attempts to measure the uv upturn at moderate redshift ( e.g. , windhorst et al . the photometric calibration is reliable at the 0.15 mag level ( baum et al . recent efforts have revised the calibration slightly ; we assume the latest revision , with an accuracy that should be better than 0.15 mag . for reference , we assume that a flat spectrum of @xmath21 erg -0.2 in 0.1 in sec@xmath16 @xmath22 @xmath16 produces one count per sec in the stis bandpass . a full description of the instrument can be found in woodgate et al.(1998 ) and kimble et al . ( 1998 ) . unfortunately , the mean temperature of the far - uv camera has increased since stis commissioning , and the dark background increases as a function of temperature . this increase appears as a `` glow '' that is strongest in the upper left - hand quadrant of the detector ( figure 1 ) , where the dark rate can be 20 times higher than the nominal 6@xmath23 cts sec@xmath16 pix@xmath16 . the glow varies from weak to strong in our 10 frames . the galactic foreground extinction toward our field was thought to be @xmath24 mag at the time these observations were planned , based upon maps ( burstein & heiles 1984 ) . the subsequent release of a new extinction map ( schlegel , finkbeiner , & davis 1998 ) , based upon an iras map of the galactic dust , revises the extinction estimate to 0.057 mag . the increased dark background and higher extinction both work to decrease the sensitivity of our observations to far - uv emission , but do not preclude constraints on the uv upturn . when accounting for this extinction , we assume the empirical model of cardelli , clayton , & mathis ( 1989 ) , which has been parameterized and tested from the uv to ir . our stis field includes four giant e galaxies ( see figure 1 ) ; three are detected at @xmath252.5@xmath26 significance . we detect three other objects from the smail et al . ( 1997 ) catalog at a significance of @xmath27 . we use the bright background spiral , dg244 ( @xmath28 ; dressler & gunn 1992 ) , to confirm the registration of our frames , and to determine the positions of the fainter galaxies . the accuracy of relative astrometry within wfpc2 or stis images is better than 0.1@xmath29 , but the accuracy of absolute astrometry can be worse than 1@xmath29 , so the positions of objects in the stis field are determined relative to this bright spiral . most of the objects in our chosen field are too red and faint to be detected by stis ; their non - detection does not suggest an unexpected lack of sensitivity , and all detected and non - detected objects are consistent with the expected range of spectral energy distributions ( seds ) . stis repeatedly images uv - bright stars ( e.g. , globular cluster ngc6681 ) , and no drop in sensitivity is apparent in observations before and after our observation of the cluster cl0016 + 16 . the stis data was reduced via methods nearly identical to those used for the hubble deep field south , to which we refer the interested reader for more details ( gardner et al.1999 ) . in brief , the images were processed via the standard pipeline , excluding the dark subtraction , low - frequency flat field correction , and geometric correction . dark frames from july through august of 1999 were processed in the same manner , and those with a strong glow ( @xmath30 cts sec@xmath16 pix@xmath16 ) were summed and fit with a cubic spline to produce a profile appropriate for our cl0016 + 16 observations ( the shape of the glow changes slowly with time , and so recent darks are required for this fit ) . a flat component and glow component to the dark background were then subtracted from each cl0016 + 16 frame , and then the flat field correction was applied . the frames were registered by integer shifts and summed via the drizzle package . the pixels in each frame were weighted by the ratio of the exposure time squared to the dark count variance , including a hot pixel mask . the algorithm weights the exposures by the square of the signal - to - noise ( s / n ) ratio for sources that are fainter than the background , thus optimizing the summation to account for the temporal and spatial variations in the dark background . the statistical errors in the final drizzled image ( cts pix@xmath16 ) , for objects below the background , are given by the square root of the final drizzled weights map scaled by the exposure time . geometric correction , which tends to smear out the pixels through non - integer shifts , was not applied . for small dithers and large extraction apertures , it is not needed for object registration . wfpc2 f814w data for the same field were obtained from the stsci archive and reduced via standard techniques , including cosmic - ray rejection and masking of problematic pixels . the total exposure in the f814w image is 16800 sec . the summed image was used to determine the restframe optical - band flux for each elliptical galaxy , and to determine the relative positions of objects in the stis field . we performed aperture photometry on the far - uv frames using idl , taking a weighted average of the flux within a 16-pixel ( 0.4@xmath29 ) radius , and a weighted average of the flux within a sky annulus of radii 80 and 100 pixels . the aperture includes the bright core of each galaxy , as they appear in the wfpc2 data . the aperture is small enough to minimize the background in the far - uv measurement , but large enough to avoid significant errors in the expected encircled energy ( due to the uncertainty in the position of the galaxies ) ; it is also relevant to measurements made of local galaxies , which use a nuclear aperture ( see [ secinterp ] ) . the weighting used the map of statistical errors ( see [ secred ] ) , in the sense that pixels were weighted less if they had less exposure ( due to masked pixels ) or higher dark count rates . this weighting did _ not _ weigh by counts in the _ data _ frame , which would obviously bias the photometry toward pixels with more source counts . we determined the local sky value from the mean instead of the median in the sky annulus , because most pixels are ones and zeros . note that alternate methods for determining the sky background ( e.g. , fitting a surface to the local residual background ) yield results well within the 1@xmath26 photometric errors . the positions of objects in the stis image were determined from the geometrically - corrected wfpc2 f814w frame , relative to the spiral galaxy dg244 ( see figure 1 ) . calculation of position in the stis frame involves a rescaling , rotation , translation , and geometric distortion , using the geometric distortion coefficients of malumuth ( 1997 ) . we tested our positioning algorithm using wfpc2 and stis images of the globular cluster ngc6681 , which was observed with stis numerous times before and after our cl0016 + 16 observations at various roll angles and positions . absolute positional accuracy for objects in the stis frame is 12 stis pixels , well within the 16-pixel radius used for aperture photometry . for the wfpc2 f814w frames , we performed aperture photometry with the iraf package phot , using a 4-pixel radius ( 0.4@xmath29 ) , and a sky annulus of radii 20 and 25 pixels , thus matching the photometry done in the stis image . note that this aperture size would produce encircled energy agreement at the 5% level for point sources , and better agreement for extended sources ( robinson 1997 ; holtzman et al . 1995 ) , so the uncertainty in encircled energy contributes less than 0.1 mag to @xmath8 . we do not perform an aperture correction because we are only interested in colors , not the absolute fluxes . photometry was performed on all objects with @xmath31 mag in the smail et al . ( 1997 ) catalog , plus a faint object ( # 696 ) that is obvious in the stis image but faint in the wfpc2 frames . table 1 gives the photometry for the four giant elliptical galaxies , plus three objects detected at @xmath27 in the stis image . the uv upturn is traditionally characterized by the restframe @xmath8 color ( see burstein et al . 1988 ) . conversion to restframe @xmath8 from our observed bandpasses depends upon the assumption of an sed . brown et al . ( 1998 ) used the spectra of three local elliptical galaxies ( ngc1399 , m60 , and m49 ) to convert their observed foc bandpasses to restframe @xmath8 , and we used the same spectra here . we redshifted the spectra of ngc1399 , m60 , and m49 to @xmath0 , and then applied a foreground reddening of @xmath32 mag ( schlegel et al . we then used the iraf calcphot routine to calculate the stis and wfpc2 countrates , giving : 0.05 in @xmath33 mag , + + 2.25 in where @xmath34 is the stis countrate ( cts sec@xmath16 ) , and @xmath35 is the wfpc2 countrate ( dn sec@xmath16 ) . this conversion gives the restframe @xmath8 values shown in table 1 . we also calculate the @xmath8 assuming the @xmath36 upper limit to the far - uv flux . note that the @xmath8 values in table 1 would be redder if the foreground extinction is closer to @xmath37 mag ( see [ secobs ] ) , or if the far - uv seds are dominated by post - agb stars ( see below ) . the @xmath8 color for the sum of the giant elliptical galaxy photometry gives an average measurement of the uv upturn in our sample . although we are looking to a significantly earlier epoch , the assumption of a model sed over an empirical sed makes little difference for the restframe @xmath38 , because the observed f814w band overlaps with restframe @xmath38 . for example , assuming that e galaxies are @xmath25 gyr old at @xmath0 , we calculated the f814w@xmath39 color using the bruzual & charlot ( 1993 ) instantaneous burst sed of age 5 gyr , and the ngc1399 sed , assuming @xmath0 with a foreground reddening of @xmath32 . the difference in f814w@xmath39 for the two seds was only 0.02 mag . the assumed sed has somewhat more effect on the far - uv . our stis bandpass spans restframe 9351290 , and the @xmath8 color is based on the average flux from 12501850 . however , we note that no model sed has been tested in the far - uv for passively - evolving galaxies at moderate redshift , and thus we prefer empirical seds . younger galaxies should have a larger contribution from relatively hot post - agb stars , instead of hb stars ( see brown et al . 1997 ) , so our conversion might systematically produce a bluer @xmath8 than the true restframe @xmath8 . alternatively , if younger galaxies are dominated by even cooler hb stars than those in local galaxies , we are calculating a redder @xmath8 than the true value . another source of systematic error is the aperture size . local e galaxies have been measured through 14@xmath29 diameter aperture ( 1.5 kpc at the distance of virgo ) , sampling only the nuclear flux ( burstein et al . the @xmath4 measurements were made through a 1.82@xmath29 diameter aperture ( 9.8 kpc at the distance of abell 370 ) , including all of the galactic light detected by the foc ( brown et al . we have measured the flux from our @xmath0 galaxies through a @xmath40 diameter aperture ( 5.4 kpc at the distance of cl0016 + 16 ) , enclosing the light from the galaxy cores . our aperture is 3.6 times larger than that used for local galaxies , but 1.8 times smaller than that used for the abell 370 galaxies . ohl et al . ( 1998 ) found that local e galaxies usually ( but not always ) become redder in @xmath8 at increasing radius ; the color gradient is not large ( 0.10.5 mag ) over radii 725@xmath29 , and the color of the integrated light in an increasing aperture will change even less , because most of the light comes from the central @xmath41 kpc even in our larger apertures . thus , the aperture effects should be small , but perhaps non - negligible ( @xmath42 mag ) . the average restframe @xmath8 color for the four giant e galaxies at @xmath0 is much redder than that observed in clusters at @xmath3 and @xmath4 . at the @xmath36 upper limit to the far - uv flux , none of the @xmath0 galaxies are as blue as the strong uv - upturn galaxies observed locally ( e.g. , m60 or ngc1399 ) . ideally , one would want to compare galaxies at similar velocity dispersion ( @xmath43 ) in each epoch , because @xmath43 can be measured in a model - independent manner , and because the uv upturn strongly correlates with @xmath43 locally ( burstein et al . measurements of @xmath43 are unavailable for some of the @xmath4 galaxies and all of the @xmath0 galaxies , but these galaxies were selected from the brightest and largest in each cluster , and would likely show strong uv upturns if observed locally . the faint far - uv emission from the @xmath0 galaxies thus demonstrates that the uv upturn is sensitive to age . we plot these uv upturn measurements in figure 2 , along with the models of tantalo et al.(1996 ) , which trace the chemical evolution of elliptical galaxies under the assumption of gas infall . note that the scatter in local uv upturn measurements is affected by the wide range of sizes and luminosities in the sample ; the moderate - redshift samples are more homogeneous , but the measurements themselves are less statistically significant . in the tantalo et al . ( 1996 ) models , the uv upturn appears at an age of @xmath44 gyr , becomes quite blue by 9 gyr , and is flat thereafter . assuming the models are correct and that the oldest stars in cluster e galaxies all formed at a common redshift , our measurements imply @xmath45 . if elliptical galaxies form through hierarchical merging , the age of the oldest stars may predate such merging ; alternatively , the age of the oldest stars may reflect the age of the galaxies themselves , if they were assembled through monolithic collapse . however , the tantalo et al.(1996 ) models rely on several parameters ( e.g. , time of onset for galactic winds , efficiency of the star formation rate , accretion timescale , etc . ) that can be tuned to delay or accelerate the onset of the uv upturn . in this sense , our results say more about the evolution of hb morphology than @xmath5 ; if galaxies form their oldest stars at @xmath46 , our data imply that a blue hb population can not arise until ages greater than 7 gyr . although the elliptical galaxies in our program are chosen through ground - based spectroscopy that confirms their passive evolution , the strong uv upturn found at @xmath4 could theoretically be due to residual star formation instead of evolved populations . this star formation would have to cease by the present epoch , because the uv emission from local cluster elliptical galaxies is clearly due to hb stars instead of star formation . if all clusters are alike , our new @xmath0 observations reinforce the case against star formation at @xmath4 , because such star formation would likely be even stronger at a larger lookback time . measurements at @xmath47 0 , 0.375 , and 0.55 demonstrate that the uv upturn rises sharply over a few gyr , and then levels off . this line of inquiry into the evolution of galaxies is still in the early stages , and further observations are needed over a range of redshifts . such measurements should become much more feasible in hst cycle 10 , once a cooler is connected to stis ( reducing the dark background to minimal levels ) . support for this work was provided by nasa through the stis gto team funding . tmb acknowledges support at gsfc by nas 5 - 6499d . we wish to thank the morphs project for making their cluster data and catalogs publicly available .

the restframe uv - to - optical flux ratio , characterizing the `` uv upturn '' phenomenon , is potentially the most sensitive tracer of age in elliptical galaxies ; models predict that it may change by orders of magnitude over the course of a few gyr . in order to trace the evolution of the uv upturn as a function of redshift , we have used the far - uv camera on the space telescope imaging spectrograph to image the galaxy cluster cl0016 + 16 at @xmath0 . our @xmath1 field includes four bright elliptical galaxies , spectroscopically confirmed to be passively evolving cluster members . the weak uv emission from the galaxies in our image demonstrates that the uv upturn is weaker at a lookback time @xmath25.6 gyr earlier than our own , as compared to measurements of the uv upturn in cluster e and s0 galaxies at @xmath3 and @xmath4 . these images are the first with sufficient depth to demonstrate the fading of the uv upturn expected at moderate redshifts . we discuss these observations and the implications for the formation history of galaxies .

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Proceed to summarize the following text: in recent years there has been an increasing interest in phase transitions between fluid phases of different electrolyte concentrations in ionic solutions . two different regimes have been identified experimentally @xcite . the `` solvophobic '' regime occurs for large solvent dielectric constants that effectively turn off coulombic interactions . consequently , solvophobic phase transitions are primarily driven by unfavourable interactions between solute and solvent . this behavior is well described by the usual theory of nonelectrolyte solutions with short range interactions , and clearly leads to ising - like critical behavior . by contrast , in the `` coulombic '' regime the solvent has a low dielectric constant and electrostatic interactions between the solute ions drive the phase separation . in this case , the phase diagrams are quite asymmetric , and apparently mean - field critical behavior has been claimed @xcite , although ising - mean field crossover has also been seen within a narrower range of temperatures around the critical than in non - ionic fluids @xcite . some of these experimental studies suggest that there exists a new characteristic length in these systems that competes with the correlation length for density fluctuations @xcite . electrolyte systems in the coulomb regime are often modelled as charged hard spheres embedded in a uniform dielectric continuum ( primitive models ) . most studied is the `` restricted primitive model '' ( rpm ) , in which the ions are of equal size . a vapor - liquid phase transition at very low temperatures and densities was predicted theoretically 25 years ago @xcite and by early computer simulation studies @xcite . improvement of computer simulation techniques has allowed an increasingly precise determination of the coexistence parameters @xcite . there have also been recent studies of primitive models with asymmetry in size @xcite and charge @xcite . very recent results suggest that the critical behavior of the rpm belongs to the ising universality class @xcite . from a theoretical point of view , different approaches have been used in order to explain the vapor - liquid transition of the primitive models . integral equations such as the mean spherical approximation ( msa ) @xcite , as well as the debye - hckel theory @xcite and poisson - boltzmann approaches @xcite have succesfully been applied to it . for the rpm , the most succesful theories are the pairing theories , that consider the ionic fluid as a mixture of bound pairs and free ions in chemical equilibrium @xcite , in the spirit of the bjerrum s ideas @xcite . however , in all these theories the transition is driven by the free ions , even when in same cases , as in the debye - hckel - bjerrum approach , the associated pairs are the dominant species . analytical @xcite and computer simulation @xcite results , on the other hand , show that the structure of the vapor phase is dominated by neutral clusters , mostly dimers and tetramers . computer simulations have also demonstrated that the phase envelope of the rpm resembles that of the charged hard dumbbell model @xcite . this result has been confirmed by theoretical studies @xcite . the question that the present work examines in detail is the influence of ionic association on the vapor - liquid transition of primitive model electrolytes . in contrast to an earlier study @xcite , which only considered tightly bound dimers , here we examine a range of models with varying values of @xmath0 , the maximum separation between ions in a dimer . @xmath2 corresponds to the stillinger - lovett description @xcite of the free ion system . the structure of this paper is as follows . in section [ sec2 ] , we examine the microscopic structure of the coexisting phases of the ionic fluid and compare the correlation functions to those of a tightly tethered dimer model . in section [ sec3 ] , an exact chemical representation of the ionic fluid as a mixture of associated pairs and free ions is introduced . the role of the associated pairs on the phase coexistence is studied in detail in section [ sec4 ] and the paper closes with discussion and conclusions . in this section we analyse the role of pairing on gas - liquid coexistence of primitive model ionic fluids . we have studied by computer simulation an 1:1 size - asymmetric primitive model , in which the ions are modelled as hard spheres of diameters @xmath3 and @xmath4 , and carrying charges @xmath5 and @xmath6 , respectively , embedded on a dielectric continuum of dielectric constant @xmath7 ( @xmath8 for the vacuum ) . the interaction potential between two ions separated by a distance @xmath9 is given by : @xmath10 where @xmath11 is the hard - core potential that takes the value @xmath12 if @xmath13 and 0 otherwise . the size asymmetry is characterized by the parameter @xmath14 , defined as : @xmath15 monte carlo simulations in the neutral grand - canonical ensemble were performed , characterized by a temperature @xmath16 and the _ configurational _ chemical potential for a pair of unlike ions @xmath17 , with @xmath18 and @xmath19 are the thermal de broglie wavelengths of the ionic species : @xmath20 as usual , cubic boxes of length @xmath21 under periodic boundary conditions are used . the long - ranged character of the coulombic interactions is handled by the use the ewald summation tecnique with conducting boundary conditions , with 518 fourier - space wavevectors and real - space damping parameter @xmath22 . the relative error due to the infinite sums truncation in the electrostatic energy is less than @xmath23 for random configurations in small systems @xcite , and this choice has been also validated by direct simulations of the rpm @xcite . in order to speed up the simulations , the basic steps of the simulations ( insertion and deletions of pairs of unlike ions ) are biased following @xcite . moreover , a fine - discretization approximation is used : the positions available to each ion are the sites of a simple cubic lattice of spacing @xmath24 . this methodology has succesfully been applied to the rpm @xcite , to 1:1 size - asymetric primitive models @xcite and to @xmath25:1 size - asymetric primitive models @xcite , and allows a speedup relative to the continuum calculations of a factor of 100 for small systems . the results are almost indistinguishable from the continuum ones for a discretization parameter @xmath26 @xcite , where @xmath27 is the unlike - ion collision diameter . this value of the discretization parameter ( @xmath28 ) was used in the present study . histogram reweighting techniques @xcite and mixed - field finite size scaling methods @xcite were used to obtain the vapor - liquid envelopes and the effective critical points , respectively . for the sake of comparison , we have also studied tightly tethered dipolar dimer systems @xcite , consisting of @xmath29 pairs of a positive and negative ion restricted to remain at separations @xmath30 satisfying @xmath31 . simulation details and some preliminary results for a range of values of @xmath14 are presented in ref . @xcite . the unlike - ion collision diameter @xmath1 provides the basic length scale appropriate for defining both the reduced temperature and reduced density via @xmath32 where @xmath33 is the particle number of each ionic species @xcite . the reduced simulation box length is defined similarly via @xmath34 , and the reduced energies and chemical potential as @xmath35 and @xmath36 . the value of the asymmetry parameter considered in this paper is @xmath37 . for that case , the gas - liquid coexistence of the ionic fluid shows a shift in both temperature and density respect to the tethered dimer fluid ( see fig . [ fig1 ] ) , which is a general feature when comparing ionic and tethered dimer systems @xcite . on the other hand the asymmetry is not high so as to favor large chain - like neutral clusters , as occurs for bigger values of @xmath14 @xcite . these features qualify this case to be a typical example for moderately asymmetric 1:1 electrolytes , including the rpm . fig . [ fig1 ] makes the similarity between the phase diagram of the ionic and the tethered dimer fluids clear . it also suggests in an indirect way that the pairing plays a decisive role on the gas - liquid coexistence in the ionic fluid . in order to clarify such a role , we have studied the ion - ion radial distribution functions @xmath38 corresponding to the ionic systems , and the corresponding ones for the tethered dimer systems . for this purpose , we have considered the gas and liquid states at coexistence for a temperature @xmath39 , with @xmath40 the corresponding critical temperature . the ion - ion radial distribution functions of the ionic systems , for @xmath41 and the coexisting densities @xmath42 and @xmath43 are plotted in figs . [ fig2 ] and [ fig3 ] , respectively . in both cases , the unlike - ion radial distribution function becomes very large close to contact , indicating the association in bound pairs of unlike ions . moreover , the like - ion radial distribution functions show maxima around @xmath44 ( in the case of @xmath45 , this peak coincides with the contact value ) , and there is a secondary maximum in @xmath46 for @xmath47 . these observations allow us to conclude that there are high correlation between pairs of associated unlike ions . we recall that the range of densities in which the gas - liquid coexistence occurs prevents packing effects , so the structure is completely given by the coulombic interactions . the ion - ion radial distribution functions for the tethered dimer fluid are plotted on the fig . [ fig4 ] for the gas branch ( @xmath48 , @xmath49 ) and fig . [ fig5 ] for the liquid branch ( @xmath48 , @xmath50 ) . the comparison between the ionic fluid and tethered dimer fluid microscopic structures confirms the qualitative similarity between both systems . a further test of this similarity is found in the comparison of the neutral cluster populations in the gas branch . we use gillan s definition of a cluster @xcite . two ions @xmath51 and @xmath52 are directly bound when the distance between them is less than @xmath53 . this condition defines mathematically an equivalence relationship , and the equivalences classes in which the ions group are the clusters . as the interaction between like ions is repulsive , the dependence of the cluster definition on @xmath54 or @xmath55 should be very weak ( if they are taken smaller than the mean distance between two like ions ) . on the other hand , the cluster definition is not very sensitive to the value of @xmath56 if that value lies between the first minimum of @xmath46 and the mean distance between aggregates . in this work we have used @xmath57 . as expected , the microscopic structure of the gas phase is dominated by the @xmath29-ion neutral clusters @xcite . their fractions @xmath58 for the ionic and tethered ion systems at the gas phase in coexistence for @xmath59 are quite similar ( see fig . [ fig6 ] ) , although the tethered dimer systems have slightly higher fractions of neutral clusters than do ionic systems , specially for large @xmath29 , in agreement with the results previously reported @xcite . despite the similarities found between the ionic fluid and the tethered dimer fluid , there are some differences that can rationalize the quantitative differences between their phase diagrams . the unlike - ion radial distribution functions of the tethered dimer fluid differ qualitatively from the ionic counterparts close to the contact value , since in the former there is a jump at the maximum allowed separation of two ions of a dimer , while the ionic @xmath46 takes smoothly higher values on that range of values of @xmath60 . consequently , the condition on the confinement of the ions that compose a tethered dimer must be relaxed in order to refine the pairing concept in the ionic fluids . in the next section we will introduce an appropriate framework for such a goal . we consider an ionic fluid ( @xmath61 ) be contained in a volume @xmath62 and in equilibrium with a reservoir at a temperature @xmath16 and chemical potentials @xmath63 and @xmath64 . for simplicity , the neutral grand - canonical ensemble , in which only neutral configurations are allowed , will be considered . this ensemble has been shown to be equivalent to the usual grand - canonical ensemble in the thermodynamic limit @xcite . the neutral grand - canonical partition function can be written as : @xmath65 where @xmath66 is the number of ions of each species , @xmath67 , @xmath68 , @xmath69 and @xmath70 are the thermal de broglie wavelengths corresponding to each species , and @xmath71 is the canonical configurational partition function corresponding to a fixed number of ions at the temperature @xmath16 and enclosed in the volumen @xmath62 : @xmath72 with @xmath73 the total potential energy and @xmath74 the hard - core contribution , equal to @xmath12 if two particles overlap , and @xmath75 otherwise ( other tempered potentials can be used , but the main results of this section will remain unchanged ) . if the fluid particles are strongly associated into bound @xmath76 pairs , as occurs in the low - temperature and low - density region in which the vapor - liquid transition occurs for the ionic fluids , the `` physical '' representation described above will be inadequate , and it can replaced by a `` chemical '' picture , in which the fluid is composed by associated pairs and free ions . first , a rule that unequivocally identifies bound pairs for each configuration of the ionic fluid ( up to a set of null measure in the configurational space ) is needed . once such a rule is defined , we can write the canonical configurational partition function as : @xmath77 where @xmath78 is the number of bound pairs , @xmath79 is the number of free ions of each species , and @xmath80 is the canonical configurational partition function corresponding to a system of @xmath78 associated pairs and @xmath81 free ions of each species : @xmath82 where @xmath83 are the positions of the ions that compose the associated pair @xmath51 , and @xmath84 corresponds to the coordinates of a @xmath85 free ion @xmath51 , respectively . the potential energy @xmath86 _ does not _ coincide , in general , with the physical potential energy @xmath87 , since different configurations of the `` chemical '' mixture can be compatible with a given ionic configuration . substituting equation ( [ decompose ] ) into the equation ( [ partition ] ) and rearranging the resulting expression , the neutral grand - canonical partition function can be written as : @xmath88 with @xmath89 and @xmath90 . these last definitions correspond to the chemical equilibrium conditions between the free ions and the associated pairs @xcite . we remark that this derivation is _ independent _ of the criterion chosen to define the pairs . however , the choice must be such that matches the microscopic structure of the fluid . as we have seen in the previous section , the vapor structure is dominated by neutral aggregates , mostly dimers and tetramers . this fact is consistent with the bjerrum ideas of pairing @xcite . consequently , a distance - based criterion seems to be the most appropriate : two unlike ions that are closer than @xmath0 ( @xmath0 being a suitable cutoff distance ) are considered as an associated pair . it is easy to see that such a criterion _ does not _ define uniquely the associated pairs when the population of tetramers and higher order neutral clusters is not negligible , as it can be seen from fig . [ fig7 ] , since there are different arrangements of associated pairs that are compatible with the same ionic configuration . hence , a more systematic criterion is needed , in order to be able to define associated pairs for _ almost every _ ionic configuration , while keeping the intuitive definition of an associated pair as encompassing unlike ions that are at the closest distances . we use a suitable modification of the stillinger - lovett pair definition on a given ionic configuration @xcite . in this prescription , all the distances between two unlike ions are computed , and then the first pair is _ defined _ as the two unlike ions at closest distance . this step is repeated , taking into account only ions that remain unpaired from previous steps for the evaluation of @xmath76 distances . we stop when the minimum distance between two unlike ions is greater than @xmath0 , considering the remaining ions as free ions ( no free ions were considered in @xcite , and consequently @xmath91 in that case ) . this protocol provides an unique configuration of associated pairs and free ions for almost each ionic configuration , since the method is ambiguous only in a subset of the ionic configurations in which at least two @xmath76 distances are equal , and the measure of such a set is null in the configurational space . obviously , different pairing prescriptions can be used . however , this protocol provides an explicit expression for the `` chemical '' potential energy @xmath92 : @xmath93 where @xmath87 is the physical potential energy due to the hard - core and eletrostatic interactions , and the other terms correspond to effective interactions of entropic origin needed to reduce the configurational space of the `` chemical '' system . the term @xmath94 depends only on the positions of the ions that compose an associated pair and confines them to be at a distance shorter than @xmath0 : @xmath95 the pairwise potential energy @xmath96 corresponds to an steric hindrance condition between two associated pairs @xcite , since the distances between two unlike ions corresponding to different pairs can not be shorter than the minimum distance between the ions that compose each pair : @xmath97 with @xmath98 . the interaction between the associated pairs and the free ions is modified by the term @xmath99 , that prevents the free ion to be closer to the unlike ion of the associated pair than its partner : @xmath100 finally , two unlike free ions can not be at a shorter distance than @xmath0 due to the term @xmath101 : @xmath102 it is not hard to see that a mixture of associated pairs and free ions with a potential energy given by the eq . ( [ hameffec ] ) is completely equivalent to the original ionic system . it is interesting to note that , except the term on @xmath103 that ties the ions that compose an associated pair , the other terms are pairwise , short - ranged modifications of the hard - core conditions , so the potential energy of an allowed mixture configuration is the same as in the corresponding ionic configuration . moreover , every configuration of associated pairs and free ions obtained from an ionic configuration by the pairing procedure described above fulfil the constrains induced by the added potentials . conversely , the mixture configuration is the same as the one obtained from the corresponding ionic configuration by the stillinger - lovett protocol . this analysis provides an exact chemical representation of the ionic system as a mixture of bound pairs and free ions in chemical equilibrium . the equivalence between representations of the system is not only at the level of thermodynamic properties , but also in the microscopic structure , unlike previous studies based on the matching of the `` chemical '' and the `` physical '' free energies , e.g. via the virial coefficients @xcite . in those studies the effective potentials given by eqs . ( [ vp ] ) and ( [ v+- ] ) could be guessed , but the other terms ( that arise from matching third and higher order virial coefficients ) were not clearly identified . furthermore , they have not being used at all in pairing theories of electrolytes . we must stress the importance of the effective new term given by eq . ( [ vpp ] ) in the potential energy , that reduces the configuration space available to the associated pairs . this effect is specially important for high values of @xmath0 . an extreme case case that illustrates the effect of missing the steric hindrance term is the loosely tethered dimer fluid , in which the ions that compose the pair can be at any relative distance greater than @xmath1 . in this case , the canonical partition function of the dimer fluid is @xmath104 , where @xmath78 is the number of dimers , @xmath105 is given by eq . ( [ partition2 ] ) and the @xmath106 factor is the number of different ways of pairing the unlike ions in each ionic configuration and lead to different dimer configurations . as a consequence , in the thermodynamic limit the helmholtz free energy per particle @xmath107 of the loosely tethered dimer fluid diverges as : @xmath108 where @xmath109 is the ionic helmholtz free energy per ion , that it is well - behaved in the case of neutral systems @xcite . the association degree of the ionic fluid in the coexistence region has been studied by computer simulation in the framework of the previous analysis . in the usual grand - canonical simulations , we have identified the associated pairs by the stillinger - lovett rule with @xmath110 , i.e. all the ions are associated , and the minimum image convention is used in order to calculate the distances between unlike ions . the probability distributions @xmath111 of having a pair an internal separation distance @xmath112 between ions is plotted in the fig . [ fig8 ] for the gas and liquid phases at @xmath113 . surprisingly , both distributions are practically identical , despite the fact that the corresponding densities are quite different . the distributions show a very pronounced peak for @xmath114 , and a local maximum around @xmath115 , approximately where @xmath46 presents the second local maximum . for large values of @xmath112 , the distributions @xmath111 decay to zero . these results confirm the strong association in the ionic fluid at low temperatures , and that the structure is weakly affected by variations in density , at least in the range in which the gas - liquid coexistence occurs . for values of @xmath112 close to the contact value , the distribution is well described by the non - interacting pair fluid probability distribution , given by the following expression : @xmath116\right)\end{aligned}\ ] ] valid for @xmath117 and an internal cutoff @xmath118 . however , for @xmath119 , @xmath111 deviates from the ideal expression as a consequence of the interaction between associated pairs . the local maximum showed by the distribution function indicates that the most bound pairs are solvated by less bound pairs , to form stable neutral tetramers ( and higher order clusters , but their inclusion does not seem to affect the conclusions of this section ) . this fact is in agreement with the features observed in the pair correlation functions in the previous section . we must stress that the structure that @xmath111 presents is only due to the coulombic interactions . it is instructive to compare our results in the coexistence region to the higher temperature ones . we have computed @xmath111 at @xmath120 and the same range of densities ( see fig . [ fig9 ] ) . the qualitative behavior of @xmath111 at high temperatures is similar to that predicted by stillinger and lovett @xcite . first , it shows a maximum localized in @xmath121 . on the other hand , @xmath111 takes significant values in a wider range than the corresponding functions at lower temperatures , decaying for large values of @xmath112 as @xmath122 , with @xmath123 . the latter prediction differs from the value @xmath124 given in ref . @xcite and requires further study to elucidate it . the probability distribution @xmath125 of having a pair an internal separation distance between ions when the cutoff distance to define a pair is equal to @xmath0 can be obtained from @xmath111 as : @xmath126 where @xmath127 is the cumulant distribution corresponding to @xmath111 . the density of associated pairs is equal to @xmath128 , and the total density of free ions is @xmath129 $ ] , where @xmath130 is the total density of ions in the physical picture . it can be seen from fig . [ fig8 ] that for @xmath131 , more than 95% of ions are associated into pairs in both vapor and liquid phases . so , if the associated pair fluid phase separate in the same @xmath132 region as the ionic fluid , it is expected than the free ions do play a mere perturbative role in the phase coexistence . actually , the free ionic subsystem is a non - additive binary charged hard - sphere mixture , where @xmath0 plays the role of the unlike - ion collision diameter . however , the interactions between like ions are purely repulsive , and it is not expected differences between the behavior of this mixture and the rpm as soon as @xmath3 and @xmath4 are less than @xmath0 . furthermore , this system is in a polar environment given by the associated pair subsystem , and consequently the effective dielectric constant @xmath133 of the background will be increased . then the free ion subsystem is expected to have the same behavior as the rpm at the effective reduced temperature @xmath134 : @xmath135 where @xmath136 is the reduced temperature . for @xmath137 , the critical temperature reduces at least to one third of the rpm reduced critical temperature . consequently , any possible free ion - driven vapor - liquid transition should happen at much lower temperatures , and thus the free ion subsystem should play no role in the vapor - liquid transition . in order to check this hypothesis , the phase diagram of the associated pair fluid has to be obtained . this issue will be addressed in the next section . the results obtained in the previous section suggest that the `` chemical '' picture introduced above is a very convenient description of the ionic fluid structure . furthermore , we can analyse the role played by the associated pairs in the phase equilibrium by eliminating the free ions . we have performed grand - canonical monte carlo simulations of the associated dimer system for @xmath37 and different values of @xmath0 . no free ions are allowed ( @xmath138 ) and the potential energy of one configuration is given by the eq . ( [ hameffec ] ) . the grand - canonical free energies corresponding to these systems will consequently provide an upper bound to the ionic fluid grand - canonical free energy at the same temperature and pair chemical potential . as for the ionic system , a fine - discretization approach with a refinement parameter @xmath28 , and ewald summation technique with conducting boundary conditions is used to take into account the long range character of the coulombic interations . the basic steps are either insertion or deletion of associated pairs ( chosen randomly with the same probability ) , biased with a boltzmann distribution that depends on the separation between the ions that compose the dimer . an associated pair is inserted in the following way : the negative ion is placed at a random place in the box , and its counterion is placed at a relative position @xmath139 ( @xmath140 ) following a probability distribution proportional to @xmath141 , where @xmath142 is the ewald potential energy between two unlike ions of unit charge . on the other hand , a pair is deleted with a probability proportional to @xmath143 , being @xmath139 the relative position of the positive ion of the pair with respect to the negative one . the value of @xmath144 has been adjusted to improve the sampling during the simulation . in this work we have set @xmath145 . the acceptance probabilities of a pair insertion @xmath146 and a pair deletion , @xmath147 are the following : @xmath148 \left[\sum_{k=1}^{n_p + 1 } \exp(\beta_0 q^2\phi_0(\mathbf{r}_\pm^k))\right]^{-1 } \right)\\ w_{ij}^{d}&=&\textrm{min}\left(1 , \frac{\exp(-\beta \mu - \beta \delta u)}{(l^*)^3 } \left[\sum_{\sigma_\pm\le |\mathbf{r}_\pm| < r_c}\frac{\exp(\beta_0 q^2\phi_0 ( \mathbf{r}_\pm))}{\zeta^3 } \right]^{-1 } \left[\sum_{k=1}^{n_p } \exp(\beta_0 q^2\phi_0(\mathbf{r}_\pm^k))\right]\right)\end{aligned}\ ] ] where @xmath51 and @xmath52 are the initial and final configurations , respectively , @xmath78 is the number of pairs at the configuration @xmath51 , @xmath67 , @xmath149 is the configurational chemical potential , and @xmath150 is the `` chemical '' potential energy variation in the movement , including the steric hindrance terms given by eq . ( [ vpp ] ) . as it can be seen , the acceptance probabilities reduce to the usual ones as @xmath151 . effective critical points for different values of @xmath152 were estimated by using mixed - field finite - size scaling methods @xcite and assuming ising - like criticality . although recent results @xcite indicate that the pressure should also enter the field mixing , our approach should be satisfactory to discern the dependence of the critical parameters on @xmath0 . moreover , a systematic study for the rpm case has shown that the assumed universality class is very likely to be the right one @xcite . we use histogram reweighting techniques @xcite to combine the histograms from different runs ( typically three ) and estimate the critical parameters and their standard errors . very long runs are needed in order to overcome the critical slowing down and the low acceptance ratios due to the low temperatures involved . defining a step as a try of a pair insertion / deletion , we have performed simulations of about @xmath153 equilibration steps and @xmath154 sampling steps for @xmath155 ; @xmath156 equilibration steps ans @xmath157 sampling steps for @xmath158 ; and @xmath159 equilibration steps and @xmath160 sampling steps for @xmath161 , in order to get good statistics for the histograms . the ( effective ) critical parameters are obtained by minimizing the deviation of the appropriately scaled mixed - field @xmath162 marginal probability distribution ( @xmath163 is the potential energy density and @xmath164 is the mixing parameter ) with respect to the corresponding critical 3-dimensional ising universal function @xcite . as also occurs in the ionic and tethered dimer systems , the matching improves as @xmath152 increases ( see fig . [ fig10 ] ) . our estimations of the critical parameters for different values of @xmath152 and @xmath165 are listed in table [ tablecritpar ] and plotted in figs . [ fig11 ] and [ fig12 ] . the smallest value of @xmath166 is the same as the maximum allowed separation in the tethered dimer system , and as expected the results obtained for the associated pairs fluid are practically indistinguishable from the tethered dimer system . as @xmath166 increases , it is observed a sharp increase in the critical temperature but the critical density remains almost unaffected . however , for @xmath167 the critical temperature reaches a maximum and then decreases , converging towards the free ion critical temperature . on the other hand , the critical density also decreases towards the free ion critical density , although the statistical uncertainties are bigger for this parameter and the exact path of convergence is less defined . we must also point out that the statistical uncertainties are big enough not to make possible an estimation of correction - to - scaling effects . for an heuristic explanation for such a behavior , we have to consider the energetically favoured configurations . it is important to consider the interaction between two different associated pairs , that has its minimum energy configuration in a square ring conformation for low values of @xmath14 , with a close - energy secondary minimum conformation for the linear @xmath76 chain @xcite . if @xmath0 is too small , by deforming the square it is possible to find energetically relevant configurations ( i.e. the potential energy difference per ion of such configurations with respect to the minimum one is of the same order as the thermal energy ) in which one of the associated pairs has an internal ionic separation less than @xmath0 , but the other one does not fulfil such a condition . consequently , a greater value of @xmath0 also increases the number of energetically relevant configurations between two associated pairs . then the effective interaction between associated pairs is enhanced , and thus an increase of the critical temperature should be expected until all the relevant configurations are allowed ( that occurs for @xmath168 ) . on the other hand , the average size of the associated pairs and their aggregates is increased , leading to a decrease on the critical density ( in order to keep the reduced density in terms of the effective particle size constant ) . these arguments are not longer valid for @xmath169 . our results show the inclusion of associated pairs with larger internal ionic separations are not crucial to the gas - liquid phase transition but also that they bar the phase transition , as it can inferred from the critical parameters decrease . as for the ionic and tethered dimer models , histogram reweighting techniques @xcite allow us to obtain the coexistence curve up to @xmath170 . as the finite - size effects are not important far from the critical point , we have considered @xmath158 . taking advantage of the wide range of densities covered by near - critical histograms , we combine them with liquid subcritical state histograms in order to extend the density range . the extra simulations involve shorter runs ( typically @xmath159 steps after equilibration ) . the gas - liquid coexistence curves for different values of @xmath166 are plotted in fig . [ fig13 ] , showing the same tendency as the observed one in the critical parameters . it is interesting to note that the vapor branches coincide ( except close to the critical point ) for @xmath171 . we conclude from this observation that is in the liquid branch where the ionic fluid differs mostly from the associated pair system . in summary , an exact `` chemical '' representation of the ionic system as a mixture of @xmath76 associated ion pairs and free ions has been introduced . this representation , closely related to the stillinger - lovett pairing procedure for the rpm , has the advantage that not only provides an exact matching of between the physical and chemical representation thermodynamics , but also from a microscopic point of view . it also avoids an entropy catastrophe that occurs for the tethered dimer model studied at large values of the tether length . the addition to the physical hamiltonian of some new pairwise hard - core interactions between the `` chemical '' components provides the suitable hamiltonian for the `` chemical '' representation . in the low temperature and low density regime in which the ionic vapor - liquid transition occurs , such a representation provides a faithful characterization of the microscopic structure of the ionic fluid , and it can be the basis of new theoretical approaches . the analysis of the phase behavior of the system only composed by associated pairs indicates strongly that the ionic fluid vapor - liquid transition is driven by them . for @xmath172 , the critical temperature increases with @xmath0 , as more energetically favoured configurations are allowed . for @xmath131 , the critical parameters converge smoothly _ from above _ towards the ionic fluid critical parameters as @xmath0 increases . this fact indicates not only that the free ions do not drive the phase transition , but also have the opposite effect in the transition . the value of @xmath173 corresponding to the maximum on the critical temperature , can be regarded as the optimal size of the associated pair . some remarks on the limitations of the present work are appropriate at this point . we have focussed only on the effect of the associated pairs in the thermodynamical properties . however , the conducting character of the ionic fluid is completely driven by the free ions . on the other hand , the remarkable success of pairing theories ( even when the `` chemical '' representation they implicitly use is not completely correct ) remains unexplained . it is possible that the associated pair solvation , in addition to the ion - ion correlations , can mimic the pair - pair interactions . further studies are needed to completely solve the origin of the ionic fluid vapor - liquid phase transition . j.m.r .- e . and l.f.r . gratefully acknowledge financial support for this research by grant no . pb97 - 0712 from dgicyt ( spain ) and no . fqm-205 from pai ( junta de andaluca ) . e . also wishes to thank an fpi scholarship from ministerio de educacin y cultura ( spain ) . azp acknowledges financial support from the department of energy ( grant de - fg02 - 01er1512 ) and acs - prf ( grant 38165-ac9 ) . 199 m. e. fisher , j. stat . phys . * 75 * , 1 ( 1994 ) ; j. phys . : matter * 8 * , 9103 ( 1996 ) . r. r. singh and k. s. pitzer , j. chem . phys . * 92 * , 6775 ( 1990 ) . h. weingrtner , s. wiegand and w. schrer , j. chem . phys . * 96 * , 848 ( 1992 ) . k. c. zhang , m. e. briggs , r. w. gammon and j. m. h. levelt sengers , j. chem . phys . * 97 * , 8692 ( 1992 ) . t. narayanan and k. s. pitzer , phys lett . * 73 * , 3002 ( 1994 ) ; j. phys . chem . * 98 * , 9170 ( 1994 ) . j. jacob , a. kumar , m. a. anisimov , a. a. povodyrev and j. v. sengers , phys . e * 58 * , 2188 ( 1998 ) . m. a. anisimov , j. jacob , a. kumar , v. a. agayan and j. v. sengers , phys . lett . * 85 * , 2336 ( 2000 ) . k. gutkowski , m. a. anisimov and j. v. sengers , j. chem . phys . * 114 * , 3133 ( 2001 ) . g. stell , k. c. wu and b. larsen , phys . lett . * 37 * , 1369 ( 1976 ) . p. n. vorontsov - velyaminov and v. p. chasovskikh , high temp . 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[ fig8],width=321 ] for the associated pairs fluid characterized by @xmath37 and @xmath185 . the solid line is the universal function corresponding to the 3-dimensional ising universality class , and the symbols correspond to the best - matching simulation results : the squares correspond to @xmath155 , the diamonds to @xmath158 and the triangles to @xmath161 . [ fig10],width=321 ]

we present a systematic study of the effect of the ion pairing on the gas - liquid phase transition of hard - core 1:1 electrolyte models . we study a class of dipolar dimer models that depend on a parameter @xmath0 , the maximum separation between the ions that compose the dimer . this parameter can vary from @xmath1 that corresponds to the tightly tethered dipolar dimer model , to @xmath2 , that corresponds to the stillinger - lovett description of the free ion system . the coexistence curve and critical point parameters are obtained as a function of @xmath0 by grand canonical monte carlo techniques . our results show that this dependence is smooth but non - monotonic and converges asymptotically towards the free ion case for relatively small values of @xmath0 . this fact allows us to describe the gas - liquid transition in the free ion model as a transition between two dimerized fluid phases . the role of the unpaired ions can be considered as a perturbation of this picture .

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Proceed to summarize the following text: in standard @xmath0cdm cosmological models , cold dark matter halos form from the gravitational collapse of dark matter particles , and they assemble hierarchically , such that smaller halos merge to form larger and more massive halos . according to the current paradigm of galaxy formation , galaxies form within halos , due to the cooling of hot gas . halos and galaxies evolve simultaneously , and the evolution of a galaxy is affected by its host halo . if the halo is accreted by a larger halo , the galaxy will be affected by it as well , and may interact or merge with the galaxies within the new host halo . such ` satellite ' galaxies in halo substructures no longer accrete hot gas , which instead is only accreted by the ` central ' galaxy in the halo . the central galaxy consequently continues to grow , while other massive galaxies may merge into it , and therefore it is expected to be the most luminous and most massive galaxy in the halo . for these reasons , current analytic and semi - analytic models distinguish between central and satellite galaxies , which at a given time are at different stages of evolution , or may have evolved differently . as galaxies evolve they transform from star - forming late - type galaxies into massive bulge - dominated galaxies with little or no ongoing star formation . it is thought that central galaxies undergo such a transformation by experiencing a major merger followed by agn feedback preventing additional gas cooling and star formation . satellite galaxies may have their star formation suppressed or ` quenched ' by a number of other processes , such as ram - pressure stripping of the cold gas reservoir , ` harassment ' by other satellites , and ` strangulation ' following the stripping of the hot gas reservoir , the latter of which appears to be the dominant process ( _ e.g. _ , weinmann et al . 2006 , van den bosch et al . galaxies in relatively dense environments tend to reside in groups and clusters hosted by massive halos . recent analyses with galaxy group catalogs have argued that many of these galaxies are very red with very low star formation rates , in contrast with galaxies in low - mass halos in less dense environments , many of which are still quite blue with significant star formation ( _ e.g. _ , weinmann et al . 2006 , berlind et al . measurements of the environmental dependence of galaxy color have found trends that are qualitatively consistent with these claims ( _ e.g. _ , zehavi et al . 2005 , blanton et al . 2005a , tinker et al . 2007 , coil et al . 2008 ) . in order to better understand galaxy and halo evolution , more models are needed that can explain the environmental dependence of color , and more measurements of correlations between color and environment are needed to better constrain such models . skibba & sheth ( 2008 ) have taken a step in this direction : they developed and tested a halo model of the color dependence of galaxy clustering in the sloan digital sky survey ( sdss ) . their model successfully explains the correlation between color and environment , quantified by the color mark correlation function , while assuming that all environmental correlations are due to those of halo mass . they distinguish between central and satellite galaxies , whose properties are assumed to be determined by host halo mass . the purpose of this paper is to further investigate these central and satellite galaxy colors , and in particular to compare the predictions of the model with measurements from recent galaxy group catalogs ( yang et al . 2007 , berlind et al . this paper is organized as follows . in the next two sections , we briefly introduce the color mark model and the galaxy group catalogs . in section [ groupcatcompare ] , we compare the satellite color - magnitude sequence of the model to that of the yang et al . catalog , and we compare the central and satellite colors of the model and both group catalogs as a function of group richness , which is a useful proxy for halo mass . we summarize our results in section [ discuss ] . our halo model of the color dependence of galaxy clustering is described in ( skibba & sheth 2008 ; hereafter ss08 ) , and we refer the reader to this paper for details . briefly , our model is based on the model of luminosity dependent clustering of skibba et al . ( 2006 ) , which explained the observed environmental dependence of luminosity by applying the luminosity - dependent halo occupation distribution ( hod ) that was constrained by the observed luminosity - dependent correlation functions and galaxy number densities in the sdss ( zehavi et al . 2005 , zheng et al . the model of galaxy colors in ss08 added constraints from the bimodal distribution of @xmath1 colors of sdss galaxies as a function of @xmath2-band luminosity . we made two assumptions : ( i ) that the bimodality of the color distribution at fixed luminosity is independent of halo mass , and ( ii ) that satellite galaxies tend to follow a particular sequence in the color - magnitude diagram , one that approaches the red sequence with increasing luminosity : @xmath3 these observational constraints and additional assumptions allowed ss08 to model the central and satellite galaxy color ` marks ' as a function of halo mass , @xmath4 and @xmath5 . ss08 used the central and satellite galaxy marks to model color mark correlation functions , in which all correlations between color and environment are due to those between halo mass and environment . the modeled mark correlation functions were in very good agreement with their measurements with volume - limited sdss catalogs , reproducing the observed correlations between galaxy color and environment on scales of @xmath6 . the two - point mark correlation function is simply the ratio @xmath7 , where @xmath8 is the traditional two - point correlation function and @xmath9 is the same sum over galaxy pairs separated by @xmath2 , but with each member of the pair weighted by the ratio of its mark to the mean mark . in practice , we measure the _ projected _ clustering of galaxies , and so we use the following analogous statistic for both the measurements and the models , the marked projected correlation function : @xmath10 where the projected two - point correlation function is @xmath11 if mark correlations are consistent with unity , then the mark is not correlated with the environment at that scale ; if the mark correlations are above unity , which is the case for galaxy luminosity and color ( skibba et al . 2006 ; ss08 ) , then higher values of the mark tend to be located in denser environments at that scale . as an example , we show the @xmath1 color mark correlation function for @xmath12 in figure [ mcfexample ] , reproduced from ss08 ( their figure 6 ) . the solid curve shows the halo model s prediction , which is in agreement with the measurements of the color mark correlations in the sdss , using petrosian colors . had we instead used @xmath13 as the mark , the resulting mark correlations would be quantitatively stronger , but with larger uncertainties the sdss measurements would have similar statistical significance and constraining power as the @xmath1 mark measurements ( see skibba et al . 2006 ) . color mark correlation function for @xmath12 . points show sdss measurement for petrosian colors , and solid curves show the fiducial model , both taken from skibba & sheth ( 2008 ) . dashed curves show the range of color mark correlations predicted by the model if color gradients in groups and clusters are included ( hansen et al . 2007 ) . ] a few assumptions were made in the model that are worth discussing . firstly , it was assumed that the central galaxies lie at the center of their host dark matter halos , as is commonly assumed in hod and conditional luminosity function ( clf ) studies . however , central galaxies are often offset from the center of the potential well ( van den bosch et al . 2005 ) , and this offset appears to weakly dependent on galaxy color ( skibba et al . 2008b , in prep . ) . secondly , it was assumed that satellite galaxies follow the dark matter profile , while satellite galaxy number density profiles have been found to be less concentrated ( _ e.g. _ , hansen et al . 2005 , yang et al . these two effects are both too weak to significantly affect the marked galaxy clustering on scales of @xmath14 , however , and if the effects were stronger , then there would have been a discrepancy with the observed unmarked correlation function @xmath15 as well , which was not the case . thirdly , we also assumed that there are not mark gradients within halos , that is , that the colors of galaxies are independent of their distance from the halo center . in contrast , hansen et al . ( 2007 ) have shown that the @xmath1 colors of galaxies are @xmath16 redder in the inner regions of groups and clusters compared to the cluster outskirts ( see their figure 10 ) . van den bosch et al . ( 2008b ) have shown a similar fractional increase in galaxy colors with decreasing halo - centric radius , down to halo masses of @xmath17 . these color gradients appear to be stronger than luminosity gradients in groups and clusters ( hansen et al . 2005 , martnez & muriel 2006 , weinmann et al . such a positional dependence of galaxy colors would only affect the color mark correlations while leaving the unmarked correlation function unchanged . in order to model position - dependent color marks in mark clustering statistics , the density profile of satellite galaxies must be replaced by a weighted profile . the calculation is done in fourier space ( see sheth 2005 , and appendix b of ss08 ) , and we replace @xmath18 , which is the fourier transform of @xmath19 , by a weighted number density profile @xmath20 where @xmath21 and @xmath22 quantifies the dependence of galaxy color on halo - centric radius , which we assume to be independent of mass because it is independent of cluster richness ( hansen et al . 2007 ; _ cf_. , van den bosch et al . the minimum and maximum amount of position - dependence of galaxy colors result in slightly stronger color mark correlations at small scales ( lower and upper dashed curves in lower panel of figure [ mcfexample ] ) . or in other words , color gradients in halos imply a slightly stronger environmental dependence of galaxy color in halo environments ( @xmath23 ) . note that color gradients in clusters are much stronger at @xmath24 ( loh et al . 2008 ) and would have a stronger effect on color mark clustering at such redshifts . finally , the analytic halo - model description of the color dependence of galaxy clustering uses the mean of the halo occupation distribution and the mean colors of central and satellite galaxies . we are assuming that at fixed halo mass , the halo occupation distribution and color distributions are approximately independent of the environment an assumption which is justified by some recent results ( blanton & berlind 2007 , van den bosch et al . 2008b , skibba & sheth 2008 ) . to explore the central and satellite galaxy color distributions predicted by the model , we construct mock galaxy catalogs . the mock catalogs are observationally constrained by the halo occupation distribution as a function of luminosity , determined from galaxy clustering measurements in the sdss ( zheng et al . 2007 ) , and the bimodal color distribution as a function of luminosity in the sdss , fit as the sum of two gaussian distributions , which we refer to as the ` red sequence ' and the ` blue sequence ' . in effect , since central and satellite galaxies have a range of colors at fixed luminosity , in the model satellite colors are drawn from the red sequence with some luminosity dependent probability and are otherwise drawn from the blue sequence ; central galaxy colors are drawn with a similar procedure , but with a different luminosity - dependent probability . the details of the algorithm are described in skibba et al . ( 2006 ) and ss08 . ss08 showed that these mock catalogs , which include not only the means of the galaxy colors but their scatter as well , yield color mark correlation functions that are consistent with the analytic model and with sdss measurements . the analysis in section [ groupcatcompare ] , in which we examine the colors of central and satellite galaxies , will constitute a test of this model , and in particular of the model s two assumptions described at the beginning of this section . we will compare the model s predictions of central and satellite galaxy colors to those of galaxy group catalogs , described in the following section . we will first compare our model to the yang et al . ( 2007 ; hereafter y07 ) group catalog , which was constructed by applying the halo - based group finder of yang et al . ( 2005a ) to the new york university value - added galaxy catalog ( nyu - vagc ; blanton et al . 2005b ) , which is based on the sloan digital sky survey ( sdss , york et al . 2000 ) data release 4 ( adelman - mccarthy et al . 2006 ) . the group finder uses halo properties as a function of the total luminosity or stellar mass of groups , and it also identifies ` groups ' with only a single member . we used only those galaxies with spectroscopic redshifts from the sdss or with redshifts taken from other surveys ( their ` sample ii')that is , we excluded fiber - collided galaxies that were not assigned fibers . including such galaxies does not affect the mean colors of central and satellite galaxies ( shown in section [ groupcatcompare ] ) , although this is not the case for the mean central galaxy luminosities ( see appendix of skibba et al . 2007 ) . y07 have used mock catalogs to account for the effects of the survey edges when estimating group halo masses ( see their paper for details ) . we only compare to volume - limited catalogs constructed from the sample ii group catalog in this work , and we have accounted for groups that overlap the redshift limits . in each group , we identify the ` central ' galaxy as the most luminous in the @xmath2 band , and the remaining galaxies brighter than the luminosity threshold are the ` satellites ' . labeling the most massive galaxy in a group , rather than the brightest , as the central one does not affect our results . we will also compare to the central and satellite galaxy colors of a volume - limited group catalog of berlind et al . ( 2006a ; hereafter b06 ) , which is drawn from the sdss large - scale structure sample ` sample14 ` from the nyu - vagc ; the sample is a subsample of sdss dr4 ( adelman - mccarthy et al . fiber - collided galaxies are included , and each collided galaxy was given the redshift of its nearest neighbor , which was shown to be an adequate correction at least for groups with ten or more members . to account for the survey edges , b06 excluded all groups whose centers lie less than 500 kpc from an edge in the tangential direction or less than 500 km @xmath25 from an edge in the radial direction . groups were identified using a halo - based friends - of - friends algorithm , which used halo occupation distribution models and the group multiplicity function . the group - finding algorithm only used the galaxy positions in redshift - space , as opposed to the algorithm used by y07 . in the following analysis , the magnitudes and colors of galaxies in the two group catalogs are petrosian , and have been @xmath26-corrected and evolution corrected to @xmath27 . the main purpose of this paper is to analyze the colors of central and satellite galaxies in groups and clusters . we compare predictions of the color mark model of skibba & sheth ( 2008 ) to measurements from the yang et al . ( 2007 ) and berlind et al . ( 2006 ) galaxy group catalogs . this analysis also constitutes a test of the model , which , as described in section [ model ] , made two assumptions in addition to those necessary to model the luminosity dependence of galaxy clustering : ( i ) that the bimodal color distribution at fixed luminosity is independent of halo mass , and ( ii ) that satellite galaxies tend to follow a particular color - magnitude sequence @xmath28 that approaches the red sequence with increasing luminosity ( equation [ csatseq ] ) . we test the latter assumption first . in figure [ censatcmd ] we show a color - magnitude diagram with the model s satellite galaxy color - magnitude sequence . it is similar to figure 2 in ss08 , but the contours are for a volume - limited catalog constructed from the y07 group catalog , with limits @xmath29 , @xmath30 , consisting of 59,085 galaxies . the figure shows the color - magnitude contours of central and satellite galaxies in the catalog ( red and blue contours , respectively ) . central and satellite galaxies appear to have somewhat similar bimodal color distributions at faint luminosities . the majority of bright blue galaxies and luminous red galaxies are centrals , however , and as we will show later , they tend to reside in less massive and more massive halos , respectively . we compare the satellite galaxy color - magnitude sequence @xmath28 ( [ csatseq ] ) of the model to the mean satellite colors in the y07 group catalog in figure [ censatcmd ] . they are in excellent agreement : the model s satellite color sequence is approximately only 0.01 magnitudes redder than that of the group catalog and within the poisson errors across most of the luminosity range . the agreement between the model and the group catalog is encouraging , and in particular it supports the use of ( [ csatseq ] ) as the satellite galaxy color sequence , which could be used as a constraint for galaxy formation models that include physical processes that quench the star formation of satellites . volume - limited sdss catalog constructed from the y07 group catalog . contours for central galaxies ( with brightest @xmath2-band luminosity ) and satellite galaxies are red and blue , respectively . the satellite galaxy color - magnitude sequence @xmath28 of the ss08 model ( [ csatseq ] ) is shown as the thick solid line ; the mean satellite galaxy colors at fixed luminosity of the group catalog are shown as the square points , with poisson errors . the color - magnitude sequence of central galaxies in the model ( [ ccenseq ] ) and the group catalog are also shown , by the dashed line and triangle points , respectively . the measured central and satellite colors ( squares and triangles ) are slightly offset , for clarity . ] we also show the color - magnitude sequence of central galaxies in figure [ censatcmd ] . in the model , this sequence is implied by the satellite sequence ( [ csatseq ] ) , the luminosity - dependent halo occupation distribution , and the observed color - magnitude constraints : @xmath31 . \label{ccenseq}\ ] ] ( see ss08 for details ) . this sequence is consistent with the measurement from the y07 catalog , at least for @xmath32 . in both the model and the y07 catalog , at fixed luminosity , central galaxies tend to be _ bluer _ than satellite galaxies . van den bosch et al . ( 2008a ) found the same result , at fixed stellar mass . however , in a given halo , the central galaxy tends to be significantly brighter and more massive than its satellites , and it also tends to be redder , as we will show in section [ ccencsatn ] . figure [ censatcmd ] showed the mean central and satellite colors as a function of luminosity , @xmath33 and @xmath28 . we now compare the central and satellite color distributions , @xmath34 and @xmath35 , in figures [ ccenlcompare ] and [ csatlcompare ] . the model predictions are determined from a mock galaxy catalog ( described in section [ model ] ) , and they are compared to measurements from the y07 group catalog . color of central galaxies as a function of luminosity . solid black histograms show the predictions of the model ( skibba & sheth 2008 ) ; dashed blue histograms show the measurements from the yang et al . ( 2007 ) group catalog . ] , but for satellite galaxies . the brightest luminosity bin is excluded because of poor number statistics . ] overall , the agreement between the model and the group catalog is quite good , especially for @xmath36 ( or @xmath37 ) . for central galaxies , the bimodality of the color distribution is stronger in the model than in the data at faint luminosities this is partly due to the fact that the double - gaussian fit to the color distribution at fixed luminosity is not perfect , and slightly underpopulates the ` green valley ' between the red and blue sequences at faint luminosities . this does not explain the discrepancy in the faintest bin , however . faint central galaxies tend to reside in low mass halos or small groups ( _ e.g. _ , skibba et al . 2007 ) , so the discrepancy could be due to inaccuracies in the model or its constraints at low mass or to inaccuracies in the group catalog in poor groups . an interesting result is that both the model and the group catalog agree that the red sequence and blue sequence are peaked at approximately the same colors at fixed luminosities for central and satellite galaxies , and that the widths of the red and blue sequences narrow similarly with increasing luminosity for centrals and satellites . the difference between the central and satellite color distributions is simply that the blue fraction at a given luminosity is lower for satellites : the red sequence of satellites becomes significantly populated at fainter luminosity than does the red sequence of centrals . we now investigate the colors of central and satellite galaxies as a function of group richness , which is strongly correlated with halo mass . we constructed a volume - limited catalog from the y07 group catalog similar to the one above , but with the following limits : @xmath38 and @xmath39 . we begin by examining the mean colors of centrals and satellites . the mean color of central galaxies in groups containing @xmath40 galaxies was defined as the mean value @xmath41 of all central galaxies of such groups ( rather than @xmath42 or @xmath43 ) . the mean satellite color in groups of @xmath40 galaxies was computed similarly ( _ i.e. _ , the mean of satellite @xmath1 ) . we have also measured the mean colors of central and satellite galaxies in the similar volume - limited catalog of b06 . this @xmath44 catalog consists of 21301 galaxies in 4119 groups having three or more members . in comparison , the corresponding y07 catalog contains 15234 galaxies in 2163 groups ( when fiber - collided galaxies are included ) , which are significantly smaller numbers considering that their catalog was drawn from a later sdss data release . the berlind et al . catalog has an overabundance of low-@xmath40 groups ( see their appendix and that of skibba et al . 2007 ) , which does not appear to be the case for yang et al . however , berlind et al . only claim to be complete for @xmath45 , and for such richnesses the two group catalogs are in better agreement . we will compare the mean central and satellite colors from both group catalogs to those predicted by the halo occupation model of skibba & sheth ( 2008 ) . the model colors are computed the same way as the central and satellite luminosities were computed in skibba et al . ( 2007 ) : @xmath46 and @xmath47 where @xmath48 is the halo mass function , @xmath49 is the halo occupation distribution , and @xmath50 is the group multiplicity function . figure [ yangberlind ] shows the results . in general , at fixed group richness , satellite galaxies tend to be bluer than central galaxies , by up to 0.1 mags . these trends are similar at fixed halo mass in the model ( not shown ) . because of the dependence of the mean satellite colors on the luminosity threshold ( @xmath51 in eq . [ csatn ] ) , the difference between the colors of centrals and satellites would be even larger if fainter galaxies were included . colors as a function of group richness , for @xmath44 . solid curves show the model predictions ( using equations [ ccenn ] and [ csatn ] ) , and red triangles and blue squares show the measurements from the b06 and y07 galaxy group catalogs , respectively . vertical error bars are the average of the poisson errors ( estimated from the number of groups in each bin ) and bootstrap errors ( estimated from the variance of 10 times as many pseudo - samples as the number of groups ) . horizontal error bars show the @xmath40 bin widths . ] the model s central and satellite galaxy colors are in very good agreement with both group catalogs , from poor groups to rich clusters . the error bars on the measurements are fairly large , but the halo - model predictions are also uncertain , due to uncertainties in the luminosity - dependent hod and in the color - magnitude constraints . the central galaxy colors are systematically redder in the yang et al . ( 2007 ) catalog than in the berlind et al . ( 2006a ) catalog . it could be due to the different cosmology assumed by yang et al . ( @xmath52 , @xmath53 ) , or it may be related to the different color - dependent clustering at fixed group mass measured from the catalogs ( berlind et al . 2006b , wang et al . 2007 ) . it is of only weak statistical significance , however . for the model , we have assumed that the bimodal color distribution at fixed luminosity is independent of halo mass . therefore , the satellite colors are only weakly dependent on group richness or halo mass because the satellite luminosities are . skibba et al . ( 2007 ) found that satellite galaxy luminosity is almost independent of group richness in their halo occupation model and in the group catalogs of yang et al . ( 2005a ) and berlind et al . ( 2006a ) . the result in the upper panel of figure [ yangberlind ] shows that this is also the case for satellite galaxy colors , and lends justification to the model s assumption that the color distribution at given luminosity is independent of mass . this result has an interesting consequence : the fact that both the luminosity and color of satellites is nearly independent of halo mass implies that the evolution of satellite galaxies is nearly independent of the environment . we discuss this further in section [ discuss ] . finally , we compare the central and satellite color distributions as a function of group richness , @xmath54 and @xmath55 , for @xmath44 , in figures [ ccenncompare ] and [ csatncompare ] . the model predictions are determined from a mock galaxy catalog ( described in section [ model ] ) , and they are compared to measurements from the yang et al . ( 2007 ) and berlind et al . ( 2006a ) group catalogs . color of central galaxies as a function of group richness , for @xmath44 . solid black histograms show the predictions of the model ( skibba & sheth 2008 ) ; dashed blue histograms show the measurements from the yang et al . ( 2007 ) group catalog ; dotted red histograms show the measurements from the berlind et al . ( 2006a ) catalog . a very large @xmath40 bin is not included , because of poor number statistics . ] , but for satellite galaxies . the three large @xmath40 bins are chosen to be the same as the @xmath56-richest , @xmath57-richest , and richest bins in figure [ yangberlind ] . ] overall , there is impressive agreement between the model and group catalogs , even for groups with few members . for central galaxies in poor groups , the b06 catalog has a slightly weaker red peak and a slightly more populated blue bump than in the y07 catalog , with the model between them . for satellite galaxies in poor groups , the red sequence is peaked at a slightly redder color in the group catalogs than in the model , and the blue bump is slightly more populated in the catalogs . these differences are very small , however . a comparison of the figures makes the following two conclusions evident . firstly , for central galaxies , while a significant fraction of them populate the blue cloud in small groups or low - mass halos , the vast majority of them in large groups or massive halos ( @xmath58 ) have moved onto the red sequence . secondly , for satellite galaxies , the color distribution has a significant blue bump even in massive groups and clusters . while there are large numbers of red satellites in such systems , of course , there are also a significant number of blue satellites as well , presumably tending to be located in the cluster outskirts , implied by the color gradients discussed in section [ model ] . moreover , the satellite color distribution is almost independent of group richness or halo mass . this occurs in the model by construction : we have assumed that the color distribution at fixed luminosity is independent of mass , and satellite luminosity only weakly depends on richness . the agreement with the group catalogs constitutes evidence in support of this assumption . to summarize , we have compared the halo model of galaxy colors of skibba & sheth ( 2008 ) to the colors of galaxies in the yang et al . ( 2007 ) and berlind et al . ( 2006a ) group catalogs . the model assumes that satellite galaxies tend to follow a particular sequence along the color - magnitude diagram ( [ csatseq ] ) , such that it approaches the red sequence at bright luminosities , and we have found that this satellite color sequence is in excellent agreement with measurements from the yang et al . group catalog . this constitutes support for the skibba & sheth model , and for the satellite color sequence itself , which could be used as a constraint for galaxy formation models on the physical processes that quench the star formation of satellite galaxies . the satellite color sequence as a function of luminosity can easily be converted into a sequence as a function of halo mass as well ( see section 2.2 of skibba & sheth ) . the agreement between the model and the group catalog suggests that a significant fraction of faint satellites ( with @xmath59 ) , which reside in low - mass halos as well as cluster - sized halos ( skibba et al . 2007 ) , are still forming stars out of a dwindling gas supply and are in the process of being quenched and transformed into ` red and dead ' galaxies . van den bosch et al . ( 2008a ) recently used the yang et al ( 2007 ) group catalog to explore the impact of various transformation mechanisms that are believed to operate on satellite galaxies . based on the colors and concentrations of galaxies at fixed stellar mass , they found that the main mechanism that causes the transition of satellite galaxies from the blue to the red sequence is strangulation , in which the hot diffuse gas around newly accreted satellites is stripped , removing its fuel for future star formation . they ruled out other mechanisms , such as ram - pressure stripping , which is efficient only in massive halos , and harassment , which alters galaxies morphologies . kang & van den bosch ( 2008 ) used these results with their semi - analytic model to show that strangulation is inefficient and takes a few gyr to operate . a similar conclusion was recently obtained by font et al . ( 2008 ) . who used a more sophisticated stripping model . cattaneo et al . ( 2008 ) have recently used another semi - analytic model to show that the observed ` archaeological downsizing ' , in which stars in more massive galaxies tend to form earlier and over a shorter period , can be reproduced if strangulation shuts down star formation only above a critical halo mass @xmath60 . gilbank & balogh ( 2008 ) came to the same conclusion when attempting to reproduce the red sequence dwarf - to - giant ratio in clusters and the field . this critical mass is slightly larger than the minimum halo mass for @xmath12 , and at this mass scale the satellite color sequence of the skibba & sheth ( 2008 ) model is indeed approaching the red sequence . we also analyzed the colors of central and satellite galaxies as a function of group richness , quantified by the number of galaxies in a group more luminous than a given threshold . we showed that at fixed richness or halo mass , central galaxies tend to be redder than satellites , and the color difference increases with richness . in contrast , at fixed luminosity or stellar mass , centrals tend to be bluer than satellites . this is simply explained by the fact that in a given halo , the central galaxy is usually the brightest and most massive galaxy . these results suggest that as central and satellite galaxies evolve , they may follow different paths along the color - magnitude diagram . for example , central galaxies become more luminous as they evolve , whether secularly or with mergers , and may continue to form stars and remain blue even after forming a significant bulge component , and then move towards the red sequence after agn feedback has operated ; on the other hand , satellites may experience star formation quenching due to strangulation and approach the red sequence while they are still faint and disk - dominated . these issues are investigated further in skibba et al . ( 2008a , in prep . ) , using morphology mark correlation functions with the sdss galaxy zoo catalog of visually classified morphologies . for the colors of both central and satellite galaxies as a function of group richness , the model of skibba & sheth ( 2008 ) and the two group catalogs of yang et al . ( 2007 ) and berlind et al . ( 2006a ) are in very good agreement . central galaxies tend to be the reddest galaxy in a halo and are much redder than the typical satellite in large groups . satellite galaxy color , unlike that of centrals , is almost independent of group richness . the distributions of central and satellite colors as a function of richness in the model and group catalogs are also in good agreement . central galaxies have a bimodal color distribution in small groups , but the vast majority of them in larger groups have moved onto the red sequence . in contrast , the satellite color distribution is almost independent of group richness . this occurs in the model by construction : we assumed that the color distribution at fixed luminosity is independent of halo mass , and satellite luminosity only weakly depends on richness . the agreement with the group catalogs supports this assumption . it is worth emphasizing that , while some authors focus on mean galaxy properties ( _ e.g. _ , martnez & muriel 2006 , conroy & wechsler 2008 ) or on red or blue fractions ( _ e.g. _ , weinmann et al . 2006 , van den bosch et al . 2008a ) , our model has predicted the mean and _ distributions _ of central and satellite galaxy colors , as a function of luminosity and richness , in agreement with the sdss data for @xmath32 . this is quite a feat , considering that our model is fairly simple , based on luminosity - dependent clustering and color - magnitude constraints , with very few assumptions . it is interesting that satellite galaxy color appears to be almost independent of host halo mass , and that this is also the case for satellite galaxy luminosity ( skibba et al . 2007 ) . since galaxy color is tightly correlated with stellar mass - to - light ratio ( bell et al . 2003 ) , this implies that satellite galaxies of a given stellar mass can also be found in halos of a wide range of masses . this suggests that what most determines a satellite galaxy s properties , and its evolution in general , is its stellar mass , not its host halo mass . in addition , galaxy color and luminosity are the primary properties that are most predictive of a galaxy s environment ( blanton et al . 2005a ) , so the flat relations of satellite galaxy color and luminosity with halo mass suggest a dearth of environmental dependence for the transformation of satellite galaxies . this point was recently made by van den bosch et al . ( 2008b ) , who used the yang et al . ( 2007 ) group catalog to show that the color and concentration of satellite galaxies are almost completely determined by their stellar mass , with only a very weak dependence on halo mass and halo - centric radius . one consequence of this is that ` pre - processing ' in groups can not be the dominant process that differentiates the cluster galaxy population from that of the field . finally , brown et al . ( 2008 ) recently completed a study of the evolution of the luminosity and stellar mass of red central and satellite galaxies , using halo occupation models . we can do a few simple comparisons between our results and theirs . they find that the stellar masses of luminous red central galaxies scales with halo mass to the power of @xmath61 . using our model s relationship between central galaxy color and halo mass ( lower panel of figure [ yangberlind ] ) with the relation between colors and stellar mass - to - light ratios ( bell et al . 2003 ) , we can estimate the relation between central galaxy stellar mass and halo mass for the model . we find that the slope of this relation approaches @xmath62 , similar to their result . brown et al . also show that approximately 50% of @xmath63 mass halos host central galaxies that are red , and this fraction increases with halo mass . our model predicts a fraction of @xmath64 at this mass , although our separation of ` red ' and ` blue ' galaxies is in terms of @xmath1 color , while theirs is in terms of @xmath65 . finally , brown et al . also conclude that the fraction of stellar mass within the satellite population increases with host halo mass , which is consistent with our results and skibba et al . i would like to thank xi kang , frank van den bosch , and ravi sheth for valuable discussions , and i thank frank van den bosch for providing the galaxy group catalog of yang et al ( 2007 ) . funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s . department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england . the sdss web site is http://www.sdss.org/. the sdss is managed by the astrophysical research consortium for the participating institutions . the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , cambridge university , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max - 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current analytic and semi - analytic dark matter halo models distinguish between the central galaxy in a halo and the satellite galaxies in halo substructures . it is expected that galaxy properties are correlated with host halo mass , and that central galaxies tend to be the most luminous , massive , and reddest galaxies in halos while the satellites around them are fainter and bluer . using a recent halo - model description of the color dependence of galaxy clustering ( skibba & sheth 2008 ) , we investigate the colors of central and satellite galaxies predicted by the model and compare them to those of two galaxy group catalogs constructed from the sloan digital sky survey ( yang et al . 2007 , berlind et al . 2006a ) . in the model , the environmental dependence of galaxy color is determined by that of halo mass , and the predicted color mark correlations were shown to be consistent with sdss measurements . the model assumes that satellites tend to follow a color - magnitude sequence that approaches the red sequence at bright luminosities ; the model s success suggests that bright satellites tend to be ` red and dead ' while the star formation in fainter ones is in the process of being quenched . in both the model and the sdss group catalogs , we find that at fixed luminosity or stellar mass , central galaxies tend to be bluer than satellites . in contrast , at fixed group richness or halo mass , central galaxies tend to be redder than satellites , and galaxy colors become redder with increasing mass . we also compare the central and satellite galaxy color distributions , as a function of luminosity and as a function of richness , in the model and in the two group catalogs . except for faint galaxies and small groups , the model and both group catalogs are in very good agreement . [ firstpage ] methods : analytical - methods : statistical - galaxies : clusters : general - galaxies : formation - galaxies : evolution - galaxies : clustering - galaxies : halos - large scale structure of the universe

You are an expert at summarizing long articles. Proceed to summarize the following text: lets consider that @xmath0 is a compact riemann surface . then we can define a green s function with respect to a metric and a divisor on @xmath0 . suppose that @xmath1 is a hyperbolic 3 - manifold with infinite volume and having compact riemann surfaces @xmath2 as its conformal boundaries at infinity . also choose a geometry on @xmath1 which bears a metric of constant negative curvature . the main new results of this paper express the green s functions on each @xmath3 in terms of the length of some certain geodesics in @xmath1 . manin has done this provided that @xmath1 has one boundary component ( i.e. n=1 ) and is uniformized by a schottky group in @xcite . in this work we will generalize manin s results in general case , @xmath1 has more than one boundary components and is uniformized by a fuchsian , quasi - fuchsian or kleinian group . + in this introduction we state the main definitions , motivation and the plan of doing the work . we start by introducing the definition of a green s function on a compact riemann surface for a divisor and with respect to a normalized volume form on it . let @xmath0 be a compact riemann surface and @xmath4 ( with @xmath5 integer number ) be a divisor on @xmath0 . we show the support of @xmath6 by @xmath7 . also lets consider that @xmath8 is a positive real - analytic 2 - form on @xmath0 . by a green s function on @xmath0 for @xmath6 with respect to @xmath8 we mean a real analytic function @xmath9 satisfying the following conditions : + ( i ) _ laplace equation : _ @xmath10 where @xmath11 , and @xmath12 is the standard @xmath13 - current @xmath14 . + ( ii ) _ singularities : _ let @xmath15 be a complex coordinate in a neighborhood of the point @xmath16 . then @xmath17 is locally real analytic . + a function satisfying these two conditions is uniquely determined up to an additive constant . and the third condition is \(iii ) _ normalization : _ @xmath18 which eliminates the remaining ambiguous constant . @xmath19 is additive on @xmath6 and for @xmath20 , @xmath21 is symmetric , i.e. @xmath22 . + lets consider that @xmath23 is another divisor on @xmath0 that is prime to the divisor @xmath6 . that means , @xmath24 . and put @xmath25 @xmath19 is additive and @xmath21 is symmetric , then @xmath26 is symmetric and biadditive in @xmath27 . + in general , the function @xmath28 depends on the metric @xmath8 , but in the case that both of the divisors are of the degree zero , from the condition ( i ) we see that @xmath29 depends only on @xmath30 . notice that , as a particular case of the general kahler formalism , to choose @xmath8 is the same as to choose a real analytic riemannian metric on @xmath0 compatible with the complex structure . this means that @xmath31 are conformal invariants when both divisors are of degree zero . also in the case that the divisor @xmath6 is principal , i.e. @xmath6 is the divisor of a meromorphic function like @xmath32 , then @xmath33 is a 1 - chain with boundary @xmath34 . the divisors of degree zero on the riemann sphere @xmath35 are principal . then this formula can be directly applied to divisors of degree zero on riemann sphere . + it is well known that the green s function of the degree zero divisors on a riemann surface of arbitrary genus can be expressed exactly via the differential of the third kind @xmath36 with pure imaginary periods and residues @xmath5 at @xmath16 when the divisor is @xmath37(see , @xcite , @xcite ) . then the generalization of the previous formula for arbitrary divisors @xmath27 of degree zero is @xmath38 in general , when the degree of the divisors are not restricted to zero , the basic green s function @xmath39 can be expressed explicitly via theta functions ( as in @xcite ) in the case when @xmath40 is the _ arakelov metric _ constructed with the help of an orthonormal basis of the differentials of the first kind on the riemann surface . the result of manin in @xcite which relates the arakelov green function on a compact riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with schottky uniformization , having the riemann surface as conformal boundary at infinity , was extremely innovative and influential and had a wide range of consequences in the arithmetic context of arakelov geometry as well as and in other contexts , ranging from p - adic geometry , real hyperbolic 3-manifolds , the holography principle and ads / cft correspondence in string theory , and noncommutative geometry . a natural question is to what extent the result of manin can be generalized to cases where , instead of dealing with a single riemann surface , one has several riemann surfaces whose union is the boundary of a hyperbolic 3-manifold , uniformized no longer by a schottky group , but by a fuchsian , quasi - fuchsian , or more general kleinian group . such a generalization is not only interesting because it is a very natural question to pass from kleinian - schottky groups to more general kleinian groups , but also for its potential applications to arakelov geometry , to the case of curves defined over number fields with several archimedean places , while manin s result was formulated for the case of arithmetic curves defined over the rationales . we have focused on the formula in manin s work , that expresses the arakelov green s function on a compact riemann surface in terms of a basis of holomorphic differentials of the first kind and of differentials of the third kind . in the case of schottky uniformization , when the limit set has hausdorff dimension strictly smaller than one , one can construct such differentials in terms of averages over the schottky group . while the same type of formula no longer holds in the fuchsian or quasi - fuchsian case , we use the canonical covering map relating fuchsian and schottky uniformization and the coding of limit sets for the fuchsian and schottky case , to express the green function in the fuchsian or quasi - fuchsian case in terms of the one in the schottky case . the approach we follow for the more general kleinian case is via a decomposition of the uniformizing group as a free product of quasi - fuchsian and schottky groups and applying the results we obtained for these cases individually . the paper consists of three chapters devoted to the cases that the hyperbolic 3-manifold @xmath1 have 1 , 2 and @xmath41 many boundary components at infinity respectively . first chapter pays to the one boundary component case and includes four subchapter devoted to the foundations and the genera 0,1 and @xmath42 respectively . also this chapter contains fundamental definitions and basic computations that are bases for the computations in the next chapters . the results of this chapter are a summary of the reference @xcite . all hyperbolic 3-manifolds with two boundary components with the same genera are uniformized by quasi - fuchsian groups ; the second chapter is devoted to this manifolds . in third chapter we consider the general case i.e. the case that @xmath1 is uniformized by some kleinian groups and have @xmath41 boundary components . finally , as a remark for the chapter three we show that for manifolds with @xmath43 many boundary components at infinity the situation is like the previous . i am grateful to matilde marcolli who suggested this problem , supported me in max - planck and hausdorff institute in bonn , and supervised me in all process . i also many thank saad varsaie for several useful and encouraging comments . in this chapter , first we gather some fundamental definitions and some useful notations that will be used in all of the paper . next we bring the computation of green s function for riemann sphere @xmath35 , the boundary at infinity of the hyperbolic space . all 3-manifolds with one boundary component that is a compact riemann surface with genus @xmath44 , are uniformized by schottky groups . the rest of the chapter is devoted to these manifolds . the results of this chapter are a summary of the reference @xcite . consider the hyperbolic space @xmath45 with the upper half space model and coordinate @xmath46 that comes from @xmath47 equipped by the hyperbolic distance function corresponding to the metric @xmath48 of constant curvature -1 . the geodesics in this model are vertical half - lines @xmath49 and also vertical half - circles orthogonal to the plane at infinity @xmath50 ( i.e. for @xmath51 ) . + if we consider the end points of the geodesics ( including @xmath43 for the ends of the vertical half - lines ) , then we can consider the riemann sphere @xmath52 ( or @xmath53 in the unit boll model ) as the boundary at infinity of @xmath45 . lets show the geodesic joining @xmath54 to @xmath55 in @xmath56 by @xmath57 ; by @xmath58 the point on the geodesic @xmath59 closest to the point @xmath54 ( i.e , the intersection point of @xmath59 and a geodesic passing through @xmath54 and orthogonal to @xmath59 ) ; by @xmath60 the distance from @xmath61 to the geodesic @xmath57 and by @xmath62 the oriented distance between two points lying on an oriented geodesic in @xmath45 ( figure [ fig1.10 ] ) . also lets show by @xmath63 the angle ( at @xmath61 ) between the semi - geodesics joining @xmath61 to @xmath54 and @xmath61 to @xmath55 , and for @xmath64 in @xmath65 by @xmath66 the oriented angle between the semi - geodesics joining @xmath58 to @xmath54 and @xmath67 to @xmath55 . in order to calculate @xmath66 we must first make the parallel translation along @xmath59 identifying the normal spaces to @xmath59 at @xmath58 and at @xmath67 . the orientation of a normal space is defined by projecting it along oriented @xmath59 to its initial end into the tangent space to @xmath65 , which is canonically oriented by the complex structure . in this case we consider that the hyperbolic manifold @xmath1 is the hyperbolic space @xmath45 with riemann - sphere as its boundary at infinity . for this case we have [ p1.5 ] for @xmath68 and @xmath69 in @xmath35 , denote by @xmath70 a meromorphic function on @xmath35 with the divisor @xmath71 . then we have @xmath72 and @xmath73 mobius transformations preserve hyperbolic distance and angel . then both sides of ( [ f1.3 ] ) and ( [ f1.4 ] ) are invariant under these transformations . hence it suffices to consider the case when @xmath74 in @xmath35 . then the geodesic @xmath75 is the @xmath76 semi - axis in @xmath77 coordinate for @xmath45 and the geodesics joining the points @xmath78 and @xmath79 normally to this semi - axis are half circles passing from these points with center in @xmath80 and normal to @xmath50 ( figure [ fig1.11 ] ) . then we have @xmath81 and @xmath82 also from the properties of cross - ratio we have @xmath83 this gives ( [ f1.3 ] ) directly . to see ( [ f1.4 ] ) , we know that angles in @xmath45 can be calculated using the euclidean metric @xmath84 and also the parallel transport along @xmath76 semi - axis coincides with the euclidean one . as it s shown in figure [ fig1.11 ] the vectors @xmath85 and @xmath86 tangent to the geodesics from the points @xmath15 and @xmath87 , are normal to @xmath88 . hence if we transport ( parallel ) the vector @xmath86 in the point @xmath89 to the point @xmath90 then both vectors are in an euclidean plane normal to @xmath76 semi - axis . then the angel from the vector @xmath85 to the vector @xmath86 can be calculated via there angels from @xmath91 axis . then we have @xmath92 this proves ( [ f1.4 ] ) . the part @xmath93 in the proposition above is the classical cross - ratio of four points on the riemann sphere @xmath35 , for which it is convenient to have a special notation @xmath94 let @xmath68 and @xmath69 be in @xmath35 . then we have + @xmath95 + from the formula ( [ f0.11 ] ) for the green s function in the case that the divisors are principal , we have + @xmath96 + then from proposition ( [ p1.5 ] ) and notation ( [ f1.5 ] ) we have ( [ f1.6 ] ) . in this case the group that uniformize the hyperbolic 3 - manifold with its boundary at infinity is a cyclic group , and there is a nice explicit formula for the basic green s function on the boundary at infinity of 3-manifold ; but because we do not use it here we do not want to point to it here . but one can find it in @xcite ( or for a physical point of view in @xcite ) . of course the process in the next part can be used for this case too . we have pointed some notes about this case in section [ s1.5 ] . consider the complex projective linear transformations group @xmath98 . _ ( i ) kleinian groups . _ a subgroup @xmath99 of @xmath98 is called a kleinian group if @xmath99 acts on @xmath45 properly discontinuously . for a kleinian group @xmath99 and a point @xmath100 in @xmath45 lets denote by @xmath101 , the orbit of the point @xmath102 under the action of @xmath99 . since @xmath99 acts on @xmath45 properly discontinuously , @xmath101 has accumulation points only on @xmath35 . they are independent of the choice of the reference point @xmath100 and are called the _ limit set _ of @xmath99 , which is denoted by @xmath103 . equivalently , it is the closure of the set of all fixed points of the elements of @xmath99 other than the identity . @xmath103 is the minimal non - empty closed @xmath99 - invariant set . the complement of the limit set @xmath104 is denoted by @xmath105 and is called the _ region of discontinuity _ of @xmath99 . + the kleinian group @xmath99 is called of the _ first kind _ if @xmath106 and of the _ second kind _ if @xmath107 . in the case that @xmath99 is of the second kind @xmath99 acts on @xmath105 properly discontinuously . therefor in this case @xmath105 is the maximal open @xmath99 - invariant subset of @xmath35 where @xmath99 acts properly discontinuously . the quotient space @xmath108 has the complex structure induced from that of @xmath105 . thus @xmath108 is a countable union of riemann surfaces lying at infinity of the complete hyperbolic 3 - manifold @xmath109 . for a torsion free kleinian group @xmath99 , a manifold @xmath110 possibly with boundary is denoted by @xmath111 and is called a _ kleinian manifold_. the interior @xmath109 of @xmath111 which admits the hyperbolic structure is denoted by @xmath112 . + _ ( ii ) loxodromic , parabolic and elliptic elements . _ an element @xmath113 in @xmath98 is called loxodromic if it has exactly two different fixed points in @xmath35 . these points are denoted by @xmath114 and @xmath115 and are called attracting one and repelling one . for any @xmath116 and @xmath117 , we have @xmath118 . if we denote by @xmath119 the eigenvalue of @xmath113 on the complex tangent space to @xmath114 then there is a local coordinate for @xmath35 that in this coordinate @xmath113 is represented by @xmath120 . for this reason @xmath119 is called the multiplier of @xmath113 , and we have @xmath121 , @xmath122 . also by definition @xmath113 is called a parabolic element if it has precisely one fixed point in @xmath35 , and elliptic if it has fixed points in @xmath45 . in fact an elliptic element fixes all points of a geodesic in @xmath45 joining two fixed points in @xmath35 . for a loxodromic or elliptic element @xmath113 the geodesic joining two fixed points in @xmath35 is called axis of @xmath113 . an element is elliptic if and only if it has finite order , then elliptic elements cause a singularity in @xmath112 . for this reason we consider kleinian groups without elliptic elements . this means that the group is torsion free or equivalently acts freely on @xmath45 . + _ ( iii ) schottky groups_. a finitely generated , free and purely loxodromic kleinian group is called a schottky group . purely loxodromic means , all elements except the identity are loxodromic . for such a group @xmath123 the number of a minimal set of generators is called the genus of @xmath123 . a marking for a schottky group @xmath123 of genus @xmath100 by definition is a family of @xmath124 open connected domains @xmath125 in @xmath35 and a family of generators @xmath126 with the following properties . for @xmath127 + a ) the boundary @xmath128 of @xmath129 is a jordan curve homeomorphic to @xmath130 and closures of @xmath129 are pairwise disjoint . \b ) @xmath131 and @xmath132 . + we say that a marking is classical , if all @xmath128 are circles . it is known that every schottky group admits a marking . in fact each schottky group admit infinitely many marking but there are some schottky groups for which no classical marking exists . + _ ( iv ) @xmath123 - invariant sets of the schottky groups and there quotient spaces_. a schottky group @xmath123 is a kleinian group , then it acts on @xmath45 properly discontinuously . this action is free too . then the quotient space @xmath133 has a complete hyperbolic 3-manifold structure this means that it is a non - compact riemann space of constant curvature -1 . topologically , if @xmath123 is of genus @xmath100 then @xmath133 is the interior of a handlebody of genus @xmath100 . + now lets consider the marking @xmath134 for the schottky group @xmath123 of genus @xmath100 and put @xmath135 the set @xmath136 is the region of discontinuity of @xmath123 and @xmath137 is a fundamental domain for the action of @xmath123 on @xmath35 . then @xmath123 acts on @xmath136 freely and properly discontinuously . so the quotient space @xmath138 is a complex riemann surface of genus @xmath100 . all compact riemann surfaces can be obtained in this way ( see @xcite ) and every compact riemann surface admits infinitely many different schottky covers . + the cayley graph of a schottky group @xmath123 of genus @xmath100 is an infinite tree with multiplicity of @xmath124 at each vertex . @xmath123 is free , then this tree is without loops and each path between two points is unique . now , by the definition of a marking for @xmath123 , each generator @xmath139 takes @xmath140 to the inside of @xmath141 and again the generator @xmath142 takes this part to the inside of the second copy of @xmath143 inside @xmath141 . this means that for each element @xmath113 in @xmath123 we can associate a path in the cayley graph that the points of @xmath140 moves along that path on a tubular neighborhood . this express the action of @xmath123 on the set @xmath136 ( figure [ fig1.12 ] illustrating this for the case @xmath144 ) . we can consider the riemann surface @xmath145 as the boundary at infinity of the manifold @xmath146 by identifying the points of @xmath145 with the points of the set of equivalence classes of unbounded ends of oriented geodesics in @xmath146 modulo the relation `` distance=0 '' . + by the definition of the limit set for a kleinian group , the complement @xmath147 is the limit set of @xmath123 . for the cyclic schottky groups ( of genus 1 ) the limit set @xmath148 consists of two points which can be chosen as @xmath149 , but for genus@xmath150 this set is an infinite cantor set ( see @xcite ) . + lets consider the irreducible left - infinite words like @xmath151 . where @xmath152 , @xmath153 , and @xmath154 is a point in the fundamental domain @xmath137 . and put @xmath155 this is a well defined point of @xmath148 and is independent of the point @xmath154 . since @xmath123 is a free group , the map @xmath156 establishes a bijection as following irreducible left - infinite words in @xmath139 + @xmath157 + ends of the cayley graph of @xmath123 , @xmath158 + @xmath157 + points of @xmath159 + denote by @xmath160 the hausdorff dimension of the set @xmath148 . it can be characterized as the convergence abscissa of any poincare series @xmath161 where @xmath15 is any coordinate function on @xmath35 with a zero and a pole in @xmath136 . for @xmath162 that re@xmath163 , this series converges uniformly on compact subsets of @xmath136 . generally we have @xmath164 ( see @xcite ) . in the next section we will consider @xmath165 to have the convergency of some infinite products defining some @xmath123 - automorphic functions . this class of schottky groups was characterized by bowen @xcite in following way : @xmath165 if and only if @xmath123 admits a rectifiable invariant quasi - circle ( which then contains @xmath148 ) . + choose marking for @xmath123 of genus @xmath100 and denote by @xmath166 the image of @xmath128 in @xmath167 with induced orientation . also for @xmath127 choose the points @xmath168 in @xmath128 and denote by @xmath169 the images in @xmath167 of oriented pathes from @xmath168 to @xmath170 lying in @xmath137 . these images are obviously closed paths and we can choose them in such a way that they do nt intersect . if we denote the classes of these pathes in 1- homology group @xmath171 by the same notations , then the set @xmath172 form a canonical basis of this group i.e. for all @xmath127 we have @xmath173 moreover the kernel of the map @xmath174 which is induced by the inclusion @xmath175 is generated by the classes @xmath166 . lets consider the cyclic schottky group @xmath176 , for @xmath177 , @xmath178 . in this case @xmath179 and consequently @xmath180 . then a differential of the first kind on @xmath180 can be written as @xmath181 where @xmath154 is any point @xmath182 . and for @xmath183 , this differential determines a differential of the first kind on @xmath167 . in general case for a schottky group @xmath123 of genus @xmath100 we can make a differential of the first kind @xmath184 for any @xmath185 on @xmath136 and @xmath167 by an appropriate averaging of this formula . lets consider a marking for @xmath123 and denote by @xmath186 a set of representatives of @xmath187 ; by @xmath188 a similar set for @xmath189 ; and by @xmath190 the conjugacy class of @xmath113 in @xmath123 . then for any @xmath191 we have \(a ) if @xmath192 , the following series converges absolutely for @xmath193 and determines ( the lift to @xmath145 of ) a differential of the first kind on @xmath145 : @xmath194 this differential does not depend on @xmath154 , and depends on @xmath113 additively . also if the class of @xmath113 is primitive ( i.e. non - divisible in h:@xmath195 $ ] ) , @xmath184 can be rewritten as following @xmath196 ( b ) if @xmath139 form a part of the marking of @xmath123 , and @xmath166 are the homology classes described before , we have @xmath197 it follows that the map @xmath113 mod @xmath198\mapsto \omega_g$ ] embeds @xmath199 as a sublattice in the space of all differentials of the first kind . + ( c ) denote by @xmath200 the complementary set of homology classes in @xmath201 as in before . then we have for @xmath202 , with an appropriate choice of logarithm branches : @xmath203 and @xmath204 where @xmath205 is @xmath206 without the identity class . for the proofs , see @xcite and @xcite . notice that our notation here slightly differs from @xcite ; in particular , @xmath207 here corresponds to @xmath208 of @xcite . lets consider the points @xmath54 and @xmath55 in @xmath140 the fundamental domain of @xmath123 and put @xmath209 . then assuming @xmath165 , we see that this series absolutely converges and because it s @xmath123 - automorphic it gives us a differential of the third kind on @xmath145 with residues @xmath210 at the images of @xmath64 in @xmath145 . moreover , since both points @xmath64 are out of the circles @xmath128 , its @xmath166 - periods vanish . now , if we consider the linear combination @xmath211 with real coefficients @xmath212 , then it will have pure imaginary @xmath166 - periods and if we find real coefficients @xmath212 so that the real part of @xmath169 - periods of the form @xmath213 vanish , we will be able to use this differential in order to calculate conformally invariant green s functions . the set of the equations for calculating the coefficients @xmath214 are as following @xmath215 for @xmath127 . the parts @xmath216 are calculated by means of formula ( [ f1.10 ] ) and ( [ f1.11 ] ) , and @xmath169 - periods of @xmath217 are given in @xcite . + now if we denote the points @xmath218 , and @xmath69 in @xmath145 and their images in the fundamental domain @xmath137 by the same notations , then we have @xmath219 and finally we have the green s function on @xmath167 as following @xmath220 all hyperbolic riemann surfaces are uniformized by fuchsian groups . also fuchsian groups act on hyperbolic space by poincare extension and uniformize some hyperbolic 3-manifolds with two boundaries at infinity with the same genera . but they do not uniformize all such manifolds ; in fact they are uniformized by an extension of fuchsian groups that are called quasi - fuchsian groups . in this chapter first we intend to extend the method in previous chapter to fuchsian groups and then for quasi - fuchsian groups . + consider hyperbolic plane @xmath221 with the upper half plane model and coordinate @xmath222 with @xmath223 , equipped by the hyperbolic distance function corresponding to the metric @xmath224 of the constant curvature -1 . the geodesics in this model are vertical half - lines @xmath225 and vertical half - circles orthogonal to the line at infinity @xmath226 ( i.e , for @xmath51 ) . + if we consider the end points of geodesics ( including @xmath43 for the end of vertical half - lines other than the end in @xmath226 ) then we can consider circle @xmath227 ( or @xmath228 in unit disc model ) as boundary at infinity of @xmath221 . each subgroup @xmath229 of @xmath230 , the general projective linear group over @xmath226 , that acts freely and properly discontinuously on @xmath221 is called a fuchsian group . similar to the kleinian groups the limit set @xmath231 and the region of discontinuity @xmath232 are defined but in @xmath233 , the boundary at infinity of @xmath221 . and the properties are similar to them . also similar to the loxodromic elements , a hyperbolic element is an element @xmath234 with two fixed points @xmath235 in @xmath233 . for a fuchsian group @xmath229 the quotient space @xmath236 is a hyperbolic riemann surface possibly with boundary @xmath237 ( if @xmath238 ) . + let @xmath229 be a fuchsian group such that @xmath239 is a compact riemann surface with the genus @xmath240 . then @xmath229 is a purely hyperbolic group of finite order @xmath124 ( for example see @xcite ) . lets denote by @xmath241 the generators of @xmath229 and by @xmath242 the fixed point set of @xmath241 . also consider that @xmath102 is the fundamental polygon of @xmath229 with @xmath243 as it s sides , such that @xmath244 and @xmath245 for @xmath246 . we represent @xmath102 as following : @xmath247 where @xmath248 and @xmath249 are the intersection points of @xmath166 and @xmath169 , @xmath250 and @xmath251 and so on . now we have @xmath252=i \rangle\ ] ] and also ( see for example @xcite ) . @xmath253 if we denote by @xmath254 and @xmath255 the images of @xmath166 and @xmath169 in @xmath0 , then @xmath256 generate @xmath257 . @xmath229 acts similarly , freely and properly discontinuously on lower half plane @xmath258 of @xmath50 too , and @xmath258 is @xmath229-invariant . lets denote by @xmath229 the poincare extension of @xmath229 on @xmath45 too ( see @xcite or @xcite ) . by this extension @xmath229 can be considered as a kleinian group . and we have @xmath259 ( see @xcite ) . and @xmath260 has a hyperbolic structure , @xcite . we know that @xmath231 the limit point set of @xmath229 is @xmath261 ( @xmath130 in unit disc model ) , @xcite . then as a kleinian group @xmath262 and the region of discontinuity of @xmath229 considering as a kleinian group is @xmath263 and also the kleinian manifold is @xmath264.\end{aligned}\ ] ] now because @xmath0 is compact then @xmath265 . consequently @xmath260 is a hyperbolic 3 - manifold with two compact boundary component at infinity @xmath266 and @xmath267 with the same genus @xmath100 . for convenience and having a simple intuition lets consider the unit disc model for @xmath221 in this section . we can code points of @xmath268 and geodesics with beginning and end points on @xmath268 as following : + lets mark semicircles including sides of p by @xmath269 in the counter clockwise direction around @xmath130 and put @xmath270 , and so on . in general for @xmath271 , @xmath272 . and label end points of @xmath273 on @xmath130 by @xmath274 and @xmath275 ( with @xmath276 ) with @xmath274 occurring before @xmath275 in the counter clockwise direction , ( see figure [ fig2.13 ] ) . and define @xmath277 @xmath278 then @xmath279 is a well defined map and is called a markov map related to the fuchsian group @xmath229 . we have the following lemma the map @xmath279 and the group @xmath229 are orbit equivalent on @xmath130 , namely except for the pairs @xmath280 , for @xmath281 ; for each x , y in @xmath130 , @xmath282 for some @xmath283 in @xmath229 if and only if there exists nonnegative integers @xmath284 such that @xmath285 . ( see @xcite ) . now label each arc @xmath286 by @xmath139 , and for each element @xmath16 in @xmath130 put @xmath287 . where the component @xmath288 is the label of the segment to which @xmath289 belongs . we know that for each generator @xmath241 of @xmath229 , the fixed point @xmath290 is in the circle @xmath291 and @xmath292 is in @xmath293 then from the definition of markov map we have @xmath294 next , if @xmath295 be a geodesic with the beginning point @xmath296 and end point @xmath297 then put @xmath298 now for each @xmath299 let @xmath300 show the geodesic arc between @xmath301 and @xmath302 ( the axis of @xmath303 ) and @xmath304 the image of @xmath305 in @xmath260 . then we have the following geodesics in @xmath260 * closed geodesics : a geodesic in @xmath260 is closed if and only if it s the projection of the axis of a hyperbolic element in @xmath229 , * images of the geodesics with beginning and end points in @xmath231 , * geodesics with beginning points in @xmath306 that are limit cycle to @xmath307 , for a generator @xmath241 of @xmath229 , * geodesics with beginning and end points in @xmath308 or @xmath309 . , * geodesics with beginning point in @xmath308 and end point in @xmath309 and vis versa . for the riemann surface @xmath0 and associated fuchsian group @xmath229 as above let @xmath310 be the smallest normal subgroup of @xmath229 including @xmath241 for @xmath311 . then it s obvious that the factor group @xmath312 is a free group generated by @xmath100 generators . if we consider the covering @xmath313 associated to @xmath310 , then from normality of @xmath310 the group of deck transformations of this cover is @xmath312 and according to the classical koebe uniformization theorem @xcite ( see also @xcite)there is a planar region @xmath314 that is a region of discontinuity of a schottky group @xmath315 and @xmath316 . @xmath317 is covering isomorphic to @xmath318 and for coverings @xmath319 , @xmath320 and @xmath321 we have @xmath322 i.e. the following diagram is commutative @xmath323 + @xmath324 + @xmath0 also we have @xmath325 for @xmath127 and @xmath326 for each @xmath283 in @xmath310 and each mappings is complex - analytic covering and @xmath123 is uniquely determined to within conjugation in @xmath327 @xcite . + for an extension of @xmath328 to a set some more than @xmath221 that we need it in the later computations , lets denote by @xmath329 the free subgroup of @xmath229 generated by @xmath330 . we know that @xmath331 is a subset of @xmath231 . @xmath331 is @xmath329 invariant and is out of the closer of the fundamental domain of @xmath329 , then all points of @xmath331 are in only the circles @xmath128 that are related to the generators of @xmath329 and their inverses . this shows that the coding of each point in @xmath331 includes only the generators of @xmath329 and their inverses . also since each generator of @xmath229 like @xmath283 is an isometry of @xmath221 , it s a composition of two reflections , then @xmath332 is not in the isometry circle of @xmath333 for each point @xmath16 in @xmath130 . then by the definition of the markov map @xmath279 all the codings of the points of @xmath130 are irreducible . this means that each point @xmath16 in @xmath331 can be coded by the irreducible formal combination @xmath334 for @xmath335 . this fuchsian coding is suitable for expressing geodesics in @xmath260 or @xmath336 when the beginning and end points of the geodesics are in @xmath331 and also can be used for extending the map @xmath328 on @xmath331 and maybe on some more . + now we want to extend @xmath328 to @xmath337 ( onto @xmath338 ) . for this , we know that each element of @xmath148 can be coded by the infinite words @xmath339 for a constant @xmath154 in @xmath136 and @xmath340 . similarly , since @xmath329 is free and purely loxodromic group then it s a schottky group too and we can use schottky coding for it too . for conveniences in some proofs we will use schottky codings for the points of @xmath331 . then each point @xmath16 in @xmath331 can be coded by the infinite words @xmath341 for @xmath335 and a constant point @xmath342 in @xmath343 ( also @xmath344 ) . then we can consider @xmath342 in @xmath345 such that @xmath346 . also from ( [ f2.10 ] ) and using the properties of a loxodromic element and invariance of the limit set under @xmath229 ( or @xmath329 ) it is not hard to show that on @xmath331 we have the following relation between fuchsian and schottky coding @xmath347 and also this is true for repelling points too because of the relation @xmath348 for each @xmath283 in @xmath229 . now , lets put @xmath349 in other words , by the definition of infinite words , the continuity of @xmath328 and the relation @xmath350 on @xmath221 @xmath351 then the map @xmath352 is well defined and onto . also from ( [ f2.9 ] ) we see that we can explain @xmath328 and consequently the functions ( spatially green s function ) on @xmath308 by the fuchsian coding , when these functions are expressed via this map . + for each point @xmath16 in @xmath331 and element @xmath283 in @xmath329 equivalent to @xmath59 ( i.e. @xmath353 , where @xmath335 and @xmath241 is replaced by @xmath354 and vice versa ) we have @xmath355 then @xmath356 and @xmath357 . let @xmath329 be the free group generated by @xmath330 and @xmath358 be a divisor with support @xmath359 in @xmath221 . put @xmath360 and again lets denote by @xmath361 a meromorphic function on @xmath35 with the divisor @xmath6 , and for an element @xmath342 in @xmath362 define : @xmath363 for the schottky group @xmath123 associated to the fuchsian group @xmath229 , if @xmath165 , then the product ( [ f2.91 ] ) converges absolutely and uniformly on any compact subset of @xmath221 , after deleting a finite number of factors that may have a pole or zero on this subset . let @xmath364 be a compact subset of @xmath221 . since @xmath229 acts on @xmath221 properly discontinuously the set @xmath365 is finite . if we delete the set @xmath366 from the index set of the product ( [ f2.91 ] ) then for each @xmath367 when @xmath332 and @xmath368 lie outside a fixed compact neighborhood of @xmath359 we have @xmath369 where @xmath370 is a constant , @xmath371 , @xmath346 and @xmath372 ( and @xmath373 ) . the last inequality comes from the equality @xmath374 . now since @xmath165 the series @xmath375 converges uniformly on the compact subsets of @xmath136 . then the product ( [ f2.91 ] ) is convergent . in @xmath376 , if we change the point @xmath342 to @xmath377 , then we have @xmath378 . where @xmath379 is a nonzero complex number that depends on the points @xmath342 and @xmath377 and @xmath380 also , for @xmath381 and @xmath382 we have @xmath383 @xmath384 is a nonzero complex number multiplicative on @xmath385 and @xmath283 and also independent of @xmath342 . we can see this as following @xmath386 then @xmath387 . when @xmath123 is a cyclic group @xmath388 . this shows that @xmath389 is not @xmath229 automorphic function on @xmath221 in general . \a ) lets denote by @xmath390 a set of the representatives of @xmath391 . then for @xmath392 $ ] in @xmath329 we have @xmath393 and by defining @xmath394 we have @xmath395 where @xmath396 is the set @xmath397 without the identity class . + b ) lets denote by @xmath398 a set of representatives of @xmath399 and by @xmath400 the conjugacy class of @xmath283 in @xmath329 . then for some @xmath377 in @xmath221 such that @xmath401 stays in @xmath402 , we have @xmath403 and this is independent of @xmath377 . lets put @xmath404 , @xmath405 and @xmath406 then we have @xmath407 the part ( [ f2.8 ] ) comes from the equation @xmath408 and the invariance of the cross - ratio on the action of mobius transformations . the second part of a ) and b ) can be proved similarly . for b ) we should consider that @xmath409 . in above theorem in the case that the divisor is @xmath71 we have @xmath410 by previous theorem @xmath411 is a meromorphic function without any poles and zeroes in @xmath221 then it is a holomorphic function on @xmath221 . also as we see in the expression , it is independent of the point @xmath377 . for each @xmath283 in @xmath329 lets put @xmath412 however @xmath389 is not @xmath229- automorphic function on @xmath221 in general but we have @xmath413 and this shows that the differential @xmath414 is @xmath229 automorphic . then @xmath415 is a differential of the first kind on @xmath308 . + according to the classical theorem of cuts we can choose @xmath136 the region of discontinuity of the schottky group @xmath123 with the marking @xmath416 with @xmath417 such that @xmath418 the image of @xmath166 for @xmath127 in @xmath308 be coincident with the image of @xmath141 . and these together with @xmath419 the image of @xmath169 for @xmath127 in @xmath308 make a canonical base for @xmath420 i.e. @xmath421 \a ) @xmath422 is a riemann s basis for the space of differentials of the first kind on @xmath308 by choosing the previous base for @xmath423 i.e. @xmath424 b ) if we denote by @xmath390 a set of the representatives of @xmath391 and by @xmath425 , the set @xmath390 without the identity class , then for @xmath426 we have @xmath427 and for @xmath428 by defining @xmath429 we have @xmath430 we have @xmath431 where @xmath432 , @xmath401 and @xmath433 . if we show this equality for @xmath434 by @xmath435 then for a ) we have @xmath436 the last equality comes from this fact that if @xmath428 since @xmath437 and @xmath438 , only for @xmath439 the first alternative and for all @xmath440 the third one is valid . if @xmath426 only the third alternative is right for all @xmath441 . we can see these from the figure 1 . + for the first part of b ) . we have shown by the point @xmath442 the intersection of @xmath443 and @xmath444 in the representation of @xmath102 , the fundamental domain of @xmath229 . then @xmath445 is the image of the part of @xmath169 that is between the points @xmath442 and @xmath446 in the fundamental domain . now , from ( [ f2.11 ] ) we have @xmath447 where the last equality comes from the definition in chapter one . similarly we can reach to the second part of b ) using ( [ f2.12 ] ) . for @xmath448 lets denote by the same words the corresponding points in @xmath449 the fundamental domain of @xmath229 ( then @xmath450 and @xmath451 are in @xmath452 the fundamental domain of @xmath123 ) and put @xmath453 . then @xmath217 is a differential of third kind on @xmath308 with the residues 1 and -1 at the images of @xmath54 and @xmath55 . also , because the points @xmath54 and @xmath55 are in the fundamental domain then they are out of the circles @xmath128 . then @xmath418 - periods of @xmath217 are zero , and from ( [ f2.13 ] ) the @xmath454 - periods are @xmath455 then we have @xmath456 now , we can reach to a differential of the third kind with pure imaginary periods by defining @xmath457 where the real coefficients @xmath458 are such that the set of the equations @xmath459 for @xmath460 are satisfied . in fact the coefficients @xmath458 kill the real part of the @xmath419 - periods of @xmath217 . notice that the new differential form @xmath461 probably have nonzero @xmath418 - periods , but since the coefficients @xmath458 are real and @xmath462 are pure imaginary then they are pure imaginary too . + finally , if we denote the points @xmath218 , and @xmath69 in @xmath463 and the images of these points in the fundamental domain of @xmath229 by the same notations , then we have @xmath464 and by ( [ t2.11 ] ) @xmath465 then the green s function on @xmath308 can be computed as following @xmath466 because of the commutativity of the diagram @xmath467 @xmath468 @xmath469 the image of the point @xmath382 and @xmath470 are the same in @xmath308 . then the above formula for the green s function on @xmath308 gives an expression via the points on @xmath308 . in fact for the points @xmath64 in @xmath232 the images of the geodesics @xmath471 and @xmath472 in the kleinian manifold @xmath473 are the same . we can see this by using the following covering spaces @xmath474 for all parts @xmath475 , and again we have @xmath476 . also since @xmath328 is extended on @xmath331 in a natural way then for each @xmath477 the image of the geodesics @xmath478 and @xmath479 in @xmath260 are the same . + one can reach to a formula for the green s function on @xmath309 similarly by replacing @xmath258 and @xmath480 instead of @xmath221 and @xmath102 . a finitely generated , torsion free kleinian group @xmath481 is called quasi - fuchsian if the limit set @xmath482 be a jordan curve and each of the two simply connected components of @xmath483 be @xmath481-invariant . given a quasi - fuchsian group @xmath481 , there exist a fuchsian group @xmath229 and a quasiconformal diffeomorphism between kleinian manifolds @xmath484 and @xmath473 . ( see @xcite ) . lets denote by @xmath485 and @xmath486 the simply connected components of @xmath483 for quasi - fuchsian group @xmath481 . then there are two fuchsian groups @xmath487 and @xmath488 related to @xmath481 such that @xmath489 is homeomorphic to @xmath490 . then each @xmath491 is isomorphic to @xmath481 . also since @xmath492 is topologically conjugate to @xmath493 ( or @xmath494 in unit disk model ) then naturally there is a markov map @xmath495 like fuchsian case . then all the statements for fuchsian groups can be extended to the quasi - fuchsian groups . in this case because the jordan curve @xmath482 is generally not smooth nor even rectifiable ( @xcite p.263 ) then the hausdorff dimension of the subgroup @xmath496 of @xmath481 ( like @xmath329 for fuchsian case @xmath229 ) may be not less than 1 in general . we have the following theorem for a finitely generated kleinian group @xmath99 the following conditions are equivalent + ( 1 ) @xmath497 is diffeomorphic to @xmath498 $ ] where @xmath0 is a component of @xmath499 ; + ( 2 ) @xmath99 is quasi - fuchsian ; + ( 3 ) @xmath105 has an invariant component that is a jordan domain ; + ( 4 ) @xmath105 has two invariant components ; ( see @xcite , page 125 ) . also two homeomorphic riemann surfaces can be made uniform simultaneously by a single quasi - fuchsian group ( theorem of bers on simultaneous uniformization ) . then we have the computations for all hyperbolic 3 - manifolds with two compact riemann surfaces with the same genus as it s boundary components . already , we have computed the green s function for the boundary components of a hyperbolic 3-manifolds with two boundary at infinity with the same genera . in this chapter we want to do the problem for the most general case that the green s function can be defined i.e. for manifolds with @xmath41 boundary components that are compact riemann surfaces probably with different genera . the case with two boundary is different genera are included in this case too . such manifolds are uniformized by some kleinian groups and for the first , we will try to find a decomposition for such kleinian groups . this , help us to find invariants of these groups that are necessary for computing the automorphic functions that will be used for computing the differentials and the green s functions . in all of this section we consider @xmath99 to be a finitely generated and torsion free kleinian group of the second kind ( i.e. @xmath501 ) . * ( the ahlfors finiteness theorem ) * let @xmath99 be a finitely generated and torsion free kleinian group . then @xmath502 is a finite union of analytically finite riemann surfaces ( closed riemann surface from which a finite number of points are removed ) . for the rest of the section we consider @xmath503 such that for each @xmath504 , @xmath3 is a compact riemann surface of genus @xmath505 and without cusped point . then we should consider that @xmath99 does nt have any parabolic elements . also consider that the component @xmath506 is the one that for @xmath507 , @xmath508 . in this case @xmath99 is a function group ( a finitely generated non - elementary kleinian group which has an invariant component in its region of discontinuity ) and there is an invariant component @xmath509 in @xmath105 . we consider two cases for @xmath509 in the proposition above : + * case1 : * the component @xmath509 is simply connected . then in this case @xmath99 is a b - group ( a function group with simply connected invariant component ) . and we know that a b - group with a compact boundary component say @xmath511 is a quasi - fuchsian group ( see @xcite page 411 ) . + * case2 : * the component @xmath509 is not simply connected . in this case we have the following theorem . let @xmath99 be a function group with invariant component @xmath509 that is not a simply connected component . then @xmath99 has a decomposition into a free product of subgroups as following @xmath512 where each @xmath513 is a b - group , @xmath514 is a free abelian group of rank two with two parabolic generators , @xmath515 is the cyclic group generated by the parabolic transformation @xmath505 and @xmath516 is the cyclic group generated by the loxodromic transformation @xmath241 . furthermore , each parabolic transformation in @xmath99 is conjugate in @xmath99 to one a listed subgroup . @xmath99 has finite - sided fundamental polyhedron if and only if all the groups @xmath513 do . this theorem shows that in our special case we have @xmath517 for b - groups if one boundary component be compact then it s a quasi - fuchsian group . and @xmath518 is a free and purely loxodromic group and consequently it s a schottky group @xcite . hence we have @xmath519 where @xmath520 is a quasi - fuchsian group and @xmath123 is a schottky group . then by van kampen theorem we have @xmath521 and if @xmath522 then @xmath523 we will denote by @xmath524 the genus of schottky group @xmath123 in the decomposition of the group @xmath99 . we intend to analyze the structure of the region of discontinuity of @xmath99 . for the first , we consider the case that we do nt have the schottky group @xmath123 in the decomposition of @xmath99 . lets denote by @xmath129 and @xmath525 the simply connected components of the quasi - fuchsian group @xmath520 . from proposition ( [ p3.11 ] ) all other components than @xmath509 of @xmath105 are simply connected . now , since @xmath531 and from lemmas ( [ l3.11 ] ) , ( [ l3.12 ] ) and the decomposition of @xmath499 we see that @xmath532 is a complete system of representatives of the equivalent classes ( @xmath533 and @xmath534 are equivalent or conjugate if and only if @xmath535 is conjugate in @xmath99 to @xmath536 ; and since @xmath537 then @xmath533 and @xmath534 are equivalent or conjugate if and only if there is a @xmath113 in @xmath99 such that @xmath538 ) of the components of @xmath105 . since @xmath509 is @xmath99 - invariant the class of @xmath509 has one member . then we have in this case we have @xmath549 + now , lets consider the general case that there is a schottky group @xmath550 ( of order @xmath524 ) in the decomposition of the group @xmath99 . again we know that all other components of @xmath105 than @xmath509 is simply connected . then @xmath551 can not be glued to the components other than @xmath506 in @xmath499 . also this means that in this case we have @xmath552 and @xmath553 in this case if @xmath554 be a marking for the schottky group @xmath555 , then the fundamental domain is @xmath556 and also for the genus of the component @xmath506 we have @xmath557 . lets denote by @xmath310 the smallest normal subgroup of the group @xmath560 including the generators @xmath561 for @xmath507 and @xmath562 and @xmath563 be the smallest normal subgroup of the group @xmath520 including the generators @xmath561 for @xmath562 . then we have @xmath564 now if @xmath310 be the smallest normal subgroup of the group @xmath99 including the generators @xmath561 for @xmath507 and @xmath562 , then by the previous lemma one can see that @xmath568 is isomorphic to the group @xmath569 then @xmath568 is a free group of order @xmath570 . + now lets consider the covering space map @xmath571 . since @xmath310 is a normal subgroup of @xmath99 then the covering space @xmath572 is regular and is between @xmath573 with the covering transformations group @xmath574 that is a free group of order @xmath570 . then similar to the previous section we have a schottky group @xmath575 ( we can arrange the generators of the group @xmath123 in this way because of the isometry ) and a complex - analytic covering mapping @xmath576 such that the following diagram is commutative and for @xmath507 and @xmath579 , @xmath580 and for @xmath581 , @xmath582 and @xmath583 ( we have considered the same notations for the generators of @xmath584 and @xmath123 ) . as a matter of fact there is a fuchsian group @xmath229 and there are normal subgroups @xmath585 and @xmath586 of @xmath229 such that the sequence @xmath587 is the composition of analytic covering maps . then we have the composition of covering maps @xmath588 and since @xmath589 is a free group , like the previous section we have @xmath590 for some schottky group @xmath123 . in fact , for the first , because of the isometries @xmath591 we can choose the fuchsian group @xmath592=i\quad \rangle\end{aligned}\ ] ] such that the equations @xmath593 for @xmath507 and @xmath594 are satisfied . where @xmath595 are the generators of @xmath99 . then like the previous section we can choose uniquely up to conjugation in @xmath596 the schottky group @xmath123 that satisfies the equations @xmath597 then since @xmath598 is onto , the relations @xmath599 and @xmath600 are satisfied automatically for @xmath507 and @xmath594 . this means that the following diagram is commutative in all loops @xmath601 @xmath602 @xmath601 this gives a proof for the existing the of covering space @xmath136 and schottky group @xmath123 satisfying the properties that we need and also shows the relations between the fuchsian group @xmath229 , kleinian group @xmath99 and the schottky group @xmath123 associated to @xmath506 . + now lets put @xmath603 . where @xmath604 is the free subgroup of @xmath520 generated by the elements @xmath605 . for the extension of @xmath328 to @xmath606 i.e. defining the map @xmath607 , we know that each @xmath604 is free and purely loxodromic then it is a schottky group . like the previous section we can code an element of @xmath608 by schottky coding @xmath609 for @xmath610 and a point @xmath342 in @xmath611 . for each element @xmath113 in @xmath612 lets denote by @xmath59 the element in @xmath123 corresponding to @xmath113 , such that @xmath613 . now put @xmath614 @xmath615 where @xmath346 . and for @xmath507 put @xmath616 @xmath617 and finally lets define @xmath618 @xmath619 + then like the previous section we can show that for each element @xmath113 in @xmath612 and @xmath59 the element of @xmath123 corresponding to @xmath113 and each @xmath16 in @xmath606 we have @xmath620 and @xmath621 . also like the previous chapter we can consider the fuchsian coding for the points of @xmath606 and from ( [ f2.9 ] ) we can explain @xmath328 and consequently the green s function on @xmath506 via the fuchsian coding . if we consider the condition @xmath165 and use the subgroup @xmath622 instead of @xmath329 in the previous section then one can bring all of the definitions like before with some changes , and compute the green s function on @xmath506 and the other parts in the same way . in this case we should notice that if @xmath623 be a set that makes a base for @xmath624 then each member of it has a class of images in @xmath509 . but we can consider that the representatives that are used in the computations are in the fundamental domain . as its shown in figure [ fig3.14 ] . in this case these images are in @xmath625 for @xmath507 and also in @xmath626 for @xmath627 . then the representatives for @xmath3 and @xmath506 are coincide if we uniformize @xmath3 by @xmath628 , i.e. identifying both components of @xmath629 . final formula for the green s function on @xmath506 is as following @xmath630 when we have infinitely many boundary components in the boundary of @xmath112 that are riemann surfaces without puncture , according to the ahlfors finiteness theorem , @xmath99 is an infinitely generated kleinian group . and hence one of the boundary components of @xmath111 is with infinity genus and is not a compact riemann surface then for this component the green s function is not defined . but for the other components we can bring the computations similar to the previous condition using some subgroups of @xmath99 that are isomorphic to the kleinian groups in the previous sections . 30 l. v. ahlfors . `` finitely generated kleinian groups '' . amer.j . math . 86 ( 1964),413 - 429 and 87(1965),759 . `` uniformization , moduli and kleinian groups '' . bull . london math . , 4(1972),257.300 . + r. bowen . `` hausdorff dimension of quasi - circles '' publ . math . ihes 50,11 - 25(1979 ) . c. series . `` markov maps associated with fuchsian groups '' . publications mathmatiques de li.h.e.s.,tome 50(1979),p.153 - 170 . g. cornel , j.h . `` arithmetic geometry '' . springer verlag ( 1986 ) . g. faltings . `` calculating on arithmetic surfaces '' . 119 , 387 - 424 ( 1982 ) . l. r. ford . `` automorphic functions '' . mcgraw - hill 1929 . d. hejhal . `` on schottky and teichmuller spaces '' . 15 , 133 - 156 ( 1975 ) . `` compact riemann surfaces '' . `` uber die uniformisierung der algebraischen kurven '' . iv , math . ann . 75 ( 1914 ) `` introduction to arakelov theory '' . new york berlin heidelberg : springer 1988 . `` three - dimensional hyperbolic geometry as @xmath43 - adic arakelov geometry '' . invent . 104 ( 1991 ) , no . 2 , 223243 . yu . i. manin , v. drinfeld , `` periods of p - adic schottky groups '' . j. reine angew . math . 262/263 ( 1973 ) , 239247 . i. manin , matilde marcolli . `` holographi principle and arithmetic curves '' . k. matsuzaki , m. taniguchi . `` hyperbolic manifolds and kleinian groups '' . oxford university press , 1998 . c. mcmullen . `` riemann surfaces and the geometrization of 3 - manifolds '' . bull . 27 ( 1992 ) , no . 2 , 207216 . a. marden . `` the geometry of finitely generated kleinian groups '' . the annals of mathematics , 2nd ser . 99 , no . 3 ( may , 1974 ) , pp . 383 - 462 . j. g. ratcliffe . `` foundations of hyperbolic manifolds '' springer verlag ( 1991 ) . p. zograf , l. takhtajan . `` on the uniformization of riemann surfaces and on the weil - petersson metric on the teichmuller and schottky spaces '' . math . ussr - sb . 60 ( 1988 ) , no.2 , 297 - 313 .

the work is motivated by a result of manin in @xcite , which relates the arakelov green function on a compact riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with schottky uniformization , having the riemann surface as conformal boundary at infinity . a natural question is to what extent the result of manin can be generalized to cases where , instead of dealing with a single riemann surface , one has several riemann surfaces whose union is the boundary of a hyperbolic 3-manifold , uniformized no longer by a schottky group , but by a fuchsian , quasi - fuchsian , or more general kleinian group . we have considered this question in this work and obtained several partial results that contribute towards constructing an analog of manin s result in this more general context .

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Proceed to summarize the following text: dwarf galaxies have come to play an increasingly important role in understanding how galaxies form and evolve . as the smallest , least luminous , and most common systems in the universe , dwarf galaxies span a wide range in physical characteristics and occupy a diverse set of environments ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , making them excellent laboratories for direct studies of cause and effect in galaxy evolution . the low average masses and metallicities of dwarf galaxies suggest they may be the best available analogs to the seeds of hierarchical galaxy formation in the early universe . a cohesive picture of the evolution of dwarf galaxies remains elusive . historically , evolutionary scenarios have often been considered in the context of a dual morphological classification , namely dwarf spheroidals ( dsphs ; we include dwarf ellipticals in this general category ) and dwarf irregulars ( dis ) . the former are classified based on a smooth morphology , with no observed knots of sf , and are generally found to be gas - poor . the latter exhibit morphological evidence for current / recent sf activity and have a high gas fraction . a third , rarer class of so - called ` transition ' dwarf galaxies , have a high gas fraction yet little to no recent sf activity ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? these galaxies may be an evolutionary link between dis and dsphs ( e.g. , * ? ? ? * ) or simply could be ordinary dis witnessed between massive star forming events ( e.g. , * ? ? ? understanding the relationship between these three types of dwarf galaxies , specifically determining if and how dis evolve into dsphs , is among the most pressing questions in dwarf galaxy evolution ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . resolved stellar populations have proven to be an incredibly powerful tool for observationally constraining scenarios of dwarf galaxy evolution . past patterns of sf and chemical evolution are encoded in a galaxy s optical color - magnitude diagram ( cmd ) . to extract this information , a number of sophisticated algorithms ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) have been developed to measure the star formation history ( sfh ) , i.e. , the star formation rate ( sfr ) as a function of time and metallicity , by comparing observed cmds with those generated from models of stellar evolution . the robustness of this method has become increasingly solidified with a variety of techniques and stellar models converging on consistent solutions for a range of galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? analysis of cmd - based sfhs have become particularly prevalent in studies of the formation and evolution of the local group ( lg ) . the lg contains @xmath2 80 dwarf galaxies ranging from highly isolated to strongly interacting ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? results from cmd - based sfhs of dwarf galaxies in the lg have revealed that both dsphs and dis feature complex sfhs , typically with dominant stellar components older than 10 gyr ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the sfhs of individual lg dwarf galaxies , as well as aggregate compilations , now serve as the basis for our understanding of the evolution of dwarf galaxies and have significantly advanced our knowledge of galactic group dynamics ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . beyond the lg , only a small subset of dwarf galaxies have explicitly measured cmd - based sfhs ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ground - based observations of resolved stellar populations in more distant galaxies are challenging , due to the faintness of individual stars and the effects of photometric crowding . however , observations from the hubble space telescope ( hst ) , particularly the advanced camera for surveys ( acs ; * ? ? ? * ) , have revolutionized the field of resolved stellar populations , producing stunning cmds of dwarf galaxies both in and beyond the lg . outside the lg , past hst observations of resolved stellar populations have been fairly piecemeal , with individual or small sets of galaxies as the typical targets . subsequent analysis often employed different methodologies or sought different science goals . this lack of uniformity also makes it challenging to place the results from lg studies in the context of the broader universe , as lg dwarf galaxies may not be representative of the larger dwarf galaxy population ( e.g. , * ? ? ? the acs nearby galaxy survey treasury ( angst ; * ? ? ? * ) was designed to help remedy this situation . using a combination of new and archival imaging taken with the hst / acs and wide field planetary camera 2 ( wfpc2 ; * ? ? ? * ) , angst provides a uniformly reduced , multi - color photometric database of the resolved stellar populations of a volume - limited sample of nearby galaxies ( d @xmath1 4 mpc ) that are strictly outside the lg . of the @xmath2 70 galaxies in this sample , 60 are dwarf galaxies that span a range of @xmath2 10 in m@xmath5 and that reside in both isolated field and strongly interacting group settings , providing an unbiased statistical sample in which to study the detailed properties dwarf galaxy formation and evolution . in this paper , we present the uniformly analyzed sfhs of 60 dwarf galaxies based on observations , photometry , and artificial star tests produced by the angst program . the focus of this study is on the lifetime sfhs of the sample galaxies , with the recent ( @xmath6 1 gyr ) sfhs the subject of a separate paper @xcite . we first briefly review the sample selection , observations , and photometry in [ data ] . we then summarize the technique of measuring sfhs in [ sfhs ] . in [ results ] , we discuss and compare the resultant sfhs in the context of both dwarf galaxy formation and evolution . we then explore our results with respect to the morphology density relationship in [ morph ] . finally , we discuss the evolution of dwarf galaxies in the context of cosmology in [ cosmic ] . cosmological parameters used in this paper assume a standard wmap-7 cosmology as detailed in @xcite . in this section , we briefly summarize the selection of the angst dwarf galaxy sample along with the observations and photometry . a more detailed discussion of the angst program can be found in @xcite . we constructed the initial list of angst galaxies based on the catalog of neighboring galaxies @xcite , consisting exclusively of galaxies located beyond the zero velocity surface of the lg @xcite . we selected potential targets with @xmath7 @xmath8 @xmath9 to avoid observational difficulties associated with low galactic latitudes . by simulating cmds and crowding limits , we found that a maximum distance of @xmath2 3.5 mpc provides the optimal balance between observational efficiency and achieving the program science goals . because the sample of galaxies within 3.5 mpc contains predominantly field galaxies , we chose to extend the distance limits in the direction of the m81 group . this extension added to the diversity of galaxies in the sample , while maintaining the goal of observational efficiency , due to the m81 group s close proximity to the fiducial distance limit ( d@xmath10 @xmath2 3.6 mpc ) and low foreground extinction values . similarly , we also included galaxies in the direction of the ngc 253 clump ( d@xmath11 @xmath2 3.9 mpc ) in the sculptor filament @xcite , further extending the range of environments probed by the sample , while maintaining strict volume limits for galaxy selection . in this paper , we analyze a sample of 60 angst dwarf galaxies from both new and comparable quality archival imaging . the galaxies range in m@xmath5 from @xmath128.23 ( kk 230 ) to @xmath1217.77 ( ngc 55 ) and in distance from 1.3 mpc ( sex a ) to 4.6 mpc ( ddo 165 ) . we have included most galaxies considered dwarf galaxies in the literature in our analysis , although the upper mass / luminosity cutoff for what constitutes a dwarf galaxy is somewhat ambiguous ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? for example , @xcite chose to exclude the lmc ( @xmath13 @xmath1217.93 ) and smc ( @[email protected] ) from the ` dwarfs ' category . analogous to @xcite , we have not included ngc 3077 ( @[email protected] ) , ngc 2976 ( @xmath13 @xmath1216.77 ) , and ngc 300 ( @xmath13 @xmath1217.66 ) , which would be among the brightest and most massive galaxies in the sample . detailed studies of the sfhs of ngc 2976 @xcite and ngc 300 @xcite are available in the literature . a full list of sample galaxies and their properties are listed in table [ tab1 ] with the distances and blue luminosities of the sample shown in figure [ sample ] . although the sample of angst dwarf galaxies is extensive , it is not complete within a fixed distance limit . galaxies at low galactic latitudes have been intentionally excluded from the original volume selection to avoid complications associated with high degrees of reddening . while sfhs can still be derived from cmds of such galaxies ( e.g. , ic 4662 ; * ? ? ? * ; * ? ? ? * ) , the effects of extreme reddening can lead to larger uncertainties and can require special analysis techniques ( e.g. , individual stellar line of sight reddening corrections ) , which detracts from a uniform approach to the data reduction . in addition there have been a number of recently discovered dwarf galaxies in the m81 group @xcite , which were not discovered in time to be included in the original angst sample . the recent distance reassignment of ugc 4879 to the periphery of the lg @xcite meant this galaxy has also been excluded from the angst sample . further , the angst sample is likely to be missing any faint dsphs which might be located in close proximity to m81 , i.e. , analogs to milky way satellites such as draco and ursa minor , as such galaxies have not yet been detected due to their inherent faintness . at a distance of m81 ( 3.6 mpc ) draco and ursa minor would have apparent blue luminosities of 18.98 and 20.13 , respectively . in comparison , sc22 ( @xmath14 @xmath15 17.72 ) has the faintest apparent magnitude in the angst sample . in this paper , we initially divide the sample of dwarf galaxies according to morphological type , @xmath16 , @xcite resulting in 12 dwarf galaxies with @xmath16 @xmath17 0 ( dsphs ) , 5 with @xmath16 @xmath15 8 or 9 ( dwarf spirals ; dspirals ) , and 43 with @xmath16 @xmath15 10 ( dis ) . for ease of direct comparison with well studied lg dwarf galaxies , we have adopted nomenclature consistent with @xcite , in which dwarf ellipiticals ( e.g. , ngc 147 , ngc 185 ) are found to be rare , and dsphs are more common . among the dis , there could be some ambiguity between bright dis and dspirals . for consistency , we defer to the @xmath16-type morphological classification scheme , but note that the distinction between a bright di and dspiral is not always clear . morphological type @xmath16 @xmath15 10 further includes transition dwarf galaxies ( dtrans ) , galaxies with reduced recent star formation but high gas fractions ( e.g. , * ? ? ? * ) , and tidal dwarf galaxy candidates ( dtidals ) , which appear to be condensing out of tidally disturbed gas . we classify subtypes , i.e. , dtidal and dtrans , as follows : the three dtidals are holmberg ix , a0952 + 069 , and bk3n , and are all located in the m81 group . for dtrans , we adopt the definition of @xcite , namely , that a galaxy has detectable gas but very little or no h@xmath18 flux . the final sample of dtrans was classified based on hi and h@xmath18 measurements in the literature @xcite . we find 12 angst dwarf galaxies that satisfy the dtrans criteria : kk 230 , antlia , kkr 25 , kkh 98 , kdg 73 , eso294 - 10 , eso540 - 30 , eso540 - 32 , kdg 52 , eso410 - 005 , ddo 6 , and ugca 438 , leaving the final tally of true dis at 28 . lccccccccccc kk230 & kkr3 & m31 & -8.49 & 1.3 & 0.04 & 10 & -1.0 & 12.0 & f606w , f814w & @xmath190.62 & 9771 + bk3n & & m81 & -9.23 & 4.0 & 0.25 & 10 & 1.0 & 18.0 & f475w , f814w & -0.32 & 10915 + antlia & & n3109 & -9.38 & 1.3 & 0.24 & 10 & -0.1 & 1.20 & f606w , f814w & @xmath191.75 & 10210 + kkr25 & & m31 & -9.94 & 1.9 & 0.03 & 10 & -0.7 & 2.50 & f606w , f814w & -0.18 & 11986 + fm1 & f6d1 & m82 & -10.16 & 3.4 & 0.24 & -3 & 1.8 & 5.00 & f606w , f814w & @xmath190.11 & 9884 + kkh86 & & m31 & -10.19 & 2.6 & 0.08 & 10 & -1.5 & 5.20 & f606w , f814w & -1.3 & 11986 + kkh98 & & m31 & -10.29 & 2.5 & 0.39 & 10 & -0.7 & 5.40 & f475w , f814w & @xmath190.46 & 10915 + bk5n central & & n3077 & -10.37 & 3.8 & 0.20 & -3 & 2.4 & 3.80 & f606w , f814w & -0.77 & 6964 + bk5n outer & & & & & & & & 3.80 & f606w , f814w & -0.70 & 5898 + sc22 & sc - de1 & n253 & -10.39 & 4.2 & 0.05 & -3 & 0.9 & 5.70 & f606w , f814w & @xmath190.30 & 10503 + kdg73 & & m81 & -10.75 & 3.7 & 0.06 & 10 & 1.3 & 12.0 & f475w , f814w & -0.38 & 10915 + ikn & & m81 & -10.84 & 3.7 & 0.18 & -3 & 2.7 & 0.58 & f606w , f814w & -0.82 & 9771 + e294 - 010 & & n55 & -10.86 & 1.9 & 0.02 & -3 & 1.0 & 4.60 & f606w , f814w & @xmath191.80 & 10503 + a0952 + 69 & & n3077 & -11.16 & 3.9 & 0.26 & 10 & 1.9 & 1.20 & f475w , f814w & -0.23 & 10915 + e540 - 032 & & n253 & -11.22 & 3.4 & 0.06 & -3 & 0.6 & 2.30 & f606w , f814w & @xmath190.10 & 10503 + kkh37 & & i342 & -11.26 & 3.4 & 0.23 & 10 & -0.3 & 3.70 & f475w , f814w & @xmath190.12 & 10915 + kdg2 & e540 - 030 & n253 & -11.29 & 3.4 & 0.07 & -1 & 0.4 & 2.70 & f606w , f814w & @xmath190.32 & 10503 + ua292 & cvni - dwa & n4214 & -11.36 & 3.1 & 0.05 & 10 & -0.4 & 5.10 & f475w , f814w & -0.37 & 10915 + kdg52 & m81-dwarf - a & m81 & -11.37 & 3.5 & 0.06 & 10 & 0.7 & 2.30 & f555w , f814w & @xmath190.37 & 10605 + kk77 & f12d1 & m81 & -11.42 & 3.5 & 0.44 & -3 & 2.0 & 0.83 & f606w , f814w & @xmath190.28 & 9884 + e410 - 005 & & n55 & -11.49 & 1.9 & 0.04 & -1 & 0.4 & 2.80 & f606w , f814w & @xmath191.70 & 10503 + hs117 & & m81 & -11.51 & 4.0 & 0.36 & 10 & 1.9 & 2.70 & f606w , f814w & -0.81 & 9771 + ddo113 & ua276 & n4214 & -11.61 & 2.9 & 0.06 & 10 & 1.6 & 1.80 & f475w , f814w & @xmath190.08 & 10915 + kdg63 & u5428,ddo71 & m81 & -11.71 & 3.5 & 0.30 & -3 & 1.80 & 1.40 & f606w , f814w & @xmath190.31 & 9884 + ddo44 & ua133 & n2403 & -11.89 & 3.2 & 0.13 & -3 & 1.7 & 0.59 & f475w , f814w & @xmath190.16 & 10915 + gr8 & u8091,ddo155 & m31 & -12.00 & 2.1 & 0.08 & 10 & -1.2 & 3.30 & f475w , f814w & @xmath190.82 & 10915 + e269 - 37 & & n4945 & -12.02 & 3.5 & 0.44 & -3 & 1.6 & 3.80 & f606w , f814w & -1.8 & 11986 + ddo78 & & m81 & -12.04 & 3.7 & 0.07 & -3 & 1.8 & 0.95 & f475w , f814w & -0.27 & 10915 + f8d1central & & m81 & -12.20 & 3.8 & 0.33 & -3 & 3.8 & 0.42 & f555w , f814w & -0.77 & 5898 + f8d1outer & & & & & & & & 0.42 & f606w , f814w & -0.90 & 5898 + u8833 & & n4736 & -12.31 & 3.1 & 0.04 & 10 & -1.4 & 5.00 & f606w , f814w & -0.23 & 10210 + e321 - 014 & & n5128 & -12.31 & 3.2 & 0.29 & 10 & -0.3 & 2.20 & f606w , f814w & -0.84 & 8601 + kdg64 & u5442 & m81 & -12.32 & 3.7 & 0.17 & -3 & 2.5 & 1.10 & f606w , f814w & @xmath190.57 & 11986 + ddo6 & ua15 & n253 & -12.40 & 3.3 & 0.05 & 10 & 0.5 & 3.00 & f475w , f814w & -0.02 & 10915 + ddo187 & u9128 & m31 & -12.43 & 2.3 & 0.07 & 10 & -1.3 & 1.60 & f606w , f814w & @xmath190.40 & 10210 + kdg61 & kk81 & m81 & -12.54 & 3.6 & 0.23 & -1 & 3.9 & 1.10 & f606w , f814w & @xmath190.33 & 9884 + u4483 & & m81 & -12.58 & 3.2 & 0.11 & 10 & 0.5 & 2.20 & f555w , f814w & -1.36 & 8769 + ua438 & e407 - 18 & n55 & -12.85 & 2.2 & 0.05 & 10 & -0.7 & 1.00 & f606w , f814w & -1.7 & 8192 + ddo181 & u8651 & m81 & -12.94 & 3.0 & 0.02 & 10 & -1.3 & 1.20 & f606w , f814w & -0.31 & 10210 + u8508 & izw60 & m81 & -12.95 & 2.6 & 0.05 & 10 & -1.0 & 2.10 & f475w , f814w & @xmath190.39 & 10915 + n3741 & u6572 & m81 & -13.01 & 3.0 & 0.07 & 10 & -0.8 & 1.60 & f475w , f814w & -0.24 & 10915 + ddo183 & u8760 & n4736 & -13.08 & 3.2 & 0.05 & 10 & -0.8 & 2.30 & f475w , f814w & -0.46 & 10915 + ddo53 & u4459 & m81 & -13.23 & 3.5 & 0.12 & 10 & 0.7 & 1.60 & f555w , f814w & -0.05 & 10605 + hoix & u5336,ddo66 & m81 & -13.31 & 3.7 & 0.24 & 10 & 3.3 & 0.71 & f555w , f814w & @xmath190.13 & 10605 + ddo99 & u6817 & n4214 & -13.37 & 2.6 & 0.08 & 10 & -0.5 & 0.58 & f606w , f814w & -0.95 & 10210 + sexa & ddo75 & mw & -13.71 & 1.3 & 0.14 & 10 & -0.6 & 0.06 & f555w , f814w & @xmath190.80 & 7496 + n4163 & u7199 & n4190 & -13.76 & 3.0 & 0.06 & 10 & 0.1 & 1.20 & f475w , f814w & -0.04 & 10915 + sexb & u5373 & mw & -13.88 & 1.4 & 0.10 & 10 & -0.7 & 0.10 & f606w , f814w & @xmath190.04 & 11986 + ddo125 & u7577 & n4214 & -14.04 & 2.5 & 0.06 & 10 & -0.9 & 0.18 & f606w , f814w & -1.3 & 11986 + e325 - 11 & & n5128 & -14.05 & 3.4 & 0.29 & 10 & 1.1 & 0.52 & f606w , f814w & -0.58 & 11986 + ddo190 & u9240 & m81 & -14.14 & 2.8 & 0.04 & 10 & -1.3 & 1.20 & f475w , f814w & -0.01 & 10915 + hoi & u5139,ddo63 & m81 & -14.26 & 3.8 & 0.15 & 10 & 1.5 & 0.34 & f555w , f814w & @xmath190.23 & 10605 + ddo82 & u5692 & m81 & -14.44 & 4.0 & 0.13 & 9 & 0.9 & 0.53 & f475w , f814w & -0.32 & 10915 + ddo165 & u8201 & n4236 & -15.09 & 4.6 & 0.08 & 10 & 0.0 & 0.50 & f555w , f814w & -0.7 & 10605 + n3109-deep & ddo236 & antlia & -15.18 & 1.3 & 0.20 & 9 & -0.1 & 0.05 & f606w , f814w & @xmath190.50 & 10915 + n3109-wide2 & & & & & & & & 0.05 & f606w , f814w & @xmath190.09 & 11307 + i5152 & & m31 & -15.55 & 2.1 & 0.08 & 10 & -1.1 & 0.10 & f606w , f814w & -1.38 & 11986 + n23661 & u3851 & n2403 & -15.85 & 3.2 & 0.11 & 10 & 1.0 & 0.39 & f555w , f814w & -0.16 & 10605 + n23662 & & & & & & & & 0.39 & f555w , f814w & -0.05 & 10605 + ho ii1 & u4305 & m81 & -16.57 & 3.4 & 0.10 & 10 & 0.6 & 0.12 & f555w , f814w & -0.32 & 10605 + ho ii2 & u4305 & & & & & & & 0.12 & f555w , f814w & -0.21 & 10605 + n4214 & u7278 & ddo113 & -17.07 & 2.9 & 0.07 & 10 & -0.7 & 0.03 & f606w , f814w & -0.48 & 11986 + i2574sgs & u5666,ddo81 & m81 & -17.17 & 4.0 & 0.11 & 9 & 0.9 & 0.05 & f555w , f814w & -0.26 & 9755 + i25741 & u5666,ddo81 & & & & & & & 0.05 & f555w , f814w & -0.55 & 10605 + i25742 & u5666,ddo81 & & & & & & & 0.05 & f555w , f814w & -0.15 & 10605 + e383 - 87 & & n5128 & -17.41 & 3.45 & 0.24 & 8 & -0.8 & 0.25 & f606w , f814w & -2.14 & 11986 + n55central & & n300 & -17.77 & 2.1 & 0.04 & 8 & 0.4 & 0.01 & f606w , f814w & -1.55 & 9765 + n55disk & & & & & & & & 0.01 & f606w , f814w & -0.42 & 9765 [ tab1 ] throughout this paper , we adopt the tidal index , @xmath20 @xcite , as a measure of a galaxy s isolation . @xmath20 describes the local mass density around galaxy @xmath21 as : @xmath22 + c , \phantom{1 } i = 1,2,\cdots , n\ ] ] where @xmath23 is the total mass of any neighboring galaxy separated from galaxy @xmath21 by a distance of @xmath24 . the values of @xmath20 we use in this paper have been taken from @xcite . negative values correspond to more isolated galaxies , and positive values represent typical group members ( see table [ tab1 ] ) . hst observations of new angst targets were carried out in two phases due to the failure of acs in 2007 . prior to the failure , we observed new targets using acs with wfpc2 in parallel mode . galaxies observed post - acs failure were imaged with wfpc2 alone , as part of a ` supplemental ' hst program ( proposal ids 11307 and 11986 in table [ tab1 ] ; * ? ? ? * ) . new acs observations used the f475w ( sdss @xmath25 ) and f814w ( @xmath26 ) filter combination , to optimize both photometric depth and temperature ( color ) baseline . a third filter in f606w ( wide @xmath27 ) is also available for most galaxies . the low throughput in the bluer filters of wfpc2 led us to only use f606w and f814w for wfpc2 observations . as described in @xcite , both new and archival observations were processed uniformly beginning with image reduction via the standard hst pipeline . using the angst data reduction pipeline , we performed photometry on each image with hstphot @xcite , designed for wfpc2 images , and dolphot , running in its acs - optimized mode , for acs observations , providing for a uniform treatment of all data . the resultant photometry for each data set was filtered to ensure the final photometric catalogs excluded non - stellar objects such as cosmic rays , hot pixels , and extended sources . for the purposes of this paper , we considered a star well measured if it met the following criteria : a signal - to - noise ratio @xmath8 4 in both filters , a sharpness value such that ( @xmath28 @xmath19 @xmath29)@xmath30 @xmath17 0.075 , and a crowding parameter such that ( @xmath31 @xmath19 @xmath32 ) @xmath17 0.1 . to characterize observational uncertainties , we performed 500,000 artificial star tests on each image . both the full and filtered ( i.e. , the ` gst ' files ) photometric catalogs , hst reference images , and cmds are publicly available on mast . definitions and detailed descriptions of the filtering criteria and the observational strategies can be found in @xcite , @xcite , and @xcite . because of the variety of distances in the sample , the acs / wfpc2 field of view does not subtend the same physical area for each galaxy . for certain comparisons among the sample ( e.g. , integrated stellar masses ) , it is important to account for these differences in coverage area . to that effect , we employ a simple areal normalization factor based on each galaxy s apparent blue surface brightness . from the measurements listed in table 4 of @xcite , we consider an effective elliptical area computed using the angular diameter and angular axial ratios at a blue surface brightness level of @xmath2 25 mag arcsec@xmath33 ( @xmath2 26.5 mag arcsec@xmath33 in the case of the faintest galaxies ) . we then calculate the normalization factor , @xmath34 , for each galaxy by taking the ratio of the angular area subtended by the acs/ wfpc2 field of view to the angular area computed from @xcite . this normalization has been specifically applied to the integrated stellar masses throughout this study . the @xmath34 normalizations are typically @xmath8 1 ( i.e. , the hst field of view exceeds the area computed by @xcite ; see table [ tab1 ] ) , and the most and least covered galaxies are bk3n ( 18.0 ) and ngc 55 ( 0.02 ) , respectively . in most cases , a single hst field was sufficient to cover the main optical body of a galaxy , ensuring the sfhs are representative of the whole galaxy . however , several of the sample galaxies required multiple observations to cover a reasonable fraction of the optical body ( ngc 2366 , holmberg ii , ic 2574 , ngc 55 , bk5n , f8d1 , ngc 3109 ) . to derive the sfhs for each of these galaxies , we first checked to see that the 50% completeness limits for each of the fields were similar ( i.e. , within @xmath350.2 mag ) . this condition was met for all galaxies except ic 2574 , ngc 55 , and ngc 3109 , which have large angular sizes and wide ranges of surface brightnesses within each galaxy . for galaxies with similar completeness limits , the photometry and false stars were first combined , and then the sfh code was run on the combined data . for galaxies with overlapping fields ( ngc 2366 , holmberg ii ) we carefully removed duplicate stars before merging the photometry . ic 2574 was a special case as it has three overlapping fields with significantly different completeness limits . in this instance , we first removed the duplicate stars from overlapping fields , ran the sfhs on each field , and then combined the results . ngc 55 and ngc 3109 have many new and archival hst observations taken over multiple visits . in each case , we selected representative non - overlapping fields , one in the center of the galaxy and one in the disk . the completeness functions were sufficiently different such that we did not combine photometry prior to computing the sfh . instead , we derived the sfhs for each field separately , and combined the resultant sfhs from each field . the fractional coverage listed in table [ tab1 ] takes into account the combined areas for galaxies with multiple fields . to ensure uniformity in the sfhs of the angst dwarf galaxies , we selected one sfh code @xcite and one set of stellar evolution models ( padova ; * ? ? ? * ) . the sfh code of @xcite provides the user with robust controls over critical fixed input variables , e.g. , imf , binary fraction , time and cmd bin sizes , as well as the ability to search for the combination of metallicity , distance , and extinction values that produce a model cmd that best fits the observed cmd . the models of @xcite combine updated asymptotic giant branch ( agb ) evolution tracks with the models of @xcite ( @xmath36 @xmath37 7 m@xmath38 ) and @xcite ( @xmath36 @xmath6 7 m@xmath38 ) . as with any models , there may be inevitable systematic biases associated with these particular choices ( e.g. , at some metallicities the padova model rgbs have red offsets from observations ; e.g. , * ? ? ? however , these systematic effects will be shared by all galaxies in the sample , making for a robust relative comparison within the sample . we discuss systematic uncertainties due to choice of isochrones in appendix [ systematics ] . here , we briefly summarize the technique of measuring a sfh based on the full methodology described in @xcite . the user specifies an assumed imf and binary fraction , and allowable ranges in age , metallicity , distance , and extinction . photometric errors and completeness are characterized by artificial star tests . from these inputs , many synthetic cmds are generated to span the desired age and metallicity range . for this work , we have used synthetic cmds sampling stars with age and metallicity spreads of 0.1 and 0.1 dex , respectively . these individual synthetic cmds are then linearly combined along with a model foreground cmd to produce a composite synthetic cmd . the linear weights on the individual cmds are adjusted to obtain the best fit as measured by a poisson maximum likelihood statistic ; the weights corresponding to the best fit are the most probable sfh . this process can be repeated at a variety of distance and extinction values to solve for these parameters as well . monte carlo tests were used to estimate uncertainties due to both random and systematic sources . for each monte carlo run , a poisson random noise generator was used to create a random sampling of the best - fit cmd . this cmd was then processed identically to the original solution , with additive errors in @xmath39 and @xmath40 introduced when generating the model cmds for these solutions . single shifts in @xmath39 and @xmath40 ) are used for each monte carlo draw , and the errors themselves were drawn from normal distributions with 1-@xmath41 values of 0.41 ( @xmath39 ) and 0.03 ( @xmath40 ) . these distributions were designed to mimic the scatter in sfh uncertainties obtained by using multiple isochrone sets to fit the data , though we prefer to use this technique due to the small number of isochrone sets that fully cover the range of ages , metallicities , and evolutionary states required to adequately model our cmds . we found that the uncertainties were stable after 50 monte carlo tests , and thus conducted 50 realizations for each galaxy . this technique of estimating error on sfhs is described in greater detail in @xcite . to minimize systematic effects for comparisons among the sample , we selected consistent parameters for measuring sfhs of all galaxies . all sfhs were measured using a single slope power law imf with a spectral index of @xmath121.30 over a mass range of 0.1 to 120 m@xmath38 , a binary fraction of 0.35 with a flat secondary mass distribution , 71 equally spaced logarithmic time bins ranging from @xmath42 @xmath15 6.610.15 , color and magnitude bins of 0.05 and 0.1 mag , and the padova stellar evolution models @xcite . we note that the difference between our selected imf and a kroupa imf @xcite is negligible , as the angst cmds are limited to stellar masses @xmath4 0.8 m@xmath38 . we designated the faint photometric limit to be equal to the 50% completeness limit in each filter ( see table [ tab1 ] ) as determined by @xmath2 500,000 artificial star tests run for each galaxy . the sfh program was initially allowed to search for the best fit distance and extinction values without constraints . initial values for the distances were taken from the trgb distances measured in @xcite , while foreground extinction values were taken from the galactic maps of @xcite . we found no significant discrepancies between the assumed values and the best cmd fit distance and foreground extinction values , i.e. , all were consistent within error . final solutions were computed using the trgb distances from @xcite and @xcite and foreground extinction values from @xcite . we placed an additional constraint on the cmd fitting process , namely that the mean metallicity in each time bin must monotonically increase toward the present . the deepest angst cmds do not reach the ancient ms turnoff , a requisite feature for completely breaking the age - metallicity degeneracy of the rgb only using broadband photometry ( e.g. , * ? ? ? * ; * ? ? ? as a result , sfhs derived from shallow cmds without a metallicity constraint can often have accompanying chemical evolution models that are unphysical ( e.g. , a drop in metallicity of several tenths of a dex over sub - gyr time scales ) . a more robust analysis of the chemical enrichment of dwarf galaxies would need to include the ancient ms , measured gas phase abundances , and/or individual stellar spectra ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? we therefore consider analysis of the metallicity evolution of the angst sample beyond the scope of this paper . as an example of a typical cmd fit made by the sfh code , we compare the observed and synthetic cmds of a representative angst dtrans , ddo 6 ( figure [ fit ] ) . @xcite determined the trgb distance of ddo 6 to be [email protected] mpc while the foreground extinction maps of @xcite give values of a@xmath5 @xmath15 0.07 and a@xmath43 @xmath15 0.06 . allowing the sfh code to search for the best fit cmd , we find best fit values of d @xmath15 [email protected] mpc and a@xmath44 @xmath15 [email protected] , both in excellent agreement with the independently measured values . examining the residual significance cmd , i.e. , the difference between the data and the model weighted by the variance ( panel ( d ) of figure [ fit ] ) , we see a good , although not perfect fit . notably , the area between the blue helium burning stars and young ms appears to be too cleanly separated in the model , which could be due to differential extinction affecting young stars in the observed cmd . additionally , the model red helium burning stars are too blue compared to the data , likely due to uncertainties in the massive star models ( e.g. , * ? ? ? * ) . however , even the most discrepant regions are fit within @xmath35 5@xmath41 , which are indicated by black or white points . overall , the model cmd appears to be in good agreement with the observed cmd , indicating that we have measured a reliable sfh . see @xcite for a full discussion of the quality measures of this cmd fitting technique . in general , uncertainties in the absolute sfhs generally are somewhat anti - correlated between adjacent time bins , such that if the sfr is overestimated in one bin , it is underestimated in the adjacent bin . however , cumulative sfhs , i.e. , the stellar mass formed during or previous to each time bin normalized to the integrated final stellar mass , do not share this property , and thus present a more robust way of analyzing sfhs . in the following sections , we consider sfhs plotted versus two different time binning schemes . for the individual galaxies ( figure [ sfh1 ] ) , we present the cumulative sfhs at the highest time resolution possible . errors in the cumulative sfhs presented at this time resolution indicate both the uncertainty in the fraction of total stellar mass formed prior to a given time , and the inherent time resolution for the sfh of that particular galaxy . a coarser time binning scheme is used for the absolute sfhs of the individual galaxies as well as the mean absolute and cumulative sfhs across the angst sample . in appendix [ appena ] , we determine that six broad time bins of of 0 - 1 , 1 - 2 , 2 - 3 , 3 - 6 , 6 - 10 , and 10 - 14 gyr provide an optimal balance between photometric depths and age leverage contained in the cmds of the angst dwarfs . the broader time bins allow us to securely average the best fit sfhs from individual galaxies . for the absolute sfhs of individual galaxies , broader time bins tend to minimize co - variance between adjacent time bins . thus applying a broader time binning scheme to the absolute sfhs of individual galaxies provides for more secure measurements of the sfrs . lccccccccc kk230 & 0.20 & 0.006 & 0.15 & 0.76 & 0.76 & 0.86 & 0.89 & 0.94 & 1.00 + bk3n & 1.40 & 0.03 & 0.37 & 0.41 & 0.41 & 0.95 & 0.95 & 0.95 & 1.00 + antlia & 0.38 & 0.10 & 1.30 & 0.16 & 0.41 & 0.83 & 0.94 & 0.95 & 1.00 + kkr25 & 0.19 & 0.03 & 0.19 & 0.58 & 0.59 & 0.90 & 0.90 & 0.97 & 1.00 + fm1 & 1.8 & 0.13 & 0.69 & 0.89 & 0.89 & 0.95 & 0.95 & 0.97 & 1.00 + kkh86 & 0.31 & 0.02 & 0.12 & 0.70 & 0.81 & 0.94 & 0.94 & 0.94 & 1.00 + kkh98 & 0.64 & 0.41 & 0.20 & 0.19 & 0.62 & 0.83 & 0.83 & 0.93 & 1.00 + bk5n & 2.0 & 0.10 & 0.46 & 0.93 & 0.93 & 0.97 & 0.97 & 0.97 & 1.00 + sc22 & 1.26 & 0.08 & 0.35 & 0.72 & 0.73 & 0.94 & 0.96 & 0.98 & 1.00 + kdg73 & 0.65 & 0.02 & 0.06 & 0.30 & 0.30 & 0.87 & 0.89 & 0.94 & 1.00 + ikn & 7.9 & 4.8 & 14.0 & 0.02 & 0.95 & 0.98 & 0.98 & 9.98 & 1.00 + e294 - 010 & 1.16 & 0.09 & 0.26 & 0.80 & 8.86 & 0.92 & 0.96 & 0.99 & 1.00 + a0952 + 69 & 0.46 & 0.13 & 0.29 & 0.40 & 0.40 & 0.68 & 0.68 & 0.68 & 1.00 + e540 - 032 & 0.24 & 0.36 & 0.76 & 0.86 & 0.86 & 0.91 & 0.91 & 0.98 & 1.00 + kkh37 & 1.64 & 0.16 & 0.31 & 0.45 & 0.48 & 0.87 & 0.87 & 0.95 & 1.00 + kdg2 & 0.34 & 0.04 & 0.09 & 0.02 & 0.02 & 0.60 & 0.68 & 0.88 & 1.00 + ua292 & 0.47 & 0.03 & 0.06 & 0.45 & 0.45 & 0.77 & 0.77 & 0.87 & 1.00 + kdg52 & 1.60 & 0.24 & 0.44 & 0.93 & 0.93 & 0.93 & 0.94 & 0.97 & 1.00 + kk77 & 4.6 & 1.94 & 3.4 & 0.47 & 0.74 & 0.96 & 0.96 & 0.98 & 1.00 + e410 - 005 & 1.42 & 0.18 & 0.30 & 0.62 & 0.79 & 0.87 & 0.89 & 0.98 & 1.00 + hs117 & 0.33 & 0.04 & 0.07 & 0.83 & 0.83 & 0.83 & 0.83 & 0.89 & 1.00 + ddo113 & 1.30 & 0.24 & 0.36 & 0.58 & 0.58 & 0.71 & 0.85 & 0.98 & 1.00 + kdg63 & 3.4 & 0.85 & 1.10 & 0.43 & 0.64 & 0.84 & 0.86 & 0.98 & 1.00 + ddo44 & 2.20 & 1.30 & 1.5 & 0.34 & 0.34 & 0.83 & 0.83 & 0.98 & 1.00 + gr8 & 0.90 & 0.10 & 0.1 & 0.59 & 0.69 & 0.88 & 0.88 & 0.95 & 1.00 + e269 - 37 & 2.00 & 0.20 & 0.2 & 0.94 & 0.94 & 0.96 & 0.96 & 0.97 & 1.00 + ddo78 & 4.75 & 1.77 & 1.7 & 0.56 & 0.56 & 0.79 & 0.79 & 0.99 & 1.00 + f8d1 & 8.35 & 3.8 & 3.2 & 0.65 & 0.65 & 0.73 & 0.89 & 0.99 & 1.00 + u8833 & 1.54 & 0.11 & 0.08 & 0.71 & 0.72 & 0.79 & 0.79 & 0.94 & 1.00 + e321 - 014 & 1.66 & 0.29 & 0.22 & 0.77 & 0.82 & 0.89 & 0.92 & 0.95 & 1.00 + kdg64 & 3.35 & 0.56 & 0.42 & 0.47 & 0.56 & 0.82 & 0.82 & 0.98 & 1.00 + ddo6 & 1.74 & 0.20 & 0.14 & 0.55 & 0.55 & 0.87 & 0.87 & 0.93 & 1.00 + ddo187 & 0.97 & 0.21 & 0.14 & 0.45 & 0.47 & 0.60 & 0.60 & 0.87 & 1.00 + kdg61 & 4.35 & 1.4 & 0.88 & 0.63 & 0.63 & 0.76 & 0.77 & 0.98 & 1.00 + u4483 & 1.57 & 0.27 & 0.16 & 0.00 & 0.91 & 0.93 & 0.93 & 0.96 & 1.00 + ua438 & 2.74 & 1.0 & 0.48 & 0.63 & 0.92 & 0.98 & 0.98 & 0.98 & 1.00 + ddo181 & 3.11 & 0.92 & 0.39 & 0.72 & 0.72 & 0.74 & 0.74 & 0.93 & 1.00 + u8508 & 2.33 & 0.39 & 0.16 & 0.58 & 0.58 & 0.73 & 0.73 & 0.93 & 1.00 + n3741 & 2.63 & 0.57 & 0.23 & 0.68 & 0.68 & 0.84 & 0.84 & 0.91 & 1.00 + ddo183 & 3.31 & 0.50 & 0.19 & 0.66 & 0.66 & 0.86 & 0.86 & 0.94 & 1.00 + ddo53 & 4.75 & 1.0 & 0.34 & 0.41 & 0.56 & 0.75 & 0.75 & 0.95 & 1.00 + ho ix & 4.04 & 1.15 & 0.35 & 0.23 & 0.72 & 0.83 & 0.83 & 0.89 & 1.00 + ddo99 & 3.12 & 4.0 & 1.20 & 0.75 & 0.93 & 0 . 94 & 0.94 & 0.95 & 1.00 + sexa & 0.59 & 3.53 & 0.75 & 0.30 & 0.46 & 0.65 & 0.67 & 0.84 & 1.00 + n4163 & 8.93 & 2.65 & 0.54 & 0.45 & 0.92 & 0.92 & 0.92 & 0.96 & 1.00 + sexb & 1.17 & 4.38 & 0.78 & 0.56 & 0.56 & 0.58 & 0.58 & 0.87 & 1.00 + ddo125 & 7.26 & 15.6 & 2.40 & 0.42 & 0.97 & 0.97 & 0.97 & 0.98 & 1.00 + e325 - 11 & 3.59 & 2.61 & 0.40 & 0.58 & 0.68 & 0.81 & 0.87 & 0.93 & 1.00 + ddo190 & 4.29 & 1.21 & 0.17 & 0.32 & 0.41 & 0.63 & 0.77 & 0.86 & 1.00 + ho i & 6.65 & 6.83 & 0.87 & 0.00 & 0.00 & 0.62 & 0.82 & 0.84 & 1.00 + ddo82 & 25.1 & 16.8 & 1.80 & 0.47 & 0.48 & 0.87 & 0.87 & 0.96 & 1.00 + ddo165 & 15.1 & 9.77 & 0.58 & 0.54 & 0.55 & 0.74 & 0.82 & 0.85 & 1.00 + n3109 & 2.64 & 20.2 & 1.10 & 0.79 & 0.79 & 0.79 & 0.79 & 0.91 & 1.00 + i5152 & 8.50 & 29.6 & 1.10 & 0.30 & 0.91 & 0.01 & 0.91 & 0.95 & 1.00 + n2366 & 28.5 & 26.8 & 0.79 & 0.67 & 0.67 & 0.74 & 0.74 & 0.89 & 1.00 + ho ii & 23.4 & 61.6 & 0.93 & 0.81 & 0.81 & 0.82 & 0.85 & 0.89 & 1.00 + n4214 & 7.02 & 82.3 & 0.79 & 0.71 & 0.95 & 0.95 & 0.95 & 0.97 & 1.00 + i2574 & 74.3 & 175 & 1.50 & 0.86 & 0.86 & 0.86 & 0.86 & 0.93 & 1.00 + e383 - 87 & 28.5 & 168 & 1.90 & 0.69 & 0.91 & 0.92 & 0.93 & 0.97 & 1.00 + n55 & 138.5 & 1210 & 6.10 & 0.63 & 0.67 & 0.91 & 0.91 & 0.97 & 1.00 [ tab2 ] the angst dwarf galaxies exhibit a wide variety of complex sfhs . in figure [ sfh1 ] we show the absolute sfhs , i.e. , sfr(@xmath45 ) , and the cumulative sfhs of the individual galaxies , sorted in order of increasing blue luminosity . a cursory inspection of the 60 sfhs reveals that often galaxies with similar luminosities , morphologies , or chemical compositions , do not have consistent sfhs , confirming the complexity of dwarf galaxy sfhs previously found in studies of the lg ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the wide variety of sfhs underlines the importance of having a large sample ; conclusions based on the sfh of a single galaxy may not be be representative of the population as a whole . this paper leverages the size of the angst sample to explore general trends seen in dwarf galaxies , e.g. , differences among the morphological types . while the sfhs of individual galaxies can provide useful insight into a particular system s evolutionary state , we generally do not consider individual galaxies in this paper . however , in table [ tab2 ] we provide data on the sfhs of individual galaxies considered in this sample , which may be useful for specific studies . lcccccccc dsph & [email protected] & 0.65@xmath46 & 0.71@xmath47 & 0.87@xmath48 & 0.89@xmath49 & 0.98@xmath50 & 1.00 + di & [email protected] & 0.54@xmath51 & 0.67@xmath52 & 0.80@xmath49 & 0.82@xmath53 & 0.92@xmath54 & 1.00 + dtrans & [email protected] & 0.53@xmath55 & 0.63@xmath56 & 0.86@xmath57 & 0.89@xmath57 & 0.96@xmath54 & 1.00 + dspiral & 74@xmath3535 & 0.59@xmath58 & 0.71@xmath59 & 0.88@xmath49 & 0.89@xmath49 & 0.95@xmath54 & 1.00 + dtidal & [email protected] & 0.38@xmath60 & 0.64@xmath61 & 0.83@xmath62 & 0.84@xmath63 & 0.85@xmath64 & 1.00 [ tab3 ] we first consider the unweighted mean specific sfhs ( i.e. , a sfh divided by the total integrated stellar mass formed ) of the angst sample grouped by morphological type ( figure [ sfh_avg ] ) . comparison among the specific sfhs shows general consistency for times @xmath4 1 gyr ago , with the exception of dtidals ( note that dtidals will be discussed in greater detail in the context of recent sfhs in @xcite ) . qualitatively , we find that a typical dwarf galaxy exhibits dominant ancient sf ( @xmath8 10 gyr ago ) , and lower levels of sf at intermediate times ( 110 gyr ago ) . the only consistent difference among the morphological types is within the last 1 gyr , where dsphs exhibit a significant drop in sfrs relative to the other types . this suggests that many of the dsphs in the angst sample could have been gas - rich as recently as 1 gyr ago , implying that the process of gas loss can occur relatively quickly and at late times . it is tempting to associate the sharp drop in the sfr of a typical dsph @xmath2 1 gyr ago with rapid gas loss . however , broad uncertainties in the agb models ( e.g. , * ? ? ? * ; * ? ? ? * ) make the precise age for the drop in dsph sfrs and the degree of synchronization uncertain . in addition , subsamples of the dsphs do appear to have dramatically lower sfrs at more intermediate ages . cumulative sfhs allow us to readily compare galaxies of different masses . like the absolute sfhs , the cumulative sfhs plotted in figure [ cum1 ] show significant diversity within each morphological class , yet converge on mean values that are broadly consistent among the morphological types . perhaps the most striking result is that the average dwarf galaxy formed @xmath4 50% of its total stellar mass by z @xmath2 2 ( 10 gyr ago ) and @xmath4 60% by z @xmath2 1 ( 7.6 gyr ago ) , independent of morphological type ( table [ tab3 ] ) . although dsphs appear to have formed a slightly larger percentage of total stellar mass than dis by z @xmath2 2 , additional consideration of systematic uncertainties , as discussed in appendix [ systematics ] , indicate that the angst data is not deep enough to make such fine distinctions . more importantly , the amplitude of the difference is significantly smaller than typical models , which assume that dsphs are dominated by ancient stellar populations and dis are consistent with constant sf over their lifetimes ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . the mean cumulative sfhs hint at divergence among the morphological types within the last few gyr ( figure [ cum1 ] ) however , the exact characteristics of differences in the sfhs during this time period are ambiguous . broad uncertainties in the agb star models ( e.g. , * ? ? ? * ; * ? ? ? * ) , our modest time resolution , and systemic effects on the mean cumulative sfh ( see appendix [ systematics ] ) all affect our ability to quantitatively examine the details of intermediate age sf . we do identify secure measurements of differences within the most recent 1 gyr . at these times , luminous ms and core blue and red helium burning stars provide excellent age leverage ( e.g. , * ? ? ? * ) , and the sfhs clearly illustrate differences among the morphological types . specifically , the typical dsph , di , dtrans , and dspiral formed @xmath2 2% , 8% , 4% , and 5% of their total stellar mass within the most recent 1 gyr . the combined findings from intermediate and recent sfhs suggest that morphological differences and complete gas loss in dwarf galaxies can be relatively recent phenomena , at least within the local volume . dsphs that are strictly old , i.e. , have no agb star populations , are known to exist in the lg ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , but appear to only represent a minority of all known dsphs in the larger volume we analyze here . this is at least in part due to the detection limit of the angst sample as discussed in [ selection ] . to further explore the differences among the morphological types , we consider the fractional sfh , i.e. , the fraction of stellar mass formed in each time bin , as a function of the total normalized stellar mass , @xmath65 ( computed by integrating the sfh over time and applying the @xmath34 area normalization ) . we show values for both the individual galaxies ( grey symbols ) and the mean values per morphological type , with error bars representative of the uncertainty in the mean in figure [ mbf_avg ] . the mean stellar masses and cumulative sfhs for each of the morphological types are listed in table [ tab3 ] . the mean stellar masses show that dspirals are typically the most massive galaxies , dtrans are the least massive , and dis and dsphs have similar mean masses . there are not strong links between patterns of sf , total stellar mass , and morphological type , confirming that sf processes among the different types of dwarf galaxies are not dramatically different . we continue analysis involving the integrated stellar masses in [ gascomp ] . galaxies in clusters , groups , and the field follow similar morphology density relationships . namely , gas - poor galaxies , i.e. , ellipticals , are generally found to be less isolated than gas - rich galaxies , i.e. , spirals ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? this same morphology density relationship has been found for dwarf galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the change in the relative fraction of gas - rich and gas - poor galaxies with environment , provides a simple test for various models of dwarf galaxy evolution . as illustrated in figures [ morphden ] and [ thf_avg ] , the angst dwarf galaxies clearly adhere to the morphology density relationship , when using tidal index , @xmath20 , as a proxy for local density . typically , dis are significantly more isolated than dsphs , in spite of having similar mean total stellar masses . dtrans , on average , have intermediate tidal indices between dis and dsphs ( see [ transition ] ) , and have a lower mean stellar mass than either . dspirals are the most massive galaxies , and are typically located in regions of intermediate isolation . these findings are in general agreement with earlier studies of lg dwarf galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . when combined with measured sfhs , the morphology density relationship can be used to gauge the reliability of dwarf galaxy evolution models , particularly processes that induce gas loss . scenarios favoring internal mechanisms ( i.e. , stellar feedback ) as the primary driver of gas loss ( e.g. , * ? ? ? * ; * ? ? ? * ) can reproduce a number of observed dwarf galaxy properties ( e.g. , surface brightness , rotation velocities , metallicities , etc ; * ? ? ? however , such models are generally unable to account for the morphology density relationship ( e.g. , * ? ? ? * ) and often predict that gas - rich and gas - poor dwarf galaxies may have different patterns or efficiencies of sf ( e.g. , * ? ? ? * ; * ? ? ? in contrast , some models that additionally factor in external effects ( e.g. , gravitational interactions , ram pressure stripping ) have been able to reproduce a wide range of dwarf galaxy properties , including a canonical morphology density relationship ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . in the following sections , we discuss results from the angst sfh analysis within the context of physical processes that can affect the evolution of dwarf galaxies . the relative masses of the gas and stellar components can provide clues to dwarf galaxy evolution . for the angst sample , we consider the gas mass and baryonic gas fraction as functions of total stellar mass and tidal index . the gas masses are based on the hi masses in @xcite , corrected by a factor of 1.4 to account for helium content , while the baryonic gas fractions , @xmath66/@xmath67 , is defined to be @xmath66/ ( @xmath66+@xmath65 ) , and does not account for the contribution due to warm / hot baryons . the stellar masses , @xmath65 , are derived from integrating the sfhs over time and applying the areal normalization , @xmath34 . we first consider the relationship between total gas mass and integrated stellar mass as shown in figure [ masses ] , where the dot - dashed line denotes @xmath66 @xmath15 @xmath65 . in this context , the most massive galaxies occupy the upper right portion of the plot , while lower mass dwarfs are located to the left . galaxies without detectable gas are located in the lower portion of the plot , and have been placed at @xmath68 3 for convenience . to illustrate the range of stellar mass covered by the angst dsphs relative to the lg dsphs , we have included select lg dsphs in grey ( placed using stellar mass - to - light values from @xcite ) . figure [ masses ] provides a concise snapshot of the current evolutionary state of nearby dwarf galaxies . for large stellar masses , this view illustrates the morphological ambiguity between dis and dspirals ; galaxies from both classes are gas - rich and can have stellar masses @xmath4 10@xmath69 m@xmath38 . hi surface density maps of some dis even reveal hints of spiral gas structure in dis ( e.g. , * ? ? ? in this same mass regime , however , there is a conspicuous absence of dsphs . in the angst sample , we not do find dsphs more massive than @xmath2 10@xmath69 m@xmath38 , which is in agreement with the lack of massive gas - poor galaxies in the lg ( e.g. , * ? ? ? * ; * ? ? ? dtrans are predominantly located at low stellar masses , yet have relatively large gas supplies . these properties suggest that some dtrans may not be significantly different from low mass dis ; we return to this point in [ transition ] . the baryonic gas fraction provides a more detailed view of the current evolutionary state of the angst dwarf galaxies ( figure [ bary ] ) . the general trend for dwarf galaxies mirrors that of massive counterparts on the hubble sequence ( e.g. , * ? ? ? * ) , namely that spiral galaxies typically have low gas fractions , while irregulars have higher gas fractions . this perspective also reinforces the view that many dtrans could be low mass dis in between episodes of massive sf ( e.g. , * ? ? ? * ) . there appear to be several outliers to the general trends in figure [ bary ] . the two most conspicuous outliers , hs 117 and ddo 113 , may simply be morphological misclassifications . hs 117 is classified as a di , but has a very low gas content , with an upper limit of @xmath70 @xmath2 10@xmath71 m@xmath38 @xcite and error bars consistent with zero . interestingly , @xcite detect low levels of h@xmath18 in hs 117 , which reinforces the di classification , but based on the lack of hi , they deem this a dsph . morphologically , inspection of the hst image further reveals that it appears to be superimposed on an hii region , leading to an erroneous di classification . we therefore suggest hs 117 is best classified as a dsph or possibly a dtrans . ddo 113 is likewise classified as a di with negligible gas content , but has a possible h@xmath18 detection @xcite . inspection of the hst based cmd suggests that ddo 113 resembles a prototypical dsph , with a handful of relatively faint blue stars , suggesting it is also likely a dsph or a dtrans . additional putative outliers include three dtrans ( eso294 - 010 , eso410 - 005 , eso540 - 032 ) and three dis ( ngc 4163 , ic 5152 and ddo 125 ) . two of the dis ( ic 5152 and ddo 125 ) are isolated and relative massive , yet have moderate gas fractions . the three dtrans have confirmed blue horizontal branch populations @xcite , which helps constrain their ancient sfhs . however , none of these galaxies show any unique features in their sfhs that would explain their outlier status . in contrast , ngc 4163 is a starburst dwarf galaxy , which may have consumed a significant amount of gas due to an intense burst of sf within the last @xmath2 1 gyr ( e.g. , * ? ? ? * ) . examining the gas fraction as a function of isolation ( figure [ thbary ] ) , we see that isolated galaxies tend to have high gas fractions , while those in high density environments have low gas fractions . interestingly , there appear to be no galaxies with _ any _ appreciable gas in dense environments ( @xmath20 @xmath4 1.5 ; hs 117 and ddo 113 have uncertain gas measurements as described above ) . we also see some evidence for separation of dtrans into groups of isolated ( @xmath20 @xmath6 0 ) and moderate / high ( @xmath20 @xmath1 0 ) density environments , which we further discuss in [ transition ] . how gas - rich galaxies _ completely _ lose their gas has long been an outstanding question in dwarf galaxy evolution ( e.g. , * ? ? ? * and references therein ) . within the angst sample , we can test the viability of various gas loss mechanisms using the observed properties of gas - poor dsphs . the angst dsphs share several notable characteristics including : ( 1 ) little sf in the most recent 1 gyr compared to gas - rich dwarfs , ( 2 ) similar total stellar masses , ( 3 ) sfhs that are generally extended and indistinguishable from dis , and ( 4 ) are located exclusively in high density environments . in what follows , we consider the impact of putative gas loss mechanisms on the evolution of a typical gas - rich dwarf galaxy ( e.g. , with both @xmath65 and @xmath66 @xmath2 10@xmath72 m@xmath38 ) in the context of figure [ masses ] . the first mechanism for gas removal is consumption through sf . this process will increase the stellar mass and decrease the gas mass of our prototypical di , moving it to the right on figure [ masses ] . though sf can consume large amounts of a galaxy s gas reservoir , the gas densities will eventually become too low to continue to form stars ( e.g. , * ? ? ? this suggests that if consumption was the only mechanism , we should observe trace amount of gas in dsphs , which is not typically the case . in the event that gas density was not a limiting factor , characteristically low sfrs and sf efficiencies in dwarf galaxies ( @xmath2 1% ; e.g. , * ? ? ? * ) imply long gas consumption timescales . in this event , gas supplies would typically not be exhausted for more than a hubble time , which suggests that dsphs would be quite rare . in addition , gas consumption does not have a dependence on the environment external to a galaxy . thus , it can not account for the observed morphology density relationship . although sf is a critical process to galaxy evolution , it can not be responsible for the complete removal of gas from dwarf galaxies . the second possible mechanism for removing gas is stellar feedback . mechanical energy due to stellar winds and supernovae provide an appealing explanation for gas removal from a galaxy via expulsion ( e.g. , * ? ? ? for the prototypical gas - rich di , this process removes gas with minimal effect on stellar mass , moving the galaxy straight down in figure [ masses ] . several feedback models can reproduce observed dwarf galaxy relationships between mass , luminosity , and rotational velocities ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? additionally , these models are generally able explain the observed mass metallicity relationship for dwarf galaxies ( e.g. , * ? ? ? although these models hold promise , there are two significant challenges to gas removal in dwarf galaxies due to feedback . first , a number of simulations have demonstrated that the energy due to stellar feedback is insufficient to completely expel cold gas ( e.g. , hi ) from the gravitational potential of a typical dwarf galaxy ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? second , stellar feedback alone can not explain the morphology density relationship . similar to the rationale for gas consumption through sf , stellar feedback is not dependent on the environment surrounding a galaxy . thus , while sf and stellar feedback are primary drivers of dwarf galaxy evolution , it is unlikely that either or both are able to account for the final transition from a gas - rich to a gas - poor state . the third mechanism we consider is ram pressure stripping . @xcite first proposed that dsphs were once gas - rich dwarfs whose gas supply has been removed by ram pressure stripping from the intergalactic medium ( igm ) . this mechanism has negligible impact on stellar mass loss , thus moving our prototypical di straight down in figure [ masses ] . satellite galaxies in higher density environments would likely encounter a denser igm , which would lead to more efficient gas loss . this provides a feasible explanation for the observed morphology - density relationship . despite the appeal of this mechanism , there is a distinct lack of observational evidence of systematic ram pressure stripping of lg dwarf galaxies . for example , @xcite and @xcite find that ram pressure stripping appears to be a localized phenomena , and only evident for a small minority of lg dwarfs . however , given that the lg is in a state of relatively passive evolution , it could be that signatures of ram pressure stripping have been erased in many satellites . an additional challenge comes from the magnitude of ram pressure stripping . @xcite demonstrate the ram pressure stripping is unlikely effective enough to completely remove the gas supply of a dwarf galaxy in the lg . it appears that ram pressure stripping is not able to account for complete gas loss in dwarf galaxies . we next consider tidal effects on gas removal from dwarf galaxies . the close passage of a dwarf galaxy to a massive companion has strong gravitational effects on the gas , stellar , and dark matter contents of the smaller galaxy . as shown in @xcite , the magnitude of the tidal force during an interaction does not appear to be enough to remove the gas content from a gas - rich di . instead , @xcite advocate a more complex approach in which a combination of ram pressure stripping and stellar mass loss due to tidal effects provide a plausible model , ` tidal stirring ' , for the transformation of gas - rich dis into gas - poor dsphs . in this scenario , the prototypical gas - rich di is able to lose both stellar and gas mass , allowing it to move down and left in figure [ masses ] , meaning a di with a high stellar mass could be transformed into a less massive dsph . predictions from tidal stirring models provide qualitative explanations for a number of trends seen in the angst sample . foremost , tidal stirring naturally produces a morphology density relationship . galaxies in lower density environments have had few ( or no ) interactions with massive galaxies , and are able to maintain high gas fractions . this is in general agreement with the trends seen in figure [ thbary ] . of the galaxies that do interact with a massive companion , tidal stirring predicts that stellar and gas mass loss happens progressively over several gyr ( a single complete orbit in the lg is typically 1 - 2 gyr ; * ? ? ? therefore , these galaxies spend the majority of their life cycling between sf and stellar feedback . by extension , this implies that many dsphs and dis should not have drastically different sfhs , in general agreement with the sfhs of angst dwarf galaxies . while the general qualitative agreement between tidal stirring predictions and the angst sample is encouraging , a quantitative comparison between predictions and observed dwarf galaxy sfhs are needed to place precise constraints on the gas loss processes in dwarf galaxies . with little evidence of recent sf , yet detectable amounts of hi , dtrans may hold clues to the transformation of gas - rich to gas - poor galaxies . synthesizing several previous studies , there are two favored scenarios for the origins of dtrans , namely that they are either in the last throes of sf or are observed during temporary lulls between episodes of massive star formation ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? dtrans in the angst sample have characteristics consistent with both scenarios . those with low gas fractions ( @xmath66/@xmath67 @xmath1 0.4 ; eso294 - 010 , es410 - 005 , and eso540 - 032 ) typically have positive tidal indices , indicating a higher likelihood of gas loss via interactions ( figure [ thbary ] ) . these three galaxies may be examples of galaxies genuinely transitioning from gas - rich to gas - poor states . in contrast , the most isolated dtrans ( kk 230 , kkh 98 , kkr 25 , and ua 438 ) typically have high gas fractions ( @xmath4 0.9 ) and stellar masses among the lowest in the angst sample ( figure [ bary ] ) . these dtrans have properties consistent with other low mass dis , and could simply be low mass dis lacking h@xmath18 , which can be attributed to a wide range of effects including temporary episodes between massive star formation ( e.g. , * ? ? ? * ) , a stochastically sampled or systematically varying imf ( e.g. , * ? ? ? * ; * ? ? ? * ) , and/or leakage of ionizing photons ( e.g. , * ? ? ? * ) . the remaining dtrans ( antlia , kdg 2 , ddo 6 , kdg 52 , kdg 73 ) appear to be have masses and tidal indices close to the genuinely transitioning group , but gas fractions similar to the low mass dis . it is possible that these galaxies may have had few interactions , resulting in less gas loss . detailed multi - wavelength studies of these galaxies could provide insight into gas loss mechanisms in dwarf galaxies . as the smallest and most pristine galaxies in the universe , dwarf galaxies occupy a critical , but poorly constrained role in cosmological models of galaxy formation and evolution ( e.g. , * ? ? * ; * ? ? ? because of their intrinsic faintness , dwarf galaxies have been difficult to directly detect at cosmologically significant redshifts ( e.g. , * ? ? ? * ) . as an alternate approach to observing high redshift dwarf galaxies , we present a comparison between the mean cumulative sfhs of the angst sample dwarfs and the cosmic sfh , as derived from observed uv fluxes in high redshift galaxies ( e.g. , * ? * ) . in the top panel of figure [ cum2 ] , we see good agreement between the cosmic sfh ( grey shaded region ) and the mean cumulative sfhs of the angst dwarf galaxies at z @xmath2 2 ( 10 gyr ago ) . at more recent times , the cosmic sfh increases sharply , while the cumulative sfhs of dwarf galaxies increase at a slower rate . this comparison suggests that the largest differences between sfrs in dwarf galaxies and massive galaxies studied at higher redshifts are at intermediate and recent epochs . this finding is generally consistent with the expectations of galaxy ` downsizing ' , where high mass galaxies preferentially stop forming stars at higher redshifts ( e.g. , * ? ? ? . the precise role of dwarf galaxies in theories of downsizing is highly uncertain , making this result difficult to interpret in the context of current models ( e.g. , * ? ? ? we also caution that the physical significance of this finding is still ambiguous due to a several factors including uncertain agb star models and redshift dependent dust corrections applied to the cosmic sfh ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . a comparison of sfhs from the entire angst sample ( including the most massive galaxies ) and the cosmic sfh is presented in @xcite . the cumulative sfhs can also provide insight into the validity of model sfhs often used to describe the evolution of dwarf galaxies . as an illustrative example , we have selected a simple exponentially declining sf model ( sfr @xmath73 @xmath74 ; a @xmath75 model ) , and show the cumulative sfhs for a range of @xmath75 values ( grey shading ) in the lower panel of figure [ cum2 ] . visual inspection suggests that the measured sfhs show significant deviations from the predicted smooth curve of single @xmath75 valued models . a @xmath76@xmath30 test between the measured cumulative sfhs and exponentially declining sfh models ( with @xmath75 varying from 0.1 to 14.1 gyr ) confirms that any single value of @xmath75 does not accurately represent the data . similarly , the mean measured sfhs also are not well matched with a constant sfh ( i.e. , @xmath75 @xmath77 @xmath78 gyr ) or a single ancient epoch of sf followed ( @xmath75 @xmath1 0.1 gyr ) by passive galaxy evolution . instead more complex and multi - component models may better describe the lifetime sfhs of dwarf galaxies . we uniformly analyzed sfhs of 60 dwarf galaxies in the nearby universe based on observations and data processing done as part of the angst program . while the sfhs of individual galaxies are quite diverse , we find that the mean sfhs of the different morphological types are generally indistinguishable outside the most recent @xmath2 1 gyr . on average , the typical dwarf galaxy formed @xmath2 50% of its stellar mass by z @xmath2 2 and 60% by z @xmath2 1 . among the morphological types , the sfhs hint at divergence within the past few gyr , although assigning a precise time to this phenomena is challenging due to uncertainties in agb star modeling and the modest time resolution afforded by the data . the clearest differences between the morphological types can been seen in the most recent 1 gyr , where the typical dsph , di , dtrans and dspiral formed @xmath2 2% , 8% , 4% , and 5% of their total stellar mass . the dwarf galaxies in the angst sample show a strong morphology density relationship . this suggests that internal mechanisms , e.g. , stellar feedback , can not solely account for gas - loss in dwarf galaxies , as the corresponding models are unable to produce this observed relationship . instead , we find qualitative consistency with the model of ` tidal stirring ' ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , which can broadly explain the extended sfhs as well as the observed morphology density relationship . a comparison with the cosmic sfh reveals general good agreement between the independently derived sfhs . the slower rate of sf for dwarf galaxies at intermediate epochs may be tentative evidence of galaxy downsizing . however , broad uncertainties in extinction corrections and agb star models may be able to explain the offset . further , the mean measured sfhs are inconsistent with single valued exponential models of sf ( i.e. , @xmath75 models ) , and may require more complex or multi - component models . we also identify 12 dtrans in the sample , based on the literature definition of present day gas fraction and sf as measured by h@xmath18 ( e.g. , * ? ? ? * ) . within this sample of dtrans , we find that galaxies with high gas fractions are associated with more isolated galaxies , while those with lower gas fractions are less isolated . this suggests that there are two mechanisms that can produce the observed dtrans characteristics : the isolated gas - rich galaxies are simply between episodes of sf due to the stochastic nature of sf in low mass galaxies , while the less isolated galaxies could be in the process of interacting with a more massive companion . drw is grateful for support from the university of minnesota doctoral dissertation fellowship and penrose fellowship . idk is partially supported by rfbr grant 10 - 02 - 00123 . the authors would like to thank stephanie ct for fruitful discussions on the nature of transition dwarf galaxies and oleg gnedin for his insightful suggestions on measures of photometric quality . this work is based on observations made with the nasa / esa hubble space telescope , obtained from the data archive at the space telescope science institute . support for this work was provided by nasa through grants go-10915 , dd-11307 , and go-11986 from the space telescope science 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with a 50% completeness value near the median of the sample ( ddo 6 ; @xmath79 @xmath2 0 ) . to test for the optimal time resolution , we constructed cmds of single age stellar populations , convolved the simulated photometry with the artificial stars and photometric limits of ddo 6 , and ran the sfh recovery program on each of these simulated cmds . this process was conducted on single age cmds of 0.5 , 1.5 , 2.5 , 4.5 , 8 , and 12 gyr in age . the purpose of this exercise is to provide a simple test for the robustness of the final time bins . while coarser time bins encompass a larger fraction of the input sf , there are generally fewer of them , which provides less information about pattens of sf over time . in this test , we deemed a good recovery as one in which the input sfh was within the uncertainties of the recovered sfh . in figure [ frac_sf ] , we show the cumulative sfhs , where the input sf is indicated by the dashed magenta line and the recovered sfh by the solid black line . uncertainties were computed using 50 monte carlo realizations as described in [ sfhs ] , including identical shifts in @xmath39 and @xmath80 . initial tests suggested that six broad time bins of 0 - 1 , 1 - 2 , 2 - 3 , 3 - 6 , 6 - 10 , 10 - 14 gyr would provide suitable time resolution across the angst sample . as shown in figure [ frac_sf ] , the general agreement between the input and recovered sfhs is good and falls within the uncertainties . clearly , the best recovery is seen in the 0 - 1 gyr bin , where the young ms and massive core and blue helium burning stars provide secure leverage on recent sf . the oldest bin also shows similarly good results . the results in the intermediate bins are generally reasonable , although not as reliable as the youngest and oldest bins . much of this can be attributed to uncertainties in agb star models ( e.g. , * ? ? ? * ; * ? ? ? although , we used the same models to generate and recover the single age populations , photometric depth , systematic stellar model uncertainties , and blurring of distinct features due to observational effects all contribute to uncertainties in the recovered sfh ( see appendix [ systematics ] ) . agb populations of different ages often have similar colors and magnitudes on optical cmds ( e.g. , * ? ? ? * ; * ? ? ? * ) , thus it can be difficult to confidently determine the epoch of sf at intermediate ages ( 110 gyr ago ) . this limitation is clearly illustrated in figure [ frac_sf ] , which shows broad uncertainties on the recovered sfhs during intermediate times . however , we still find that the input sfhs are generally within the recovered sfh error bars . additionally , @xmath4 50% of the sf is recovered in the correct time interval . although it would be possible to combine bins at intermediate times to increase the accuracy of the recovered sfrs , we believe that this scheme will permit future comparisons with sfhs derived using improved agb star models ( e.g. , * ? ? ? for this time binning scheme , we find that reported uncertainties in the sfrs typically decrease for cmds deeper than ddo 6 , and increase for shallower cmds , in agreement with expectations . we thus conclude that this scheme is adequate for comparison of sfhs across the angst sample . photometric depth is an important consideration when interpreting the accuracy of a measured sfh . varying photometric depths impact both the number of stars on a cmd and determine the presence of age sensitive cmd features ( e.g. , ancient ms turn - off , horizontal branch ) . intuitively , a cmd with a brighter photometric limit has less information available than a deeper cmd , and the derived sfh is thus more uncertain . however , quantifying the precise impact of these uncertainties , such as the amplitude and ages affected , is challenging as varying photometric depths can amplify effects of uncertainties in the stellar models used to measure sfhs . in this section , we explore the effects of photometric depth on the accuracy of sfh recovery . we demonstrate the effects in two regimes : one where the only variable is photometric depth , and the other where we vary both photometric depth and stellar evolution models . we first analyze the accuracy of recovered sfhs as a function of _ only _ photometric depth . that is , we construct synthetic cmds at select photometric depths , then attempt to recover the input sfh using identical parameters ( e.g. , imf , binary fraction , filter combination ) and , in this case , the same stellar evolution models . more concretely , we constructed cmds at six different photometric depths ( @xmath81 @xmath194 , @xmath192 , @xmath191 , 0 , @xmath121 , @xmath122 ) , using a constant sfh ( @xmath827.4 - 10.15 , with a time resolution of @xmath2 0.1 dex ) , a fixed metallicity , and the basti stellar evolution models . each cmd was populated with @xmath2 10@xmath83 stars in order to minimize the contribution of random uncertainties due to the number of stars used to measure the sfr in each time bin ; for sfhs measured from observed cmds , the uncertainties for poisson sampling of the cmd are already well characterized by our monte carlo tests . we then recovered the sfh of each cmd using the basti stellar evolution models , to ensure that the only variable being tested is photometric depth . in figure [ basbas_sf ] , we compare the input ( black lines ) and recovered ( colored lines ) cumulative sfhs at each photometric depth . overall , the recovered cumulative sfhs are in excellent agreement with the input sfhs at all photometric depths . the maximum deviation between the input and recovered sfhs at any photometric depth is @xmath2 4% , which is consistent with the expected poisson precision of 1/@xmath84 , where @xmath85 is the number of stars used to measure the sfr in a given time bin . this exercise demonstrates that if all the underlying stellar models are known exactly , then the accuracy of the recovered sfh only depends on the number of stars in the cmd , and not the photometric depth . the same results are found when using different stellar evolution models , e.g. , padova , dartmouth @xcite , to conduct this exercise . systematic uncertainties are introduced into sfh measurements by uncertainties in the selected stellar models . the luminosity , color , and number density of specific cmd features provide leverage on the sfr at different epochs . however , stellar models are not always self - consistent when trying to model multiple observed features ( e.g. , reproducing colors consistent with observations for both the horizontal branch and rgb ; e.g. , * ? ? ? the effect on a measured sfh is that the sfr may be systematically shifted into a particular time bin , depending on the stellar model used and photometric depth of the cmd ( i.e. , which particular age sensitive cmd features are available ) . we refer the reader to @xcite for a comprehensive review on the adequacy of stellar evolution models for reproducing cmd features . to test for systematic effects on measured sfhs , one ideally wants to test for differences between the stellar models and ` truth ' , that is , stellar population characteristics based on the exact physics governing stellar evolution in nature . however , current stellar evolution models represent the best physical descriptions we have of nature . by measuring how different stellar models impact a measured sfh , we can get a sense of the amplitude of systematic uncertainties for sfh recovery . to examine the magnitude of systematic effects , we again construct six cmds at selected photometric depths each containing @xmath2 10@xmath83 stars , assuming a constant sfh , fixed metallicity , and the basti stellar evolution models . however , the sfhs are now recovered with the padova stellar models . thus , in this case the differences between the input and recovered sfhs are indicative of the systematic effects introduced by choice in stellar model . in figure [ padbas_sf ] , we see the differences in the input ( black lines ) and recovered ( colored lines ) mean cumulative sfhs are more substantial than when identical models were used . for the deepest cmd considered ( @xmath86@xmath194 ) , the agreement between the input and recovered cmd is excellent . here , the ancient ms turnoff provides a reliable constraint on the ancient sfh , meaning the systematics in progressively younger time bins are also known more precisely . for shallower cmds that do not include the ancient ms turn - off , we see significant deviations between the recovered and input sfhs . these difference arise primarily in the treatment of such features of the horizontal branch , red clump , and rgb ( see @xcite for a more detailed discussion ) . we quantify the magnitude of the discrepancies between the padova and basti models by examining the absolute value of the differences in input and recovered cumulative sfhs ( figure [ diff_sf ] ) . of interest are the general order of magnitude variations , and not a specific point by point analysis . in the case of the shallowest cmds ( @xmath86@xmath122 , @xmath121 , 0 ) , the ancient sf is preferentially recovered at intermediate ages , with a typical magnitude in the difference of input and recovered sfh of @xmath2 20% . for the deeper cmds ( @xmath86@xmath191 , @xmath192 ) , sf at the oldest times tends to be over - estimated by up to @xmath2 40% . at all photometric depths , the systematic effects within the last @xmath2 2 gyr are @xmath1 10% . the exercises we have conducted demonstrate the importance of systematic uncertainties in sfh recovery . we caution that our results are actually measuring the systematic differences between the padova and basti models , and only serve a a proxy for the difference between a given model and ` truth ' , i.e. , observed cmds . that being said , the methodology of this type of analysis provides a framework for exploring the effects of systematic uncertainties . the primary limitation to this type of analysis is the number of models available that span the entire age / metallicity range needed to derive sfhs in nearby galaxies . as the number of models that sample a broader parameter space increases , it will be possible to gain more leverage on the effects of systematic uncertainties .

we present uniformly measured star formation histories ( sfhs ) of 60 nearby ( @xmath0 @xmath1 4 mpc ) dwarf galaxies based on color - magnitude diagrams of resolved stellar populations from images taken with hubble space telescope and analyzed as part of the acs nearby galaxy survey treasury program ( angst ) . this volume - limited sample contains 12 dsph / de , 5 dwarf spiral , 28 dirr , 12 dsph / dirr ( transition ) , and 3 tidal dwarf galaxies . the sample spans a range of @xmath2 10 in @xmath3 and covers a wide range of environments , from highly interacting to truly isolated . from the best fit sfhs we find three significant results : ( 1 ) the average dwarf galaxy formed @xmath4 50% of its stars by z @xmath2 2 and 60% of its stars by z @xmath2 1 , regardless of current morphological type ; ( 2 ) the mean sfhs of dis , dtrans , and dsphs are similar over most of cosmic time , and only begin to diverge a few gyr ago , with the clearest differences between the three appearing during the most recent 1 gyr ; and ( 3 ) the sfhs are complex and the mean values are inconsistent with simple sfh models , e.g. , single bursts , constant sfrs , or smooth , exponentially declining sfrs . the mean sfhs are in general agreement with the cosmic sfh , although we observe offsets at intermediate times ( z @xmath2 1 ) that could be evidence that low mass systems experienced delayed star formation relative to more massive galaxies . the sample shows a strong density - morphology relationship , such that the dsphs in the sample are less isolated than dis . we find that the transition from a gas - rich to gas - poor galaxy can not be solely due to internal mechanisms such as stellar feedback , and instead is likely the result of external mechanisms , e.g. , ram pressure and tidal stripping and tidal forces . the average transition dwarf galaxy is slightly less isolated and less gas - rich than the typical dwarf irregular . further , the transition dwarfs can be divided into two groups : interacting and gas - poor or isolated and gas - rich , suggesting two possible evolutionary pathways .